Lecture 1 | Modern Physics: Special Relativity (Stanford)

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Lecture 1 of Leonard Susskind’s Modern Physics course concentrating on Special Relativity. Recorded April 14, 2008 at Stanford University.

This Stanford Continuing Studies course is the third of a six-quarter sequence of classes exploring the essential theoretical foundations of modern physics. The topics covered in this course focus on classical mechanics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.

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this program is brought to you by Stanford University please visit us at stanford.edu this quarter we're going to learn about field theory classical field theory fields such as the electromagnetic field gravitational field other fields in nature which I won't name right now propagate which means they change according to rules which give them a wave-like character moving through space and one of the fundamental principles of field theory in fact more broadly nature in general is the principle of relativity the principle the special printless the the principle of special relativity in this particular case the principle of special relativity well let's just call it the principle of relativity goes way back there was not an invention of Einstein's I'm not absolutely sure when it was first announced or articulated in the form which I'll spell it out I don't know whether it was Galileo or Newton or those who came after them but those early pioneers certainly had the right idea it begins with the idea of an inertial reference frame now inertia reference frame this is something a bit tautological about an inertial reference frame Newton's equations F equals MA are satisfied in an inertial reference frame what is an inertial reference frame it's a frame of reference in which Newton's equations are satisfied I'm not going to explain any further what an inertial reference frame is except to say that the idea of an inertial reference frame is by no means unique a reference frame first of all was a reference frame in tale of a reference frame first of all entails a set of coordinate axes in ordinary space X Y & Z and you know how to think about those but it also entails the idea that the coordinate system may be moving or not moving relative to whom relative to whomever we sitting here or you sitting here in this classroom here define a frame of reference we can pick the vertical direction to be the z axis the horizontal direction along my arms here to be the x axis X plus that way X my X is minus in that direction and which one have I left out I've left out the y axis which points toward you from me so there are some coordinate axes for space XY and Z and I didn't this in addition to specify a frame of reference one also imagines that this entire coordinate system is moving in some way relative to you sitting there presumably with a uniform velocity in a definite direction if your frame of reference is an inertial frame of reference in other words if when you throw balls around or juggle or do whatever is supposed to do in an inertial frame of reference if you find yourself in an inertial frame of reference then every other frame of reference that's moving with uniform velocity relative to you now remember what uniform velocity means it doesn't just mean with uniform speed it means with uniform speed in an unchanging direction such a frame of reference is also inertial if it's accelerated or if it starts standing still and then suddenly picks up some speed then it's not an inertial frame of reference all inertial frames of reference according to Newton and also I think also Galileo Galileo was often credited with the idea but I never read enough of Galileo to know whether he actually had it or not neither did I read enough of Newtons they both wrote in languages that I don't understand what was I saying oh yes right according to both Newton and anybody else who thought about it very hard the laws of physics are the same in all inertial reference frames laws of physics meaning F equals MA the forces between objects all the things that we would normally call laws of nature or laws of physics don't distinguish between one frame of reference of and another if you want a kind of pictorial example that I like to use a lot when I'm explaining this to the children or to grownups I like to think about the laws of juggling there are very definite procedures that you train your body to do uh in order to be able to juggle balls correctly now you can imagine yourself being in a railroad car moving with perfectly uniform velocity down the x axis and trying to juggle do you have to compensate for the fact that the train is moving and for particular when you throw a ball up into the air that you have to reach over to the right to compensate for the fact that the train is moving to the left my left your right the answer is no you don't the laws of juggling are the same in every reference frame and every inertial reference frame whatever you do in one reference frame you do exactly the same thing and you'll succeed or fail depending on whether you're a good juggler or not but it will not depend on whether you're moving with uniform velocity so the laws of juggling are the same in every inertial reference frame the laws of mechanics are the same in every inertial reference frame the laws Newtonian laws of gravity are the same in every inertial frame according to Newton what about the laws of electrical phenomena well there there was a clash the clash had to do with Maxwell's equations Maxwell's equations were the field equations the field theory that governed the electromagnetic field and the way that it propagated and sent waves electromagnetic waves that we ordinarily call light or radio waves or so forth and the fundamental dilemma as you all know I'm sure you all know the fundamental dilemma was both according to well here was the dilemma Maxwell's equations said light moves with a certain velocity if you take the various constants that appear in Maxwell's equations and put them together in the right way you get the velocity of waves moving down an axis and that velocity comes out to be a certain number out of Maxwell's equations you have two choices one is to believe that Maxwell's equations are true laws of nature as good as any other laws of nature in which case the principle of relativity says they should be the same in every reference frame but if it follows from Maxwell's equations that the speed of light is three times ten to the eighth meters per second which is about what it is if it follows from Maxwell's equations that light moves that fast and if Maxwell's equations are laws of physics fundamental laws of physics and if the laws of physics are the same in every reference frame then the speed of light must be the same in every reference frame but that was very hard to swallow because if a light beam is going down that axis and you chase it and run along with it that lets say three-quarters of the speed of light then you want to see that light ray moving much more slowly than three times ten to the eighth meters per second relative to you on the other hand the light ray going in the other direction since you're sort of running into it you should see going even faster so all these possibilities could not simultaneously be correct that the laws of nature are the same in every reference frame and that Maxwell's equations are laws of physics in the same sense that Newton's laws of physics namely the same in every reference frame something had to give well the point was of course that they were good laws of nature and that they were the same in every reference frame the thing that had to give is our concepts of velocity space and time and how we measure velocity especially velocities were up which are up near the speed of light now I'm not going to spend the full amount of time that I did previously on the special theory of relativity that can be found on lectures from how long ago and there on the Internet I believe relativity and electromagnetism I think that was maybe about three quarters ago I've lost track yeah they're up there they're on the net and they're the lectures on relativity special relativity and electromagnetic theory we're just going to cut through it real fast we're going to cut through the basic ideas of relativity a little more mathematically than I would do if I were teaching it for the first time I teach it the first time I tend to teach it the way Einstein first conceived of it how do you measure distances how do you measure velocities how do how does the propagation of light influence these things instead I'm going to take a more mathematical view of it and think about the properties of various kinds of coordinate transformations coordinates now consists not only of XY and Z but also time T so imagine every event in the world is characterized by just like every particle would be characterized by a position x y&z every event taking place in space-time is characterized by four coordinates X Y Z and T let's suppress for the moment y&z let's just forget I forget them for the moment and concentrate on X and T that would be appropriate if we were mainly interested in motion along one axis let's focus on that motion along the x axis let's suppose there is no motion along y&z then we can forget y&z for the moment momentarily we'll come back to them and think of motion along X and T and the various reference frames that might be moving along the x axis alright here's here's time vertically is space horizontally physicists always draw space horizontally and time vertically I found out that mathematicians are at least certain computer scientists always draw time going horizontally I didn't know that and I got into an enormous argument with a quantum computer scientist which was ultimately resolved by the fact that he had time going horizontally and I had it going vertically these are traditions I guess traditions grow up around subjects but time is north and X is east I guess or at least time is upward yeah yeah yeah that's what that that that's the point that is the point yes they're thinking of time is the independent variable and everybody knows that it's a law of nature that the independent variable should be horizontal ok all right now let's in let's imagine a moving observer moving down the x axis with a velocity V let's take his origin of spatial coordinates his origin of spatial coordinates at time T equals zero is just the same let's assume that my I'll be the moving observer I move down the x-axis I am my own origin there's nobody who was your origin that seat is vacant over there so that absent a human over there is the center of the x-coordinates in your frame I'm the X prime coordinates and of course I being very egocentric will take my x-acto is origin to be where I am there I do I move down the x-axis we pass each other our origins pass each other at t equals 0 so that means at T equals 0 your axis and my axes are the same or your origin in my origin is the same but then as I move down the x axis my core my coordinate center moves to the right most of the right that's supposed to be a straight line that's as good as I can do under the circumstances that's a straight line and it's moving with velocity V which means it's X prime equals SR it means x equals VT but it's also that's the way you describe it in terms of your coordinates my centre you described by saying x equals VT how do I describe it I just say X prime my coordinate X prime is 0 X prime equals 0 is the same as x equals VT all right what's the relationship between X Prime and X and T well it's easy to work out if you believe this picture the X prime coordinate is the distance from my origin the x coordinate is the distance from your origin so one of these is X the other is X prime the upper one here is X prime the low and here is X and the relationship between them is that they differ by an amount VT in particular X is equal to X prime minus VT or X prime is equal to X plus VT will have it wrong yes I do X prime is X minus BT and X is X prime plus VT yeah I think I have that's correct now all right what about time itself well according to Newton and according to Galileo and according to everybody who came afterward up until Einstein time is just time is just time is just time there was no notion that time might be different in different reference frames Newton had the idea of a universal time sort of God's time God upon his cloud ticking off with his with his super accurate watch and that time was universal for everybody no matter how they were moving and so everybody would agree on what on the time of any given event in this map of space and time here and so the other equation that went with this is that T prime is equal to T let's forget the top equation here let's just forget it one might say that this was the Newtonian or the Galilean transformation properties between X and T your coordinates and the coordinates that I ascribe to a point in space-time now let's examine a light ray moving down the plus x axis if it starts at the origin here then it moves along a trajectory which is x equals CT C being the speed of light now shortly I'm going to set C equal to 1 we're going to work in units in which C is equal to 1 but not quite yet incidentally once you understand a bit of relativity working in coordinates in which C is not equal to 1 is about as stupid as using different units for x and y are if we used yards for x and feet for y then we will have all kinds of funny factors in our equations which would be conversion factors from X which is measured in feet to Y which is measured in our yards the cycle has its uses log scale has its uses no long skilling long scale well let common interest yep I'm not sure we good but okay I'm just saying it is quite often in practical circumstances that one uses different scales yeah you sometimes you might there might be a good reason I mean um it wouldn't be totally unreasonable for a sailor to use different units for horizontal direction and vertical direction hmm I mean he's used to moving around horizontally he might use what miles miles versus fathoms or something nautical miles versus paddles yeah Persian is relative but um when you talk about a frame of reference you need to specify a period of time because obviously goes that 15 billion years there is no yeah we're ignoring now the fact that the universe began at some time and we're imagining now as Newton did and as the early Einstein did that the universe has just been here forever and ever and ever unchanging totally static and space and time have properties which don't change with time now of course that's incorrect in the real world and at some point we will take up the subject of cosmology and find that's not right but as long as we're interested in time intervals which are not I suspect this is what you're getting at as long as we're interested in time intervals which are not too long in particular time intervals over which the universe doesn't expand very much and so forth we can mainly say the properties of space don't change over a period of time and so everything just stays the same as always was is that what you're asking it seems that that this assumption if it is made it needs to what you're describing so well so the question is without imagining to some point as it doesn't lead it doesn't lead to what I'm describing where is this this room for different formulas here this is a formula which is based on an assumption the assumption being that time is universal that's what Einstein found was wrong basically what he found is that when you're in a moving frame of reference to different the observers will not agree about what time a particular event takes place this is the culprit here this one and some modifications to this one but in any case to see what's wrong let's go to Maxwell's equations Maxwell's equations say that light always moves with this velocity C being some numbers in meters per second okay 3 times 10 to the 8th meters per second we will later as I said say C equals 1 let's imagine a light beam moving down the x axis let's describe how X prime sees it in other words you see the light move this way to the right how do I see the light well let's see what I see let's just work it out X prime will be X which is CT for that light ray minus VT which is the same as C minus VT all this says is that I see the light moving with a diminished velocity a velocity C minus V why is that because I'm moving along with the light so naturally I see it move slowly the slow compared to what you see it what about the light going in the other direction supposing it was a light beam going in the other direction then how would you describe it you would describe it as x equals minus CT and if I do exactly the same thing I will find that X prime is equal to X that's minus CT – VT which is the same as minus C plus V times T so what this says is that I will see the light moving also in the negative direction that's the minus sign but I'll see it moving with an enhanced velocity C plus V if this were the right story and if these were the right transformation laws for space and time then it could not be the case that Maxwell's equations are laws of physics or laws of nature in the sense that they were true in every reference frame they would have to be corrected in moving frames just like the juggler who had to reach to the right who didn't actually but who thought he had to reach to the right to collect the ball when train is moving the physicist interested in light beams would have to correct things for the motion of his reference frame now it's an experimental fact that this is not the case that you don't have to correct for motion was the famous Michelson Morley experiment Einstein he just rejected he just felt this can't be right Maxwell's equations were much too beautiful to be relegated to the approximate or to the contingent on which reference frame and so he said about to find a framework in which the speed of light would be the same in every reference frame and he basically focused on these equations and after various very very beautiful Gedanken experiments thought experiments about light and about measuring and so forth he came to a set of formulas called the Lorentz transformations I'm going to explain them the Lorentz transformations in a more mathematical way not fancy mathematics but just get we want to get right to the heart of it and not spend the three weeks doing it the best way is to a mathematical problem but before I do let me set up a different mathematical problem which is for most of you you've seen me do this before but nonetheless let's go through it again the problem of rotation of coordinates we're going to do this quickly let's just take spatial coordinates now for the moment two dimensional spatial coordinates let's forget X and T and just concentrate on X&Y two coordinates in space instead of events in space-time concentrate on a point in space a point in space has coordinates and we can determine those coordinates the x and y coordinates just by dropping perpendicular to the x axis in the y axis and we would describe this point as the point at position let's just call it X Y now there's nothing sacred about horizontal and vertical so somebody else may come along some crazy mathematician a really nutty one who wants to use coordinates which are at an angle relative to the vertical maybe a couple of beers and you don't know the difference between vertical and worth worth worth we should give this direction a name oblique yeah all right the oblique observer the blue observer can blue be seen everybody can see blue okay good ah the blue observer also characterizes points by coordinates which he calls X Prime and Y Prime the X Prime and the Y prime coordinates are found by dropping perpendicular to the X Prime and the Y prime axis so here's X prime is y prime and given a point X Y there's a role it must be a role if you know the value of x and y you should be able to deduce the value of X I'm in y-prime if you know the angle between the two coordinates between the x coordinate and the X prime coordinate and the formulas simple we've used it least in these classes many times I'll just remind you what it is that's X prime is equal to x times cosine of the angle between the two frames between the two coordinate systems minus y times sine of the angle and Y prime is equal to minus plus I think X sine of theta plus y cosine theta I just want to remind you about a little bit of trigonometry all of trigonometry is encoded in two very simple formulas I've used them this signs on these signs of are on the right let's Ella and X prime is bigger than X for small theta since ours here are all so it's Auto Expo Rhine is bigger than it is is it yeah let's see if you rotate it to the next so that y is y prime is zero it's further out X prime rook will have it backward yeah what's your gift I'm not gonna fit nobody so let's say just make sure the links take survive is the little perpendicular there no my life primary so that's y prime y prime is this is why I'm here right right that's why I'm in X prime is bigger than X so there has to be a plus sign on the second you know its prime is bigger than X let's see um yeah X prime is bigger than X yeah X prime is bigger than X looks like that's probably right probably sign but then this one must be man negative yeah okay there's an easy way to correct for it another way to correct for it just call this angle minus theta that would also do the trick because cosine of minus theta is the same as cosine of theta and sine changes sign when you change theta 2 minus theta so if instead of calling this angle theta I called it minus theta then my previous formulas would be right it's true true but the it's an excuse all right what do we know about sine and cosine it's important to understand sine and cosine everything you ever learned about trigonometry can be codified in two very simple formulas if you know about complex numbers the two very simple formulas are that cosine of theta is e to the I theta plus e to the minus I theta over 2 and sine of theta is e to the I theta minus e to the minus I theta over 2i those two formulas contain everything about trigonometry you don't have to know any other formulas other than these for example I will assign you the homework problem of using these two formulas to find cosine of the sum of two angles but the way you would do it is just write the sum of two angles in here and then reexpress the Exponential's in terms of cosine and sine that's easy to do e to the I theta is equal to cosine of theta plus I sine theta and e to the minus I theta is cosine of theta minus I sine theta so work through these formulas get familiar with them they're extremely useful formulas once you know them you will never have to remember any trigonometric formulas again the other thing to know is that e to the I theta times e to the minus I theta is 1 all right e to the anything times e to the minus the same thing is one those things characterize all trigonometric formulas in particular as was explained to me by Michael a number of times if we multiply e to the I theta times e to the minus I theta we will get one on this side but on this side we will get cosine squared of theta plus sine squared of theta naught minus sine squared but plus sine squared cosine squared and then ice minus I squared sine squared that gives us cosine squared plus sine squared cosine squared theta plus sine squared theta so that's equivalent to the fact that e to the I theta times e to the minus I theta is 1 all right now the most important fact that again follows from the simple trigonometry is that when you make the change of coordinates from XY to X prime Y prime something is left unchanged namely the distance from the origin to the point XY that's something which is you know you count the number of the molecules along the blackboard from here to here and that doesn't change when I change coordinates so the distance from the origin to the point XY has to be the same independent of which coordinate axes we use well let's take the square of that distance the square of that distance we know what it is let's call it s squared I'm not sure why I use s but s for distance s s for distance s for space I think it must be for space that I'm using it for the spaces for the spatial distance from the origin to the point XY we know what that is it's Pythagoras theorem x squared plus y squared but as I said there's nothing special about the XY axes we also ought to be able to calculate it as X prime squared plus y prime squared well it's not too hard to work out that X prime squared plus y prime squared is x squared plus y squared it's easy to use do X prime squared plus y prime squared will have x squared cosine squared theta it will also have x squared sine squared theta when you add them you'll get x squared plus y squared you know you know the rigmarole so it follows from cosine squared plus sine squared equals 1 that X prime squared plus y prime squared equals also equal is equal to x squared plus y squared work that out make sure that you have this on the control that you understand why from the trigonometry not from the the basic physics of it or the basic geometry of it is clear make sure that you understand that you can see that from the trigonometry okay one last thing about sines and cosines if I plot on the blackboard for every angle if I plot sine or cosine along the horizontal axis supposing I plot cosine of theta along the horizontal axis and sine of theta along the vertical axis then if I plot all possible angles they will correspond to a bunch of points that lie on a unit circle Y on a unit circle because sine squared plus cosine squared equals 1 so one might call the properties of sine and cosine the properties of circular functions circular in that they're convenient for rotating they're convenient for describing unit circles points on unit circles are described in terms of coordinates which are cosines and sines of angles and so forth it's natural to call them circular functions these are these are not the functions that come in to the transformation the new transformation properties first of all these are wrong and I don't want to use X what's X ya ya now just wrong Newton had it wrong Newton or Galileo however it was postulated who postulated it Einstein modified it now we're going to have to make sure that Einstein's modification doesn't change things in situations where Newton knew where Newton's equations were good approximations the situations where I'm Stan's modifications are important is when we're talking about frames of reference moving very rapidly up near the speed of light before the 20th century nobody or nothing had ever moved faster than a hundred miles an hour probably well of course some things did light did but for all practical purposes light didn't travel at all it's just when you turned on the switch the light just went on so light didn't travel nothing and anybody's experienced direct experience traveled faster than 100 or 200 miles an hour and well I should say nothing travels faster than 100 miles an hour and then live to tell about it so all of experience was about very slow velocities on the scale of the speed of light on the scale of such velocities newton's formulas must be correct they work they're they're very useful they work Nutan got away with it so there must be good approximations it better be that whatever einstein did to the equations in particular to these two equations here had been a reduced to newton's equations in the appropriate limit okay let's come back now to light light according to the Newton formulas doesn't always move with the speed of light but let's let's try to figure out what it would mean of a better formula of a replacement for this but light always moves with the speed of light first of all let's set the speed of light equal to one that's a choice of units in particular it's a choice of the relation between space units and time units if we work in our light years for spent for a distance and years for time then light moves one light year per year the speed of light is one if we use seconds and light seconds it's also one whatever whatever scale we use for space if we use for time the time that it takes light to go that distance one unit of space if we use that for time units then the speed of light is equal to one now from the ordinary point of view of very slowly moving things those are odd units but if we were electrons with neutrinos and whizzing around like photons they would be the natural units for us speed of light equals one so let's set the speed of light equal to one as I said it's just the choice of units and then a light ray moving to the right just moves along a trajectory x equals T C is just equal to one a light ray moving to the left is x equals minus T how can we take both of these equations and put them together sorry x equals minus T can I write a single equation which if it's satisfied is a light ray either moving to the left or to the right yes here's an equation x squared equals T squared it has two solutions x equals T and X equals minus T the two square roots or x squared equals T squared is equivalent to either x equals T or x equals minus T in other words this equation here has the necessary and sufficient condition for describing the motion of a light ray either to the right or to the left supposing we found a replacement for this equation which had the following interesting property that whenever let's let's write it this way X square minus T squared equals 0 this is even better for our purposes x squared minus T squared equals 0 that's the necessary and sufficient condition to describe the motion of a light ray supposing we found a new set of rules a new set of transformation properties which which um had the property that if x squared minus T squared is equal to 0 then we will find that X prime squared minus T prime squared is equal to 0 in other words supposing this implied this and vice-versa then it would follow that what the unprimed observer you and your seats see is a light ray the primed observer me moving along also see as a light ray both of us agreeing that light rays move with unit velocity now this doesn't work for Newton's formula here it just doesn't work if X is equal to T it does not follow that X prime is equal to the T prime in fact it says something quite different okay so the form of these equations must be wrong let's look for some better equations now at this point let's in fact let's even be a little bit more ambitious it turns out being a little bit more ambitious actually simplifies things let's not only say that when X square minus T squared is equal to zero then X prime squared minus T prime squared is equal to zero let's say something even bolder let's say the relation between XT and X prime T prime is such that x squared minus T squared is equal to X prime squared minus T prime squared in other words pick any X and any T and calculate X square minus T squared then take the same point except reckoned in the primed coordinates in other words we take a certain event a light bulb goes off someplace you say that corresponds to X and T I say it corresponds to X Prime and T Prime but let's require just to try it out see if we can do it let's look for transformations so that X square minus T squared will always be equal to X prime squared minus T's prime squared that would be enough to ensure that everybody will agree about the speed of light why if x squared minus T squared equals X prime minus T prime squared for all X and T and so forth then when X square minus T squared equals zero X prime minus T prime squared will be zero and then if this is a light ray so is this a light ready everybody get the logic ok good so let's assume now that let's ask can we find transformations which have this particular property now it's not so different from looking for transformations which preserve x squared plus y squared equals x prime squared plus y prime squared it's just a little minus sign other than a minus sign here X square minus T squared look of these two is very similar and the mathematics is quite similar here are the transformations which preserve x squared plus y squared what are the transformations which preserve x squared minus T squared well they are the Lorentz transformations they are the fundamental transformations of the special theory of relativity they're not this but they're closely related or perhaps one should say closely analogous to these equations here but we have to substitute for circular trigonometry hyperbolic trigonometry so let's go back and remember a little bit about hyperbolic functions instead of circular functions well I didn't want to erase that all right these are the basic rules governing circular functions cosine theta this sine theta is equal to this and the e to the I theta in terms of cosine and sine all right let's see if we have a yeah we do have a blank blackboard here let me write whoops what did I do here I erased something I didn't mean to erase incidentally does everybody see how I got this side from the side you just add and subtract the equations appropriately and you isolate it to the I theta e to the minus R theta that's elementary exercise alright hyperbolic functions what are hyperbolic functions alright those are functions of the form hyperbolic cosine cosh hyperbolic cosine first of all the angle theta is replaced by a variable called Omega which I will call Omega Omega is called a hyperbolic angle it doesn't go from zero to two pi and then wind around on a circle it goes from minus infinity to infinity goes from minus infinity to infinity so it's a variable that just extends over the entire real axis but it's defined in a manner fairly similar to cosine and sine cosh Omega is by definition you're not allowed to ask why this is definition e to the Omega plus e to the minus Omega over 2 all we do is substitute for theta or for Omega theta I theta substitute Omega and that gives you hyperbolic functions likewise or similarly there's the hyperbolic sine and that's given by e to the Omega minus e to the minus Omega over 2 essentially you throw away all eyes out of that formula out of the top formulas just throw away all Sun all eyes the equations on the right-hand side become e to the Omega equals hyperbolic cosh Omega plus sin Chi Omega and e to the minus Omega equals cosh so mega- cinch Omega I think that's right is it right gosh – cinch it is yeah it is right okay now what about the analog of cosine squared plus sine squared equals one that simply came by multiplying this one by this one so let's do the same operation multiplying e to the Omega by each by e to the minus Omega gives one and now that gives cosh squared minus cinch squared you see we're getting a minus what we want we want that minus the minus is important we want the well somewhere is under here was a formula with a minus sign yeah we want to get that – into play here that's cos Omega squared knockouts Prakash squared Omega minus sin squared Omega so it's very similar everything you want to know about hyperbolic trigonometry and the theory of these functions is called hyperbolic trigonometry everything you ever want to know is codified in these simple formulas these in these and they're more or less definitions but there are the useful definitions now yeah go ahead yeah not only is it worth mentioning I was just about to mention it so I squared minus y squared is what hyperbola yeah right exactly so if I were to play the same game that I did here namely plot on the horizontal and vertical axis the values not of cosine of theta and sine of theta but cosine cosine cosh of that of Omega and since Omega what's in other words on the x-axis now we're going to plot cos Omega and on the y-axis cinch Omega then this is a hyperbola not a circle but a hyperbola and it's a hyperbola with asymptotes that are at 45 degrees you can see let me show you why why the asymptotes are at 45 degrees when Omega is very large when Omega is very large then e to the minus Omega is very small right when Omega is very large e to the minus Omega is very small and that means both cosh and cinch are both essentially equal to e to the plus Omega in other words when Omega gets very big cosh and cinch become equal to each other and that's this line here cash equals cinch along this line here so when Omega gets very large the curve asymptotes to to a curve which is a 45 degrees it's not hard to see that in the other direction when Omega is very negative that that it asymptotes to the other asymptotic line here so that's why it's called hyperbolic geometry it the hyperbolic angle the hyperbolic angles the caches the cinches play the same role relative to hyperbolas as sines and cosines do two circles any questions No so cosh Omega equals zero how would you plot that hi purple okay show me hmm Oh cos squared minus sin squared equals zero no that's no no cos squared minus sin squared equals one in the same sense that sine squared plus cosine square it never equals zero I think what I think you want to ask a different question I think oh well since Omega equals zero is the horizontal axis the costume a equals zero is the vertical eyebrows right okay well this is the x-intercept yeah it's it's the vertex I just think here's one point on a minute oh man the x-intercept there is one yeah because Kostroma cost of zero is one to see that just plug one r 0 in here 1 plus 1 divided by 2 is 1 at least it was yesterday yeah stores okay so now we we're sort of starting to cook a little bit we're starting to see something that has that nice minus sign in it but what's it got to do with X and T and X Prime and T prime we're now set up to make let's call it a guess but it's a guess which is based on the extreme similarity between hyperbolas and circles cautions and cosines and so forth he is the guess I'm going to make and then we'll check it we'll see if it does the thing we wanted to do my formula instead of being this has gotten with and we're now going to have instead of x and y we're going to have x and t time and x later on we'll put back y&z we're going to have to put back y&z but they're very easy okay so let's start with X prime X prime is the coordinate given to a point of space-time by the moving observer namely me and I'm going to guess that it's some combination of X and T not too different but not the same as where is it X prime equals X minus VT I'm going to try cosh Omega X let's write X cos Omega minus T sin Omega sort of in parallel with this I could put a plus sign here but you can go back and forth between the plus and the minus by changing the sign of Omega just as you did here so this let's do it this way X cos Omega minus T sin Omega and T prime going to look similar but without the extra minus sign here this you know the relation between sines cosines and cautious and cinches is one of just leaving out an eye you go from sines and cosines the clashes and cinches by leaving out the I well if you track it through carefully you'll find that this minus sign was really an I squared it's not going to matter much I will just tell you it was really came from some I squared and if you leave out I I squared just becomes one squared is no minus sign so here's the guess for the formula connecting X prime T Prime with X and T it equals let's say X since Omega – no – plus T cos Omega in this case there are two minus signs in this case there was only one minus sign okay but but let's check what do we want to check we want to check that X prime squared minus T prime squared is equal to x squared minus T squared your ask you're probably asking yourself what is this Omega what does it have to do with moving reference frames I'll tell you right now what Omega is it's a stand-in for the velocity between the frames we're going to find the relationship between Omega and the relative velocity of the reference frames in a moment there has to be a parameter in the lower end these are the lines in these are the Lorentz transformations connecting two frames of reference in the Lorentz transformations as a parameter it's the velocity the relative velocity that parameter has been replaced by Omega it's a kind of angle relating the two frames a hyperbolic angle but we'll we'll come back to that for the moment let's prove that with this transformation law here that X prime squared minus T prime squared is equal to zero ah is equal to X square minus T squared I'm getting to that point in the evening where I'm going to make mistakes all right this is easy you just work it out you use all you have to use is that cosine squared minus sine squared is 1 you can work that out by yourself but we can just see little pieces of it here X prime squared will have x squared cos squared Omega t prime squared will have x squared sin squared Omega if I take the difference between them I'll get a term with an x squared times cos squared minus sin squared but cos squared minus sin squared is one fine so we'll find the term with an x squared when we square take the square of the difference between the squares of this and this and likewise will also find the T squared the cross term when you square X Prime you'll have XT cost cinch when you square T Prime you'll have XT costs inch when you subtract them it'll cancel and it's easy to check that's our basically one liner to show that with this transformation here x prime squared minus T's prime squared is x squared minus T squared which is exactly what we're looking for let me remind you why are we looking for it if we find the transformation for which the left-hand side and the right-hand side are equal then if x squared equals T squared in other words if the right-hand side is 0 the left-hand side will also be 0 but x squared but x equals T that's the same as something moving with the speed of light in the X frame of reference if this being 0 is equivalent to the left hand side being 0 it says that in both frames of reference the light rays move with the same velocity so that's the basic that's the basic tool that we're using here X prime squared minus T prime squared is equal to x squared minus T squared all right that does follow by a couple of lines using cos squared minus N squared equals 1 but what I want to do let's take another couple of minutes now let's take a break for five minutes and then come back and connect these variables Omega with the velocity of the moving frame of reference somebody asked me a question about the ether and what it was that people were thinking somehow Einstein never got trapped into this mode of thinking um well what were they thinking about when they were thinking about the ether what exactly was the michelson-morley experiment well I'll just spend the minute or two mentioning it certainly Maxwell understood that his equations were not consistent with with Newtonian relativity he understood that but his image of what was going on is that the propagation of light was very similar to the propagation of sound in a material or water waves propagating on water and of course it is true that if you move relative to the atmosphere or move relative to the substance that sound is propagating in you'll see sound move with different velocities depending on your motion if you're at rest in a gas of material isn't there's a natural sense in which is a particular rest frame the rest frame is the frame in which on the average the molecules have zero velocity if you're in that reference frame then first of all light has the same velocity that way as that way number one and it has a velocity that's determined by the properties of the fluid that the sound is moving in okay Maxwell more or less thought that light was the same kind of thing that there was a material and the material had a rest frame and that particular rest frame was the frame in which light would move with the same velocity to the left as to the right and he thought that he was working out the mechanics or the behavior of this particular material and that we were pretty much at rest relative to this material and that's why we saw light moving the same way to the left of the right one would have to say then that Maxwell did not believe that his equations were a universal set of laws of physics but that they would change when you moved from frame to frame just happened by some luck we happen to be more or less at rest relative to the ether to this strange material um of course you could do an experiment with sound if you're moving through the sound you can check that the velocity in different directions is different you do let's not worry exactly how you do that that's what the Michelson Morley experiment was Michelson and Morley I suppose said look the earth is going around in an orbit maybe at one season of the year we just happen to be at rest relative to the ether by accident and some other season six months later we're going to be moving in the opposite direction and we won't well we won't be at rest only at one point in the orbit could we be at rest relative the–this or at any other point in the orbit we wouldn't be so if we measure in November that light moves the same than all possible directions then in what's what's the opposite of November May then in May we should find that light is moving with great with the different velocities in different directions and he tried it and a very fancy and sophisticated way of measuring the relative velocity in different directions but he found that there was no discrepancy that the light traveled the same velocity in every direction at every time of year there were all sorts of ways to try to rescue the ether but none of them worked none of them work and the result was one had to somehow get into the heart of space and time and velocity and mid distance and all those things in a much deeper way in a way that didn't involve the idea of a material at rest in some frame of reference that that propagated the light okay oh where are we I forgotten where we were when we stopped somebody remind me whoo-hah Omega yeah what is Omega forgotten Omega Oh how Omega is really metal speed of light but to the velocity of the moving reference frame here we have two reference frames X T and X Prime and T prime what's the relationship between them well the whole goal here was to understand the relationship between frames of reference moving with relative velocity between them Omega is not exactly the relative velocity but it is closely related to it okay let's say let's see if we can work out the relationship we know enough to do it let's see if we can work out the relationship between Omega and the velocity of the moving frame all right again let's go back to this picture there's a frame of reference moving let's redraw it here's my origin moving along okay what does it mean to say that from your perspective my frame of reference so my origin is moving with velocity V well by definition this is not a law now this is a definition and says that this line here has the equation x equals VT that's the definition of this V here my origin moves relative to your origin it moves with a uniform constant velocity that's an assumption that we're talking about two inertial frames of reference and you in your frame of reference will write x equals VT that's the definition of V if you like what will I call it I will call it X prime equals zero all along there I will say X prime is equal to zero it's my origin of coordinates okay now let's come to this transformation law here and see if we can spot how to identify V well X prime equals zero that's this trajectory moving at an angle with a velocity V X prime equals zero is the same as saying X cos Omega equals T sin Omega X prime equals zero set this side equal to zero and that says that X cos Omega equals T sin Omega all right so whatever the connection between velocity and Omega is it must be such that when X prime is equal to zero X cos Omega equals T sin Omega well let's look at that equation it also says that X is equal to sin CH Omega over cos Omega times T well that's interesting because it's also supposed to be equivalent to x equals VT now I know exactly how to identify what the velocity is as a function of Omega the velocity of the moving transformation the moving coordinate system must just be sin Chi Omega over cos Omega that's the only way these two equations can be the same x equals VT x equals sin Chi Omega over cos Omega times T so now we know it we know what the relationship between velocity and Omega is write it down the velocity of the moving frame now this is not the velocity of light it's just the velocity of the moving frame must just be cinch Omega over cos omega well actually i want to invert this relationship i want to find sin and cos omega in terms of the velocity i want to rewrite these Lorentz transformations where are they i want to rewrite these Lorentz transformations in terms of the velocity that's the familiar form in which you learn about it in in elementary relativity books X prime is equal to something with velocities in it to exhibit that all we have to do is to find Cinch and cosh Omega in terms of the velocity that's not very hard let's let's work it out the first step is to square it and to write V squared is equal to cinch Omega squared over cosh Omega squared that was easy next I'm going to get rid of since Omega squared and substitute where is it I lost it one is equal to cos Omega squared minus cinch Omega squared alright so wherever I see cinch Omega squared I can substitute from here namely cosh squared Omega minus one is equal to sine squared Omega so here we are this is just equal to hash of Omega squared minus one divided by cost of Omega squared or let's multiply by what I want to do is solve for cost Omega in terms of velocity I want to get rid of all these cautions and cinches of Omega and rewrite it in terms of velocity so first x cost Omega squared we have cosh squared Omega times V squared equals cosh squared Omega minus one or it looks to me like this is cosh squared Omega times one minus V squared equals one what I've done is transpose yeah cos squared times V squared minus cos squared itself that gives you cos squared 1 minus V squared equals 1 change the sign can everybody see that the second line follows from the first I'll give you a second yeah yeah yeah it's clear ok finally we get that cos Omega is equal to 1 divided by 1 minus V squared but now I have to take the square root cos Omega / one minus V squared and then take the square root and that gives you cos Omega now we've all seen these square roots of 1 minus V squared in relativity formulas here's where it begins the kayne we begin to see it materializing what about sin Chi Omega let's also write down sin Chi Omega well from here we see that sin Chi Omega is just equal to V times cos Omega this is easy since Omega equals V times cos Omega sorrow sin Chi Omega is V divided by square root of 1 minus V squared let's go back to these Lorentz transformations over here and write them getting rid of the trigonometric functions the hyperbolic trigonometric functions and substituting good old familiar velocities let's get rid of this and substitute the good old ordinary velocities ok so we have here X prime equals x times cos Omega and that's divided by square root of 1 minus V squared then this minus T times sin Omega which is V over the square root of 1 minus V squared or if I put the two of them together and combine them over the same denominator it's just X minus VT divided by square root of 1 minus V squared I think most of you have probably seen that before maybe slightly different let's let's clean it up a little bit X prime equals X minus VT divided by the square root of 1 minus V squared what about T prime T Prime is equal to t minus V X over square root of 1 minus V squared T prime is equal to T times cos cost is just 1 over square root and then x times sin CH that gives us the extra V in other words the formulas are more or less symmetrical and those are all good old Lorentz transformations now what's missing is the speed of light let's put back the speed of light the put back the speed of light is an exercise in dimensional analysis there's only one possible way the speed of light can fit into these equations they have to be modified so that they're dimensionally correct first of all one is dimensionless has no dimensions it's just one velocity is not dimensionless unless of course we use dimensionless notation for it but if velocity is measured in meters per second then it's not dimensionless how do we make V squared dimensionless we divide it by the square of the speed of light in other words this V squared which is here which has been defined in units in which the speed of light is 1 has to be replaced by V squared over C squared likewise over here V squared over C squared now velocity times time does have notice first of all the left hand side has units of length the right hand side this is dimensionless X has units of length but so does velocity times time so this is okay this is dimensionally consistent as it is but over here it's not the left hand side has dimensions of time that's all right 1 minus V squared over C square that's dimensionless this has units of time but what about velocity times X velocity times X does not have units of time in order the given units of time you have to divide it by C square okay let's check that velocity is length all the time times length divided by C squared that's length square R which gets correct but it's correct all right this is probably familiar to most of you who've seen relativity once or twice before these are the equations relating to different moving coordinate systems moving relative to the x axis but you see the deep mathematics or the mathematical structure of it in many ways is best reflected by this kind of hyperbolic geometry here and you know most physicists by now never write down the Lorentz transformations in this form much more likely to write them in this form easier to manipulate easier to use trigonometry or or hyperbolic trigonometry it's a little exercise it's a nice little exercise to use this the hyperbolic trigonometry to compute their to compute the compounding of two Lorentz transformations if frame two is moving relative to frame one with velocity V and frame three Israel moving relative to two with velocity V Prime how is three moving relative to one the answer is very simple in terms of hyperbolic angles you add the hyperbolic angles not the velocities but the hyperbolic angles the hyperbolic angle of three moving relative to one is the hyperbolic angle of three moving relative to two plus two moving relative to one and then you use a bit of trigonometry or hyperbolic trigonometry to figure out how you do the inches and kosh's of the sum of 2 hyperbolic angles very straightforward and I'll leave it as an exercise to see if you can work that out much easier than anything else ok so there there we have the Lorentz transformations yeah oh oh absolutely yes that's that's that's a good point yeah when we that's right if we have frame 1 let's call this x1 and y1 x2 and y2 and finally x3 and y3 well then the angle of – let's call F of 3 relative to 1 let's call it theta 1 3 is just equal to theta 1 2 plus theta 2 3 the angle connecting frame one with frame 3 is just the sum of the angle theta 1 2 plus theta 2 3 so in that respect the Lorentz transformations are much simpler in terms of the Omegas it's the Omegas which combined together to add when you add velocities now how different is omega from the velocity let's work in units in which the speed of light is equal to 1 where is our formula for velocity all right let's take this formula over here what a cinch Omega 4 small Omega let's put the C squared there a let's not put the C square there or not put the C square there since Omega is essentially Omega when Omega is small just like sine is omega where is theta when theta is small the cinch function the cost function looks like like this the cinch function looks like this but it but it crosses the axis with a slope of 1 for small Omega cinch Omega is proportional to Omega for small velocity one minus V squared is very close to 1 if the velocity is a hundredth of the speed of light then this to within one ten-thousandth is just 1 if we're talking about velocities a millionth of the speed of light then this is very close to 1 and so since Omega and velocity are very close to each other it's what's going on here Thanks okay so for small velocities Omega and velocity are the same the actual correct statement is that V over C is like Omega the dimensionless velocity over the speed of light is like Omega for small Omega and small velocity so for small velocity adding velocities and adding omegas are the same things but when the velocities get large the right way to combine them to find relationships between different frames is by adding Omega and not adding velocities when you add Omega like compounding velocities as you've got it there I guess you won't go greater than 45 degrees that guess because that would be faster than light no but Omega no more you see this bit the speed of light is V equals one that corresponds to Omega equals infinity yeah yeah so Omega Omega runs over the whole range from minus infinity to infinity but when it does V goes from minus the speed of light to the speed of light so you can add any omegas and still add any omegas Omega that's right there's no there's no speed limit on Omega is this like we just go on that diagram it looks like it's greater than 45 degrees if here where where I make a and I guess they use the definition of state along the hyperbola yeah that's right sorry where are we right there today I guess that's theta though isn't it this is Theta that's a good oh god yeah right right yeah Omega is the distance along hyperbola that's right distances that's right Omega is a kind of distance along the hyperbola all right now let's let's talk about that a little bit all right now that we've established the basic mathematics structure of the transformations I think we should go back and talk about some simple relativity phenomena and derive them oh one thing which is important which I yeah well let's see we're here are my Lorentz transformations over here I said we should we ought to at the end make sure that our transformations are not too dissimilar from Newton's in particular when the velocities are small they should reduce to Newton that's all we really know that's or at least that's all that Newton really had a right to assume that when the velocities are smaller than something or other that his equations should be good approximations isn't adding velocity good enough isn't velocities adding good enough in fact you're right in fact you're right but let's just look at the transformations themselves all right as long as the velocity is a small percentage of the speed of light an ordinary velocities are what a hundred miles an hour versus 186,000 miles an hour what is that it's small right and it's doubly small when you square it so for typical ordinary velocities even the velocities of the earth around the Sun and so forth fairly large velocities what 60 kilometers per second or something like that 60 kilometers per second is pretty fast that's the that's the orbital earth around the Sun it's pretty fast but it's nowhere near 300,000 kilometers per No yeah looks here on a thousand meters per second we're I'm sorry three times ten to the eighth no three times three hundred thousand kilometers per second right 60 kilometers per second three hundred thousand kilometers per second small fraction and then square it so for ordinary motions this is so close to one that the deviation from one is negligible so let's start with the top equation for the top equation this is negligible and it's just x prime equals X minus VT the bottom equation here you have a C squared in the denominator whenever you have a C squared in the denominator that's a very very large thing in the denominator this is negligible compared to T so here the speed of light is also in the denominator just forget this and it's just T but it's just T prime equals T it's just D prime equals T so in fact Newton's formulas are essentially correct for slow velocities no no significant departure from Newton until the velocities get up to be some some appreciable fraction of the speed of light okay let's talk about proper time proper time and then let's do a couple of relativity examples yeah question the bottom equation when X is very large yes that's right when X is exceedingly large you get a correction but that correction that X has to be very large look let's let's discuss before we do anything else let's let's let's talk about that a little bit X minus VT one minus V squared over C squared yeah let's alright in my drawings I'm going to sitt C equal to one but in the equations you can leave the C there okay this equation we understand apart from this one minus V squared over C squared in the denominator it's just this x equals V T or X minus V X minus X minus VT that's Newton let's look at this one over here okay let's look at the surface T prime equals zero T prime equals zero is the set of points that I in my moving reference frame call T call time equals zero it's what I call the set of points which are all simultaneous with the origin T prime equals zero is just everyplace in space-time which has exactly the same time according to my frame of reference and I will therefore call all those points synchronous at the same time what do you say about them if T prime is equal to zero that says that T is equal to V over C squared X now let's set C equal to one for the purpose of drawing just for the purpose of drawing I don't want this huge number C squared to distort my drawings too much it says the T equals V X what does the surface T equals V X look like it looks like this T equals V X which is also X is equal to 1 over V T so it's just a uniform line like that all of these points are at different times from your reckoning this ones later this ones later this ones later and so forth according to my reckoning all these points are at the same time so we disagree about what's simultaneous this was this was the hang-up incidentally this was the basic hang-up that took so long to overcome that took Einstein to overcome it the idea that simultaneity was the same in every reference frame nobody in fact it was so obvious that nobody even thought to ask a question is simultaneous does it mean the same thing in every reference frame no it doesn't in more in your reference frame the horizontal points are all simultaneous with respect to each other in my reference frame what I call horizontal what I call simultaneous you do not okay so simultaneity had to go let me point out one more thing about these equations I'm not going to solve them for you but I will tell you the solution anyway how do you solve for X and T in terms of X Prime and T Prime well think about it in the case of angles supposing I have a relationship like X prime is equal to X cosine theta what is it plus plus y sine theta and y prime is equal to X minus X sine theta plus or Y cosine theta and supposing I want to solve for x and y in terms of X Prime and Y Prime you know what the solution is just change theta 2 minus theta and write that X is equal to X prime cosine of minus theta but what's cosine of minus theta right cosine theta plus y sine of minus theta what's sine of minus theta minus sine theta times y and likewise for y prime Y prime is equal to minus x times sine of minus theta so that becomes plus X sine theta plus y cosine of minus theta which is cosine theta you don't have to go through the business of solving the equations you know that if one set of axes is related to the other by rotation by angle theta the second one is related to the first one or vice versa the first one is related to the second one by the negative of the angle if to go from one frame to another you rotate by angle theta and to go from the second frame back to the first you rotate by angle minus theta so you just write down exactly the same equations interchange Prime and unprimed and substitute for theta minus theta same thing for the Lorentz transformations exactly the same thing if you want to solve these for X and T write down the same equations replace primed by unprimed and change the sign of omegas to minus the sines of omegas change sinus rgn of all the sign all the cinches okay in other words just send Omega 2 minus Omega and that will solve the equations in the other direction yeah yes it's also the same as changing V 2 minus V yes the way to see that is to go right what was it what do we have cosh Omega yep yeah that's right via sign yes that was correct yeah you just well you change Omega 2 minus Omega it has the action of changing V 2 minus V you can just check that from the equations good alright let's let's talk about proper time a little bit proper time if you're doing ordinary geometry you can measure the length along a curve for example and the way you do it is you take a tape measure and you you know sort of take off you take off equal intervals equal equal little separations you can think of these separations as differential distances DS squared small little differential distances and that differential distance is d x squared plus dy squared with the x squared and the y squared are just the differential increments in x and y DX and dy this is d s alright so that's the way and you add them up you add them up that's the way you compute distances along curves it's quite obvious that if you take two points the distance between those two points depends on what curve not the same for every curve so I'll measure the longer curve you have to know not only the two points but you have to know the curve in order to say what the distance between those points are of course the distance between its longer straight line that's that's well-defined but the distance along a curve depends on the curve in any case D s squared equals the x squared plus dy squared is the basic defining notion of distance between two neighboring points if you know the distance between any two neighboring points in a geometry you basically know that geometry almost essentially completely so given this formula for the distance between two points you can compute if you like the distance along a curve because you've got to take the square root of this and then add them up don't anhedonia the squares add the differential distances all right the important thing is here that square root of DX squared plus dy squared which is the distance between neighboring points doesn't depend on your choice of axes I could choose X Y axes I could choose X prime y prime axes if I take a little differential displacement the X and the y or I just take two points two neighboring points don't even give them labels and measure the distance between them the distance between them should not depend on conventions such as which axes are used and so when I make rotational transformations the X square plus dy squared doesn't change the X and the y may change but the x squared plus dy squared does not change the same thing is true in relativity or the analogous thing we don't measure distances along the paths of particles let's say now that this curve here is the path of a particle moving through space-time there's a particle moving through space-time and we want some notion of the distance along it the notion of distance along it another example would just be a particle standing still as a particle standing still particle standing still is still in some sense moving in time I wouldn't want to say that the distance between these two points and space-time is zero they're not the same point I wouldn't like to say it's zero I would like to say there's some kind of notion of distance between them but it's quite clear that that distance is not measured with a tape measure this point and this point are the same point of space boom here at this point of space and that at a later time boom again at the same point of space two events at the same point of space how do I characterize and some nice way the distance between those two events that occurred in the same place you don't do it with a tape measure all right what do you do with a clock a clock you take a clock and you start it at this point tic tic tic tic tic tic tic a stopwatch you press it at this point tic tic tic tic tic it picks off intervals and then you stop it at that point and you see how much time has evolved that's a notion of distance along a particle trajectory it's not the distance the particle moves in space it's a kind of distance that it's moved through space-time and it's not zero even if the particle is moving standing perfectly still in fact what it is is it's the time along the trajectory what about a moving particle well you can imagine that a moving particle carries a clock with it of course not all particles carry clocks but we can imagine they carry clocks with them as they move and we can start the clock over here and then the clock over here what is the time read off by this moving clock the time read off by a moving clock is much like the distance along a curve measured by a tape measure in particular it should not depend on the choice of coordinates why not this is a question that has nothing to do with coordinates I have a clock made in the standard clock Factory the standard clock Factory and I don't know we're in Switzerland someplace makes a certain kind of clock that clock gets carried along with a particle and we ask how much time evolves or how much time elapses or how much the clock changes between here and here that should not depend on a choice of coordinates it shouldn't depend on a choice of coordinates because it's a physical question that only involves looking at the hands of the clock in fact we can ask it for little intervals along along the trajectory we could ask how much time elapses according to the clock between here and here well the answer again should not depend on what coordinates you use which Lorentz frame you use and there's only one invariant quantity that you can make out of the D X's and DTS describing this point describing these two points there's a little interval DT and there's a little interval DX now we're in space and time not ordinary not ordinary space and the quantity which is invariant there's really only one invariant quantity that you can make out of it it is DT squared minus DX squared it's the same quantity x squared minus T squared for a whole you know for a whole interval the T squared minus DX squared that's the quantity which is invariant it's minus D it's the negative of what I wrote over here x squared minus T squared okay this quantity is equal to the X prime squared minus DT power sorry DT prime squared minus the X prime squared the same algebra goes into this as goes into showing that X prime squared minus T prime squared equals x squared minus T squared incidentally this is the same as saying T prime squared minus X prime squared equals T squared minus x squared doesn't matter which way you write it all right so that suggests that suggests that the time read off the invariant time read off along a trajectory between two points separated by DX and DT is just the square root of DT squared minus DX squared why the square-root incidentally okay you're going to integrate in detail I can integrate DT yeah well alright why not just DT square minus the x squared for the time between here and here is it here's an answer supposing we go to you two intervals exactly the same as the first one we go an interval over here DX and DT and then we go another DX in DT what happens when we double the interval to DT squared minus DX squared it gets multiplied by four because everything is squared well I wouldn't expect a clock when it goes along you know when it goes along a trajectory for twice the the interval here to measure four times the the time I expected to measure twice the time so for that reason the square root is the appropriate thing here okay that's called D tau squared the tau squared the proper time along the trajectory of an object you're right that's just the towel or D tau squared being the x squared minus DT squared the Tau is called the proper time let's go I think we'll let's see the towel is called the proper time and it is the time read by a clock moving along a trajectory it's not just DT that's the important thing it's not just DT the T squared minus the x squared let's do one last thing let's just do the twin paradox in this language I think I think I've had it I'm going to finish you can do the twin paradox in this language all you have to do is to compute the proper time along two trajectories one that goes out with a uniform velocity turns around and comes back with the same uniform velocity versa a trajectory which just goes from one point to the st. the another point along a straight line and it's no more weird it's no weirder really from this perspective than saying the distance from one point to another along two different curves do not have to agree the proper time along two different curves in general will not agree what is a little bit weird is that because of this minus sign the proper time this way is less than the proper time this way that's the consequence of this minus sign here moving with some DX decreases the proper time all right we'll do a little bit more next time but then I want to get to the principles of field theory and and connect some of this with field equations for interesting wave fields the preceding program is copyrighted by Stanford University please visit us at stanford.edu

Why can't you go faster than light?

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One of the most counterintuitive facts of our universe is that you can’t go faster than the speed of light. From this single observation arise all of the mind-bending behaviors of special relativity. But why is this so? In this in-depth video, Fermilab’s Dr. Don Lincoln explains the real reason that you can’t go faster than the speed of light. It will blow your mind.

over the past hundred years or so scientists have pushed our understanding of the universe into some extreme conditions for example the world of the very small the realm of very high speeds and under the frigid conditions of near absolute zero while each of us have developed an intuition about how the world works it's very important to remember that this intuition only applies to a very limited set of conditions for instance there's absolutely no reason to expect that matter will act the same in the center of the Sun as it does here on earth on a bright and sunny day however that last statement is hard for some people to accept and judging by my email INBOX the extreme realm that causes people the most difficulty is what happens when things are going super fast in 1905 Albert Einstein published his theory of special relativity it predicts all sorts of mind-blowing things for instance distance shorten and clocks slowed down I made another video about how clocks act at high speed it turns out that all of those seemingly crazy implications originate from a single cause or maybe two if we take it slow so first let me tell you what this video isn't it doesn't tell you about the postulates that Einstein used to build this intuition and it certainly doesn't derive as equations instead this video tries to tell you the key insights that make it easier to develop a relativistic intuition I hope to teach you why it is impossible to go faster than the speed of light if you're not a physics groupie hearing that there's a maximum speed in the universe might surprise you but it's true and if you are a groupie you've probably heard that the reason that you can't go faster than light is due to the fact that mass increases when you speed up it turns out that the explanation of mass changing as you go faster is a wrong one I know that statement is going to confuse some people including those with fairly sophisticated understandings of relativity but it's true however that then leaves an open question just why is it that you can't go faster than speed of light it turns out to be due to a combination of a deep and fundamental property of the universe and fairly simple geometry so let me explain how that all works the first two the two crucial insights is that Einstein taught us the space and time were not separate entities but rather they are two components of a bigger idea called space-time I'll give you a helpful visual way to think about this in a moment but for right now just trust me on this then we need to combine that insight with the observation that everybody sees the speed of light to be the same no matter how fast they're moving with respect to one another let's start with an analogy and then come back to relativity to understand the analogy you need to imagine a car driving on a huge flat surface further you need to imagine that the car can only move at one speed say 60 miles per hour or so the comments don't fill up with a metric snobbery hate-mail 100 kilometers per hour now let's put a couple of arrows on the screen to point out north and east well we know the overall speed the car is going we don't know how much of it is in the east direction and how much of it is in the north direction so let's take a closer look at that the car can move entirely in the eastward direction which means that it has no motion in the northward direction or the car can move entirely northward and not at all eastward or we can live dangerously and move towards the Northeast in this case we see that the car is moving in both the east and north directions with neither direction getting all of the motion so that's the core analogy and hopefully it's very clear now let's bring in relativity and relativity we don't have the east and north directions instead we have space-time let's imagine that the horizontal direction of space and the vertical direction is time so suppose that there is a single and fixed speed that we can travel through space-time this happens to be true so it's not a ridiculous supposition we can therefore mix these ideas with our earlier analogy an object can move vertically in that case there moving through space and they're moving entirely through time that's probably what you're doing right now you're sitting and watching this video so your position in space isn't changing however you are experiencing time you aren't moving through space but you're moving through time on the other hand what happens as you start moving through space that's a fancy way to say that you've gained some velocity well we see here that what starts to happen is that as you begin to move through space you move less through time and eventually when you move only through space you don't move through time at all and this is basically what relativity says as you move faster and faster your clocks slowed down and as you get very close to the speed of light your clocks very nearly stopped we've scientifically proven that this is what happens and I direct you to my video on time dilation so you can see one way that we've tested that so this brings us to our fundamental realization of relativity the reason that we can't move through space faster than the speed of light is because we're constantly moving through space time at a single speed the speed of light if we aren't moving through space we experience time in the fastest way and if we start moving through space we experience time slower and slower finally since we're moving through space time at a single speed that means when we're only moving through space there's no more speed to gain we move through space at the speed of light and that's it this observation wasn't made by Einstein it was made by his mentor Hermann Minkowski Minkowski was one of Einstein's mentors and he was a better mathematician two years after Einsteins a seminal 1905 paper Minkowski appreciated the geometrical underpinnings of special relativity and had determined this deep and fundamental explanation why we can't travel faster than light through space there are two final important points first while Minkowski showed why Lightspeed is the maximum speed through space what he didn't explain was why we move only at one speed through time to this day nobody really knows it seems to be a fundamental property of space-time maybe it will take another person as smart as Einstein to figure out that particular conundrum the second point is more technical and I mention it only for the real physics nerds in my analogy I connected space and time as being similar to east and north and there's a lot of merit in that morphing from motion through time to motion through space was like turning a car from moving north to moving east however this analogy is also technically inaccurate from a mathematical point of view it uses the geometry of circles well the proper geometry is that of hyperbolas I only bring this up because I want you to know my analogy is imperfect and you shouldn't push it too far otherwise you might come to a numerically incorrect conclusion and think that you've made a new discovery if you want to dig into this more deeply be sure to use the full and proper Minkowski mathematics still even with the limitations I mentioned the core point is valid the reason that you can't move faster through space than the speed of light is because every object moves through space-time at one and only one speed the speed of light once you've embraced that central idea and the fact that space and time are just like two directions of space-time then all of those seemingly weird observations of relativity just click into place and special relativity makes total sense so I don't know about you but I think this insight about relativity is just about the coolest thing ever if you liked this video be sure to LIKE subscribe and share let's get those numbers up and let me know what you think in the comments I'll see you next time and keep on physics

General Relativity & Curved Spacetime Explained! | Space Time | PBS Digital Studios

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The Final Installment of our General Relativity Series!!!

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We’ve been through the first few episodes of our crash course on general relativity, and came out alive! But it’s officially “time” for CURVED spacetime. Join Gabe on this week’s episode of PBS Space Time as he discusses Newton and Einstein’s dispute over inertial frames of reference. Is Einstein’s theory inconsistent? Is gravity even a force??? Check out the episode to find out!

Previous Installments of the General Relativity Series:

“Are Space And Time An Illusion?”:

“Is Gravity An Illusion?”

“Can A Circle Be A Straight Line?”

“Can You Trust Your Eyes In Spacetime?”:

Let us know what topics you want to learn more about:

well we're finally here a synopsis of general relativity that builds on these previous four episodes if you haven't seen them then pause me now go watch them in order and meet me back here after the music to hear about curved space-time mutants and Einstein's dispute over gravity comes down to competing notions of what constitutes an inertial frame of reference Newton says that a frame on Earth's surface is inertial and relative to that frame a freely falling Apple accelerates down because it's pulled by a gravitational force but Einstein says nah it's the apples frame that behaves like a frame in deep space so the apples frame is inertial and the earth frame is actually accelerating upward you just get a false impression of a gravitational force downward for the same reason that a train car accelerating forward gives you a false impression that there's a backward force so who's right well between our gravity illusion episode and your comments we've seen that Einstein's position seems internally inconsistent remember that inertial frame in Zeke's face well the Apple accelerates relative to it so even Urschel frames define the standard of non acceleration how can both of those frames be inertial today we're finally going to show how curved space-time makes einstein's model of the world just a self-consistent as newton's step one is to express both Newtons and einstein's view points in geometric space-time terms since that's the only way to compare them in a reliably objective way remember humans experience the world and talk about the world dynamically as things moving through space over time but even in a world without gravity we already know that clocks rulers and our eyes can all mislead us so to be sure we're talking about real things as opposed to just artifacts of our perspective we have to translate dynamical statements into tense lists statements about static geometric objects in 4d space-time let's start with Newton he says that space-time is flat just think about it on the flat space-time diagrams of inertial observers the world lines of other inertial observers are straight indicating constant spatial velocity this captures Newton's idea that inertial observers shouldn't accelerate relative to other inertial observers Newtonian gravity would just be an additional force we introduced like any other that would cause some world lines to become curved II's facially accelerated this is a bit oversimplified but for today it'll do now for Einstein's position this is actually more subtle and it'll be easier to explain if I first set up an analogy using our old friend the 2-dimensional ant on the surface of the sphere a tiny patch at the equator looks like a plane and within that patch two great circles both look straight but suppose the ant believes that he lives on an actual plane and decides to draw an XY grid on a large patch of the sphere with its x-axis along the equator and the y axis along a longitude line relative to this grid the second grade circle looks bent so the ant concludes that it's not a geodesic but you see the ants mistake right his grid is distorted you can't put a big rectangular grid on a sphere without bunching it up try it with some graph paper and a basketball it doesn't work stated another way a sphere can accommodate local Euclidean grids and tiny patches but not global ones so the ant can use his axes as rulers and protractors within a patch but not between patches flat space definitions of straightness apply over small areas but not big ones okay Einstein's position is that Newton is making the same mistake as the ant inertial frames that means axes plus clocks are the spacetime equivalent of ants XY grid if space-time is curved then those frames are only valid over tiny space-time patches so when an observer in deep space says that the falling Apple is accelerating he's pushing his frames past the point of reliability just like the ant did in other words global inertial frames don't exist in space-time however global inertial observers do their observers that have no forces on them their world lines will be geodesics and their axes and clocks can serve as local inertial frames provided that we think of them as being reset in each successive space-time patch and by the way pictures like this are not intended to make literal visual sense on the contrary they're designed to break your excessive reliance on your eyes so that your brain becomes more free to accept what reality isn't remember no one can really see or draw space-time there is no spoon now the world line of a falling Apple turns out to be a geodesic it has no forces on it so there's no need to invent gravity okay but what about two apples in a falling box like at the end of our gravity illusion episode remember they get closer as the box fault now according to mutant that happens because the apples fall radially instead of down but according to Einstein it happens because the apples are on initially parallel geodesics that since space-time is curved can and do cross just like on the sphere in contrast the world line of a point on Earth's surface is not a geodesic it has a net force on it and it's really accelerated so does that mean that Earth's surface has to be expanding radially well be careful in order to compare distant parts of Earth you'd need a single frame that extends across space-time patches but that frame can't be inertial so any conclusions you base on it have to be interpreted with a heavy grain of salt okay so Einstein's gravity free curved space-time sounds like it's self consistent but then again so does Newton's flat space-time picture that has gravity injected as a kicker so once again which of them is right the answer is whoever agrees better with experiments and there's over a century of experiments to refer to now we haven't probably fleshed out all of general relativity yet but there's one experimental fact that I can use to show you that space-time must be curved just based on what we've seen in this series of episodes so far it's a cool argument originally presented over 50 years ago by physicist Alfred shild and it goes like this fire a laser pulse from the ground floor of a building up to a photon detector on the roof now wait five seconds and then do it again on a flat space-time diagram the world lines of those photons should be parallel and congruent without making any assumptions about how gravity affects light that would be true even if it turned out that gravity slowed photons down and bent their world lines since both photons would be affected identically now space-time is flat then clocks on the ground and on the roof should run at the same rate they're both stationary thus the vertical lines at the ends of the photon world lines should also be parallel and congruent but if you actually do this experiment you find that photons arrive on the roof slightly more than five seconds apart the excess time is less than a nanosecond but any discrepancy means that clocks are running at different rates in which case the opposite sides of this parallelogram aren't congruent and that's geometrically impossible if space-time is flat thus the very existence of gravitational time dilation regardless of its degree requires that space-time be curved and that means game over for Newton in fact to the extent that we can speak about space sometimes separately at all most of the everyday effects on earth the Newton would attribute to gravity are due to curvature in time the 3d space around earth is almost exactly Euclidean those pictures that you see of Earth deforming a grid the way a bowling ball deforms a rubber sheet or even the pictures we sometimes use on this show they all suggest spatial curvature only so they're somewhat misleading remember a frame consists of axes and clocks and around Earth space-time curvature manifests itself seam clocks much more than in rulers so even though it's hard to visualize it's curved time that makes the freefall orbits of satellites look spatially circular in frames of reference that cover too big a space time patch so why is space-time curved in the first place unfortunately the math gets heavier here and good analogies are harder to come by but here's the flow chart level answer consider a region of space-time and remember that means a collection of events not just locations its curvature and geodesics are determined by how much energy is present at those events via a set of rules called no surprise the Einstein equations so for example say you stick the energy distribution of the Sun into the Einstein equations and turn a crank what comes out is a map of the geodesics in the sun's space-time neighborhood now when you translate those geodesics into 3d spatial and temporal terms what you find is planetary orbits or spatially straight radially inward trajectories along which you would see spatial speed increase or pretty much anything else that you would otherwise attribute to a gravitational force it's pretty amazing I want to conclude with a question once asked by one of our viewers Evan Hughes if there's no gravity and gravity is not a force and why do we keep using that word well physicists are still human as far as I know most of us have no special ability to visualize or directly experience 40 space-time so we often think in Newtonian gravitational terms because it's easier and because the resulting errors are usually small we just remind ourselves that it's just a crutch that we have to use with caution but even when people are referring to relativity or string theory or whatever it's just a lot easier to say the word gravity than to say curvature or four-dimensional space-time you

What is a Black Hole? — Black Holes Explained

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BREAKING NEWS: First image of a Black Hole captured by the Event Horizon Telescope!
If you’re looking for a video to explain Black Holes to kids, we recommend this excellent video from Socratica Kids:

Black Holes are super dense regions of space. They have such immense gravity that anything close by is sucked in, never to escape. Even LIGHT! That’s why we call the BLACK HOLES.

Some black holes are small. Some are HUGE! There are stellar black holes, about the size of a dozen of our suns. Then there are SUPERMASSIVE black holes, that are the size of MILLIONS and MILLIONS of our suns! Scientists think there are supermassive black holes in the center of each galaxy. The supermassive black hole in the center of our Milky Way galaxy is called Sagittarius A. Recently, the Event Horizon Telescope was able to image the supermassive black hole in the center of the distant galaxy Messier 87. This black hole is known as M87, and was the first black hole ever to be imaged using the giant radio telescope EHT.

In this video, we talk about how Black Holes are formed and how we can detect them using various methods.
You can jump to chapters in our video here:

1:03 History of Black Holes
1:49 Theory of Relativity by Einstein – Spacetime
3:12 Are black holes real?
3:48 Finding black holes
4:13 Binary Stars
4:54 Schwarzschild Radius and the Event Horizon
5:24 Finding the mass of a black hole
5:56 Stellar and supermassive black holes
6:44 Angular momentum
7:37 The Ergosphere
8:09 Electric Charge of a black hole
8:42 No Hair Theorem
9:04 Inside a black hole
10:09 Current research topics

We highly recommend the Astronomy textbook “Universe” by Roger Freedman et al:
Universe 9th Edition
Universe 10th Edition

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#BlackHoles #Astronomy #EventHorizon

from faraway stars our tiny points of light but up close stars are massive seething fiery balls of burning gas this fierce display does not last forever eventually the nuclear fusion which powers the star will burn all its fuel gravity then collapses the remaining matter together for very large stars what happens next is a display of extremes first the star explodes in a supernova scattering much of its matter throughout the universe for a brief moment the dying star outshines its entire galaxy for once the light fades and darkness returns the remaining matter forms an object so dense that anything that gets too close will completely disappear from view this is a black hole the idea of a black hole originated hundreds of years ago in 1687 Isaac Newton published his landmark work known as the Principia here he detailed his laws of motion and the universal law of gravitation using a thought experiment involving a cannon place on a very tall mountain Newton derived the notion of escape velocity this is the launch speed required to break free from the pull of gravity in 1783 the English clergyman John Mitchell found that a star five hundred times larger than our Sun would have an escape velocity greater than the speed of light he called these giant objects dark stars because they could not emit starlight this idea lay dormant for more than a century then in the early 20th century Albert Einstein developed two theories of relativity that changed our view of space and time the special theory and the general theory the special theory is famous for the equation e equals MC squared the general theory painted a new and different picture of gravity according to the general theory of relativity matter and energy Bend space and time because of this objects which travel near large mass will appear to move along a curved path because of the bending in space-time we call this effect gravity one consequence of this idea is that light is also affected by gravity after all of space-time is curved then everything must follow along a curved path including light Einstein published his general theory of relativity in 1915 and while Newton's theory of gravity could be expressed using a simple formula Einstein's theory required a set of complex equations known as the field equations only a few months after Einstein's publication the German scientist Karl Schwarzschild found a surprising solution according to the field equations an extremely dense ball of matter creates a spherical region in space where nothing can escape not even light a curious result but did such things actually exist at first the idea of a black sphere in space from which nothing could escape was considered purely a mathematical result but one which would not really happen however as the decades passed our understanding of the life cycle of stars grew it was observed that some dying stars became pulsars another exotic object predicted by theory this suggested that dark stars could actually be real as well these strange spheres were named black holes and scientists began the hard work of finding them describing them and understanding how they are created but how do you find an object in space that is completely black luckily because black holes have a large mass they also have a large gravitational field so while we may not be able to see a black hole we can observe its gravity pulling on its neighbors with this in mind astronomers looked for a place where a visible star and the black hole were in close proximity to one another one such place is binary stars a binary star is a system of two stars orbiting one another we can spot them in several ways you can look for stars that change position back and forth ever so slightly alternatively if you observe a binary star from the side the brightness will change when one star passes behind the other so it's possible that somewhere in space there's a binary star consisting of a black hole and a visible star in fact such binary systems have been observed astronomers have found stars orbiting an invisible companion from the size of the visible star and its orbit astronomers calculated the mass of its invisible neighbor it fit the profile of a black hole since we can't see a black hole is there a way to find its size from Einsteins field equations we know that given the mass of a black hole we can determine the size of the sphere that separates the region of no escape from the rest of space the radius of the sphere is called the Schwarzschild radius in honor of Karl Schwarzschild the surface of the sphere is called the event horizon if anything crosses the event horizon it's gone forever hidden from the rest of the universe this means once you know the mass of a black hole you can compute its size using a simple formula and it's actually quite easy to measure the mass of a black hole just take a standard-issue space probe and shoot it into orbit around the black hole just like any other system of orbiting bodies like the earth orbiting the Sun or the moon orbiting the Earth the size and period of the orbit will tell you the mass of the black hole if you don't have a space rope handy then compute the mass and orbit of a companion star and use that to find the Schwarzschild radius black holes come in many sizes if it was made from a dying star then we call it a stellar mass black hole because its mass is in the same range as stars but we can go bigger much bigger and to do so we're going to visit the center of a galaxy galaxies can contain billions and billions of stars all orbiting a central point scientists now believe that in the center of most galaxies lives a black hole which we call a supermassive black hole because of its tremendous mass the size can vary from hundreds of thousands to even billions of solar masses for example at the center of our own Milky Way galaxy is a supermassive black hole with a mass 4 million times that of our Sun black holes have another property we can measure their spin just like the planets stars rotate and different stars spin at different speeds imagine we can adjust the size of the star but keep the mass constant if we increase the radius the spinning slows down if we decrease the size the spinning speeds up but while the rotational speed can vary the angular momentum never changes it remains constant even if the star were to collapse into a black hole it would still have angular momentum we could measure this by firing two probes into opposite orbits close to the black hole because of their angular momentum black holes create a spinning current and space-time the probe orbiting along with a current will travel faster than the one fighting it and by measuring the difference in their orbital periods we can compute the black hole's angular momentum this space-time current is so extreme it creates a region called the ergosphere where nothing including light can overcome it inside the Ergo sphere nothing can stand still everything inside this region is dragged along by the spinning space-time the event horizon fits inside the Ergo sphere and they touch at the poles so in one sense black holes are like whirlpools of space-time once inside the Ergo sphere you are caught by the current and after you cross the event horizon you disappear one final property of black holes we can measure is electric charge while most of the matter we encounter in our day-to-day lives is uncharged a black hole may have a net positive or negative charge this can easily be measured by seeing how hard the black hole pulls on a magnet what charged black holes are not expected to exist in nature this is because the universe is teeming with charged particles so a charged black hole would simply attract oppositely charged particles until the overall charge is neutralized there are three fundamental properties of a black hole we can measure mass angular momentum and electric charge it is believed that once you know these three values you can completely describe the black hole this result is humorously known as the no-hair theorem since other than these three properties black holes have no distinguishing characteristics it's not a blonde brunette or a redhead we now have a good idea of a black hole from the outside but what does it look like on the inside unfortunately we can't send a probe inside to take a look once any instrument crosses the event horizon it's gone but don't forget we have Einstein's field equations if these correctly describes space-time outside the black hole then we can use them to predict what's going on inside as well to solve the field equation scientists consider two separate cases rotating black holes and non rotating black holes non rotating black holes are simpler and were the first to be understood in this case all the matter inside the black hole collapses to a single point in the center called a singularity at this point space time is infinitely warped rotating black holes have a different interior in this case the mass inside of black hole will continue to collapse but because of the rotation it will coalesce into a circle not a point this circle has no thickness and is called a ring singularity black hole research continues to this day scientists are actively investigating the possibility that black holes appeared right after the Big Bang and the idea that black holes can create bridges called wormholes connecting distant points of our universe we know a great deal about black holes but there are still many mysteries to be solved it's a little-known fact that all YouTube videos are stored in the special fabric called play time when you watch a video it sends ripples of energy throughout play time and when you subscribe to a channel it creates a teeny tiny black hole so if you like black holes then you know what it is

Einstein's Theory Of Relativity Made Easy

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… Albert Einstein’s Theory of Relativity (Chapter 1): Introduction.

The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word “relativity” is sometimes used in reference to Galilean invariance.

The term “theory of relativity” was coined by Max Planck in 1908 to emphasize how special relativity (and later, general relativity) uses the principle of relativity.

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Special relativity is a theory of the structure of spacetime. It was introduced in Albert Einstein’s 1905 paper “On the Electrodynamics of Moving Bodies” (for the contributions of many other physicists see History of special relativity). Special relativity is based on two postulates which are contradictory in classical mechanics:

1. The laws of physics are the same for all observers in uniform motion relative to one another (principle of relativity),
2. The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the source of the light.

The resultant theory agrees with experiment better than classical mechanics, e.g. in the Michelson-Morley experiment that supports postulate 2, but also has many surprising consequences. Some of these are:

• Relativity of simultaneity: Two events, simultaneous for one observer, may not be simultaneous for another observer if the observers are in relative motion.
• Time dilation: Moving clocks are measured to tick more slowly than an observer’s “stationary” clock.
• Length contraction: Objects are measured to be shortened in the direction that they are moving with respect to the observer.
• Mass-energy equivalence: E = mc2, energy and mass are equivalent and transmutable.
• Maximum speed is finite: No physical object or message or field line can travel faster than light.

The defining feature of special relativity is the replacement of the Galilean transformations of classical mechanics by the Lorentz transformations. (See Maxwell’s equations of electromagnetism and introduction to special relativity).


General relativity is a theory of gravitation developed by Einstein in the years 1907–1915. The development of general relativity began with the equivalence principle, under which the states of accelerated motion and being at rest in a gravitational field (for example when standing on the surface of the Earth) are physically identical. The upshot of this is that free fall is inertial motion; an object in free fall is falling because that is how objects move when there is no force being exerted on them, instead of this being due to the force of gravity as is the case in classical mechanics.

This is incompatible with classical mechanics and special relativity because in those theories inertially moving objects cannot accelerate with respect to each other, but objects in free fall do so. To resolve this difficulty Einstein first proposed that spacetime is curved. In 1915, he devised the Einstein field equations which relate the curvature of spacetime with the mass, energy, and momentum within it.

Some of the consequences of general relativity are:

• Time goes slower in higher gravitational fields. This is called gravitational time dilation.
• Orbits precess in a way unexpected in Newton’s theory of gravity. (This has been observed in the orbit of Mercury and in binary pulsars).
• Rays of light bend in the presence of a gravitational field.
• Frame-dragging, in which a rotating mass “drags along” the space time around it.
• The Universe is expanding, and the far parts of it are moving away from us faster than the speed of light.

Technically, general relativity is a metric theory of gravitation whose defining feature is its use of the Einstein field equations. The solutions of the field equations are metric tensors which define the topology of the spacetime and how objects move inertially.

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relativity relativity is just a method for two people to agree on what they see if one of them is moving and since we all move about pretty regularly we can find many examples of how useful relativity is in everyday life even if we don't call it by name one miracle of modern life is the global positioning system or GPS it is pretty amazing that the GPS can pinpoint your location anywhere on earth to within a few yards and this magic depends entirely on the existence of the two dozen satellites 12,000 miles above the earth and a little relativity briefly here's how it works the GPS receiver gives a timing signal from several different high flying satellites and using Einstein's theory of relativity it calculates the distance from each satellite throw in a little triangulation and I'll come to your location simple and concept but to do this successfully the timing signals must be accurate to a few billions of a second so that the distance calculations can be accurate to a few yards but with all this motion going on time and distance must be reconciled carefully without an Stein's version of relativity the accuracy of the global positioning system would drift more than seven miles every day but of course relativity was not a new concept with Einstein the problem of how two people reconcile their observations about the world if one of them is moving has been addressed for centuries let's easier way into relativity with some common experiences if you are travelling in a car on a smooth straight stretch of highway there's no sensation of motion at all you mean I could read a book or a drink flip a coin and everything looks and feels the same as if the car we're sitting still that's because relative to the car view of the book the drink and the coin are not moving notice that this works only if the car is not changing direction or speed so if the car accelerates or turns pouring that drink becomes a real problem but constant motion feels just like sitting still and if you want to know what it feels like to move at a thousand miles per hour just look around because of the Earth's spin we zip along our time zone at a speedy 1,000 miles per hour and because of its motion around the Sun the earth carries us through space about 67,000 miles per hour and because of the motion of our solar system about the center of our galaxy we are moving at more than half a million miles an hour but it's not enough to ask how fast am i moving we must ask how fast am i moving relative to some other thing let's make up a simple rule that allows two observers to agree on how fast something is moving we begin at a moving walkway at the airport the walkway is moving at a brisk 3 miles per hour so if Susan simply stands on the walkway she is moving at 3 miles per hour relative to Sara who is standing still but not on the walkway if Susan walks on the walkway at 3 miles per hour she can accurately say she is walking at 3 miles per hour but Sara sees her moving at 6 miles per hour and if Susan walks against the walkway at 3 miles per hour Susan can still say she's walking at 3 miles per hour but now Sara sees her as standing still zero miles per hour so our first conclusion is that two observers can simply add or subtract their speed with respect to each other to any measurement of velocity they make this idea is the basis of classical relativity here's another scenario suppose there's a truck moving down the road at a constant speed of 50 miles per hour on the back or a baseball pitcher a catcher and their pitching coach armed with the speed gun as long as the truck doesn't speed up or slow down or hit any large bumps they can conduct pitching practice just the same as they would on the baseball field and when the pitcher throws a 100 mile-per-hour fastball the coaches speed gun will read 100 miles per hour the ball is indeed moving 100 miles per hour relative to the pitcher the catcher the coach and the truck but suppose an observer standing by the side of the road plucks the speed of that same baseball what speed would this observer measure for the ball well the ball would already be moving at 50 miles per hour when the pitcher was just holding it so this observer would measure a speed of a hundred and fifty miles per hour for the pitch the speed of the ball relative to the truck plus the speed of the truck relative to the observer the example of adding velocities in the bullet and plane example is classical relativity at its finest this classical version of relativity simply add in the velocities worked perfectly well for centuries for describing horse carts and ships or baseballs and trucks even airplanes and rockets and bullets but the relativity of classical physics is merely a very close approximation to reality at very very fast speeds classical relativity breaks down but this wouldn't be clear until scientists began flying Sopwith camels and examining the nature of the fastest known thing light

Time Is But a Stubborn Illusion – Sneak Peek | Genius

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Watch an exclusive sneak peek from the first episode of Genius, starring Geoffrey Rush as the older Einstein and Johnny Flynn as the younger.
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From Executive Producers Brian Grazer and Ron Howard, National Geographic’s first scripted anthology series, GENIUS, will focus on Nobel Prize-winning physicist Albert Einstein. The all-star cast includes Geoffrey Rush, Johnny Flynn, and Emily Watson.

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Time Is But a Stubborn Illusion – Sneak Peek | Genius

National Geographic

What is time? A deceptively simple
question, yet it is the key to understanding relativity. It is sort of the reason
my hair is going gray. [laughter] When we describe
motion, we do so as a function of time, 10 meters
per second, 100 miles per hour. But the mathematical
description of velocity is moot unless we
can define time. Is time universal? In other words, is there
an audible tick-tock throughout the galaxy, a
master clock, so to speak, forging ahead like
Mozart's metronome? The answer my friends is no. Time is not absolute. In fact, for us, the living
physicists, the distinction between the past,
present, and future is but a stubborn illusion. [music playing] A lot to consider, I know. I know. [laughter] But understanding time is
essential to understanding relativity. Now, I want you all
to close your eyes. Not to worry, I don't bite. But I am on the
lookout for a new pen. [laughter] Go on close your eyes. To truly grasp the idea of
time, we must take a step back and ask, what is light? So journey with me to the Sun. Light travels from the Sun to
the Earth through space, yes. When I was your age,
I wanted to know how can something, light,
travel through nothing, space? Let us isolate a light beam
and travel alongside it. But let us go faster. You're there with me. Faster. Faster! What is time? [thud] PROFESSOR WEBER: Herr,
Einstein, wake up! I wasn't sleeping, sir. I was thinking. Oh, really. About what exactly? The secrets of the
cosmos, I suppose. I suggest you think
about trigonometry instead, with your eyes open. And sit up! Laws of sines and cosines? c squared equals a squared
plus b squared, subtract 2 [inaudible] cosine b. PROFESSOR WEBER: The
area of a triangle? STUDENTS: The area equals
b squared times a times b over 2 times c. PROFESSOR WEBER: What is
the solution [inaudible] differential equation? Herr Einstein, are you
still too busy contemplating the secrets of the cosmos
to solve this equation? Oh, no sir. I've already solved it. PROFESSOR WEBER: Leave, now. On what offense? PROFESSOR WEBER: Your mere
presence spoils the respect of the class for me! That is not an
objective reason. Out! [music playing] The natural log of a
constant multiplied by x equals the natural log
of 1 plus v squared. And since v equals y
over x, that gives us the final function, x
squared plus y squared minus c x cubed equals 0. And speaking truthfully,
sir, your mere presence spoils my respect for the
future of Prussian mathematics. Out. [door slamming]

Why You Can Never Reach the Speed of Light: A Visualization of Special Relativity

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This video is an entry for the Breakthrough Junior Challenge 2015 which gives a unique visualization of Special Relativity using hyperbolic geometry.

This idea was inspired by the famous woodcut by M.C. Escher Circle Limit III.

I liked the idea of making a video on Special Relativity because I had already explored the use of M.C. Escher’s woodcut Circle Limit III as a teaching tool for explaining the hyperbolic geometry of Minkowski spacetime. The main intuition is that the principle of Relativity asserts that the manifold of frames of reference is homogeneous and isotropic, and there are exactly three geometries associated with this: Sphere (which exists as rotations through space), Plane (which represents Galilean Relativity) and the Hyperbolic Plane (which exists as rotations through spacetime).

I wanted to find a way to use this intuition without overwhelming the audience with technical terms. When the timelike unit vectors in 2+1 spacetime (which correspond to velocity four-vectors) are projected onto a disk (see Wikipedia: “Hyperboloid model”), the result is the Poincare Disk Model, which is (almost) the geometry portrayed in Circle Limit III. I had never seen anyone to use this as a visualization of Special Relativity, and so I decided to make the video.

Public Domain Image Sources (in order of appearance):
pulse of light:
wall clock:
Electric Current:
water waves:
cosmic lake:
Lorentz: :
Minkowski: :
star cluster:
trash can:
Einstein and Lorentz:
atomic cloud:

Software used to create this video includes iMovie, Keynote, Pixelmator, MathType, and Quicktime Player.

the theory of special relativity which is commonly associated with Albert Einstein has proven to be one of the most successful theories in all of physics surviving any test we can think of to try to prove it wrong however it has a reputation for only being possible to learn through headache-inducing thought experiments where the student has to keep up with pulses of light multiple clocks and moving observers all while trying to adapt to a new concept of space and time while all of this is fascinating and I really encourage you to look into the felt experiment development of special relativity my goal in this video is to present you with more direct intuition for how special relativity works without using as much math let's start with explaining Galilean relativity where the outcome of any experiment will be the same regardless of the frame of reference that is to say if all relevant parts of the experiment are given some boost with the same speed and direction the experiment will come out the same the concept of a boost will be important later on for this discussion think of a boost as a push into a new frame of reference imagine throwing a ball straight up in your driveway it will fall straight down now imagine that you and the ball are in a car that is moving at a constant 60 miles per hour rest from the frame of reference of the ground if you throw the ball up it will move upward but also westward at 60 miles per hour to form an arc if I asked what the ball looks like to you you would quickly tell me that it looks like it is moving straight up and straight down just like in your driveway while this seems obvious it is only because of symmetry that we can make this assumption the idea behind Galilean relativity is that you can skip this step where you consider the frame of reference from the ground because of the symmetry between all frames of reference in fact the ground is just Earth's surface which is rotating at some thousand miles per hour relative to the axis but even the axis orbits the Sun which in turn orbits a galaxy it therefore follows that all friends of reference are equally valid or in other words there is no way to determine if you are moving or at rest in any absolute sense only the relative motion between different objects is physically meaningful this is called the principle of relativity okay now we need to switch gears for a minute to talk a little bit about electromagnetism and waves electromagnetism is the theory describing the one unified interaction between charges currents magnets and light in the 19th century it was realized that the equations for electromagnetism had a class of solutions that corresponded to physical waves in the electromagnetic field it did not take long for experiments to reduce these electromagnetic waves in the laboratory and confirm that light is just a small subset of these waves with a particular range of frequencies the electromagnetic field was thought to exist in the ether with its own state of motion through which electromagnetic waves of light traveled at relative speed of 300 million meters per second this however poses a problem for Galilean relativity to see why we are going to use an analogy water waves imagine you are floating in a lake and you drop a baseball in the water the disturbance created in the height and motion of the water will propagate outward as an expanding ring this ring will let us determine if we are at rest with respect to the water as long as you are in this lake you can determine if you are at rest by dropping a baseball in the water and checking to see if you remain in the center of the expanding ring shaped wave around you if you are moving you will be able to see this as the center of the expanding waves moves away from you you don't stay in the center of the expanding ring giving such a test to determine if you are moving or at rest may seem to contradict the principle of relativity or remember you are only measuring your motion relative to the water and its associated frame of reference so according to the ether idea we should be able to do the same thing with an electromagnetic or light wave as we did in the lake which would mean that we should be able to see our movement in the ether by measuring light waves physicists Albert Michelson and Edward Morley tried an experiment to measure this what they found is that it does not appear that there is any motion at all through the ether even at times of year when the earth is moving at a velocity up to 60 kilometers per second difference this experiment showed that the ether theory was wrong and that the electromagnetic field did not behave like some cosmic lake with a fixed background frame instead the electromagnetic field treats all frames of reference equally which means that the speed of light is 300 million meters per second and every frame of reference this is called the principle of constancy of the speed of light but wait how can this be consistent with Galilean relativity Galilean relativity says that if a galileo galilei is moving toward you at 1 million meters per second and he's shines a light at you that leaves him at 300 million meters per second then the light will arrive to you moving at 301 million meters per second breaking the principle of constancy of the speed of light this inconsistency let theoretical physicists on Rippon Caray Albert Einstein and Hendrik Lorentz as well as mathematician Hermann Minkowski to develop a new theory of space and time somehow it had to be true that there is no preferred frame of reference and the speed of light is the same in all frames of reference instead of following the original thinking of Einstein or Lorentz I'm going to use the fact that everyone thinks they are in the center of the universe no really all observers see themselves as staying at the center of a pulse of light emitted from them regardless of their state of motion the resolution found by the fathers of special relativity required replacing Galilean relativity so we need a symmetry that respects the principle of relativity and also the principle of constancy of the speed of light inter Lorentzian relativity which was discovered by Lorentz and Einstein a visualization of the principle of relativity will require a geometry in which every point in every direction is essentially the same there is a theorem in mathematics that there are exactly three such geometries to begin with let's look at a geometry that does not satisfy this property on the cube not any two points are the same for example while these two points are geometrically equivalent these two are not the first geometry that does fit our criteria is the plane since the plane is infinite any point could call itself the center the next one is the sphere since the sphere is perfectly round any point could call itself the center or the pole in both of these geometries each point is basically the same the third and final geometry is one that is less familiar since there is no perfect way to represent it in three-dimensional space it is called the hyperbolic plane and it is what we will use to understand relativity in our visualization the regular plan will represent Galilean relativity and the hyperbolic plane will represent Lorenzi and relativity so here we go imagine a car is moving at some speed through a large parking lot according to Newtonian physics the car can have any speed in any direction the speed of the car can be represented as a point on this grid of tiles to move from one tile to an adjacent one would require a standard boost say 10 miles per hour remember that a boost is just a push into a new frame of reference each point on this circle represents a velocity of 10 miles per hour in a different direction and each point on this circle represents a velocity of 20 miles per hour this grid represents Galilean relative notice that no particular tile is special we can easily shift the grid around and put any tile in the center now I am going to state something that will become clear later nothing can move faster than the speed of light therefore the velocities that the car can move out are represented inside the circle for example this point represents half the speed of light and north if you imagine that this circle is the size of a dinner plate any speeds that would practically be obtained by a vehicle on earth stay in a tiny circle only about a thousand atoms in diameter so the new grid of tiles looks like this the most important thing to understand is that the tiles become more and more compressed near the boundary each tile is still the same amount of boost but they shrink in size as you approach the edge this explains why you can never get to the speed of light to increase your speed you need to undergo a certain boost moving to the next tile but as you get near the speed of light each tile takes you less and less distance with the same boost so while the speed of light is only a finite number of meters per second away it is still an infinite number of tiles that you must cross to get there so you can see that when you compare our two grids that in the first one which represents Galilean relativity the velocities are additive each tile adds the same amount of boost or change in velocity regardless of which frame of reference you are observing it from in the second grid which represents Lorentzian relativity each tile adds the same amount of boost but it looks completely different depending on where you are viewing it from if you are viewing a change that is far away the boost or tile appears very small but if you are in the center or close by the tile appears very large with a bit of visualization you can imagine that the grid could be shifted so that this arc moves to this arc with all of these tiles getting compressed over here and these tiles expanding to fill up all of this space so you can see that any tile could reasonably call itself the center tile and so it follows each tile sees the speed of light as a circle around it now maybe you can see why with lorenzi and relativity each observer sees light emitted from them moving out in a perfect sphere regardless of their state of motion which in our visualization is represented by which tile they are on one last thing to note is that Galilean relativity is an approximation to Lorentzian relativity as long as the speeds involved are small for example the 1000 ad immed I am circle representing typical speeds of human vehicles is hardly affected by the distorting near the speed of light the theory of special relativity found by Einstein gave rise to a whole new picture of space and time which became incorporated into some of the most profound and successful theories in all of physics I wish I could go on to talk about these ideas of space-time and mass energy but unfortunately I'm out of time I hope I've given you a taste of special relativity why you can never reach the speed of light it's not using all this I want to encourage anyone interested to find material on the topic of special relativity and to look at the woodcut circle limit 3 by MC Escher which inspired this video my next video will be on Schrodinger

Lecture 1 | Quantum Entanglements, Part 1 (Stanford)

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Lecture 1 of Leonard Susskind’s course concentrating on Quantum Entanglements (Part 1, Fall 2006). Recorded September 25, 2006 at Stanford University.

This Stanford Continuing Studies course is the first of a three-quarter sequence of classes exploring the “quantum entanglements” in modern theoretical physics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.

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this program is brought to you by Stanford University please visit us at stanford.edu I almost always begin with the same sermon about especially when teaching about quantum mechanics or relativity the sermon is always the same it's the fact that we as animals have inherited through the process of evolution certain intuitive ways of thinking about the physical world and if you don't believe it you think that maybe ordinary animals are not physicists you watch a lion chasing an antelope and you notice that that lion the minute that the antelope that the relative velocity between the Antelope and the lion changes sign the lion just stopped dead somehow he did some calculation or she it's usually as she the lion did some calculation some physics calculation involving some very complicated concepts of velocity Direction all kinds of complicated computations like that a a primitive cro-magnon man I'm not a cro-magnon man and they end the thal who comes to a cave and sees that the cave was blocked by a boulder and tries to push the boulder and can't push the boulder decides to aim his body that way hmm why so that he gets a bigger component of force in that direction has he ever heard of force is you ever heard of components where'd he get this idea of components that he know about sines and cosines yes somehow he did know about sines and cosines these are things which were inherited are biological in origin and they are the basis of our intuitions about physics our intuitive picture of the world much of physics has to do with those things in fact all of modern physics everything in modern physics has to do with those things which are beyond the intuitions that we were able to get from the ordinary world has to do with with ranges of parameters which are way outside the range of parameters that humans or animals ever experienced for example it's not too surprising that human beings didn't know how to deal with velocities approaching the speed of light but they got the wrong ideas about how to add velocities when nobody in 1900 had ever probably had never probably had never moved faster than 50 or 60 or 100 miles an hour well there are they probably did when they were falling off cliffs but they didn't live to talk about it maybe they got up to 200 miles an hour maybe but nobody ever had experienced anything like the velocities approaching the speed of light and so it was not surprising that their intuitions that their way of thinking about adding velocities and so forth a theory of relativity was and how you synchronize clocks all that stuff all that good stuff that Einstein did that it was outside the framework of their ability to think about through intuitive pictures to intuitive mathematics arm they had to invent new mathematics the new mathematics was abstract meaning to say you couldn't visualize it four-dimensional space-time hmm you can I can't visualize four dimensions I've learned tricks to visualize it so physicists to some extent rewire themselves or people who learn physics through a process of rewiring themselves to some extent to develop intuitions to be able to deal with these new ranges of parameters but still they're foreign they're alien the peculiar even to me quantum mechanics deals with a range of phenomena which is also outside the experience of ordinary humans for which evolution simply didn't provide you the means to visualize evolution did not provide you the means to visualize an electron to visualize the motion of an electron to visualize the uncertainty principle when you think of a particle moving what is a particle a particle is a thing with a position at every instant of time it has a position if at every instant of time it has a position it has a trajectory if it has a trajectory you can calculate the velocity along that trajectory just by knowing the separation between points and what the time interval is you can calculate the velocity and that's the intuitive picture of a particle and where does it come from it comes from thinking about rocks throwing rocks shooting arrows all kinds of things that human beings normally do so we never developed the need it would have it very bizzare if our brains have been wired to understand the uncertainty principle why would Darwin have given us the part of the incidentally if you prefer to think of the intelligent design or go right ahead I prefer I prefer to think about Darwin but why would either of Darwin's ideas or the intelligent designer have provided us with the ability to understand the uncertainty principle when it's never anything that's part of our ordinary experience the answer is it didn't and so quantum mechanics for that reason appears extremely weird to us physicists as I said rewire themselves and developed ways of thinking about it which are intuitive but still quantum mechanics is much much more unintuitive incidentally than the special theory of relativity and what we're going to try to do here is expose some of the weirdness of quantum mechanics the weirdness of the logic of quantum mechanics the weirdness of how quantum information works this is not a class a conventional class in quantum mechanics a conventional class in quantum mechanics would stress such things as the Schrodinger equation and waves and how particles sometimes behave like waves and so forth we may or may not get to a bit of that but that's not the important subject that we're going to concentrate on well going to concentrate on is the basic logic of quantum mechanics the basic logic of quantum information theory physics is information when you say something about a physical system you're saying something some you're giving some information about it you give the information in various forms usually in the form of numbers in classical physics you often give I will give you some examples but you you usually give it in the form of real numbers the position the velocity is set of real numbers and quantum mechanics sometimes you use real numbers but very very often you give discrete information discrete information such as yes or no or up or down or male or female well that's probably not such a good example the difference is because yeah that's prevalent up I think I withdraw that heads or tails heads or tails do I have a coin no Gedanken coin thought coin head tail head tail right when I flipped the coin head that's a piece of information it could be tail it's a two valued system either yes or no up or down heads or tails or sometimes they're logically all the same of course they're logically all the same whether we're talking about heads or tails up or down or whatever it is they're logically the same and they're simply decisions which have to possible or questions which have two possible answers and a bit of information which has two possible answers is called a bit it's called a bit and it can either be a classical bit or a quantum bit all real bits in nature are quantum bits obviously since nature is made out of quantum mechanics but sometimes the quantum aspects of it don't manifest themselves in an ordinary computer the quantum aspects of the bit don't really manifest themselves for reasons that we'll come to and it's just called a classical bit a classical bit of information this head the coin flip yes or no the quantum bit is all is the quantum analog of the flip coin the yes or no type question but it is much much more subtle and the first thing we're going to want to explore is what is a quantum bit now but before we do that let's talk about classical bits classical bits can be described either by writing down a 0 or a 1 these are we could also use 1 and minus 1 or we could use 5 and 15 doesn't matter but 0 and 1 is a convenient notation for the two possible our values 0 could stand for heads 1 could stand for tails and so forth so we're thinking over thinking about this we're thinking about some physical system when we're thinking about information we're thinking about a physical system such as a coin and this is the information contained in that coin either a 0 or a 1 there's a notation this seems like a ridiculous and redundant notation its importance will only become clearer when we start to think about quantum bits but we're going to use the Dirac notation the Dirac notation describes the state of a bit not whether it's California or Aragon but the configuration of the bit and it's usually labeled with the notation 0 or 1 or whatever other information whatever other way you decide to think about the bit these are the two states that a bit can have either a 0 or 1 and it's represented I don't know that I don't know if I drew that well let's draw it again 0 or 1 these are the two states of a bit all of this extra junk here is excess you don't need it it doesn't tell you anything it just says that you're putting it inside the bracket incidentally this point the bracketed object over here the thing that contains the information on the inside is called a ket it's called a ket because it's the second half of something which we will later learn is a bra ket or a bracket there's another half that we haven't exposed yet now what about multi bits supposing you have more than one bit and we're talking now classical physics so far we're not talking about anything quantum mechanical supposing we have several coins and I line them up I label them so we know which one is which in fact just in order to not confuse coins let's make sure the different coins penny nickel dime quarter half dollar silver dollar okay so we have a bunch of coins we can't confuse them and we can lay out some information by saying head tail tail head head tail that would be some information about a collection of bits all right how would you label that well you would label it with a string of zeros and ones so for example if we let's take 0 always to stand for head it's easy to remember 0 stands for head and one stands for tail for obvious reasons umm right so my string of coins heads head head head tail head tail head I would label 0 for head 0 for another head 1 0 1 1 for example that's that's a configuration of 1 2 3 4 5 6 coins right let's say it was 6 coins right that's a configuration of a multi-bit system in this case 6 bits again for reasons that and absolutely nothing to this description we're going to stick it inside our cat and stick it inside a cat which is just a kind of notation it's might be a good idea to put some commas between these but maybe not maybe it's just best to leave it that way that's a specification or it could be the specification of the bits of information inside a computer it could be just the series of heads or tails or so forth but before we do anything else with this let's ask a very simple question how many possible configurations how many possible states are there of well let's start with one bit if there's only one bit then there are only two states what if there are two bits well then you can have up up up down down up down down for two times two so for two bits we have two squared what if we have what if we have a hundred bits the answer is 2 to the 100th power 2 times 2 times 2 times 2 100 times so if you have an in bit system the number of possible classical figure configurations is just 2 to the N let's write that down let's so let's put a notation let's write the number of states the number of states n sub s the number of states of a system of little n bits is 2 to the N let's suppose we are let's invert that first of all let's invert that little N is what little n is the number of bits big in is the number of states little n is the number of bits ok so if the number of bits is 4 then the number of states is 2 to the 4 which is 16 and so forth we can invert this and we can write if we knew the number of states in a system then we can take the logarithm of this equation log to the base 2 is particularly convenient if we take the log to the base 2 of the number of states that's equal to the number of bits you can generalize this not every system has as its number of states to to a power supposing I have a state a system um a die you know the things you use in Las Vegas to throw away your money with it's got six possibilities one through six that is not to to any particular power it's just six but we can still generalize this definition of the number of bits of information in fact the number of bits of information that a system can contain is by definition the logarithm to the base 2 of the number of states which for the die would be log to the base 2 of 6 what is log to the base 2 of 6 is it an integer no it's some stupid irrational number I don't even what is it how big is it about the 2 to 2 to the 2 to the 2 is 4 2 to 3 is 8 2 to the 2 point 5 3 7 9 8 6 14 or whatever so the amount of information which is always the logarithm of the number of states does not have to be an integer but we're going to be considering systems which are made up out of some number of bits each of which has two states so for the simplicity we're going to be talking about systems the number of states is always 2 to a power that's just the simplicity there's nothing special about but almost every system can be represented that way or approximately represent it that way let me give you an example supposing we have some question of physics which has as its answer a real number but we're only interested in that real number to a certain approximation the temperature the temperature in the room I'm interested in the temperature in the room Co after to a certain number of significant figures I can represent the temperature or any other number for that matter by writing it as a number in base two right if what's the temperature in this room incidentally it's about three hundred degrees from absolute zero so it's three hundred I can write three hundred not as three hundred which is what is 300 min 300 you know it means it means three times ten to the two buts nothing times 10 to the 1 plus zero times 10 to the zero but we write it two things in base two all right I don't know what 300 looks like in base two somebody can figure it out base to you everybody know how to rhythmic taken base to anybody not know arithmetic in base 2 okay so everybody knows arithmetic in base two we write out any number that we like as a series of zeros and ones one zero zero one zero one zero zero one that's some number to the some particular integer it's an integer so if I'm interested in the temperature and I'm not interested in being too careful to define fractions I want to know whether it's 72 degrees or 73 degrees I don't care about seventy two point four oh six nine I can write it as an integer and that integer can be represented as a sum of bits not a sum of bits but as a collection of bits every number every single number if you're willing to truncate the number of decimal places approximate that number and say I'm interested in that number only 235 decimal place or whatever or 2:35 places to base to that that number simply is both is represented by and represents a collection of bits so anytime and incidentally if you want to have a finer grained description of the temperature than then integers in centigrade you just use a more refined notion of degree you go down to ask how many degrees is it but not in centigrade units but in units of ten to the minus 100 centigrade again you can give it as an integer and integers can always be represented as sequences of zeros and ones so almost any information in physics can be represented in terms of bits in particular the measurement of quantities such as temperature for example let me give you another example this is a more complicated example of the same thing supposing I'm interested in a field a field means a thing which can vary throughout space all right a thing which can well the temperature can vary throughout space the temperature is a field it varies throughout space it's not one of the more interesting fields from the point of view of fundamental of fundamental particle physics or anything but that certainly is a field it varies from place to place and how can we represent that can we represent that in terms of bits yes if we're willing to tolerate certain approximations and we're always willing to tolerate some degree of approximation what we do is we break up the room into a lot of little tiny cells I won't try to draw a three-dimensional room in my notes I drew a three-dimensional room it took me about a half an hour to put in all the lines just a two dimensional room and here's what we do we first of all order the cells we make the cells small enough so that the temperature doesn't vary very much because from cell to cell so we might fill this room with several billion cells label the cells this is the first cells – one two three four five six seven or up to a thousand thousand one thousand two thousand and three thousand and four and we can label all of the cells and list them once we've listed them we can write the temperature of the first cell there's the temperature of the first I'm putting a little comma in just as distinguish between cells then we can write the temperature in the next cell 0 0 1 and 1 2 3 4 5 6 7 8 9 I've kept 9 decimal places in you in in the basis in arithmetic and base 2 1 1 0 1 however till I'm come on finished then I go to the next cell do the same thing temperature there is 1 1 0 1 0 0 and so forth eventually all I have is a list of zeros and ones this long list of zeros and ones if somebody knows how to use it is equivalent to knowing the temperature at every point in the room the same is true of the electric field the magnetic field anything which varies from place to place so almost everything that I can think of in physics can be represented in terms of bits so if you know everything about how bits work you'll basically know everything about how physics works of course you may not know what the rules are the myth I'm manipulating these things but this is the basic setup of physics information in the form of a series of questions each of which can be yes or no now of course you may want to refine your description to refine your description you may might want to add more decimal places to the temperature to the specification of temperature and you might want to make your lattice finer that's just making a better approximation so the right thing to say is that most physical systems that we know about as far as I know all physical systems can be represented at least approximately and perhaps two always increasing approximation by a series of bits that's why we get to use computers to do physics if this weren't true we couldn't use a computer we couldn't use a digital computer in any case to do physics we have to use analog computers or something okay so that's a let me give you another example another example of how you might use bits to represent another these are all so far classical systems as I said I don't want to redraw the lattice but I do want to get rid of its top row here since I've already mutilated it here's a lattice and what I'm interested in is the motion of particles this lattice is just an artificial imposed lattice that I've imposed on the room here just so that I've divided the room into mathematical cells and what I'm interested in is the motion of particles moving around in this room at any given instant I can ask the question let's take a very simple case let's take the case where a particle where you can't squeeze more than one particle into one of these cells we can imagine that the cells are about as big as a particle in which case you can't squeeze in more than one then every cell either has a particle or it doesn't have a particle we can label the cells that have particles with an X we can label the cells that don't have a particle with nothing or better yet we can label the cells that have a particle with a1 and the ones that have no particle with a zero in that case this becomes a specification of where the particles are in the lattice so the longer the temperature but the same long sequence of zeros and ones now the number of zeros and ones would just be equal to the number of number of cells in the lattice what would this number mean it would mean that in the first cell there's a particle and the second cell is no particle and the third cell is no particle in the fourth cell is a particle and the fifth cell no particle in the sixth cell particle and so forth and so on and so given such a string of numbers you are given a specification of where the particles are in this room in that way again motion of particles most of the fields temperature just about anything in physics can be represented in terms of bits any questions right a bit it a bit is by definition a question about a system which has only two possible answers which you can always take to be yes or no used to be a game twenty questions didn't where somebody would think of a category and then you would stand there and ask yes/no questions and until you try to figure out what the category or what the what the category was so that was using the idea of death yes question oh I just I just arbitrarily said supposing we're interested in the temperature to a certain degree of accuracy right so I'm interested in the temperature to accuracy but now I'm not speaking about temperature I'm just giving another example these are just examples intended to show you something which is which is more or less clear otherwise we could not use computers to to simulate physical problems classical figure physic pop yes right but you need to provide for the general real number you need an infinite number bits right any rational number can be represented by a finite number of bits and the rule well that's not quite true you have to you have to remember to repeat them but yeah if it's rational it's going to repeat after some point right so but if it's an irrational number then you need an infinite string of bits but in general we will allow infinite strings of bits although not in a genuine computer well so so far remember we're doing classical physics all right so far no quantum mechanics so I will come on let's see where yes we were going to come to that very very shortly let me tell you how very quickly an electron first of all we're not talking about motion yet we're talking about configuration configuration means the state of the system at a given instant of time okay so the presence of an electron at a given instant of time let's suppose the nucleus is known to be right over here and we're not going to ask about the nucleus the nucleus just sits there it's a lump on all right so um we could say at instant number one when we begin the experiment the electron is over here in that case we would write down a string of zeros with a 1 someplace pure zeroes one electron pure zeroes except for one splice in the in the sequence where there's a 1 now if we wanted to describe the motion of the electron we would say starting with this configuration we move and let's use this symbol here to indicate that at the next end we we could we've broken up space into a lot of little individual cells we could also break uptime I thought I had my watch but I don't we could also break up our watch into a digital watch which the witch digitizes time just again as either a convenience or an approximation and we could say if at digital time number one the electron was or the system was described by one electron located at this location then what happens next next it moves did some no configuration in this case it might move over one place 1 2 3 4 5 6 it moves over to the sixth place one two and so forth so the motion of a system is described by a rule of updating of updating information how you update it from one instant to the next all right so physics basically consists of two a physical system consists of two things it consists of a collection of possible States which can be labeled by a collection of bits and it consists of an time evolution which is an updating which tells you how to take one collection of bits and replace it by another collection of bits at a slightly later instant of time I don't know if that answer to actually work out an orbital motion orbiting around here gets confusing because when you jump from one layer to the next if this is one and this is a hundred then 101 is over here so you don't jump from a hundred to 101 you might jump from a hundred to over here which would be 200 so it can be complicated the updating procedure it can look complicated but nevertheless it's an updating procedure that that just updates your your state of knowledge at each instant of time that's classical physics now there are some rules and we're going to come to them but before we do let's define the space of states this is and I want to emphasize we are still doing classical physics there is nothing quantum mechanical even though we're talking about discretizing systems and making out of them systems of individual bits so far we are dealing with what should be called classical bits see bits I think they're called as opposed to qubit skew bit as a quantum bit this these are classical bits so far okay so let's take all of the configurations and just abstractly in a purely abstract way we take all of the configurations incidentally what it is about ten by ten is roughly a ten by ten lattice ten by ten lattice has a hundred sites how many states does it have if we I'm not talking about one particle now I'm talking about any number of particles can be on this lattice how many different configurations are there two to the hundred a very very very big number two to the hundredth power that's how many different ways we can arrange zeros and ones on this lattice or specify whether there's particles in various positions a very large number of possible states but let's just abstractly think about all these states and just draw them as points if there are a 10 to 100 I have to draw a 10 to the hundred points which I'm not about to do these are the various states these are not the lattice points these are the various states for example for one bit if I had only one bit then the space of states would consist of only two points up and down and I would just draw two points this would be the space of states of a simple one bit system now let's ask our what are the possible laws of updating in other words what are the laws of motion laws of motion are the laws for updating configurations what are the possible laws of updating well here's one possible law of updating this could stand for heads this could stand for tails let's let's think about it in terms of coins for the moment this could stand for heads this could stand for tails if we start with heads if I had a coin we would do it as it tails heads tails all right one possibility is very simple if you start with heads it stays heads nothing happens if you start with tails it stays tails nothing happens that's a law of updating it's not a very interesting law of updating how would you draw that well here's how we'll draw it heads goes to heads will make an arrow if we start with heads it stays heads and we start with tails it stays tails so we draw an arrow from what you start with to what you end with what's another possible law of updating here's the law of updating if its tail's it becomes heads that it becomes tails then it becomes heads then it becomes tails that's a that's a little more not very much but a little more interesting a slightly more interesting system it just flip-flops back and forth how would we draw that we would draw that again heads tails heads tails if you start with heads you go to tails and sour tails you go to heads so the law of updating in this case is just described by such a diagram basically a diagram which tells you if you started a given state what it will be in the next instant of time is that clear alright so this is one way of describing the laws of physics write down all the states keep in mind what they stand for of course remember that in this case one stands for heads one stands for tails or whatever it happens to stand for if it's that male female this could be an interesting case of this wouldn't that would be an interesting this this would be a very interesting the law of motion in that case I don't think I want to I don't think I want to explore that any further if this was my undergraduate class I would never have brought that up this is more likely this is the more likely law of updating for sexuality female male so you see simple laws sometimes apply sometimes they're a little more complicated can you think of an interesting system that flip-flops like this Oh hand I can't think of anything I mean it's obvious that it applies to a lot of things but I offhand I can no no no right right right right but this is the thing yes but if you just tell you're not intervening I don't want you to intervene this is the system by itself if you had some peculiar lights which by itself we're back and forth and back and forth and back and forth in a regular way but that's stay and you know this is the Excel which well a lot of states to the pendulum in between but yes you've got the idea it's hard to think of a simple example but I bet by that if we'll go home and we came back next week every one of us would have an example of a we could call this the flip-flop this is a flip-flop motion this is the the the uh the uh notion well we can extend this if we know what the space of configurations is and we lay them all out either abstractly in our mind or actually just write them on the blackboard then the motion of the system can be represented by a series of arrows where I'm getting tired but and so forth and so on yeah um let's do let's do the possible let's think of some possible motions of a two-bit system a two-bit system simply has four states that's all we have to know it has four states well here's one possible motion if we start with this configuration we move to that configuration if we start with this configuration we move to that configuration and so forth if we watch the what actually happened with time the system would move from one configuration to the next around the closed loop now the closed loop is not necessarily a closed loop in space it's a closed loop in the logical space of possibilities here logical space of configurations that's one possible thing that K here's another one perfectly good what this is is it's a pair of systems a pair of it's a pair of systems which are separately undergoing flip-flops each one undergoing flip-flops this one is flipping and flopping this one is simultaneously flipping and flopping if we start over here let's see what that stands for that stands for example for both heads could same for both heads then we go to both tails then we go to both heads then we go to both tails or we could start with one head one tail and do this that's what this is this is a pair of systems flipping and flopping there are other possibilities so there are different laws of motion that the system whatever it happens to be could have so when you specify a system you not only have to specify with the states of the system R but you'll have to specify how it moves and how it moves is a rule for jumping from one configuration to the next now let me give you an example of a logically perfectly sensible rule but which is defective from the physics point of view never happens in physics we can do it I think well let's do it with four with with four states with four states let's see how this went yeah here it is if you start here you go here if you go here start here you go here excuse me one moment for some odd reason in my notes I've drawn instead of a diamond shape I've drawn a square let me go back there's my four states okay if you start here you go here if you start here you go here if you're over here you go over here and if you're over here you go over here all right so now we can say what happens wherever you start if you start over here you jump to here you jump to here you jump to here you jump to here you jump to here you go here you go here you go here you go here you go here notice you never come back to here okay with this particular law there's something different about this law then there is about the other examples and all the other examples well can anybody spot what's wrong with this well not what's wrong with it but what's different about it well it doesn't consist of loops this is true you can't figure out necessarily where you came from you may be able to tell where you go next but you can't always tell where you came from for example if you find yourself over here you don't know if you came from here or whether you came from here if you're over here you have lost a piece of information this is a motion which loses information it loses information in the sense that you can't tell where you came from there's no way to reconstruct the past but you can reconstruct the future or construct the future wherever you are you're told where to go next but wherever you are you don't know how to get back if you're over here well you know you came from here but then you don't know whether to go back here or to go back here so this is what is called an irreversible history it's a history or a law which loses information and at the fundamental level of physics fundamental level where you're really keeping track of everything not where you're not where your coarse graining or not looking carefully but we are carefully looking at every degree of freedom of a system classical physics never allows the loss of information like this there is a unique future point wherever you are and there is a unique past point wherever you are okay that is one of the laws it's not necessarily a law of logic it is something which is true of all physical systems that they are reversible in that sense yeah let's say I change state twice all right I'm over here one two or one two I don't know if I came from here or here right we could give this property a name we give it the uniqueness of the future point and the uniqueness of the past point we could invent a name for we could call it uni parity do you know what the quantum version of unique parity is it has a name it's called unitarity unique parity is a name I just made up unitarity is the quantum equivalent which tells you that you can always reconstruct the past from the future of the state of a quantum system you can either run forward uniquely or run backward uniquely and you'll come to some unique previous state or future state and that's called unitarity in quantum mechanics we haven't done quantum mechanics nothing's quantum mechanical yet it's a kind of time it's a kind of time reversal symmetry or it's actually not a time reversal okay so it's not a time reversal symmetry exactly it's a time reversibility I would say this diagram has a sense of orientation to it if I start something over here it goes around this way it definitely does not go around this way unless I reverse it unless I look at it backward in time so it's it's not precisely what you would call time reversal symmetry time reversal symmetry means that you could either go in going into the future you could go either way but in this case you only go one way but it's it's it's the reversibility of the laws that that you can find the reverse law give it a law you can find a reverse law which will take you in the backward direction I think that's I think that's right yes yes well two arrows coming away from a point since I don't know which way to go oh boy I go that way or do I go that way so it's it's it's clear that that's not a law of motion okay no branching ratios oh well let's we do at the moment we're doing quantum mechanics so I mean classical mechanics so it gets more complicated with the branching ratios in the classical mechanics is less fundamental in quantum mechanics all real systems are quantum mechanical the question why some of them suppress the quantum mechanics and you don't see it is a question which we'll try to answer as we go along but we might ask it'll at the quantum level and at the quantum level I think we can give a better answer but ultimately at the end of the day it seems to be a law that that basically says that forward in time and backward in time are our I won't say equivalent to each other but that there's no preferred really preferred sense in which forward in time is different than backward in time even though it feels like there is well this this this is this is the question that took 20 years to answer and the answer is in my opinion no it is not possible but it was one of the great questions of physics that took a long time to answer and I'm not going to get into it now but it might be an interesting thing for us to explore toward the end after we've talked about quantum mechanics we we have to talk about quantum mechanics before that makes sense that question yeah I think they're all Newtonian in a sense in the sense Newtonian to me simply means that there's a definite state for a system that it evolves with time according to a definite law of a deterministic deterministic that's that that's the right word yeah but yeah I think you can think of more complicated situations you just can start drawing some diagrams yourself and and see what makes sense what what's reversible what's not reversible and but yes you're right that is true it loses the information as the way you came from the system yep we mean it's not part of the system the system started here at went to here I went to here aren't they yeah whatever hmm you won't find that point ever again right right but the main point is you've lost the distinction between two possible starting points whereas in all the other situations if you know where you are and you know how many steps you made you can say where you were well I think I think for a long long time mr. Stephen Hawking thought that this is the way black holes work so not so clear not so clear okay yeah yeah good good so let's let's talk about that um if I take a bunch of molecules in a bathtub I don't what's a good example well let's take the molecules in this room and I sell to start them all out in a certain configuration very definite configuration I put them all up in the left-hand corner of the room there and I let them go after a while the room will be full of air just like it is if I put them up in that corner of the room over there and let them go after a while same thing put it up in that corner after a while same thing so it looks like we've lost information but in fact that's not true if we followed every single molecule and we followed it in infinite detail with infinite precision which we don't do of course ah then then we could reconstruct by running everything backwards we can reconstruct the fact that the molecules may have come from that corner of the room it's prohibitively impossible to do in practice but in principle following every single detail of every molecule know what really happens in the real world is we lose information because we lose the ability to follow the details not because the information gets lost but because we lose the because we lose the ability to follow the information that's where the second law comes when you start losing the ability to distinguish different states so we don't distinguish whether in our coarse-grained picture we don't distinguish the different detail at the level of a molecular detail and so it looks like different configurations become the same configuration but that's only because we simply don't look carefully enough it's because we're lazy do you need an infinite number of bits – ah ah ah you mean in the you know in a real room like this no because of quantum mechanics because of quantum mechanics no but if it were not for quantum mechanics yes you would need an infinite number of bits now what does that mean that means that you have to specify a bunch of real numbers precisely with infinitely with tremendous precision you have to precisely prescribe the locations and also the velocities but in particular the locations of every single molecule with a tremendous amount of precision and the longer that you want to track the system the more precision that you need so ultimately to track a system for a long time you need to specify with infinite precision the exact positions of every point wherever every molecule that means you'll have to give a set of real numbers a set of real numbers involves as you say an infinite number of bits so the answer is for for a collection of real particles moving around that you really try to follow classically depending on how long you want it to follow it you would need more and more bits to describe it oh oh yes yes yes that's right if room were really sealed let's let's let's idealize this room so that nothing can get into or out of the room all particles bounce off reflect off the walls of the room so that it's an entirely sealed up room then the room can be described discreetly because of quantum mechanics at least up to some energy if we know that the energy isn't arbitrarily high then we can describe it by a discrete collection of variables that has no exit so let's say yeah so we could so to make such a thing we could just reverse all the arrows here is an example no this one has no exit well then if I wanted to exit to itself I have to do this we could do that but as I drew it it has no exit but but let's let's think about what it means I'm not too interested in we're here the question is what happens when you're over here all right when you're over here you have two ways that you could go and you don't know which way to go so it's not deterministic it doesn't know whether to go this way or this way it might go half the times this way half the times this way you might need some statistical rule 50% of the time or 30% of the time it goes this way 30% so it's our 70% of the time with random statistics that would be non-deterministic okay so it seems that the real real laws of nature are both deterministic forward in time and backward in time that's the implication of not having loose ends floating around like this that they're deterministic either way so that wherever you are you can either trace forward uniquely or backward uniquely and that is all of classical physics in a nutshell you now think in a complete course in classical physics or there's nothing that does not fit that pattern at least two arbitrarily high degree of approximation let's take a let's take a seven-minute break well I was going to jump to quantum mechanics but before I do I want to do a little bit of mathematics elementary mathematics most of you know it but nevertheless let's lay it out matrices and vectors I'm not at the moment I'm not going to mathematically define a vector in any sort of sensible method you know even approximately rigorous way or abstract way I'm just going to tell you a vector is a sequence of numbers a finite sequence of numbers and you can represent it in a variety of ways but I'll give you two ways to represent a sequence of numbers the first way is to write them one after another let's just give them names I don't want I don't want to koala and my numbers now I at the moment I mean real numbers as opposed to a complex numbers I don't mean zeros and ones I mean arbitrary sets of real numbers they could be zeros and ones but zeros and ones are fine but just general numbers so I just lay my what should we call them hmm yeah yeah the called components but I want to I want to letter for them a a a is good so a a 1 a 2 a 3 a 4 and just put something around them to surround them so that we know this would be a four dimensional vector all right why four dimensional because it has four components we get don't try to visualize vectors now there's no value at all for our present purposes and trying to visualize these as pointing in space or anything like that they're just lists of numbers okay lists of that's one way okay there's another way that we can list the same set of numbers put them in a column a 1 a 2 a 3 a 4 same information in him I mean then I'm not talking about information in the abstract sense that I used before same thing sometimes it's useful to write it this way sometimes it's useful to write it that way you'll find out as we move along when it's written in this form it's called a row vector when it's written in this form it's called a column vector well we're actually talking about now is notations neat notations for for doing certain arithmetic 'el operations involving collections of numbers when we get the complex numbers we will then use complex conjugate notation yes but for the moment let them just be real numbers ok now there's another concept now called a matrix and think of a matrix as the following way a matrix is a thing which acts on a vector to give another vector all right so it's a kind of machine you put the vector into the machine and out pops another vector according to a particular rule oh no sorry before we do that before we do that let's imagine a particular column vector and another different row vector different row vector has different entries not the same set of numerical entries but a different set of the miracle enter entry so let's call them B B 1 B 2 B 3 B 4 these could be six point oh one five point nine seven three point oh four and A one could be seven point eight a two they none of them could be the same or they might not be the same the A's and the B's this is some particular row vector and some particular column vector there's a notion of multiplying a row vector by a column vector the notion of multiplying a row vector by a column vector is as simple as a following simple operation you take the first entry Oh incidentally the dimensionality of the row vector and the dimensionality of a column vector should be the same that means that they should have the same number of entries not necessarily four could be five six seven in which case they would be five dimensional vector spaces six dimensional vector spaces this extends the any number of entries into the columns and rows but the rows and the columns should have the same number of entries all right there's the notion of the product of a row vector and a column vector it's called the inner product and it's very simply constructed you take the first entry of the row and multiply it by the first entry of a column you add to that the second entry times the second entry plus the third entry times the third entry plus the fourth and three times the fourth entry so the product of these two which you could just write as B next to AE that product the inner product is B 1 a 1 plus B 2 a 2 plus B 3 a 3 plus B 4 a 4 it's a number it's not itself the product of these two vectors the inner product is not another vector it's not a matrix it is just a number the numerical value has just gotten by adding up the column though sorry the row times the column in just this form P 1 a 1 plus P 2 a 2 plus B 3 a 3 plus B 4 a 4 is that clear don't ask me why that's definition yeah yeah if we were talking about ordinary vectors in space it would be the dot product yeah yeah more abstractly for abstract vector spaces it's called the inner product but yes it is the same as the dot product for our three dimensional ordinary vectors in space where these would be the components of the vector yeah ok now there's the concept of a matrix and a matrix as I said is it it's an operation that you can do on a vector to give a new vector all right but it's not any old operation there particular family of operations that are characterized by matrices a matrix is represented by a square array of numbers let's call the entries M all right so in the first place we put m 1 1 to indicate that it's in the first row in the first column then m 1 2 then M 1 3 then M 1 for M what should I call this 1 2 1 it's in the second row for the first column this is in the second row second column second row third column next one m3 1 m3 2 M 3 3 m3 4 and m41 m4 to m4 3 & 4 for now as I said I've chosen four dimensions just arbitrarily four is about as many as bigger as I want to handle on the backboard and it's big enough to be a little abstract so that it's general enough to see what's going on alright that's what a matrix is that's all it is now you can think of it you can think of it you can think of each column as a column vector whose components are labeled by the first entry here okay each one of these can be thought of as a column vector where the first entry labels the column entry or you can think of it as a collection of row vectors where it's the second entry which labels the component either way you can think of it both ways at the same time a collection of column vectors or a collection of row vectors but altogether it forms a matrix now matrices can multiply vectors so let's put a vector over here a1 a2 I should line them up more carefully a 1 a 2 a 3 a 4 and when you I don't know I've done a reasonable job of keeping rows and columns underneath and next to each other but if you like draw some imaginary lines to separate them into rows and columns all right this matrix acts on this vector to give a new vector what is the new vector and he is the rule I've made the vector wide because each entry is going to be a fairly complicated expression but it is just another vector it's another single it's another column it's a column which I've had to draw wide in order to be able to fit in everything I want to write down here's what you do if you want to find the first entry into this column I'm sorry into this row into this row you take the first row and you multiply it by the column the inner product of the first row with a column here so what is that that's M 1 1 times a 1 plus M 1 2 times a 2 plus M 1 3 times a 3 plus M 1 4 times a 4 in other words you take all of this and you multiply it by this according to the inner product rule and that gives you the first row M 1 1 a 1 plus M 1 2 a 2 plus M 1 3 a 3 plus M 1 4 a 4 now you want the second entry into this new vector over here done exactly the same way except you go to the second row and you take the second row and multiply it by the column that's going to give you and I'm only going to do two of these the rest you can do yourself n – 1 M 2 1 again times a 1 plus M 2 2 times a 2 plus M 2 3 a 3 plus M 2 4 times a 4 and the other two entries you can figure out you get them by multiplying the next row by the column and finally the third row by the column that gives you a new vector it's a way of processing a vector to produce a new vector I will give you some examples as we go along it's a rule of multiplication which is very useful the reason it's defined is because it's useful and we're going to see how it's useful by using it let me give you an example of how a matrix how the idea of a matrix can represent the time evolution of the configuration of a system supposing again we have a our our configuration space let's label let's label them let's take a let's label them the first configuration the second configuration the third configuration the fourth and the fifth configuration these are not points of space these are configurations of a system which has five distinct States and let's take a very very simple law of evolution the first one if you start here you go to here and you've got here you go to here if you start here you go to here if you come here you go here and what do I do it from here go back now well you know that one that's no good that's that's disallowed I think is that this allowed I think that's this allowed I think that's this allowed yeah that's this allowed because if you find your yeah that doesn't that's not reversible that's not reversible that's not what I wanted to hear what I wanted to hear is that you go back to here so this is just a 1 goes to 2 2 goes to 3 3 goes to 4 4 goes to 5 5 goes back to 1 it's a cycle here's another way to represent the same thing we can represent the state of the system by a column vector and the column vector we simply insert a 1 someplace if I want to represent the first state over here I put a 1 and then a bunch of zeros 1 2 3 4 5 1 2 3 4 5 this simply represents the first state what about the second state the second state are represent by 0 1 0 0 0 the third state by 0 1 and so forth so the states of a system can be represented by a column but a particular kind of column a column with all zeros and a one someplace where is the one namely whichever state you're focusing on if you're focusing on the 5th state put the 1 in the 5th entry here ok now what is this rule of evolution the rule of evolution says that if you have a 1 someplace then in the next instant of time the one moves down so if you start here the 1 moves down to here and the next instinct it moves down to here next instant it moves down to here next instant moves down to here then where does it go up to the top right so there's a procedure that you do on this column to tell you where the system goes in the next instant of time that process can be represented by a matrix so let me show you the matrix that represents that the matrix is an operation on a vector which you can think of in this case as the updating operation the operation which updates the vector so here it is let's see we put 0 1 0-0 this is 5 dimensional so I need 5 0 0 1 0 0 0 0 0 1 0 0 0 I'm sorry I'm gonna make this for dimension I'm getting sick on I don't like 5 dimensions 5 too many for me right 1 0 0 0 let's try it out let's try it out on this vector right over here this represents the third state what happens if we act with this matrix on the 3rd state let's just try it out let's see what we get well the first entry up on the top is gotten by taking the top vector and multiplying by the column 0 times 0 plus 1 times 0 plus 0 times 1 plus 0 times 0 what's the answer 0 next place 0 times 0 0 times 0 1 time 1 whoops whoops whoops again ok yeah instead of going down it's going to go up by it's okay up down we just turn the whole thing over would you prefer let's let's let's just let's let's get it right let's get it right 0 0 0 0 with a habit be flat over here 1 1 one one and then up here one okay so let's start over again what's up on the top 0 times 0 0 times 0 0 times 1 1 times 0 we still okay 0 next 1 1 times 0 0 times 0 0 times 1 0 times 0 still 0 what about the third place oppa please please please God 0 times 0 1 times 0 0 times 1 0 times 0 it's still 0 but now in the last place I have 0 times 0 0 times 0 1 times 1 0 times 0 so 1 column has moved down one step now you can check for yourself here's your homework check that any place that you put this one it will move down by one step till it gets to the bottom and then it will recycle and go up to the top ok so that's a little thing to check in fact you can put in any numbers here only zeros and ones way make sense but we could put in any number a b c d and what will come out over here is everybody will move down the step a b c but then d will move up to the top so if you put a 1 in any one of the places it will slide down 1 unit and then we appear at the top the point is that the evolution of systems can be represented by matrices matrices of a particular kind bunch of in classical physics in this kind of classical physics there always just one here quantum mechanics is more complicated and more difficult but in classical physics but sprinklings of zeros and ones so as to make the each state shift into the next one that's that's an example of the use of matrices in classical physics so far no quantum mechanics just pure classical physics there isn't there is an interesting well all right this this this will do for the time being will come we'll come back to it so that's an example of matrix algebra matrices multiplying vectors what about matrices multiplying matrices here why might we want to multiply matrices by matrices well here's the idea supposing we wanted to upgrade or update a second time to update a second time what we would do would be to apply the same matrix to the resultant that we got in other words let's write it this way let's write it abstractly we have a matrix M which we multiply by a vector V to get a new vector V Prime ok that's just abstract notation for writing a matrix and a vector and getting a new vector that's updating the vector V to a new vector let's update it again let's go one more interval of time how do we do that well what we do is we write M times V prime equals V double prime we would do the same updating trick except now update the prime instead of V and we would get V double prime V double prime being the state of the system after two units of time but we could also write that by realizing that V prime is M times V we could write this as M times M times V is equal to V double Prime this just means we apply the matrix twice we can also think of it as squaring the matrix M and then multiplying it by V so how do you square a matrix or how do you multiply one matrix by another matrix this is what you would do if you would want to update twice once with one matrix and then once with another matrix or the same matrix how do you multiply matrices and the answer is basically the same kind of rule I will do it now for two by two matrices because it's getting too complicated even for 4×4 matrices for a two by two matrix we have M 1 1 M 1 to M 2 1 M 2 2 let's call it some other matrix n n 1 1 n 1 2 & 2 1 & 2 2 the result of multiplying a matrix by a matrix is another matrix it's another matrix and we do it in a very similar manner supposing we want 1 1 entry here we get the 1 1 entry by taking the first row and multiplying it by the first column M 1 1 times N 1 1 plus M 1 2 times n 2 1 same kind of inner product and we put it over here now supposing we want the next entry for the next entry we take the first row because after all we're interested in the first grow up here we take the first row but multiply it by the second column over here so what would be over here would be M 1 1 times N 1 2 plus M 1 2 times n 2 2 I'm not going to write it all out now we will move down to the bottom the bottom if we wanted this entry we would take the bottom row and multiply it by the first column if we want the last entry over here we would take the bottom row and multiply it by the last column so we multiply matrices by the same kind of pattern that we multiplied matrices times vectors we can simply think of it as multiplying this matrix by this vector putting it over here multiply this matrix by this vector put it over here okay so there's a notion of multiplying matrices and what multiplying matrices does is it gives you a new matrix which updates you not by one interval of time but updates you by two intervals of time if you wanted to short-circuit are the problem of updating and you want it to update the state of a system five units of time what you would do is multiply the matrix together five times that's you do it in sequence first the first time the next result times the next one times the result times the next one and you can you can work out what the matrix is which would take you from the state of the system and an instant of time to a state of the system five instants later so matrix multiplication multiplying matrices by matrices is also an important concept one last example of matrix algebra involves row vectors supposing you have a row vector and you want to multiply it by a matrix the rule is you write the row matrix first B 1 B 2 B 3 B 4 and then you write the matrix M 1 1 M 1 2 m 1 3 and so forth M 1 4 dot dot dot dot dot dot dot dot dot dot tired of writing M's well what's the result going to be the result is going to be a row vector and here's the way you get the entries of the row vector the first entry of the row vector you get by taking the original row vector and multiplying it by the first column vector over here that product is the first entry then you take the original row and multiply it by the second column that gives you the second entry then you take the original row and you multiply it by the third column that gives you the third entry over here and so forth you see the pattern it's always multiplying rows by columns and putting them putting the result in the right place in the right row and column in this case a row vector times a matrix is another row vector a row a matrix times a column vector it is another here it is a matrix times a column vector is another column vector and a matrix times a matrix is another matrix um get familiar with that work out some examples work out some examples of your own devising just put some numbers in multiply row vectors times matrices matrices times column vectors and matrices times matrices and get the experience of working out how these things work do it for 2 by 2 for 3 by 3 matrices and you'll get familiar with it because we will use it over and over and over again in fact that's the primary mathematical operation of quantum mechanics is multiplying rows and columns times matrices if you know how to do that and you're familiar with it and you can read off the answers easily you've got all of the basic mathematics of quantum mechanics it would help to have a little bit of calculus to go with it but the basic new thing is matrix multiplication and column vectors and row vectors so please practice with it a little bit that should have made up some examples for you to do but you can make up your own they're very straightforward ok ah we're getting close to 9 o'clock are there any questions next time we're going to start talking about qubits quantum bits and how quantum bits are very different than classical bits question yes my name Leonard Susskind or Susskind you if you like you know polishing the Apple for the professor you can call me Leonardo I like that very much well you mean let's say the other way what restriction does reversibility place on em yeah yeah that it have an inverse right there yeah yeah but I mean in the more if I were earth in a more abstract sense the answer is that it should have an inverse that the matrix should have an inverse the inverse of course is the thing that takes you back not all matrices have inverses so and you want you known inverses yeah okay then we'll look if you don't we'll come to it any other question yes that is a good question yeah the final exam is buying me lunch right that's a lot of lunches out there boy every look nobody asked me about grading the class no but I mean a lot of you have been here before so you know my policies my policies are you here to learn physics there is nobody here who is here for a degree or if you are then I'll be glad to give you a numerical grade if you need one in fact if everybody needs a numerical grade I know that there's an enormous difference in the level of preparation of different people here and to compare to compare you in an A and an exam setting wouldn't make sense because I do know that there's an enormous difference I know that everybody here is here because you want to be here and you want to learn physics and not because you have to be here so my policy is to either not grade the course at all or if somebody needs a grade in order for some particular purpose to give it d-minus low lowest possible grade lowest possible passing behavior all right so it is it's I didn't tell you what it's for yet I gave you that's right I gave you an example of a ha way you can use it to implement the idea of updating a vector from one from one instant tone to another it's one example and but I haven't told you yet why we're doing this I often spend I often spend an hour talking about qualitative aspects of physics in this case it was how do you abstractly think about deterministic physics abstractly in terms of bits and so forth and then spend some time doing some mathematics which really I want to tell you what it's for until the next time but I want to make sure since I'm going to start doing some quantum mechanics the next time I want to make sure that everybody will recognize the little algebraic little bit of manipulations that we'll do and have have the the mathematics for the next time so it's really for the next time that that I set this up yeah I think I think you will see I think it will be clear I think it will be clear uh right I think it will be clear yes I do promise to tell you why ya know

Which Way Is Down?

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Links to sources and to learn more below!

my twitter
my instagram

Thanks to Eric Langlay ( for producing, editing, and animating this episode with me. Thanks also to Henry Reich ( for his advice and guidance.

Universe Sandbox² :

Mass vs weight:

Great Veritasium video:
two other great videos: and

baseballs coming together under gravitational attraction can be simulated in Universe Sandbox 2. More math behind it can be found here:

Weight to mass (on surface of Earth) convertor:

pencil and Earth falling numbers:

NASA HD footage:


Earth’s spin and its effect on ‘down’:

The measurement of Earth and its gravity:

Movement of Earth’s center of mass:

you get heavier before you get lighter as you descend into Earth:

vertical deflection:

practical uses of measuring gravity:


Interactive Earth geoid:

your weight when moon is overhead:

Hammer and feather drop on moon:

Why things fall at the same rate:

Wolfram Alpha cone geodesic tool:

General Relativity:

simple animation showing geodesic on cone and how it causes motion DOWN in space:
GREAT pbs spacetime video (watch the whole channel):

time and gravity:

tests of general relativity:


great introductory texts:

“Relativity Visualized” by Lewis Carroll Epstein

There’s also this PDF that takes Epstein’s diagrams into more detail:

“Relativity Simply Explained” by Martin Gardener

great intro to the math of general relativity:

“A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity” by Peter Collier

Requires some background in relevant math topics (see above) but very very good:

“Spacetime And Geometry: An Introduction To General Relativity” by Sean Carrol

مرحباً مشاهدي Vsauce مايكل هنا بالأسفل هنا ولكن إلى أي اتجاه يكون الأسفل؟ وكم وزن أخف شيء ينزل للأسفل؟ حسناً, الريشة تزن 0.01 جرام لكل سنتيميتر مكعب خفيفة جداً مما يسمح للطيور بالطفو على سطح الماء ولكن مجرّد أن تتركها .. فإنها تسقط للأسفل إذاً هذا هو اتجاه الأسفل إنه الاتجاه الذي تسحبنا إليه الجاذبية. حسناً, بالنسبة لشخص في الجهة الأخرى من الأرض,
فإن اتجاه الأسفل عندي هو اتجاه الأعلى عنده ولكن أين تذهب الأجسام التي تسقط؟ ولماذا تسقط؟ هل تٌسحب أم تُدفع أو أنها تسقط بفعل السفر عبر الزمن؟ لنبدأ بالأولويات .. دعونا نحوّل الشمس إلى ثقب أسود يمكننا فعل ذلك بواسطة لعبة Universe Sandbox 2 هذا المُحاكي سيُدهشك .. يروقني جداً أحبه لدرجة وضعت كود لتحميل اللعبة بشكل مجّاني في Curiosity Box إن لم تشترك معنا في Curiosity Box فقد فاتك الكثير. حسناً, بناءً على الهدف من هذا الفيديو نريد المجموعة الشمسية ..
وها هي .. لاحظ أن كل شئ يتحرك بسرعة حول الشمس والسبب أننا ضبطنا إعدادات اللعبة بحيث كل ثانية تمرّ بالنسبة لنا عبارة عن 14 يوم تقريباً داخل اللعبة إن قمت بتغيير هذه إلى ثانية واحدة, سوف ننظر للمجموعة الشمسية بنفس سرعة وقتنا قد لاحظتم أن الكواكب بالكاد تتحرّك رغم أن الأرض تدور حول الشمس بسرعة 30 كم/ثانية, إلا أنها تبدو بالكاد تتحرّك .. وهذا يعطينا انطباع عن أن الكون شاسع أحببت هذه الحركة .. الآن انظروا للشمس حتى الآن لم نجعلها كثقب أسود, لكن يمكننا تغيير ذلك.
ما نحتاج فعله هو أن نقوم بضغط الشمس. إذاً دعونا نقفل الكتلة, لتبقى ثابتة حتى عندما نقوم بجعل قطرها أقصر. دعونا نقوم بتصغير قطرها لأصغر طول ممكن ثم, أوه, أين اختفت؟
حسناً هي لازالت موجودة .. لكنها أصبحت ثقب أسود هذا مرعب قليلاً,
لكن الآن دعونا ننظر لبقيّة عناصر المجموعة الشمسية لم يتغيّر شيء, أعني تغيّرت حاجات صغيرة. أصبح بارد ومظلم, لكن الكواكب لا تطفو بحرّية في الفضاء
ولم يمتصّها الثقب الأسود. مثل ما شاهدتم, عندما قلّصنا حجم الشمس,
لم يتغيّر اتجاه الأسفل بالنسبة للكواكب. يتم سحبهم بشكل دائم بواسطة الجاذبيّة باتجاه الوسط ..
وتقليصنا لحجم الشمس لم يحرّك الشمس. أيضاً, قوّة الجذب التي كان تسحبهم لاتجاه الشمس بقيت كما هي هذا يعطينا دليل حول اتجاه الأسفل. الدليل هو الشيء الآخر الذي لم نقم بتغييره: الكتلة. الكتلة هي مقياس لكم من الصعوبة يمكنك جعل جسمٍ ما يتسارع; تغيّر سرعته. الآن, هذه الكرتين ليس عليهم أي تأثير منّي بخصوص الحركة. لكن عندما أدفع هذه الكرة البلاستيكية المجوّفة بشكل متكرّر عمليّة سهلة جداً, لكن عندما أفعل نفس الشيء مع الكرة الفولاذية الصلبة العمليّة أصعب بشكل كبير. الجاذبيّة والوزن ليس لهما أي علاقة في هذا. الجاذبيّة تسحب للأسفل, وليس عكس اتجاه الدفع الذي قمت به بالطبع, الجاذبيّة تؤثر على الاحتكاك,
لكن الاحتكاك يؤثر عكس الاتجاه عندما أبدأ بتحريك الكرة, لكنه يؤثر معي بنفس الاتجاه عندما أقوم بإيقاف الكرة. والكرة الفولاذية أصعب في إيقافها من البلاستيكية الاختلاف هو الكتلة.
الكرة الفولاذية أكثر كتلة أكثر مقاومة لتغيير حركتها. الكتلة هي خاصية باقية على طبيعتها;
لا تعتمد على ما حولها ولا تتغيّر من مكان لآخر. يمكن تعريفها أحياناً أنها مقدار المادة الموجودة في جسمٍ ما كتلك ثابتة بغض النظر عن مكانك,
على القمر .. الأرض .. أو حتى حُراً في الفضاء ولكن يُقال أن الكتلة تهتمّ بما حولها على ما يبدو. الكتلة تحب الصحبة. الأجسام التي لديها كتلة (أو طاقة)
تتجاذب فيما بينها بواسطة قوّة نسميّها الجاذبيّة. شعور الجاذبيّة هو فقط عبارة عن
نت والأرض تتجاذبون لبعضكم كل نقطة في أي جسم لديه كتلة,
تنجذب إلى نقطة أخرى في جسم آخر. متوسط عمليات السحب هذه
هي جاذبيّة بين مراكز الكتلة لهذه الأجسام. الأشياء الضخمة مثل الأرض تبذل قوة سحب واضحة,
لكن كل شيء يقوم بذلك فعلاً. حتى كرة البيسبول. هذه الكرات تسحب بعضها البعض بواسطة جاذبيّة كلٍّ منهما. باستثناء أن كتلة كلٍّ منهم صغيرة جداً, لذلك القوّة ضئيلة,
ولا يمكنها التغلّب على قوّة الاحتكاك أو دفع الهواء عن طريقها. لن تنجذب هذه الكرتين لبعض لكن .. إن وضعت كرتي بيسبول لمسافة متر واحد بينهما
في وسط الفضاء ..حيث لا يوجد قوّة تؤثر عليهما سوف تسقط كلٍّ منها للثانية وتتصادم حرفياً. سوف يستغرق ثلاثة أيام, لكنه سيحدث. إسحاق نيوتن وجد أن كميّة القوة التي تجذب جسمين لبعض تساوي:
حاصل ضرب كتلتيهما مقسوماً على المسافة بين مركزي الكتلة للجسمين مربعاً,
مضروباً في ثابت الجاذبية الأرضية إن قمت بجعل أحد الجسمين أو كلاهما أكثر كتلة أو قرّبتهما لبعض,
كميّة القوة سوف تصبح أكبر قوّة الجذب هذه .. هي ما نسميّها: الوزن إذاً, الكتلة تبقى على طبيعتها بينما الوزن يعتمد على ما حوله شيء غريب يحصل عندما تقيس وزنك على أغلب مقاييس الوزن الوزن عبارة عن قوّة
لكن مقاييس الوزن تُظهِر باوند وكليو جرام مما هي وحدات للكتلة,
ماذا يحدث؟ أن الميزان يعمل بواسطة القوّة أي قوّة ليس شرطاً أن يكون سببها الجاذبيّة. الميزان يعرض كميّة الكتلة حول سطح الأرض التي تنجذب للأرض
بواسطة القوّة التي يقيسها. حسناً, بما أن مقاييس الوزن يُعرف أنها تستخدم على سطح الأرض بواسطة أشخاص لا تؤثّر عليهم قوّة غير الجاذبيّة الأرضيّة فإنها لا تخطئ في الغالب,
ولكن يمكن خداعها بسهولة لأنها لا تفرّق بين الكتلة والوزن لاحظ أن الوزن خاصيّة متبادلة. تقوم الأرض بجذبك للأسفل
بنفس القوّة التي تقوم أنت فيها بجذب الأرض للأعلى. بناءً على الميزان أنا أزن 180 رطلاً على الأرض والأرض تزن 180 رطلاً عليّ ولكن لأن كتلة الأرض أكبر من كتلتي أو لأن كلما كان الجسم أكثر كتلة,
كلما كان أقوى في مقاومة أن يتحرك قوّة الجذب المتساوية بيننا
تقوم بجعلي أتسارع أكثر بكثير من الأرض إن قمت بإسقاط قلم رصاص من ارتفاع 6 أقدام,
فإن القلم لا يسقط على الأرض وحسب بشكل أدقّ: كليهما ينجذبان لبعض ينجذبان لبعض بواسطة قوى متساوية لكن القوّة نفسها تحرّك القلم أكثر بكثير من الأرض عندما تترك القلم,
فإن الأرض حرفياً تنجذب للأعلى باتجاه القلم بواسطة جاذبيّة القلم لمسافة تساوي تقريباً
جزء من 9 تريليون من عرض البروتون. هذه القوّة نفسها تحرّك القلم للمسافة المتبقيّة
والتي هي بالطبع 6 أقدام لو كنت على مدار محطة الفضاء الدولية,
فإنك والأرض تنجذبون لبعضكما بنسبة 10% أقل من لو كنت على سطح الأرض,
سوف تصبح كتلتك أقل بـ8.8 مرات, لكن ليس صفر لهذا السبب .. فإن روّاد الفضاء عند انعدام الجاذبيّة,
هم ليسوا بلا وزن وليسوا في منطقة انعدام الجاذبيّة قوّة أوزانهم لا تنجح في انجذابهم للأرض لأنهم يتحرّكون أفقياً بشكل سريع حيث أنهم يسقطون بنفس سرعة انحناء سطح الأرض عنهم رغم أنهم يشعرون بـ90% من الجاذبيّة التي نشعر بها نحن الآن ولهذا هم لا يحلّقون بعيداً لا يوجد قوى جاذبية تقاوم أوزانهم, حيث كل شيء حولهم يسقط أيضاً قوة المقاومة التي تشعر بها بسبب وزنك ما لا نراه على روّاد الفضاء هو الوزن الظاهري كذلك أيضاً بالون الهيليوم له وزن أعني أنه مصنوع من مادة,
من الواضح أن له كتلة .. لذا البالون ينجذب للأرض دعونا نقيس قوة وزنها حسناً, لديه وزن ظاهري سلبي وهذا لأن انجذابه للأرض
أضعف من قوّة قابليّة الطفو من الهواء حوله الذي يدفعه للأعلى بينما يتحرك للأعلى يضغط على جزيئات الهواء للأسفل, لكنها تنقل هذه القوّة بشكل واسع.
ليس مباشرة باتجاه الأسفل على الميزان سبب قوى قابليّة الطفو
هو حقيقة أنه عندما تكون مغموراً بمائع كالماء أو الهواء الجزئيات السفلى تتعرّض لضغط أكبر حيث أنه يتم ضغطها بواسطة أوزان الجزيئات التي فوقهم جميعاً
والأقرب للأرض لذا يتم سحبها بواسطة قوّة أكبر حسناً كون لديها ضغط أكبر يعني أنه يمكنها تكوين قوّة تصادم أكبر. لذا أفقياً, هذه التصادمات تلتغي أما عمودياً, قوّة التصادم الكبرى اختفت لتقدّم قوّة رفع تمكّن البالون على الطوفان هذا يحدث أيضاً مع جسمك, من خلال السطح الخارجي للجسم
الهواء يرفعك للأعلى بقوّة مقدارها 1 نيوتن مما يساوي قوة وزن تفاحة حسناً, إذا وزنت نفسك في الفراغ بلا هواء سوف يكون وزنك أكثر من وزنك الحالي بمقدار وزن تفاحة لكن هذا ليس كل شيء, عندما تدور الأرض حول نفسها فإنها تتسبب في انتفاخ عند خط الاستواء لذا, كل ما كنت أقرب لخط الاستواء ستكون أبعد عن مركز كتلة الأرض وسيكون وزنك أقل اتجاه الأسفل يتغيّر دائماً أعني أين هو مركز كتلة الأرض؟ سيكون نفس مركز الشكل الهندسي للأرض
في حالة كان شكل الأرض منتظماً لكن الأرض عليها كمّيات من الصخور الضخمة بارتفاعات مختلفة ..
وماء وجبال وبداخلها أشياء تتحرّك باستمرار,
والهواء, والثلوج الموسمية وبرغم أن هذه العناصر بعيدة, إلا أنه الجاذبية تستمر للأبد لذا .. فإن القمر, والشمس, والنباتات .. جميعها تجذبك بمقدار معيّن لا يُذكر لكن حقاً, سوف يكون وزنك أقل بمليون مرّة من وزنك الحالي
لو كان القمر فوقك مباشرة هذا التغيير المستمرّ في التوازن للمواد على الأرض وفي الكون بشكل عام يعني أن اتجاه الأسفل يتغيّر باستمرار بالإضافة لهذا, دوران الأرض حول نفسها
يسبب انحراف ما نسميّه اتجاه الأسفل عن مركز كتلة الأرض لأنه كما يبدو أن الدفع الذي تتعرض له بسبب دوران الأرض
يتسبب في ارتفاعك قليلاً وتقليل وزنك الظاهري ويحني الأرض باتجاه خط الاستواء النتيجة هي وزن ظاهري أقل بحوالي نصف بالمئة
على خط الاستواء إن كان وزنك يظهر على الميزان 200 رطل على القطبين فإنه عند خط الاستواء سيظهر 199 باوند الرقم 9.8 الذي يستخدم في الفيزياء كقوّة جاذبيّة الأرض
تم حسابه عن طريق تأثير هذه العوامل على شخصٍ ما على خط عرض 45 درجة كل هذه التأثيرات على اتجاه الأسفل
يصبح نتيجتها انحراف عمودي هذا أكثر مقدار ثوان قوسية على الأرض "ثانية قوسية يساوي حجم قطعة نقود معدنية من على بُعد
4 كيلومترات (2.5 ميل) ضئيل جداً لدرجة لا يمكن الشعور به,
لكن تغيير الاتجاه والقوة يمكن استخدامه لدراسة شكل قاع البحر تحديد ما يكون تحتك عندما تكون على سطح البحر
أو حتى مساعدتك لاكتشاف أشياء مدفونة قديمة المقصود هو أنه ليس كل اتجاهات الأسفل لدينا
هي خطوط مستقيمة ومتوازية باتجاه مركز الأرض اتجاهات الأسفل هي خطوط متعرجة بما أن الأجسام الصلبة لا تتحرك, فإنها لا تتأثر بهذا الموضوع
بينما الماء يتأثر لذا, مع تجاهل عوامل مؤثرة مثل الرياح والمد والجز,
فإن أسطح المحيطات والبحيرات عموديّة على اتجاه الأسفل لو كان الماء يجري خلال اليابسة,
أو لو كان سطح الأرض مغمور بالمياه سوف تكون الجاذبيّة نفسها على
سطحها الوعر سطح مثل هذا يُطلق عليه اسم
(مجسم أرضي) يمكن رسمه على أي خط عمودي لو أردت صناعة طاولة تغطي كل سطح الأرض
سوف يكون سطحها مموّجاً بعمق 100 متر تقريبأ في بعض المناطق
من أجل أن تكون مستوية بحيث لو وضعنا كرة في أي مكان على الطاولة
لا تتدحرج هذا مجسم أرضي مضاعف إلى 1000 مرة سوف تزن حوالي 100% أقل من وزنك لو كنت هنا عن لو كنت هنا,
حيث الجاذبيّة أقوى المقصود هو أن قوّة واتجاه الأسفل يتغيّران
باختلاف المكان والتغييرات التي تحصل عبر الزمن إذاً, الأسفل هو قوّة موجهة متغيّرة.
هذا سهل جداً لكن لماذا يجب على مادةٍ ما
أن تجذب مادة أخرى من الأساس؟ كان (إسحاق نيوتن) قادراً على وصف الجاذبيّة,
لكنه لم يفسّرها اقتربت البشرية من تفسيرها عندما قدّم (ألبرت أينشتاين)
نظريّة النسبيّة العامّة فكّر أينشتاين كثيراً حول حقيقة سقوط الأشياء إلى الأرض بنفس المعدّل مهما كانت كتلة الجسم كبيرة, فإنه إن تم إسقاطه
سوف يتسارع باتجاه الأرض بمقدار 9.8 متر بالثانية لكل ثانية أثناء سقوطه هذا يعني أن مطرقة ذات كتلة كبيرة
وريشة ذات كتلة ضئيلة عندما يتم إسقاطهما معاً من نفس الارتفاع
سوف يرتطمون بالأرض بنفس الوقت حسناً, ما حدث للتو.. كان خطأ بسبب الهواء لكي تسقط الأشياء من خلال الهواء
فإنها يجب أن تدفع الهواء عن طريقها لكن إن كان لديها سطح واسع وقوّة وزن خفيفة
فإنه سيكون لديها الكثير من الهواء لدفعه عن الطريق لكنها لن تستطيع دفع الهواء بشكل سريع في الفراغ, تسقط الأشياء بنفس معدل التسارع
بغض النظر عن الكتلة تم وصف هذه الحالة على القمر بواسطة قائد رحلة أبولو 15 ديفيد سكوت ريشة مطرقة هذا عجيب أليس كذلك؟
أعني إن كان جسم ذو كتلة كبيرة يتم سحبه بقوّة أكبر ألا يجب أن يسقط بشكل أسرع؟
حسناً, تفسير نيوتن كان بسيطاً: تنجذب الأجسام ذات الكتل الكبيرة بواسطة قوى كبيرة أيضاً سوف تتطلّب قوّة أكبر ليتم تحريكها
مقارنةً مع الأجسام ذات الكتل الأقل جسم أكبر في الكتلة من الثاني بمئة مرة
سوف يتطلّب قوّة أكبر بمئة مرة ليتم تحريكه نفس الحركة وسيتم سحبه بواسطة الجاذبيّة أكثر بمئة مرّة أيضاً إذاً, كل شيء يسقط على الأرض بنفس المعدل يالها من صدفة ممتعة, أليس كذلك؟ ربما لا أدرك أينشتاين أنه يوجد طريقة أخرى لإظهار
الأجسام تسقط بنفس السرعة بغض النظر عن كتلها تخيّل ريشة ومطرقة في غرفة في الفضاء
إن تحرّكت الغرفة للأعلى بشكل مفاجئ بسرعة 9.8 متر بالثانية سوف تصدم المطرقة والرشية بالأرض في نفس الوقت سواءً كانت الغرفة ارتفعت لتصدم بهما,
أو أنه تم تشغيل الجاذبيّة بشكل مفاجئ لن يشعر أي جسم بأي قوّة تدفعه لا يوجد طريقة لمعرفة أيّ من هذه حدث هذا هو مبدأ التكافؤ الذي طوّره أينشتاين اعترف مرّة أن أحد أعظم أفكاره هي عن رجل يسقط من السطح أثناء سقوطه, لن يشعر بأي قوّة تؤثر عليه
برغم أنه يتسارع السقوط الحرّ لا يمكن تمييزه عن العوم الحر في الفضاء
من جانب أنك لا تشعر بأي قوّة مؤثرة وأنه لا يتم تحريكك ماذا لو لم تكن الجاذبيّة قوّة على الإطلاق؟
ماذا لو لم تكن الأشياء تسقط بسبب أنه يتم دفعها أو سحبها؟ لكن بسبب أنه لا يتم دفعها ولا سحبها. لنرى كيف يمكن هذا, نحتاج للحديث عن الخطوط المستقيمة لديّ هنا حامل بطاقة مع خيط قابل للسحب هذه طريقة رائعة لاختبار المسارات المستقيمة لأنه يتم الحفاظ على الخيط مشدود دائماً البطاقة مرسوم عليها خطّين مستقيمين وإن أبقيت الخيط مستقيماً أثناء سحبه للخارج
فإنه سيبقى بين الخطّين سأعرف أنني لم أثنيه أثناء سحبه لأنه أي ثني يتم تعريفه على البطاقة بشكل زاوية مختلفة بين الخطّين والحبل حسناً, إن وضعت اثنان من هذه على طاولة مسطحة
وقمت بسحبهما, أستطيع ضمان أنهما يسيران بشكل مستقيم لن يتقاطعا أبداً, وسيكونان متوازيين للأبد. لكن الآن دعونا نضعها على جسم كروي مرة أخرى, أقوم بسحب الخيطين معاً للأمام,
مع التأكد أنني أسحبهما بشكل مستقيم, بلا ثني انتظر .. لقد تقاطعا حسناً, إنهما لم ينثنيا .. انظر ربما هناك قوّة من نوع غريب كانت تسحب يديّ
مثل الجاذبيّة, لم أشعر بها, لكنها حدثت لا ما حدث لم يكن نتيجة قوّة,
بل كان مجرّد نتيجة طبيعية لـ؟ الانحناء قد تقول انتظر, هل هذه حقاً خطوط مستقيمة؟
انظر, لا يبدوان مستقيمين بالنسبة لي. أيضاً, ماذا لو نقلتهما إلى خطوط العرض؟
هكذا لن تتقاطع أبداً, وتبدو مستقيمة أيضاً لكنهما ليسوا كذلك!
الخط المستقيم لا ينثني
وبرغم أن خطوط العرض تبدو مستقيمة للوهلة الأولى عندما نريد تتبّع أحدها
يتطلّب منا الثني للعثور على مسارات خط مستقيم على أسطح
سواءً كانت مسطحة مثل هذا أو منحنية أنا أفضّل اختبار الشريط
يمكنك استخدام شريط من القماش لكني وجدت شريط ورقي, يعمل بشكل أفضل,
دعونا نلقي نظرة على هذا المسار هنا مستقيم في البداية, لكنه بعد ذلك ينحني,
الآن, لو لدينا شخصين يمشون على هذا المنحنى ويريدون البقاء معاً, الشخص الذي بالداخل سوف يقطع مسافة أقصر
من الشخص الذي بالخارج بما أن كلا الجانبين من هذا الشريط الورقي لا يمكن تغيير أطوالها سوف يساعدنا هذا في العثور على مسار مستقيم
إن كان يمكننا وضع الشريط الورقي بشكل مسطح سنعرف أننا قد وجدنا خط مستقيم,
وكما تشاهدون يمكننا وضع الشريط بشكل مسطح على الجزء المستقيم من هذا المسار لكن عندما نصل إلى المنحنى, من أجل تتبع المسار الآن وبما أن الشريط لديه زيادة في الجزء الداخلي من المسار, وهذا الجزء ينثني ويرتفع عن السطح,
لذا نعلم أن هذا الجزء من المسار ليس مستقيماً دعونا نستخدم اختبار الشريط للعثور على خطوط مستقيمة
على سطح مخروط حسناً, بمحاذاة مباشرة من القاعدة إلى القمة,
يبدو أن الاختبار يُظهر لنا خطوط مستقيمة على سطح المخروط نعم, الشريط ينبسط بشكل مسطح على هذا المسار,
لكن ماذا عن حلقة حول المخروط؟ لا, لا تعمل
المسافات القصيرة حول قمة المخروط
تعني أنه هناك زيادة في الشريط في الأعلى لذا لا تنبسط بشكل مسطح دعونا نرى ماذا أيضاً هناك
سوف أبدأ هنا, ثم أسمح للشريط بأن ينبسط حسناً, حصلت على شكل يبدو متعرّج قليلاً أقول: يبدو متعرّج, لأنه بالنسبة لشخص في قاعدة هذا المسار
قد يبدو أن المسار يرتفع قليلاً يتباطأ قليلاً, ثم يغيّر الاتجاه ويسقط بشكل أسرع وأسرع
بما أن الشريط على سطح مستوي مثل هذا في الحقيقة أنه بالنسبة للأشخاص الذين على سطح المخروط مستقيم تماماً,
لو تتبّعنا مسار الشريط على المخروط يمكننا أن نرى بوضوح أن المخروط يمكنه تسوية خط مستقيم على سطح منحني
وهذا ما يسمّى بالجيوديسية ( تعميم للخط المستقيم على الأسطح المنحنية) هنا جيوديسي على شكل كرة
خط الاستواء هو واحد هنا خط آخر,
خطوط العرض ليست جيوديسية ليس خط مستقيم,
لنرى لماذا, دعونا نحاول تتبع الخط بواسطة الشريط أتعلم؟ يجب أن أبقى أرفعه نعم, انظر
المسافات حول الكرة تصبح أقصر كلما ارتفعنا للأعلى لذا, يوجد زيادة في الجزء العلوي من الشريط وهي ترتفع عن السطح,
هذا المسار يحتوي منحنيات ومن أجل أن نثنيه,
يجب أن يكون هناك قوّة تؤثر عليه. إن لم يكن هناك أي قوّة مؤثرة,
هذا هو المسار الذي سوف يتخذه لاحظ أن الشريط يبدأ بالتحرّك من الشرق,
لكنه بعد ذلك يسقط باتجاه الجنوب يسقط أدرك أينشتاين أن الانحناء يمكن أن يتسبب في جذب الأشياء إلى بعضها البعض بدون الحاجة لاختراع وجود قوى مثل الجاذبيّة لكن التجاذب يحدث فقط إذا تحرّكت الأشياء على سطحٍ ما, لكن إن بقيت ثابتة فإنها لا تتجاذب لبعضها حسناً بالنسبة لشيءٍ ما ساكن
كيف يبدأ السقوط؟ أعني أن الأشياء يجب أن تتحرّك إلى هذا الاتجاه,
لكنها الآن في حالة سكون, صحيح؟ نعم لكنها فقط في حالة سكون في الفضاء,
وهذه ليست القصة بأكملها أعلى .. أسفل .. أمام .. خلف .. يسار .. يمين هي كل ما تحتاجه لوصف مكان وقوع حدثٍ ما, ولكن لوصفٍ كامل تحتاج أيضاً إلى وصف:
متى هذه الأبعاد الأربعة معاً
تشكّل الإعداد الذي يجعل كل شيء يحدث في عالمنا الزمكان وبما أننا استطعنا الحديث عن سقوط القلم باستخدام بُعد مكاني واحد:
فوق وأسفل يمكننا استخدام ورقة لاستعراض الزمكان لسقوط القلم حسناً, لدينا أعلى وأسفل,
لكن يجب أن نضيف اتجاه آخر يتحرّك فيه القلم الزمن الآن إن لم تؤثر أي قوّة على القلم
فإنه لن يتحرّك عبر المكان سوف يمر الوقت وهو ساكن,
وكما تشاهدون, إن بقي ساكناً مع مرور الزمن فإنه لن يسقط إن كان الزمكان مسطحاً,
فإني عندما أترك القلم فإنه لن يتحرّك لكن دعونا الآن نسمح للأرض والتي هي بطبيعتها ضخمة
للتلاعب بالزمكان لكن لنفرض أنها مخروط الآن مع عدم وجود أي قوّة مؤثرة على القلم,
فإن كل جزء من القلم يتبع خط مستقيم ولكن على مخروط, كما شاهدنا سابقاً
مثل هذا المسار سوف يُظهر سقوط القلم وذلك لأن المسافات تكون أقصر حول المخروط كلما ارتفعت للأعلى الوقت يمضي بشكل أسرع ولكن ليتحرّك القلم بشكل مستقيم وبدون انحناء,
يجب أن يقطع كل جزء من القلم مسافة متساوية في الزمكان هكذا فقط عندما يصطدم القلم بالأرض, فإن التنافر بين الألكترونات المتبادلة يسبب قوّة تدفع القلم من الخط الجيوديسي بالنسبة للأرض, فإن الزمن هو عبارة عن سلسلة شرائح من هذا التطوّر, جيوديسية القلم الحرّة من أي قوّة مؤثرة
هي السبب في سقوطه ليس بسبب قوّة دفع أو سحب,
فقط نزعة طبيعية للمتابعة على خط مستقيم حتى يؤثر عليه شيء آخر, حسناً, نحن لم نستخدم إلا بُعداً واحداً فقط من المكان,
وواحداً فقط من الزمان لأنه عمليّة تصوّر الأكوان, ثلاثة أبعاد من المكان, وواحداً من الزمان
سيكون خارج حدود ما يمكننا عرضه على الورق والشاشات ولكن الرياضيات يمكن أن تأخذنا هناك النسبيّة العامة تتيح لنا حساب مقدار الكتلة والطاقة
التي تسببت في انحناء الزمكان وقد كانت تستخدم لشرح أشياء مثل نظرية نيوتن القديمة للسقوط
كنتيجة أن القوى التي لا تؤثر مثل الشذوذ في مدار كوكب عطارد الذي يدور قريباً من الشمس,
وبالتالي فهو الأكثر تأثراً من الشمس على الزمكان وقد أكدت العديد من التجارب الأخرى صورة نظرية النسبيّة العامة للكون مما يوصلنا إلى استنتاج أنه لا يوجد جاذبيّة! هناك فقط .. الزمكان كونه منحني, وكوننا نعيش فيه مثل المقولة الشهيرة لـ(جون ويلر):
الزمكان يُملي على الكتلة كيف تتحرّك, والكلتة تُملي على الزمكان كيف ينحني بالنسبة للأرض .. نحن لا نتحرّك بسرعة كبيرة, حتى الطائرات النفّاثة
تتحرّك بشكل لا يُصدّق بسرعة قريبة من سرعة الضوء لذا, بالنسبة إلى الأرض, نحن نتحرّك تقريباً
خلال الزمن فقط على هذا النحو, نحن أكثر تأثراً بالطريقة التي ينثني بها الزمان بواسطة الكتلة أكثر من مقدار انحناء المكان. وقد دفع هذا الكثير إلى الادّعاء بأنك في معظم الأحيان
تشعر وكأنك تُدفع إلى الأرض ليس بسبب قوّة تسمّى الجاذبيّة ولكن لأن الوقت يتحرّك بشكل أسرع بالنسبة لرأسك عن قدميك اتجاه الأسفل نسبيّ ودائم التغيّر,
ولكنه موجود بسبب وباتجاه الزمن الأبطأ (برتراند راسل) سمّى هذا بـ(قانون الكسل الكوني) كل شيء موجّه إلى حيث يكون الوقت أبطأ,
هذا نسمّيه الذهاب للأسفل لذلك لا يجب عليك التحفّظ على شيء الوقت سوف يتولّى هذا الأمر وكالعادة, شكراً لمشاهدتكم ترجمة: عبدالرحمن التميمي
[email protected] تذكّر أنه يمكنك دعم Vsauce والبحوث الخاصة بمرض ألزهايمر عن طريق الاشتراك في (Curiosity Box) الاشتراك الحالي يأتي مع كود نسخة مجانية للعبة Universe Sanbox 2 وهذا مدهش ومجموعة كاملة من الألعاب العلميّة الأخرى
والأدوات التي اخترتها أنا و(جيك) و (كيفين) أحبكم, وأيضاً آمل أن أراكم في عروض (Brain Candy) المباشرة.
نحن قادمون إلى العديد من المدن قريباً نأمل أن يكون أحد هذه العروض بالقرب منكم,
بالذهاب للعرض سوف تروني أنا و(آدم) نفعل أشياء قد لا تكونوا رأيتمونا نفعلها من قبل,
نستكشف العلم والمفاهيم الخاطئة الشائعة وراء كل شي ربما قلت كثيراً, ربما لا.
آمل أن أراكم هناك
وكالعادة .. شكراً لمشاهدتكم


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Relativity is not physics but “metaphysics”, or maybe metaphysics, but then it should not present itself as presenting the scientific method.

Speed has technological [and legal :)] limits but not a scientific, logical limit. Relativity depends on the allegation that there is a scientific, logical limit to speed. From there it proceeds logically to absurdities, falsehoods.

1. I’ve heard it said that Relativity says a person could travel to yesterday and replace himself there. But there should be two of them there, unless he displaces himself.

And I’ve heard it said that Relativity says that that person however cannot change history, meaning the subject matter of the field of knowledge known as History. But in the field of knowledge known as Science, everything in the past is history, and there is no distinction for the history subset that is the subject of the field of knowledge known as History. Relativity takes the truth that we cannot change history and arbitrarily gives it a limited meaning, unscientifically.

Just by travelling to the past, a person already changes history by displacing molecules. Further interactions there are further changes to history.

If after a day he returns again to yesterday, would there be two of them there?, or three? But Relativity says time is the fourth dimension needed to identify a unique point in space-time.

And in fact if a person travels to yesterday, it wouldn’t be yesterday because someone from today would be there. In the first place, molecules would have been displaced; it wouldn’t be yesterday.

2. From the allegation that there is a scientific, logical limit to speed, Relativity says that in a situation wherein a person A is moving relative to a person B, time would be moving slower for A relative to B, and faster for B relative to A. For a footnote to this, watch and see the description below that video and my pinned comment below that description.

This applies to everything in the universe, so that for each unit of matter in the universe of relative motions, each of the rest would potentially have its own unique time relative to it.

3. From the allegation that there is a scientific, logical limit to speed, Relativity proceeds logically to an internal contradiction.

It says that in a situation where a person A is moving relative to a person B, time would be moving slower for A relative to B, and faster for B relative to A.

But A moving relative to B is equivalent to B moving relative to A. This is a principle of Relativity itself, and a valid principle by the way.

With B moving relative to A, time would be moving slower for B relative to A, and faster for A relative to B.

This would be a contradiction; time could not be moving slower and faster at the same time for A relative to B, and faster and slower at the same time for B relative to A.

And in fact the mathematics returns us to the truth. Slower by a certain magnitude + faster by the same magnitude equals there is no difference in the way time moves for A and B.

Related videos and a playlist by other channels:

Making sense of string theory | Brian Greene

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In clear, nontechnical language, string theorist Brian Greene explains how our understanding of the universe has evolved from Einstein’s notions of gravity and space-time to superstring theory, where minuscule strands of energy vibrating in 11 dimensions create every particle and force in the universe. (This mind-bending theory may soon be put to the test at the Large Hadron Collider in Geneva).

TEDTalks is a daily video podcast of the best talks and performances from the TED Conference, where the world’s leading thinkers and doers give the talk of their lives in 18 minutes. TED stands for Technology, Entertainment, Design, and TEDTalks cover these topics as well as science, business, development and the arts. Closed captions and translated subtitles in a variety of languages are now available on TED.com, at

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المترجم: Fady Alshaar
المدقّق: Anwar Dafa-Alla في العام 1919، اقترح عالم رياضيات ألماني مغمور يدعى تيودور كلوتزة اقترح فكرة غاية في الجرأة، و بالأصح، غاية في الغرابة. اقترح أن كوننا هذا قد يحتوي على ما هو أكثر من الأبعاد الثلاثة التي ندركها جميعنا. أبعاد أخرى تضاف الى الأبعاد المألوفة، طول، عرض، ارتفاع، اقترح كلوتزة بأنه ربما يكون ثمة أبعاد إضافية للمكان و لسبب ما لا نراها بعد. الأن، عندما يقوم شخص بطرح فكرة جريئة و غريبة، أحيانا تكون هذه هي — مجرد فكرة جريئة و غريبة، و لكن لا علاقة لها بالعالم من حولنا. و لكن، هذه الفكرة، بالتحديد — وبالرغم من أننا لانعلم بعد مدى صحتها، وفي نهاية المحاضرة، سأناقش تجربة علمية، و التي من المحتمل أن تثبت فيما إذا كانت هذه الفكرة صحيحة أم لا في غضون السنوات القادمة — إلا أن لهذه الفكرة أبلغ التأثير في الفيزياء خلال القرن الأخير و لازالت تكشف الكثير من خلال الأبحاث الدقيقة الجارية. لذلك أود أن أخبركم شيئا عن قصة هذه الأبعاد الإضافية. أذن أين نمضي؟ كبداية، لابد من معرفة القليل عن خلفية هذه القصة. بالعودة إلى العام 1907. حيث كان أينشتاين يتوهج شهرة لإكتشافه النظرية النسبية الخاصة و قراره أن يواصل تقدمه في مشروع جديد — في محاولة منه لفهم القوى الشاملة و النافذة للجاذبية. في تلك المرحلة، كان هنالك الكثير ممن إعتقد بأن المشروع قد إكتمل و حلت المعضلة. فقد قام العالم نيوتن بوهب العالم نظريته في الجاذبية في أواخر 1600 حيث كانت النظرية أنذاك ممتازة، في شرحها لحركة الكواكب، لحركة القمر و غيرها، لحركة سقوط التفاح من على الأشجار، المشكوك في صحتها مصطدمة برؤوس الناس. كل ذلك تم شرحه باستخدام نظرية نيوتن في الجاذبية. و لكن أينشتاين أدرك أن نيوتن قد أغفل شيئا في نظريته، لأنه حتى نيوتن نفسه كان قد كتب أنه و بالرغم من فهمه لكيفية حساب تأثير الجاذبية، إلا كان غير قادر على فهم كيفية عملها فعلا. كيف للشمس، التي تبعد 93 مليون ميل عن الأرض، أن تؤثر في حركة هذه الأخيرة؟ كيف لتأثير الشمس أن يصل بعيدا، عبر فضاء فارغ و خال، ليطبع تأثيره هناك؟ و هذه هي المهمة التي خصص أينشتاين نفسه لأجلها — ليدرك حقيقة عمل الجاذبية. دعوني هنا أريكم ما توصل إليه أينشتاين. إكتشف أينشتاين بأن الوسط الذي ينقل قوة الجاذبية هو الفراغ نفسه. الفكرة تبدو كما يلي: تخيل أن الفراغ هو عبارة عن طبقات يتواجد عليها كل شي في الكون. حيث قال أينشتاين بأن الفضاء مستو و أملس، في غياب وجود المادة. و لكن في وجود جسم مادي في المحيط، كالشمس، فإنها تسبب إنحناء و تقوسا في الفراغ . و هذا ما يجعل الجاذبية تمتد في الفراغ. حتى أن الأرض تحني الفراغ حولها. الآن أنظر إلى القمر. حسب هذه الأفكار، فإن القمر أسير في مدار، لأنه يتدحرج في واد محفور في البنيان المقوس المنحني و الذي تشكل بسبب وجود الشمس و القمر و الأرض في الفراغ. لننتقل إلى إطار المشهد كاملا. الأرض نفسها أسيرة في مدار لأنها تتبع تقعرا في الفراغ بسبب تواجد الشمس. و بهذا، تكون هذه هي الفكرة الجديدة عن كيفية عمل الجاذبية. هذه الفكرة إختبرت عام 1919 من خلال الرصودات الفلكية. و هي صحيحة فعلا، حيث أوضحت البيانات. و هذا ما أكسب أينشتاين الشهرة حول العالم. و هذا ما استرعى إهتمام كلوتزة. فقد كان كلوتزة كما كان أينشتاين، منهمكا في البحث بما نسميها "نظرية الحقل الموحد." و هي عبارة عن نظرية واحدة جامعة من المحتمل أن تكون قادرة على وصف جميع القوى في الطبيعة من خلال مجموعة من الأفكار، مجموعة واحدة من المبادئ، أو من خلال معادلة رئيسية واحدة، إذا كنتم ترغبون بهذه التسمية. لذلك، فقد قال كلوتزة في قرارة نفسه، أينشتاين استطاع شرح الجاذبية بوصف الفراغ بنيان قابل للتعقر و الإنحناء — في الواقع الفراغ المكاني الزماني، توخيا للدقة. يمكن أن أعزف على نفس الوتر، فيما يتعلق بالقوة الأخرى المعروفة، والتي، كانت تعرف حينذاك، بالقوة الكهرومغناطيسية — نحن نعلم بوجود قوى أخرى الآن، و لكن في ذلك الوقت كانت تلك القوة الوحيدة التي استرعت إهتمام الناس. تعلمون، أن هذه القوة هي المسؤولة عن التجاذب الكهربائي و المغناطيسي و إلى ما هنالك. و لذلك قال كلوتزة قد أمضي على نفس المنوال لأصف القوة الكهرومغناطيسية على شكل تقعر و انحناء. ما يبرز سؤالا هنا: في أي وسط سيكون هذا التقعر و الإنحناء؟ أينشتاين كان قد استخدم الفضاء الزمكاني مسبقا، على شكل إنحناء و تقعر، لشرح الجاذبية. لايبدو أن ثمة وسط آخر لينحني و يتقعر. لذلك فقد قال كلوتزة، حسن، ربما يوجد أبعاد إضافية أكثر للفراغ. حيث قال: إن كنت أريد شرح قوة إضافية أخرى، ربما أحتاج لبعد إضافي آخر. لذلك، فقد تخيل أن الفراغ له أربعة أبعاد، لا ثلاثة، و تخيل أن القوة الكهرومغناطيسية ما هي إلا إنحناء و تقعر في ذلك البعد الرابع. و هنا المهم: عندما قام باستنتاج المعادلة التي تصف هذا الإنحناء و التقعر في كون ذو أربعة أبعاد، و ليس ثلاثة، حصل على المعادلة القديمة التي إشتقها أينشتاين مسبقا، في الأبعاد الثلاثة — و التي خصصت للجاذبية — ولكنه حصل على معادلة إضافية بسبب البعد الإضافي الآخر، و حالما نظر إلى المعادلة. أدرك أنها لم تكن إلا تلك المعادلة التي عرفها العلماء لفترة طويلة، و التي تصف القوة الكهرومغناطيسية. مدهش — كيف انبثقت بتلك الطريقة. و قد كان شديد الحماس بهذا الإكتشاف لدرجة أنه أخذ يجوب منزله راكضا، مصيحا "وجدتها!" — لعلمه بأنه إكتشف نظرية المجال الموحد. من الواضح، أن كلوتزة كان رجلا نظريا بحتا. في الواقع — هنالك قصة تروى عنه، أنه عندما أراد أن يتعلم السباحة، قام بقراءة كتاب، مقالة في السباحة — (ضحك) — و من ثم غطس في المحيط. هذا نوع من الأشخاص الذين يعتمدون على النظريات في حياتهم. و لكن بالنسبة للبعض منا ذوي التفكير العملي فإن، سؤالين سيطرحان مباشرة نتيجة هذه المقاربة. السؤال الأول: إن كان بالفعل ثمة أبعاد أكثر، فأين هي؟ لا يبدو أننا قادرين على رؤيتها. و السؤال الثاني: هل هذه النظرية تتحقق فعلا عندما يتم تطبيقها على أرض الواقع، عندما نحاول إسقاطها على العالم من حولنا؟ السؤال الأول تمت الإجابة عليه في عام 1926 من قبل زميل يدعى أوسكار كلاين. حيث أشار بأن الأبعاد يكمن أن تتشكل في نمطين مختلفين — إذ هنالك أبعاد كبيرة، و التي تسهل رؤيتها، و من جهة أخرى هنالك أبعاد غاية في الضآلة، ملتوية ملتفة على بعضها، ملتفة بشكل دقيق جدا، و بالرغم من أنها حولنا في كل مكان، إلا أننا لا نستطيع رؤيتها. دعوني أوضح لكم هذا الأمر بصريا. تخيلوا أنكم تنظرون إلى جسم ما كالسلك الذي يدعم إشارة المرور الضوئية. في منطقة منهاتن، أنت في (سنترال بارك) — خرجنا من الموضوع قليلا — و لكن هذا السلك يبدو أحادي البعد من المسافة التي ننظر إليه، و لكن أنت و أنا جميعنا نعلم أن هذا السلك له سماكة. من الصعوبة بمكان رؤيتها، من هذه المسافة البعيدة. و لكن إن توجهنا مقتربين منه، لرؤيته من وجهة نظر، لنقل، نملة صغيرة تتجول في محيط السلك — النملات صغيرات الحجم جدا لدرجة أنه بمقدورها الوصول لجميع الأبعاد — البعد الكبير (الطولي)، و أيضا البعد الصغير (الملتف)، ذو الإتجاه الدوراني، مع أو بعكس إتجاه عقارب الساعة. و أتمنى أن تقدروا هذا الذي ترون. فقد استغرق منا الكثير من الوقت لدفع هذه النملات للقيام بذلك. (ضحك) و لكن ما رأيتموه يوضح حقيقة أن الأبعاد يمكن أن تكون ذات نوعين: كبيرة و صغيرة. و الفكرة المطروحة هنا أن الأبعاد الكبيرة حولنا ربما تكون الأبعاد التي يسهل علينا رؤيتها، و لكن ربما يوجد أبعاد إضافية أخرى ملتوية و ملتفة، كنوع ذلك البعد الدوراني الملتف للسلك، و هي غاية في الضآلة إلى درجة أنها بقيت حتى الآن غير مرئية. دعوني أريكم كيف قد تبدو عليه هذه الأبعاد. إذن، إن ألقينا نظرة، لنقل، على الفضاء المكاني نفسه — بالطبع، يمكنني أن أريكم عرضا، على شاشة ذات بعدين. بعضكم سوف يطور تقنية لتجاوز هذا الأمر يوما ما، و لكن كل ما هو ليس بمسطح على الشاشة يعد بعدا جديدا، البعد يصغر، و يصغر، و يصغر، و في النهاية، في عمق سحيق شديد الضآلة و الصغر في الفراغ نفسه — تبدو الفكرة على النحو التالي: يمكن أن يكون هنالك أبعاد إضافية ملتوية. هنا شكل دائري صغير — و هذه الأبعاد شديدة الصغر لدرجة أنها مستعصية على الرؤية. و لكن إن بدونا على شكل نملة شديدة الضآلة تتجول في المكان، فسيكون بقدورنا أن نتجول في البعد الكبير الذي ندركه جميعنا — و الذي يبدو هنا في الجزء المخطط — و لكنك أيضا سيكون بمقدورك الوصول إلى البعد الملتف الصغير و الذي لايمكن رؤيته لا بالعين المجردة لضآلة حجمه و لا حتى بأدق الأجهزة التي نستخدمها الآن. و لكن ما إن نبحر عمقيقا في النسيج المكاني الفراغي نفسه، فإن الفكرة تقول بإمكانية وجود أبعاد إضافية، كما لاحظنا. هذا شرح لكيفية أن الكون قد يحتوي على أبعاد إضافية أكثر مما نراه. و لكن ماذا بشأن السؤال الثاني الذي طرحته: هل تتحقق هذه النظرية فعلا عندما نحاول تطبيقها على أرض الواقع؟ حسنا، هذا يعود بنا إلى عصر أينشتاين و كلوتزة و غيرهم الكثير حيث عملوا جاهدين على بلورة هذه الرؤية و تطبيقها على الفيزياء الكونية كما كانت مفهومة في ذلك العصر، و لكنها لم تتحقق فعلا في تفاصيلها. في تفاصيلها، على سبيل المثال، لم يتمكن العلماء من الحصول على كتلة الإلكترون بما ينسجم مع هذه النظرية. و العديد قد حاول العمل على هذه النظرية، و لكن مع الأربعينات، تحديدا الخمسينات من القرن الماضي، فإن هذه النظرية الغريبة و المتحدية عن كيفية توحيد قوانين الفيزياء قد ذهبت أدراج الرياح. إلى أن حصل شيء رائع في زماننا. في عصرنا، فإن مقاربة جديدة لتوحيد قوانين الفيزياء تم السعي وراءها من قبل فيزيائين من أمثالي، و من أمثال آخرين من أنحاء المعمورة، تسمى نظرية الأوتار الفائقة. و المدهش في الموضوع أن نظرية الأوتار الفائقة و من النظرة الأولى لا علاقة لها بفكرة الأبعاد الإضافية، و لكن ما أن ندرس نظرية الأوتار الفائقة، حتى نجد أنها تبعث من جديد فكرة هذه الأبعاد و لكن بشكل جديد متألق. لذلك اسمحوا لي أن أسرد لكم كيف تتبلور هذه النظرية. نظرية الأوتار الفائقة — ما مفهومها؟ حسنا، إنها عبارة عن نظرية تحاول الإجابة عن السؤال التالي: ما هي المكونات الأساسية الأولية الغير قابلة للتجزئة الغير قابلة للتقسيم و التي يتركب كل شيء منها في هذا العالم من حولنا؟ الفكرة تبدو على النحو التالي. فلنتخيل أننا ننظر لجسم مألوف لدينا، شمعة على حامل، و لنتصور أننا نريد معرفة مم يتكون منه هذا الحامل. نرتحل في رحلة عميقا داخل هذه الجسم لنتعرف على الوحدات الأساسية المكونة له. عميقا جدا في الداخل — حيث نعلم جميعنا أننا في الموضع المناسب عميقا، سنرى الذرات. جميعنا نعلم أن الذرات ليست نهاية القصة. في هذه الذرات يوجد إلكترونات تدور حول نواة مركزية و هذه النواة تتكون من وحدات هي النترونات و البروتونات. حتى النترونات و البروتونات تتكون من دقائق أصغر في الداخل تعرف باسم الكواركات. هنا حيث تتوقف الفكرة التقليدية. الفكرة الجديدة لنظرية الأوتار تقول. عميقا في أي من هذه الجسيمات، يوجد شيء آخر. هو عبارة عن خيوط من الطاقة التي تهتز. و التي تبدو كأوتار مهتزة — و من هنا إشتق إسم هذه النظرية. و كما هو الحال في الأوتار المؤلفة لآلة التشيلو الموسيقية و التي تهتز في أطوار مختلفة، فإنه الحال كذلك مع هذه الأوتار من الطاقة، التي تهتز قي أطوار مختلفة. و لكن بدلا من إصدار نغمات موسيقية. فإنها تصدر التوليفة التي تؤلف الجسيمات التي يتكون منها الكون من حولنا. لذا، في حال صحة هذه الرؤى، فإن الأرضية المتناهية الصغر للكون ستبدو على هذا الشكل. مكونة من عدد هائل من هذه الخيوط الطاقية المهتزة المتناهية الصغر، تهتز ببترددات مختلفة. و هذه الترددات المختلفة هي التي تكون الجسيمات الأولية المتنوعة. و هذه الجسيمات الأولية هي المسؤولة عن الغنى و التنوع في العالم من حولنا. و هنا يمكنك أن ترى نوعا من توحيد المجال، لأن الجسيمات المادية، كالإلكترونات و الكواركات، و الجسيمات الموجية، كالفوتونات، و الغرافيتونات، جميعها مكونة من وحدة بناء واحدة. و بهذا فإن المادة و القوى التي تعمل في الطبيعة جميعها تم وضعها تحت عنوان واحد هو الأوتار المهتزة. و هذا ما قصدناه بعبارة نظرية موحدة. و الملفت في الموضوع أنه. عندما تقوم بدراسة البنية الرياضية لهذه النظرية، فستجد أن النظرية لا تصلح للعمل في كون ذو ثلاثة أبعاد فراغية فقط. لا تصلح للعمل في كون ذو أربعة أبعاد أيضا، و لا حتى ذو خمسة أو ستة أبعاد. أخيرا، يمكنك دراسة المعادلات، و التي تظهر أن النظرية صحيحة فقط في كون ذو عشرة أبعاد مكانية و بعد زماني واحد. مما يعيدنا مجددا إلى فكرة كلوتزة و كلاين — القائلة بأن عالمنا عندما يشرح بشكله الصحيح، فسيكون له أكثر من الأبعاد التي نراها. ربما الآن تفكرون بهذا و تقولون، حسن، إن كان هنالك أبعاد إضافية، و هي ملتفة على بعضها بشكل كبير، نعم، ربما لن يكون بمقدورنا رؤيتها و هي في تلك الضآلة من الحجم. و لكن إن كان ثمة حضارة صغيرة مكونة من مخلوقات خضراء تتجول في ذلك الفراغ، و هم من الضآلة بمكان بحيث يستحيل رؤيتهم أيضا، هذا صحيح. هذه إحدى تكهنات نظرية الأوتار — كلا، هذه ليست إحدى التكهنات الناتجة عن نظرية الأوتار. (ضحك) و لكن هذا يطرح السؤال: هل ما نحاول فعله هنا هو محاولة لإخفاء هذه الأبعاد الإضافية، أم أن هذه الأبعاد تخبرنا بشيء عن الكون؟ في الوقت المتبقي، أود أن أخبركم عن ميزتين من ميزات هذه الأبعاد الإضافية. الأولى هي، الكثير منا يعتقد بأن هذه الأبعاد الإضافية تحمل الإجابة لما قد يكون أحد أعمق الأسئلة في الفيزياء النظرية, و العلم النظري. هذا السؤال هو التالي: عندما ننظر من حولنا، كما فعل العلماء في القرون القليلة الماضية، فسيبدو لنا حوالي عشرين من الأرقام و التي بحق تصف كوننا. هذه الأرقام من مثل قيم كتلة الجسيمات، مثل الإلكترونات و الكواركات، و أيضا شدة قوة الجاذيية، شدة القوة الكهرومغناطسيسية — و هنالك قائمة بحوالي عشرين رقما و التي تم قياسها بدقة متناهية للغاية، و لكن أحدا لم يكن لديه شرح عن سبب إمتلاك هذه الأرقام لقيم محددة بهذا الشكل. الآن، هل توفر نظرية الأوتار إجابة؟ ليس بعد. و لكننا نعتقد بأن الجواب عن سبب إمتلاك هذه الأرقام لتلك القيم بالتحديد ربما يكمن في شكل الأبعاد الإضافية. و المدهش في الموضوع، إن كان لهذه الأرقام قيما أخرى مختلفة عن القيم التي نعرفها، فإن هذا الكون، لن يكون هو نفسه بالصورة، التي نعرفها الآن. هذا سؤال جوهري. لماذا هذه الأرقام مضبوطة بدقة متناهية بما يسمح بتوهج النجوم و تشكل الكواكب، إذ عندما نعلم بأننا إن عبثنا بهذه الأرقام — لو امتلكت 20 قرصا مدرجا هنا و سمحت لك بالعبث و تغير قيم هذه الأرقام، فإن أي تغير نجريه على الأقراص سيجعل الكون يختفي. فهل من الممكن شرح هذه الأرقام العشرين؟ تقترح نظرية الأوتار بأن هذه الأرقام العشرين لها علاقة بالأبعاد الإضافية. دعوني أوضح لكم كيف. إذن، عندما نتحدث عن أبعاد إضافية في نظرية الأوتار، فهي ليست مجرد بعد إضافي واحد فقط، كما في أفكار كلوتزة و كلاين. هذا ما تطرحه نظرية الأوتار فيما يتعلق بالأبعاد الإضافية. لهذه الأبعاد هندسة معقدة متداخلة مع بعضها. هذا مثال لشكل يدعى شكل كالابي-ياو — الإسم ليس مهما جدا هنا. و لكن كما ترون، فإن هذه الأبعاد الإضافية مطوية على بعضها البعض و متداخلة في ما بينها في نموذج مثير للإهتمام، و بنية أخاذة. و الفكرة هي أنه إن لهذه الأبعاد الإضافية هذه الهندسة، فإن الفراغ المتناهي الصغر للكون من حولنا سيبدو على هذا النحو. عندما تلوح بيدك، فإنك ستتحرك ضمن هذه الأبعاد الإضافية مرارا و تكرارا، و لكنها من الضآلة بمكان بحيث لا ندرك ذلك. إذن ما هو المضمون الفيزيائي، المتعلق بهذه الأرقام العشرين؟ تمعن في ما يلي. إذا نظرت إلى آلة موسيقية، البوق الفرنسي، لاحظ أن إهتزاز تيار الهواء يتأثر بشكل الآلة الموسيقية. في نظرية الأوتار، فإن جميع الأرقام ما هي إلا نتيجة لطريقة إهتزاز هذه الأوتار. لذلك و كما هو التيار الهوائي في الآلة الموسيقية الذي يتأثر بتغير شكل الآلة الموسيقية، فإن الأوتار نفسها ستتأثر بالنموذج الإهتزازي في الهندسة الفراغية التي تتواجد فيها هذه الأوتار. لنحضر بعض الأوتار هنا. عند مشاهدتك لهذه الأوتار تهتز في الأنحاء — ستظهر على الشاشة خلال لحظات — هنا تماما، لاحظ أن الطريقة التي تهتز فيها تتأثر بشكل مباشر بالهندسة الفراغية للأبعاد الإضافية. فإذا علمنا بالضبط كيف تبدو عليه الأبعاد الإضافية — لا علم لدينا بعد، و لكن إن علمنا — فسيكون بمقدورنا حساب النغمات الممكنة، و الأنماط الإهتزازية الممكنة. و إن تمكنا من حساب الأنماط الإهتزازية الممكنة، فسيكون بمقدورنا حساب هذه الأرقام العشرين. فإذا كانت الإجابة التي نحصل عليها من حساباتنا متوافقة مع قيم هذه الأرقام المحددة مسبقا من جراء التجارب و القياسات الدقيقة، فسيكون هذا و بشكل كبير الشرح الجوهري الأساسي الأول عن سبب تشكل الكون في بنيته التي يبدو عليها بهذا الشكل. القضية الثانية التي أود أن أختم بها هي: كيف يمكننا أن نختبر هذه الأبعاد الإضافية بشكل مباشر؟ هل ما تحدثنا عنه مجرد بناء رياضي مثير للإهتمام يستطيع وصف بعض المظاهر الغامضة للعالم، أم أنه بإمكاننا فعلا فحص هذه الأبعاد الإضافية؟ نعتقد — برأيي هذه أمر ممتع جدا — أنه و بغضون حوالي السنوت الخمس القادمة قد يكون بمقدورنا أن نتحقق من وجود هذه الأبعاد الإضافية. و إليكم كيف يتم ذلك. في سيرن، جنيف، سويسرا، آلة يتم بناؤها الآن تدعى مصادم الهايدرونات العملاق. و هي عبارة عن آلة تقوم بإرسال جسيمات حول قناة، في اتجاهات متعاكسة، بسرعة تقارب سرعة الضوء. و كثيرا ما تتجه هذه الجسيمات باتجاه بعضها البعض، محدثة تصادما رأسيا مباشرا. الأمل هو أنه إن كان لهذا التصادم طاقة كافية، فقد يسمح بقذف بعض الشظايا من هذا التصادم من أبعادنا، مجبرة هذه الشظايا على الدخول إلى أبعاد أخرى. كيف سنعرف هذا الأمر؟ حسن، سنقوم بقياس كمية الطاقة بعد التصادم، و نقارنها بكمية الطاقة قبله، فإذا كان هنالك نقص في الطاقة بعد التصادم عما كان عليه قبل التصادم، فسيكون هذا دليلا على أن الطاقة قد انتقلت لبعد آخر. فإذا كان هذا الإنتقال مطابقا للنموذج الموافق لقياساتنا، فسيكون هذا برهانا على وجود الأبعاد الإضافية. دعوني أوضح لكم هذا الأمر بصريا. تصوروا أنه لدينا نوع محدد من الجسيمات يدعى غرافيتون — و هو نوع من تلك الشظايا التي نتوقع أن تقذف بعد التصادم إذا كانت فكرة الأبعاد الإضافية حقيقية. هنا يبدو لنا كيف تعمل هذه التجربة. نأخذ هذه الجسيمات، نصدمها عنيفا ببعضها. نصدمها عنيفا ببعضها، فإن كنا على حق، فإن بعض الطاقة الناتجة عن التصادم ستغدو على شكل شظايا تتنتقل بعيدا إلى هذه الأبعاد الإضافية. إذن هذا نوع من التجارب التي سنتطلع إليها في غضون الخمس، السبع، إلى العشر سنوات القادمة. و إذا أتت هذه التجارب أكلها، إذا استطعنا أن نرى ذلك النوع من الجسيمات المقذوفة عن طريق ملاحظة النقص في الطاقة الموجودة في أبعادنا الحاصل بعد التصادم، فإن هذا سيبرهن على أن الأبعاد الإضافية حقيقية. و بالنسبة لي فإن هذه القصة على درجة عظيمة من الأهمية، و فرصة نادرة. بالعودة إلى زمن نيوتن حيث الفضاء ذو قيمة مطلقة — لا يوفر أي شيء سوى أنه حلبة، ساحة تجري فيها أحداث الكون. أتى بعدها أينشتاين بقوله، حسن، المكان و الزمان يمكن لهما أن يتقعرا و ينحنيا، و هذا ما يولد الجاذبية. و الآن فإن نظرية الأوتار لتقول، نعم، الجاذبية، الميكانيك الكوانتي، الكهرومغناطيسية — جميعها توضع في محتوى واحد، بشرط أن يكون للكون أبعادا أكثر من تلك التي نراها. و هذه عبارة عن تجربة يمكن أن تؤكد وجود هذه الأبعاد في زماننا. واعدة بإحتمالات مذهلة. شكرا جزيلا لكم. (تصفيق)

Light ☀

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Relativity is not physics but “metaphysics”, or maybe metaphysics, but then it should not present itself as presenting the scientific method.

Speed has technological [and legal :)] limits but not a scientific, logical limit. Relativity depends on the allegation that there is a scientific, logical limit to speed. From there it proceeds logically to absurdities, falsehoods.

1. I’ve heard it said that Relativity says a person could travel to yesterday and replace himself there. But there should be two of them there, unless he displaces himself.

And I’ve heard it said that Relativity says that that person however cannot change history, meaning the subject matter of the discipline known as History. But in the discipline known as Science, everything in the past is history, and there is no discrimination against the history subset that is outside the subject of the field of knowledge known as History. Relativity takes the truth that we cannot change history and arbitrarily gives it a limited meaning, unscientifically.

Just by travelling to the past, a person already changes history by displacing molecules. Further interactions there are further changes to history.

If after a day he returns again to yesterday, would there be two of them there?, or three? But Relativity says time is the fourth dimension needed to identify a unique point in space-time.

And in fact if a person travels to yesterday, it wouldn’t be yesterday because someone from today would be there. In the first place, molecules would have been displaced; it wouldn’t be yesterday.

2. From the allegation that there is a scientific, logical limit to speed, Relativity says that in a situation wherein a person A is moving relative to a person B, time would be moving slower for A relative to B, and faster for B relative to A. For a footnote to this, watch  and see the description below that video and my pinned comment below that description.

This applies to everything in the universe, so that for each unit of matter in the universe of relative motions, each of the rest would potentially have its own unique time relative to it.

3. From the allegation that there is a scientific, logical limit to speed, Relativity proceeds logically to an internal contradiction.

It says that in a situation where a person A is moving relative to a person B, time would be moving slower for A relative to B, and faster for B relative to A.

But A moving relative to B is equivalent to B moving relative to A. This is a principle of Relativity itself, and a valid principle by the way.

With B moving relative to A, time would be moving slower for B relative to A, and faster for A relative to B.

This would be a contradiction; time could not be moving slower and faster at the same time for A relative to B, and faster and slower at the same time for B relative to A.

And in fact the mathematics returns us to the truth. Slower by a certain magnitude + faster by the same magnitude equals there is no difference in the way time moves for A and B.

Related videos and a playlist by other channels:

Philosophy of Physics

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From Newton and Maxwell to General Relativity, Quantum Mechanics, Dark Matter, and Dark Energy. The nature of fundamental physical laws.

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التصور الحالي لطبيعة الكون من حولنا هي على نحو مختلف تماماً عن تصور أسلافنا خلال مسيرة تاريخ البشرية كان الإعتقاد السائد أن هنالك جملة من قوانين الفيزياء التي تحكم الأحداث هنا على سطح الأرض وجملة أخرى مختلفة من القوانين لأحداث الفضاء الخارجي الاعتقاد السائد كان يتلخص بأن سلوك النجوم والكواكب التي تسبح في ملكوت السماوات يختلف بشكل جوهري عن سلوك الأجسام هنا على سطح الأرض لكن وجهة النظر هذه تغيرت مرة واحدة وإلى الأبد مع بزوغ فجر قوانين العالم الإنجليزي "إسحاق نيوتن" لقد آمن "نيوتن" أن قوانين الحركة الثلاثة التي صاغها تصلح على سطح الأرض وفي سائر أرجاء الكون،
بشكل متساو وبدون أي إنحياز … أي أنها قوانين عالمية مطلقة وفقاً ل" نيوتن"، قوة الجاذبية التي تسبب سقوط الأجسام نحو الأرض في حياتنا اليومية هي نفسها المسؤولة عن دوران الكواكب في أفلاكها حول الشمس بحسب "نيوتن" لا يوجد هنالك مجموعتان منفصلتان من قوانين الفيزياء واحدة خاصة بالأرض والأخرى للسماوات،
وإنما مجموعة واحدة من القوانين العالمية تنطبق في كل مكان هنالك قصة توحيدية أخرى مشابهة في فرع آخر من علوم الفيزياء في القديم كانت قوانين الكهرومغناطيسية تصف سلوك المغناطيس وتدفق التيار الكهربائي في الدوائر الكهربائية من جهة أخرى مختلفة، قوانين الضوء تختص بالبصريات وسلوك الضوء خلال مروره بالعدسات أو انعكاسه من المرآة لاحقاً اكتشف "جيمس كلارك ماكسويل" من خلال نظرية الكهرومغناطيسية أن الضوء في واقع الأمر عبارة عن أمواج كهرومغناطيسية قوانين "ماكسويل" تكونت من مجموعة واحدة من القواعد
و لم تكتف بوصف الظواهر الكهرومغناطيسية
وإنما فسرت البصريات وسلوك الضوء خلال تاريخ الفيزياء وبغض النظر عن الموضوع قيد البحث
تكرر هذا النمط مرات ومرات مع توسع و تعمق معرفتنا نجد أن عدداً أكبر من الظواهر المتباعدة يمكن وصفها من خلال عدد أقل من القوانين هذه النزعة جعلت البعض يعتقد بإمكانية وصف كل الظواهر الطبيعية والكونية من خلال مجموعة واحدة من القوانين نحن ندرك الآن أن قوانين الحركة والجاذبية ل "نيوتن" وقوانين الكهرومغناطيسية ل "ماكسويل" ليست مطلقة الصحة فوفقاً لنظرية النسبية العامة … الجاذية ليست قوة
ومردها يعود للإنحناء في نسيج الزمكان بسبب كتلة المادة وبحسب نظرية ميكانيكا الكم لا وجود للمجال الكهربائي والمغناطيسي، أما الضوء فيتشكل من جسيمات "فوتونات" … وليست أمواج نظرية النسبية العامة تنص على أن الأجسام تسير في مسارات مستقيمة لكن في نسيج زمكاني منحني … وهذ ما يخلق وهم الجاذبية ميكانيكا الكم تشير إلى أن الشحنات الكهربائية تتبادل فيما بينها فوتونات وهذا ما يخلق القوى الكهربائية وليس وهم المجال الكهرمغناطيسي فنتائج نظرية نيوتن حول الجاذبية العامة يمكن الوصول إليها وفق نظرية النسبية العامة في الجاذبية لألبرت أينشتاين والتي يمكنها تفسير ظواهر أخرى إضافية تستعصي على الفيزياء التقليدية مثل ظاهرة تباطؤ الزمن بفعل الجاذبية … كما توقعت وجود الثقوب السوداء حيث يتجمد الزمن تماماً على نحو مماثل، لقد فسرت ميكانيكا الكم نتائج الكهرومغناطيسية الكلاسيكية كما أنها تتعداها وتفسر ظواهر أخرى إضافية،
مثل الأثر الكهربائي للضوء، والنفق الكمومي، وسلوك الذرات والجزيئات بشكل عام نظريات نيوتن وماكسويل بدت لنا صحيحة لأنها فسرت نتائج وقدمت توقعات قريبة جداً من رصداتنا في أغلب الظروف التي نستطيع مشاهدتها.
أي أنها حالة خاصة ومجرد نموذج تقريبي للحقيقة ولكي لا نقع في نفس الخطأ مرتين،
حتى نظرية النسبية العامة وميكانيكا الكم ما هي إلى نماذج تقريبية لقوانين الطبيعة التي تحكم الكون إذا كانت الأمور على هذا النحو،
فهل يمكن لهذه المجموعة من قوانين الفيزياء الأساسية أن توفر أساس نظري موحد لتفسير كل الظواهر الكونية؟ بعض الفلاسفة الحتميون (الماديون) يقولون أن كل ما في الكون من مادة – بما في ذلك المجتمعات البشرية – تخضع لقوانين الفيزياء وقوانين الفيزياء هذه قادرة على تفسير، وتوقع، كل الظواهر الكونية واحتمالاتها نزولاً من سيرورة التاريخ البشري … وحتى سلوك الفرد الإنساني الآخرون يعارضون وجهة النظر هذه بشدة وفقاً لهؤلاء، بعض الظواهر لا يمكن تفسيرها من خلال المادة التي تخضع لقوانين الفيزياء … مثال ذلك ظاهرة العقل الواعي بغض النظر عن أي الوجتهين هو الصحيحة، وحتى لو لم تتمكن قوانين الفيزياء من تفسير كل مناحي الحياة في الكون فإن هذه القوانين قادرة على تفسير الكثير من الظواهر وتوقعها نحن نستطيع حالياً تفسير الكثير من العمليات الحيوية في الخلية في ضوء فهمنا للتفاعلات الكيميائية بين الذرات والجزيئات كما يمكننا وضع تفسير لهذه التفاعلات الكيميائية مبني على فهمنا للسلوك الفيزيائي للذرات والجزيئات المشاركة فيها لذا عندما ندرس الفيزياء فإننا ندرس أسس الكيمياء والأحياء
وكل ما حصل في الماضي … وسيحصل في المستقبل وبأقل تقدير … إن لم يكن شيء
فتقريباً كل شيء من الممكن للبشرية أن تمضي في الحياة بدون الفيزياء في مجالات العمارة والهندسة حيث ابدعت الحضارات لسنوات حتى قبل ان تسمع بنيوتن وماكسويل وأينشتاين كما أن المقاربة الفيزيائية لظاهرة ما قد تخطئ الخطى … فتضللنا بعيداً عن فهم هذه الظاهرة الكونية كمثال، الحاسوب قائم على أساس البوابات المنطقية، ومدخلاتها – كما مخرجاتها – ثنائية
إما "عالية" أو "منخفضة" … 0 أو 1 هذه البوابات المنطقية مبنية على الترانزستورات
التي تخضع لقوانين فيزياء الكم مكمن المشكلة أنه حتى لو فهمنا كل ما يتعلق بالجزيئات داخل هذه الترانزستورات من حيث الموقع والعزم وفقاً لقواعد ميكانيكا الكم إلا أنه قد يتعذر علينا الوظيفة المنطقية التي تخدمها هذه الدائرة الكهربائية (خوارزميتها) على نحو مشابه، لو درسنا بالتفصيل كل الذراات والجزيئات في أضواء لوحة إعلانية … كل هذا لايضمن لنا معرفة فحوى الرسالة في هذه اللوحة وبرغم ذلك، عدم دراسة الفيزياء سيجعلنا في موقف ضعف،
حتى في الواجبات التي نتوهم أننا نتقنها كمثال، لنفترض أننا مهتمين بعملية جمع رقمين إحدى الطرق أن نغطي كل الإحتمالات ونحفظ الجواب لحاصل جمع أي رقمين،
هذه الطريقة تكون عملية في حالة الاحتمالات المحدودة نسبياً وليست الكبيرة هذه الطريقة تزداد صعوبة بزايدة عدد الأرقام موضع الإهتمام لحسن الحظ، هنالك طريقة أخرى عملية أكثر هي عملية الجمع الطويل وبمجرد تعلم مبادئ وأسس عملية الجمع فإنه يمكن تعميمها على أي رقمين نصادفهما بنفس الطريقة، في كثير من فروع المعرفة بذل الناس جهوداً مضنية للتعرف على النتائج المحتملة في كل السيناريوهات لكن الأسهل هو تعلم القوانين الأساسية والمبادئ التي يمكن تعميمها بمجرد فهم هذه المبادئ والسيطرة عليها، كل السيناريوهات تصبح حالة خاصة وتطبيق مباشر لهذه المبادئ وأكثر هذه القواعد عمومية وقابلية للتطبيق والتعميم
هي القوانين الفيزيائية والرياضية (الأكثر إختزالاُ وتجرداً ) وحدها مبادئ الفيزياء هي التي مكنتنا من تطوير فروع التكنولوجيا التي تفوق الخيال كما أشرنا سابقاً، عالم الحاسوب والألكترونيات الرقمية قائم على الترانزيستورات وفهم آلية عمل الترانزيستورات مبني على فيزياء الكم كمثال آخر، الأقمار الاصطناعية المرتبطة بنظام التموضع العالمي (GPS) تعمل بشكل دقيق إذا وفقط إذا أخذت بعين الاعتبار تأثير الجاذبية الأرضية على سريان الوقت وفق نظرية النسبية العامة هنالك أسباب أخرى تدفع باتجاه فهم مبادئ وقوانين الفيزياء الأساسية بمعزل عن فوائدها التطبيقية وهذا يتجلى بالشغف والفضول لفهم طبيعة الكون من حولنا هذا الفضول يقودنا في كثير من الأحيان إلى اكتشافات أغرب بكثير مما قد يجنحه إليه خيالنا كمثال،
قوانين الحركة الخاصة بنيوتن توقعت بدقة ووصفت مسار المقذوفات على سطح الأرض، وحركة الكواكب في أفلاكها وهذا الكلام ينطبق على كل الكواكب باستثناء إثنين:
أورانوس … وعطارد فمدار أورانوس كان يجنح قليلاً عما كانت تتوقعه قوانين الفيزياء الكلاسيكية هذا الأختلاف لم يدفع الفيزيائيين إلى التشكيك بصحة قوانين نيوتن ولكنهم أرجعوا هذه الاختلافات إلى عدم دقة القياسات
وأفترضوا وجود كواكب أخرى خارج مدار أورانس تؤثر عليه بجاذبيتها وبالإعتماد على قوانين جاذبية نيوتن جرى حساب المكان المتوقع لهذه الكواكب الجديدة الضرورية لتفسير الاضطراب في مدار أورانوس المفاجئة السارة هي أن الرصدات الفلكية كانت متوافقة تماماً مع الحسابات النظرية … وتم اكتشاف الكوكب الجديد نيبتون كما توقعت الحسابات الفيزيائية بدقة لقد كانت هذه سابقةً في تاريخ البشرية
فالحسابات المبنية على أسس نظرية سبقت الرصدات التي أكدتها هذه النتائج زودت تأكيداً آخر إضافياً لصحة نظرية الجاذبية لنيوتن لكن بقيت معضلة عطارد عصية على الحل وبالرغم من ذلك، الإيمان بصحة نظرية نيوتن لم يخفت،
فكما تمكن العلماء من إيجاد تفسير لمدار أورانس متوافق مع نظرية نيوتن،
الكل تأمل بتفسير مشابه لمدار عطارد يوماً ما في المستقبل المجتمع العلمي انتظر ملياً لهذاالتفسير … لكن دون جدوى وفي بداية القرن العشرين، طور العالم ألبرت أينشتاين نظرية النسبية العامة في الجاذبية بمعزل عن معضلة مدار عطارد نظرية أينشتاين كانت قادرة على الوصول لنفس التوقعات التي سبقتها لها نظرية نيوتن للجاذبية، لكنها تفوقت عليها بقدرتها على تفسير سلوك مدار كوكب عطارد لم تكن نظرية أينشتاين قادرة على الوصول لتوقعات أكثر دقة فحسب، بل أنها بينت أن حقيقة نسيج المكان- زمان هي أكثر غموضاً عما كان يتصوره حتى كتاب الخيال العلمي في السابق على سبيل المثال، نظرية أينشتاين لا تعترف بوجودالزمان المطلق، وكل مراقب قد يكون له زمانه الخاص و تعريفه الخاص لما يحدث الآن وقد يختلف الراصدون، بحسب مراجعهم الإسنادية، حول آنية حصول حدثين متزامنين، دون أن يكون لأحدهما الأفضلية دون اغيرهم ويمكن لراصدين مختلفين في مراجعهم الإسنادية أن يرى كل منهما أن ساعة الآخر تتباطئ مقارنة بزمانه الخاص، ويكون كلاهما محقاً وإذا كان مرجع اسناد الراصد قصورياً قد يتخيل أنه ساكن فيما الكون من حوله هوالذي يتحرك بسرعة ثابتة في الإتجاه المعاكس وقد جرى التحقق من نتائج النسبية العامة و إختبارها في مرات عديدة وبظروف متباينة كمثال، جرى مقارنة الساعة الذرية في مدارات حول الأرض بنظرياتها على مستوى الأرض كما جرى مقارنة زمن التحلل لنظائر تسير بسرعة قريبة من سرعة الضوء واخرى في حالة السكون وقد كانت نتائج التجارب على الدوم متطابقة مع نظرية أينشتاين لكن كما كانت الفيزياء الكلاسيكية عاجزة عن تفسير مداري أورانوس وعطارد، هنالك ظاهرتان مستعصيتان على نظرية أينشتاين دون تقديم تفسير شاف لهما هاتان الظاهرتان هما … المادة المظلمة … والطاقة المظلمة المادة المظلمة وضعت كمقترح لتفسيرالفرق بين كمية المادة القابلة للرصد في كل مجرة والتي تقل كثيراً عن كتلة المادة المطلوبة حتى تبقى المجرة متماسكة تحت تأثير قوة الجاذبية آخذين بعين الأعتبار معدل دوران المجرة حول نفسها أما الطاقة المظلمة فقد اقترحت كتفسير لمشاهدة أن الكون يتمدد على نحو متسارع وليس على نحو متباطئ كما ينبغي قد نتمكن يوماً ما من تفسير هاتين الظاهرتين ضمن إطار نظرية أينشتاين كما حصل مع مدار أورانوس مدار أورانوس لم يتعارض مع الفيزياء الكلاسيكية النيوتونية، مصدر الخلل كان في المعطيات الناقصة وليس في النموذج النظري.
بعد اكتشاف نيبتون وأخذ كتلته بعين الإعتبار تم اصلاح الخلل على نحو مماثل، اذا ثبت وجود مادة مضادة مثل جسيمات دون ذرية غير مرئية بين المجرات، يمكن عندها أخذ أثر كتلتها بالحسبان دون مخالفة نطرية النسبية من جهة أخرى، يمكن أن المادة المظلمة والطاقة المعتمة مثل مدار عطارد معضلة مدار عطارد كشفت أن الخلل في النموذج الفيزيائي النظري لنظرية نيوتن هو خلل بنيوي في فهم الزمان والمكان … وليس متعلق بنقص المعطيات أو دقة الرصدات على نحو مشابه، قد تكون معضلتي المادة المظلمة والطاقة المظلمة أدلة بحوزتنا على أن حقيقة والواقع هي أكثر روعة وغموضاً من النموذج الذي تقدمه حالياً النسبية وفيزياء الكم على سبيل المثال،المادة المظلمة يمكن تفسيرها بوجود أكوان متوازية تتفاعل فيما بينها فقط من خلال الجاذبية المادة في الأكوان المتوازية لكوننا تتفاعل بواسطة قوة الجاذبية مع المادة في كوننا لذلك تتشكل المجرات في الأكوان المتوازية على مقربة من المجرات في كوننا التفاعل الثقالي بين المادة في المجرات في الأكوان المتوازية والمادة في كوننا يعوض عن النقص في المادة المرئية القابلة للرصد في كوننا هذه الأكوان المتوازية تشكل بمجموعها ما يعرف بالكون المتعدد كل كوكب في كوننا له مناخه الخاص المميز لكنها جميعها تخضع لقوانين الفيزياء ذاتها … الخاصة بكوننا على نحو مشابه، يمكن أن يكون للأكوان المتوازية قوانينها الفيزيائية الخاصة بها، لكنها جميعها تخضع لمجموعة من قوانين الفيزياء الأساسية التي تحكم منظومة الأكوان المتعددة بالمجمل وكما أن نسبة ضئيلة من الكواكب في كوننا تمتلك ظروف مناخية على سطحها موائمة للحياة كما نعرفها فإن هذه الحجة قد تنطبق أيضاً على الأكوان المختلفة يبقى هنالك إحتمالية وجود أشكال أخرى للحياة غير الشكل الذي نعرفه ونظرية الأكوان المتعددة تحتمل وجود أشكال وأنماط مختلفة لحياة ذكية تفوق الخيال كل هذا هو مجرد تخمين، ولم نتمكن بعد من رصد أي من هذه الأكوان وبرغم ذلك، يجب أن نتذكر أن عمر الحضارة البشرية يمتد لألاف السنين، أما تاريخ العلم ممثلاً بفيزياء نيوتن وماكسويل، فلا يتعدى بضعة قرون ولم يتم اكتشاف نظريتي النسبية وميكانيكا الكم إلافي القرن العشرين لقد قطعنا شوطاً كبيراً منذ كنا نعتقد أن قوانين الفيزياء و الحركة على الأرض تختلف عن قوانين الفلك لكن ما يزال أمامنا الكثير لنكتشفه الأجيال المستقبلية قد تنظر لمنظورنا في فهم الفيزياء تماماً كما ننظر نحن لسذاجة النماذج السابقة التي كانت تعتقد أن الأرض هي مركز الكون وأنها مستندة على ظهر سلحفاة عملاقة

Lecture 1 | Modern Physics: Quantum Mechanics (Stanford)

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Lecture 1 of Leonard Susskind’s Modern Physics course concentrating on Quantum Mechanics. Recorded January 14, 2008 at Stanford University.

This Stanford Continuing Studies course is the second of a six-quarter sequence of classes exploring the essential theoretical foundations of modern physics. The topics covered in this course focus on quantum mechanics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.

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this program is brought to you by Stanford on iTunes you at Stanford University please visit us at itunes dot stanford.edu I just want to do one thing actually before we start the the material of the class a whole bunch of people a lot of people particularly people from Europe who actually follow these things on the internet seem very puzzled they send me email even some of my friends in Europe who are physicists – a following if are very curious about what's going on here they don't see you people they only see me and they know that I'm teaching a class in this or that special special topic in classical mechanics or whatever and they can sense that there's something a little bit different about it they can sense that it's not a standard undergraduate or graduate course in these subjects and they ask me what is this what is this about oh who are these people you're teaching – they can tell it there and most of them don't know what continuing education means they didn't know what it meant and so they asked me I thought I would tell them over the over the internet what this class is it's continuing education which means education for people not from Stanford people not oh they can be from Stanford some of the people probably are either employed by Stanford maybe were students at Stanford but almost everybody in this class is a little bit too old to be an undergraduate student even a little bit too old to be a graduate student the people from the community from Palo Alto from Mountain View from Huell is further than Mountain View ok what's the furthest there you come for this where force the city ok so that's 20 miles away so people from within a certain radius I don't know whatever the radius is are allowed to come and take courses at Stanford they pay a bit of money for it and a professor teaches them now this is not it is most definitely not your standard freshman physics there's another guy on the internet who teaches freshman physics totally does it very well that's not what this is this is the real McCoy theoretical physics at the full-scale level we use equations we not only use equations we sometimes use hard equations but we tend to try to use the simplest equations that will do the job basically we try to keep it minimal just out of curiosity I I more or less know the answer or just by looking but what the age distribution of people here is by comparing is there anybody here under 40 yeah there's there's maybe one handful of people under 40 incidentally for those who can't see the audience it's probably about a hundred people in the room I'm not sure it's a it's a small theater with a lot of seats it's a lecture theater more or less filled up and maybe four or five or three or four people who are under 40 let's see let's go to the other end how many people here over 70 a good deal more maybe 10 or 12 anybody over 80 I got a couple of people over 80 anybody over 90 this time oh there have been people of a 90 there have been people over 90 in this class so you see the age distribution is such that basically people want to get to the basic ideas fast they don't have a hell of a lot of time they want to get there fast so I try to tell them in some minimal way what the basic things you really have to know in order to get on to the next thing sometimes the basic things that you need to know are a little more difficult a little more elaborate we do them anyway we do them as efficiently and as straightforwardly and with the minimal amount of stuff to get them right to get them really right not metaphor not the not the analogy but equations when necessary what else can I say about this class I guess it has various names quantum mechanics 25 I'm inclined to call it quantum mechanics for old people including myself anyway the first course in the series let me just say what the outline of the whole course of courses is these series of courses will consist of about 610 lecture series the first of which was in classical mechanics the classical mechanics is the basics basis of all classical physics classical means before quantum mechanics anything that doesn't involve quantum mechanics or which ignores quantum mechanics or which is in a range of parameters where quantum mechanics can be replaced by basic classical logic that's called classical physics so the first class with classical mechanics which is in some sense the basis for all the physics the motion of objects the energy of objects the momentum of objects what the characteristic behavior of systems is as they evolve with time and so forth we discussed that last quarter and anybody who is thinking of following these classes should begin with that the next class which is already on the internet was called I believe quantum entanglement now this is also a class on quantum mechanics and it will be self-contained but I would strongly advise anybody who's following to go the course number two was occult quantum entanglement does anybody remember I think was called quantum entanglement that is also on the internet there was the first one that was on the Internet and to get that under your belt first before attempting to do the full scale quantum mechanics that we're going to do here although what I'm going to do is pretty self-contained I'm going to start with this evening with some basic thoughts about the deep differences in the logic of classical mechanics and quantum mechanics or classical maybe even mechanics is too strong a word classical physics and quantum mechanics there are some very queer phenomena in quantum physics that don't exist in classical physics now one of them is the fact that quantum mechanics is based on statistical thinking randomness a certain degree of not imprecision that's not the right word a certain degree of non-deterministic or in deterministic behavior unpredictability that's the word I'm looking for unpredictability but it's a special kind of unpredictability Einstein famously said God does not play dice Niels Bohr told him Einstein don't tell god what to do but in fact in a sense God I don't know if it's God when I use the word God in the same sense the physicists always use it meaning the laws of physics or something like that the laws of physics don't play dice at least not in the standard sense let me imagine for you a theory that's based on a bit of Statistics which is not the ways of quantum mechanics works and then I'll illustrate it for you by showing you some of the differences imagine that there was a element of randomness in Newton's law whatever which law F equals MA together with the law of of gravity all right so F equals MA together with the law of gravity tells us how objects move let us say how the moon moves around the earth and it predicts with infinite precision if you could work out the equations with infinite precision and if you could account for every detail of the earth and the moon and all the material that's in between and so forth it would predict with enormous detail deterministically the motion of the moon which means if you know where the moon starts and you know how the moon is moving in the beginning then you can predict forever after exactly how the moon moves around the earth that's the deterministic classical mechanics now let me imagine a modification modification involves a little bit of randomness for example okay so let's imagine God sitting on his throne throwing dice every tenth of a second God throws the die and if he gets snake-eyes what he does is he gives the moon a little extra push in one direction if he gets a seven he pushes the moon a little bit in the other direction a degree of randomness based on the random throwing of dice and really random a really random number generator being used to put a little bit of fluctuation into the motion of the moon that sounds like it introduces the kind of uncertainty the kind of thing that one talks about in quantum mechanics uncertainty in non predictability unpredictability but it is not anything like the randomness and unpredictability of quantum mechanics the randomness and unpredictability of quantum mechanics is exceedingly special exceedingly special and accept quite different and it's that that we want to get our head around and learn and understand the difference between these things by the time this class is finished but also we want to learn how to use quantum mechanics a little bit to calculate some things let's notice one thing about this law which includes a little bit of randomness the throwing of dice in order to either kick the moon a little bit or inhibit the moon's motion a little bit one of the things that would do is to add a little bit of energy or subtract a little bit of energy from the motion of the moon randomly if you randomly give the moon and knock this way and then a kick that way in a bunk that way and keep doing it over and over again each time you do it this on the average it may not change the energy but each little increment is randomly going to either increase the energy or decrease the energy of the moon and if you do that randomly eventually that will build up to a statistical randomness in the energy in other words energy would not be exactly conserved in a world in which the laws of motion included a little bit of classical randomness I call it classical randomness to distinguish it from quantum randomness in quantum mechanics you prepare a system the same way that you might prepare the moon in an initial situation you let it go for a while and then you look at it and indeed you discover at the end of it that what you measure is a little bit unpredictable but you also find that energy is exactly conserved no hint no remnant at all of energy being knocked this way and not that way if you start it with a given energy with a given precise energy and you let it evolve for a while and you measure the energy later the energy is exactly the same as you started with right so there's something funny about this randomness it seems to affect some things and not other things and doesn't work the way you might expect the randomness in classical physics to work let me give you two ho I think three other examples of the oddness of this randomness the first comes from an experiment which to my mind shows the weirdness of quantum mechanics in the easiest and most straightforward least difficult way much easier for my money than bells inequalities and all that sort of stuff it's called a two-slit experiment all of you know about it or most of you know about it if not you learn about it right now but it is extremely odd when compared with classical randomness all right so classical random is just being this idea that every now and then you give a system a little knock the two-slit experiment involves a source of particles those particles could be photons they could be electrons they could be neutrons they could be bowling balls except the effect for bowling balls is so minut that you'd never be able to measure it so when we speak about particles we think about things which are very light and because they're light the quantum effects associated with them are significant and measurable all right so we have a source of some kind it could be a laser shooting out photons but imagine the photons are coming through one at a time very small number of them are quanta of light one at a time one at a time one every five minutes if you like I don't know just to go to a some extreme situation these are the photons coming out of a photon gun and the photons pass through a obstacle with a little hole in it all right this is a two-dimensional diagram of a three-dimensional situation raishin this blue object over here is intended to be a disc with a tiny hole in it so the photons can go through and they come out the other side of the hole and when they come out the other side of the hole they eventually get to a screen over here that screen records the photon by a flash a flash of light at the screen not the flash of the original photons but another flash of energy appears at a point on the screen and records or it could be just the blackening of a photographic a photographic plate something that records the position of the photon when it goes through here now first let's think classically are but classically with a bit of randomness the bit of randomness that we can imagine is that when the photon goes through here a little random kick might influence the photon and either kick it upwards or downwards in fact by a random amount what would we expect then well if there was no randomness then the photons would go straight through and illuminate a single point be completely deterministic the photons would always arrive at exactly the same point if the hole was small enough and if the beam of photons was narrow enough but now we can imagine a random kick what is the random kick do well it changes the direction of the photon of the photon not the whole photon beam but one photon at a time it might kick the first photon up a little bit the second photon down and so forth and so on eventually what you will see is a blob of illumination on the screen over here on the average the photon might go straight ahead so the blob might be most intense at the center it might be highly improbable to knock the photon through a 60 or 70 degrees so the signal would fade as you moved away from the center you would see a blob with a maximum intensity near the center and thinning out as you moved far away and you might describe it by a probability function a probability function being the probability that the photon arrives in different places okay now we go oh now we do the same thing in real quantum mechanics in other words in the real world we see essentially the same thing the photons go through the hole with no ability to control the situation we find out that the photons again create a blob like this but now we're going to do something a little different and I think everybody here more or less more or less everybody knows what I'm going to do I'm going to open a second hole okay but let's think about what classical randomness would do classical randomness would simply mean that we and we'll also imagine that this beam of photons is at its origin a little bit uncertain and a little bit random so that some photons begin a little bit upward some photons a little bit downward some photons go through the upper hole some photons go through the lower hole if a photon goes through the upper hole it may or may not get a random kick and get knocked off of course if it goes through the lower hole it also gets a random kick now we're imagining the photons come through one at a time we could even imagine a okay I do that all the time where was I yes the photons get a kick we're imagining that the photons come through one at a time very sparsely and so what one photon does doesn't influence a later photon because the later photon comes through so much later that whatever gave the first Photon kick is already finished happening and it's waiting for the next throw of the dice in fact the next photon may come through a hundred throws of the dice later and so we expect the next photon to be random statistically independent of the first photon under those circumstances what we would expect well let's serve let's let's decide what we would expect supposing only one hole is open if only one hole was open then we would see a blob of illumination like that with a profile it might look something like that supposing we closed up the first blob the first hole and opened the second hole close the first one open the second one all right so first we begin with just one hole then we close the first hole and open the second hole what would you expect to see what you would expect to see under those circumstances is a different blob slightly displaced from the first one the blue blob wouldn't be there because we closed the first hole the green blob would be there because we open the second hole now what happens if we open both holes what happens if you open both holes in classical mechanics is the probability for a photon to get to the screen at any given point is the sum of the probabilities for it to get there by either root the photon photon can either go through the upper route or it can go through the lower route if both holes are open the probability to get to this point over here let's call it the green point over here the probability to get over there is the probability to go through the upper hole and arrive at the green point plus the probability to get to go through the lower hole and arrive at the green point so the result is in classical physics you would always see the signal over here the profile over here just being the sum of the two probability distributions and it would look I wish I had another color they never leave me enough colors all right we would see something which would look like just a higher I don't know did much we would be bigger than that it would look like that and in particular if there was any point over here such that the photon could arrive either from one hole or from the other hole or both then we would find illumination at that point for sure by opening both holes that's what classical logic let's let's call it by the right name classical logic that's what classical logic classical statistics classical probability would dictate for a series of particles coming through here one at a time when they come through one at a time they make blip-blip blip-blip but the average probability of the distribution of blips would be a distribution which would be the sum of the two distributions what happens if you really do this experiment you find what's called an interference pattern the interference pattern looks like this well let's see we get it right in particular well first of all what is this figure this figure is a probability distribution and it tells you the horizontal axis here tells you what the probability of a photon getting to a particular point at a particular height here but in particular it says that there are no photons which arrive at that point there are no photons which arrive at that point there are no photons which arrive at that point this is odd if you opened only one hole then you would find a probability distribution which wasn't zero in other words you would find illumination at that point you open the other hole you still find illumination at that point you open both holes and all of a sudden no photon gets to that point even though they're coming in coming through once every 20 minutes or once every twenty years and therefore how can they know about each other nevertheless if you open both holes there will be places where no photons can get to despite the fact that photons arrived at those points when only one hole was opening that is um you might be able to sit down and work up some in some interesting but rather elaborate mechanism to make this happen you could imagine elaborate complicated mechanisms where somehow this this screen here some degrees of freedom inside the screen remember how many photons went through and they remember what they're supposed to do but it would be a rather elaborate mechanism just for this one purpose this phenomenon of interference of destructive interference this is called destructive interference that the probabilities cancel instead of adding at certain points that's a very generic property in in quantum mechanics and so it requires a kind of explanation which is not some detailed mechanical complicated explanation it requires a a broad new idea about how statistics works and how about how the logic of of quantum mechanics works so that's the first really weird thing that happens in quantum mechanics now let me give you another example if you remember in the last course we talked a little bit about reversibility we talked about the laws of physics in particular we talked about the laws of physics for discrete systems for example we discussed the possible laws of physics deterministic laws of physics for a coin which can either be heads or tails now of course flipping the coin that introduces a level of uncertainty level of a statistics probability in two things I want to think about the deterministic laws the truly deterministic laws are the ones that whatever the coin is doing it will tell you what the coin is doing next so when we discussed this last quarter we talked about two possible laws of physics the first was that if you find heads in the next instant when you look at it after an instant of time you'll find heads again if tails then you find tails again then your laws of physics just permit two possible evolutions heads heads heads heads heads heads heads or tails tails tails tails tails and the other possible laws of physics or law physics was that when you see a head in the next instant microsecond or whatever your unit of time is you will see the opposite tails then the two possible laws of physics are the two possible laws of physics the two possible evolutions one of them begins with heads and goes heads tails heads tails heads tails and the other one begins with tails and goes tails heads tails heads tails heads whether you know it or not those two different things were different they began one with heads one with tails or that was that was the basic idea of the deterministic law of physics now you can have of course more complicated systems in there's just two state systems we imagined for example a six state system a six state system was a die a die that can be one two three four five six but let's just take a simpler thing let's take a coin with three sides if you can't imagine a coin with three sides then you'll have to do this using abstract mathematics okay a coin with three sides has heads tails and what so they're ELLs feet heads tails and feet all right heads tails and feet all right and there are two alright let's let's consider a simple law of physics a simple law of physics could tell you that whenever you have heads it goes to tails whenever you have tails it goes to feet whenever you are feet it goes to heads again and then wherever you start you just cycle around endlessly forever and ever now there's a certain sense in which this is revert in which this is what does it mean to say it's deterministic what does it mean to say information is not lost in this process what it means is that no matter how long the system evolves let's suppose you started with heads and you let it go a million units of time it will just go around and around the cycle if at the end of that you reverse the law of physics now reversing the law of physics just means having it go in the opposite direction if you could somehow press a button or turn a knob which have the effect of reversing the law of physics in other words changing the direction of every arrow and think about actually if you could really do this by pressing a button change the law of physics then if you allow the system to evolve for any length of time at the end of that time reverse it and let it evolve for the same length of time again guess what it magically comes back to the same original configuration that's what it means to say that physics is reversible or that's what it means to say that information is never lost in physics that that no matter how long you keep going if you reversed if you could find a way to reverse the laws of physics and run them backward for the same exact length of time you'll come back to the starting configuration now to do that you might not even need to know the laws of physics you might need to know very little the only thing you would need to know is how to reverse the law you mean you needn't even know what the law is whatever the law is if you can find a button to push that reverses it then you can test the determinism of physics by simply starting someplace letting it evolve for a long period of time and then letting it evolve with the reverse law of physics if you come back to the same place every time then your law office is deterministic now what about a little bit of classical probability a little bit of deity playing dice so let's suppose with some very very small probability the deity does something different that's not prescribed by the laws of physics for example and some random way with a probability of one in a million the law might say don't move instead of moving the way the diagram says stay still if you allow the system to evolve for a short period of time then and then run it backward it will go back to the same point but if you allow it to run long enough that there's a significant statistical probability for a fluctuation for something to happen which is not deterministic in other words if it was one in a million that you stand still but you let the system run for ten million units of time then guess what no chance or not no chance you have a chance that you'll come back to the same point but you'll also have an equal chance that you'll come back to any of the other points in other words this test will fail it'll fail one-third of the time you will come back to the same point two-thirds of the time you will come back to two different points so a little bit of classical randomness destroys the what I called last quarter the conservation of probability nor the conservation of information information gets lost what about quantum mechanics is information lost well there is an element of statistical things in quantum mechanics when a electron goes through a hole like this it has a probability for getting kicked up let's just take the case let's simplify now this is the one slit experiment we don't even need the two slit experiment let's start with a one slit experiment the electron goes through the slit it's aimed toward the slit sometimes it more or less go straight through sometimes it gets kicked up a little bit sometimes it gets kicked down a little bit and you might think that this is more or less like throwing dice in fact if you look for the electron afterwards if you look for the electron out here after you've given it time to pass through at the moment now let's just send one electron through one electron if we send one electron through it may get kicked up it may go straight through or may get kicked down if we do the same experiment repeatedly which is of course is the same thing as sending many electrons through but if we do the same experiment repeatedly we will find some go up some go down some but now let's ask the following question supposing after a period of time we send an electron through one electron after a certain period of time having given it enough time to get to the other side but not but let's remove the screen over here let's remove the screen send the electron through it goes through comes out someplace else but we don't look at it we don't we don't bother detecting where the electron is instead we just reverse the law of physics we do what we did over here reverse the direction of time if you like can you really do this can you real is there really a but a button that you can push is there really something that you can do to a system that will reverse the motion yes in many systems there is in many systems we actually know how to manipulate the system how to change magnetic fields how to do things to a system of electrons so as to run it backward so let's take it as a as a given that after a certain amount of time somebody can press the button that reverses the law of physics that reverses the direction of time so to speak and runs the system backward for the same length of time what happens does that electron we're not going to look at it but do we later after after the end of the experiment we allow it to evolve for a time T and then we allow it to evolve backward for another time T to t all together do we find the electron moving backward along the original trajectory or do we find the probability the fluctuations compounding and that after we turn it around there's even worse fluctuation particle comes through gets knocked up we run it backward it either gets knocked down and locked up which is it does it does it reverse precisely along the original trajectory every single time or does the statistical the imprecision the the unpredictability in one direction add to the unpredictability in going backward and make it even more unpredictable afterwards than it was to begin with the answer is very curious the answer is that if we don't look at the system in the intermediate stage after it's gone through the hole over here does don't look at it means don't interfere with the electron in any way take that screen away that might have converted that electron into a little pulse of light just remove any influence on the electron over here completely do not look at it do not interfere with it do not do anything to disturb it but just run the law of physics backward then we will exactly every single time detect the electron running back along the reverse trajectory on the other side over here in particular we'll find the electrons is going right back into the gun what if somebody does look at the electron when it goes through to the side over here for example supposing somebody sets up an electron detector which detects where the electron is and then lets it go having reversed the law of physics we look at it and then reverse the law of physics now in this case over here the case we were talking about classical coins looking at a thing doesn't disturb it very much if I have a coin where's my coin I've lost my coin it's okay it's only a Chilean peso I lost my Chilean peso I think it was actually a hundred pesos okay now here's my Chilean peso I put it down and I put it down heads don't look at it now I look at it it would be rather amazing if just the process of looking at it was capable of flipping it from heads to tails now of course if I look at it with an intense enough a high frequency light beam a beam of very very energetic photons sure enough an energetic photon could could hit the coin knock it into the air and spin it over that's true but you could but in classical physics you can look at an object and determine its state determine the heads or tails nosov it with an arbitrarily gentle interaction and so by looking at a system in classical mechanics and looking at it with a very very gentle apparatus or a very gentle photon or whatever you like it does not entail disturbing the system so if you look at it after you allowed it to go a million times and then reverse the law it will have no effect no detrimental effect on the experiment and you will come back to the same to the same point that you started with all right so just looking at the system in between has no effect no does not necessarily have an effect on it even though whoever looks at it can determine after a million units of time where the system is here looking at it has no effect on what happens after you run it backward exactly the opposite in quantum mechanics if you detect the object if you do anything to detect the object electron in this case and then run the law of physics backward what you'll find out is that the probabilistic character of it gets compounded so the fluctuation that sent it up here when you run it backward and you now know that it came out up here and you run it backward it's likely to come out down here or up here and it will disturb the system in such a way that the the reversibility will fail so this is curious that that whatever quantum logic is the questions of the kind we're asking are deeply dependent on a ridiculous question did somebody look at the system during the course of its evolution and as I said of course looking at a system can disturb it but in classical physics we can look at a system without disturbing it we can look at a system detect the system measure the system as gently as we like arbitrarily gently and have arbitrarily small effect on it so they're running it backwards will be exactly as if we didn't look at it not so in quantum mechanics determining the state of a system is never a small thing to do to the system and this is an example that completely destroys the experiment the two-slit experiment has also a similar a similar story that goes with it the story that I told you a moment ago about the interference pattern and the destructive interference is only true if nobody records which way the electron went through now when I say nobody I don't mean a human being necessarily I mean that nothing in the environment of the experiment records and remembers which way the electron went through in other words after the experiment there is nothing in the environment of this experiment which has recorded which way the electron went through nothing in here only no this hasn't this also has been recorded where the electron went through if nothing records where the electron goes through then there's an interference pattern but if you were to put a little demon over there a real physical demon now real physical demon could be another electron it could be some some gas in the apparatus some gas molecules in the apparatus whatever it is in such a way that something record which way the electron went through something remembers it afterwards perhaps a molecule over here gets disturbed gets excited if the electron goes through the upper hole gets excited and changes the character of that molecule if it goes through the lower hole it changes the character of a different molecule so looking afterwards after the electron went through you can tell which hole the electron went through then the interference pattern is destroyed and the answer is exactly the same as the classical answer namely the probabilities add just exactly as in classical physics so again there's no way to record whether the electron went through the upper hole or the lower hole without seriously disturbing the experiment without so seriously disturbing the experiment that you drastically continue to change the conclusion that's a quantum that's roughly speaking the general character of quantum mechanics that you cannot do measurements on systems without disturbing them and disturbing them can change completely the character of a question yeah yeah haha that is yeah it's a very very good question that's an excellent outstanding question I wish I knew the answer now I know the answer the answer has to do with the probability distribution for the position of the of the of the let's call this the detector let's call this the detector okay now if the detector is very well localized in space as it would be if it were a heavy massive classical detector then by the uncertainty principle which we haven't talked about yet if we're going to let me skip ahead then imagine that we have talked about the uncertainty principle that's a sophisticated question so I suspect you you've thought about a little bit if the position of this detector the up/down position of this detector Delta X is very very small that means that the uncertainty in the momentum of the detector is large okay that means if I were to plot the probability distribution of the momentum of the detector let's plot the probability distribution of the momentum of the TEC detector it's rather broad because for a heavy detector its location is so well-defined all right now the electron comes through and kicks the play-doh to pick this thing a little bit it gives it a small kick and what does it do it shifts this probability distribution a small amount but unless the probability distribution has been shifted by something approximately equal to its width then you can't tell afterwards whether it got a kick or not you have to be in order to be able to be certain that it got a kick you'll have to kick it by an amount large by comparison with the uncertainty so it's the uncertainty principle that comes in and rescue rescue rescue you or rescues me make sure that the interference pattern is not destroyed okay uncertainty principle itself so let's go to the uncertainty principle since it's been raised since I raised it the uncertainty principle is another factor in quantum mechanics which is extremely different than physics in classical physics conceptually very different I know we haven't talked about there what the uncertainty principle is yet but let's let's discuss it anyway let's jump ahead my main motivation now is to explain the strangeness of not to explain the strangeness of quantum mechanics but the points and fingers at the strangeness of quantum mechanics so that you see that it really is fundamentally logically different than classical mechanics logically different whole logic of quantum mechanics is different okay it's easy to imagine a bit of uncertainty in classical physics as I said the the coin throwing or the dice throwing beauty who sticks his finger into the system and gives it a push this way a push that way and so forth and that way things after a short period of time to come on uncertain and it's easy to imagine that that uncertainty can affect both position and momentum and if you wait a little while both the position and the momentum may be uncertain in fact if you wait a little bit while more than a tiny fraction of a second for a particle you might discover that inevitably both the position and the momentum get jostled about and so that there's a good deal or a certain amount of uncertainty in position and uncertainty in momentum but you would be unlikely to say that that uncertainty in position and momentum is um what's the word I'm looking for well first of all that's fundamental in any sense you would say it was a result of hitting the system randomly but I think most of us would agree that under those circumstances it was a bit of laziness that didn't allow us to watch the particle carefully enough to see what exactly the momentum was and the position was we always imagined in that kind of classical context an example of that classical context would be the random walk of a Brownian moving particle right so if we were if we watch a whole bunch of particles a whole bunch of particles might form a cloud and that cloud might spread and perhaps even the velocity of the cloud or the cloud describing the velocities might also spread but every one of those particles if we cared to we could look at with better precision better and better precision we could look at it gently as gently as we liked in classical physics and determine both its velocity and its position simultaneously certainly that would be true of a classical Brownian motion particle that was being knocked around by by ordinary collisions with some gas or something like that in principle we can just get ourselves a better microscope better a better accuracy better precision better resolving power and do it very gently so as not to disturb the system when we measure the position we don't want to jostle a momentum when we measure the momentum we don't want to do something funny to the position and we just measure it gently enough so that we measure both the position and the momentum that would be expected to be true for a Brownian particle so the uncertainty in position and momentum is something which in a certain sense is due to our own laziness and not spending enough money and buying a good enough detector and so forth to to be able to detect both the position and the velocity at the same time on the other hand in quantum mechanics there's a really fundamental obstruction a logical obstruction the deep obstruction to knowing to ever being able to measure both the position and the velocity of a particle I'm going to work it out for you and show you how it works when I show you how Heisenberg first thought about it well he first thought about it entirely through abstract mathematics for us that's going to come later but then when questioned by Bohr what are you talking about that you can't measure the position in the velocity simultaneously your mathematics is a crock of baloney don't tell me that X times P is not equal to P times X come on it's like saying three times five is not five times three did I write it right yeah not equal to don't tell me such nonsense stories give me some physics and so Heisenberg cooked up his experiment to show to illustrate the fact it's not show but illustrate the fact that there was a good deep consistent reason why it's impossible to ever ever under any circumstances simultaneously determine the position and the momentum of an object so let's go through that a little bit it has a similar character to some of the other illustrations that are given here what Heisenberg and Bohr and Einstein and all those people knew around 1926 was the property of photons in fact whenever they thought about measuring the properties of a particle they were always thinking roughly speaking of looking at it under a microscope detecting it by bombarding it with photons now it doesn't matter whether they were photons or not it's just that they were thinking about microscopes and they were thinking in a language way you optically look at things doesn't have to be optical but this is what they were thinking so Heisenberg imagined putting his particle under a microscope and detecting its position and its velocity and seeing what he could learn okay the measurement involved interacting with the particle with a photon the photon would be the thing which would be used to determine the position of momentum in other words looking at it really meant hitting it with some light letting the light scatter off and then focusing the light waves focusing the light waves in order to see exactly where the particle was just as you would do it for a billiard ball or for anything else focus the light on your retina focus the light on some screen and reconstruct the position of the particle that was what they were thinking about okay now here's what I'm Stein and the Broglie and others have taught them about photons first of all I'm Stan had told them that the energy of a photon is equal to Planck's constant times the frequency of the light describing the photon this is equivalent to plunks other constant h-bar times the angular frequency with F stands for the frequency of a wave number of cycles per second measured in hertz okay usually a frequency of a light wave is a lot of Hertz how many Hertz how many 10 to the 15th for ordinary light that's just the number 10 to the 15 Hertz Omega is the same as the frequency except multiplied by 2 pi it's the angular frequency instead of the instead of the number of cycles per second all right and h-bar is just H divided by 2 pi so these are the same expressions I'll use this one over here for the moment energy is equal to H times frequency that's something that was known by Einstein now here's another fact about a beam of light beams of light have not only energy but they also have momentum you can take a beam of light and shine it on something and it will warm it it will heat it that tells you it has energy but you can also take a beam of light and you can shine it at that door if the beam of light has intensity it'll just push the door open in other words the beam of light has momentum survey point it when the door absorbs the beam of light door gets a kick just as if just as if you threw a ball at the door and the door collide through the wall it has momentum okay now what is the relationship between the energy of the beam of light and momentum and I just tell you what it is this is this is classical Maxwell theory of light if a beam of light moving in a particular direction assume it's moving in a particular direction has energy E then it also has momentum and the relationship is that the energy is the speed of light times the momentum incidentally let's just check the units of that equation in another way let's compare that with Newton's theory of momentum and energy for an ordinary particle the energy is P squared over 2m that's nonrelativistic the momentum P squared over 2m young which is equal to P times P over m and an extra 1 over 2 one-half momentum underneath them divided by velocity now what's will make them divided by velocity tell you let them divide by mass velocity so this is equal to 1/2 momentum times velocity well it's almost the same for a photon dizzer it's not the same because we have to use the theory of relativity sorry but the correction from the theory of relativity is not so so enormous this is 1/2 the velocity of the particle times its momentum this is just equal to the velocity of the particle times the momentum so for highly relativistic particles the formula is very similar except 1/2 goes away among other things this argument tells you that the units are right energy is velocity times momentum um let me just solve that momentum is energy divided by the speed of light we can now plug in and find that the momentum of a photon or momentum of any new piece of light let's do a single photon now the energy of the photon is h times f r so the momentum is h times the frequency divided by the speed of light now if you have a wave that's moving with the speed of light the other way of moving with the speed of light it's moving down and it has a certain frequency and a velocity there's a connection between the frequency and the between the frequency and the wavelength think about it for a moment here's a wavelength let's call the wavelength lambda lambda standard a standard notation for the wavelength of light wavelength of anything how far does the wave move in one cycle the answer is lambda that's what lambda is it's the distance that the wave moves in one cycle if you stand there with your nose watching that light ray and it's moving past you all right it moves past you one wavelength per cycle one wavelength per unit first recycle how long does a cycle paint how long does one cycle take what's the period that goes with one cycle the inverse of the of the frequency so the time that it takes the time that takes to move distance lambda is 1 over the frequency let's just write that down time to go 1 cycle is equal to 1 over the frequency the distance that it goes in that same time isn't lambda so what's the velocity of the wave the velocity of the wave is the distance that it goes divided by the time that it travels distance all the time is velocity C is equal to lambda divided by T which means lambda times the frequency so this is a general formula relating velocity lambda and frequency and let's see let's plug it into here now let's get rid of the frequency frequency is C divided by lambda so frequency is C divided by lambda and then is another C down here and we get the broglie's equation that the momentum is platz constant divided by the wavelength the smaller the wavelength the larger the momentum if you want a team of particles with very high momentum give it a small wavelength or say it the opposite way if you want to do a short wavelength particle it's at the cost of having that particle have a large momentum momentum and wavelength are inverse to each other so now let's let's go back to once the measurer wants to get me roughly speaking he wants to get a photograph of the electron with the electron is not fuzzy on scales larger than Delta X so once I take a photograph of the electron oh it doesn't have to be an electron it could be a golf ball with matter where it is once they get a photograph of it and once the photograph should be non fuzzy on a certain scale no facts well every photographer knows that or anybody who understands anything about waves and images and so forth knows that the form an image which is precise for non fuzzy the size Delta X you have to use wavelengths that are shorter than Delta X if you try to make an image of a golf ball with a radial wave a radio wave of let's say 10 meters the golf ball will look fuzzy on the scale of 10 meters if you try to do it with a wavelength of a tenth of a tenth of a golf ball a tenth of a centimeter the golf ball will look pretty good so the rule is that lambda the wavelength of the light must be less than Delta X if you want to get an image with precision Delta X now I erased an equation the equation that I erased is that the momentum is equal to Planck's constant divided by lambda so now Heisenberg was caught in a bar if he wants to measure the position to a high accuracy Delta X is gotten to use a he's got to use a high momentum electron he's a short wavelength electron if he wants Delta X to be smaller than a centimeter then he's got to be using a wavelength smaller than a centimeter but if he's using the wavelength smaller than a centimeter that means the momentum of the photon has to be larger than H over one centimeter so the smaller Delta X the larger the momentum of the photon that he has to use to make the image shorter the wavelength of the photon made' has to used by the image if lambda is small then P is going to be large well now what does that mean that means that we're going to wind up bombarding this object with a high momentum Photon the high momentum photon they make a very good image of the position but then it's going to collide with this and knock it off in some random direction knock it off in a random direction with a momentum uncertainty about uncertainty mentum of order of magnitude of this momentum here this particle will come in and just like the photon hitting the slit it will get knocked in some random direction and so the conclusion will be that immediately after you try to measure the position immediately afterwards the momentum has become very uncertain it has been kicked hard having been kicked hard if you measure is four make them afterwards it will have nothing to do with the momentum beforehand so you cannot determine both the position and the momentum at the same time measuring the position necessarily imparts a random momentum a kick a random momentum kick to the particle result is whatever the momentum is before was beforehand it won't be that afterwards and that's another example of the fact that there's no such thing as a gentle determination requirement well we can do a gentle determination but it will be a very imprecise determination in classical physics incidentally Oh keep in mind why what about classical physics why is it different in classical physics the reason is because light doesn't come in discrete packets and classical projects and we've used the fact that the light comes in discrete indivisible quantum the street indivisible photons and if we have a wavelength it is a minimum amount of energy that can be that can go with our middle amount of momentum that can go with that wavelength namely one photon you can't have less than a single photon in classical physics energy does not come in discrete multiples of some basic unit and so you can do the same experiment with as small energy as you like you can so in classical physics you could subdivide that photon into arbitrarily small units of you take just one of them the same wavelength and form an image with it in quantum mechanics you're always stuck by the fact that the energy of a light wave comes in these discrete packets and at the screen packet that has wavelength lambda will have a momentum H over lambda and therefore give this inevitable kick of a mountains over laptop now that's this is several examples of the same kind of thing that doing experiments in quantum mechanics is different than doing experiments in classical mechanics you can always imagine in classical mechanics doing a very gentle experiment that disturb the system and then just go on from there and do a later experiment and do a later experiment the earlier experiments not having influenced the outcome of the later experiments so for example you can measure the position and that little not not affect the outcome of a later detection of the velocity which is not for employment events these are a lot of examples but the examples add up to a a if you want a notion that the basic logic of classical mechanics is incorrect basic underlying logic is not sufficient to understand measurement process in quirements the whole set up the whole set up not just not to take you look at each experiment you can go and analyze it but try to figure out what's wrong with it and try to correct for it no the whole underlying structure of classical physics is inadequate to discuss point mechanical phenomena let's take a break for five minutes then start quantum mechanics proper okay somebody asked me about how do you measure the velocity of a particle the answer is gently first of all here's a simple is a simple conceptual way to do it velocity or momentum momentum let's suppose we know the mass of the particles of a pen that by measuring its velocity we also measure its momentum a simple conceptual way to measure velocity is to measure location at two different times and then take the difference of location that's how far it travels and divide by the time between measurements and that's the velocity that's the way you would measure velocity now you have to be careful you want to measure the you want to measure the position twice in succession but you don't want to measure the position with such with such good with such a good determination that it gives a random kick to the velocity which is what you're trying to measure you wouldn't want to measure the velocity by beginning the experiment with a random kick which were n demises the velocity and sends it off in some direction very different than it was moving with to begin with direction and velocity so your two measurements of velocity I'm sorry of position should be very gentle measurements that don't change the velocity very much or the don't change the momentum very much that means that they must be done with photons of very long wavelength if you don't want your first initial detection of the location of this particle that to to to give it a good whack then you want to do it with a very long wavelength photon a long wavelength photon will then tell you only that the electron is in some region of size lambda okay so what you know then is the position of the electron X let's say plus or minus lambda meaning to say that there's an uncertainty of magnitude lambda then you wait a really really long time until the electron has moved a long long ways has some velocity it started out with some velocity even you've changed the velocity only by a very small amount by using a very long wavelength photon so we take the wavelength of the photon to be so long that there's been an unappreciative and then in a much later time we discover that the particle is at X plus or minus lambda maybe even plus or minus 2 lambda plus VT it's velocity times the time between the two measurements so true there's a little bit of sloppiness in the measurement of the positions a large sloppiness excuse me the measurements of the positions and that's going to lead to a sloppiness in the dis speak that the particle moves what's the sloppiness the sloppiness the distance that the particle will move will be VT plus a sloppiness of Auto lambda and we take lambda to be very big so that so that we don't disturb the velocity very much then how do you find this is this is how far the particle moves let's call it D distance that it moves in order to find a velocity we have to divide the distance that it moves by the time so let's divide it by the time this isn't this is now a measurement in the sloppiness of the velocity the sloppiness of the velocity measurement is lambda divided by T no matter how big lambda is if we wait long enough if we let T be very very large then the sloppiness and the velocity can be made small so the way we measure the velocity is by taking a long time to do it through very very gentle measurements of the position so that we don't lack the velocity and in that way we can measure the velocity with great precision however it's been at the cost of knowing where the particle is the particle is not been determined to a precision better than lambda the velocity has been determined to a precision of something like lambda over T so whatever you do you'll never be able to determine both the position and the velocity in the same experiment to do one and not the other one or the other kind of quantum mechanics yeah yes well this arm reverse it again yeah ah let's say you wait a very long time so that that's the that's because with your taking this to be the uncertainty and of velocity but there's also an uncertainty from here that has to do with the original setup let me think about it yeah I understand your question let me think about it and come back to you with an answer yeah it's a good let me come back to it yes yeah I see I see yes you can um you can determine the position of this wall by averaging over a long period of time with photons BAM many many photons banging off it yeah yeah I'm uh I'm trying to remember what I was going to talk about we'll come back to the uncertainty principle for sure but I'm I've lost the thread of my thought all right so I forgot what I was going to say but let's just move on so the fundamental logic of quantum mechanics is not the same as the fundamental logic of classical mechanics and that shows up at the earliest possible stage namely what do you mean by the state of a particle what do you mean by the configuration what do you mean by knowing everything that there is to know about a system or everything that can be known about a system in classical mechanics the space of states what we call the phase space we had some various versions of it in one version it was just a set of points I use the word set and I use the word set on purpose a set of points the states of a system heads tails or one two three four five six for a die in particle mechanics or mechanics or more that's a little bit less primitive than this very simple system here it's phase space the states of the system are phase space peas and X's peas and axis points in a phase space are the states of a system but again the collection of possible states of the system form a set they form a set they form a set of points and the basic logic of classical physics of classical phase space is set theory sets of points describe the possible states of a system and transitions or motions from one point of the in that set to another point in that set describe the mechanics or the dynamics of a system in phase space it's the flow through phase space what exactly is a state a state is a point in that set a member of the set an X and a p4 particle a point H or T for heads and tails or a bunch of peas in X's for a general system of many particles but a point or a member of a set now you could think of something a little more general and possibly call it a state in which you introduce a bit of statistics a bit of a probability or a bit of uncertainty from the beginning you might say look I my apparatuses do are not sufficient to determine with infinite precision the position and velocity of a particle so instead of doing that I will say the particle is somewheres in some little region here or I might assign a probability distribution a probability distribution as a function of X and P which might be peaked at the center here and fall off as you move away from it and so forth a probability distribution on phase space might even be a more general version of what you mean by the state of a system but if I told you that the notion of a state of a system is a probability distribution I think most of you will come back and say yeah but you know if I look at more carefully system if I look more carefully I can always reduce that probability distribution and I can always get as close as I like by doing delicate experiments and again in classical physics doing experiments which don't disturb the system terribly much doing experiments on the system which will get it to be closer and closer to a point so when you speak about a statistical distribution your as being a state you're not talking about the maximum amount of knowledge that you could have about a system you're talking about an impractical limitation because of the coarseness of your apparatuses and so forth might force you to use a probability distribution and only talk about things to within a precision that you can measure them but you wouldn't think that that was very fundamental and eventually you would say with sufficient accuracy in your apparatuses that that the maximum you can know about a particle corresponds to a point in the phase space and in that sense states in classical physics are points in a set points in a set and you can know everything that's implied by knowing a point in the set in particular in this case knowing a P and an X in quantum mechanics states are not sets do not form sets the natural way of manipulating States and asking questions about them is not set theory as not state is not set theory states are vectors in a vector space in quantum mechanics quite a different mathematical object than a set a vector space is mathematically extremely different than a set and in order to understand quantum mechanics this is not some abstraction that is really sort of unnecessary to understanding the subject this is so central that that to not talk about if we would completely miss the basis of the basics of quantum mechanics what in order to understand the logic of quantum mechanics we have to understand the mathematics of vector spaces which is forced to it there's no way around it if I try to do things without it I'd be faking so let's talk about vector spaces now the use of the term vector oh these are vector spaces linear vector spaces every vector space is linear so linear is a redundant word but their vector spaces over the complex numbers that may mean nothing to you right now but it will as we go along vector space the space of states the state space space of states is not a set but it is a vector space now the use of the word vector can be Invictus a vector space over the complex numbers over C C stands for the complex numbers and I'll tell you exactly what this means if you don't know what it means don't worry because you will know okay first of all most of us are familiar with the notion of vector from the classical notion of a pointing of a thing pointing in a direction in ordinary three-dimensional space with a certain length what I'm talking about is not vectors in space in that sense these are abstract vectors in abstract vector spaces which have nothing to do with your naive concept of a vector in space as an arrow pointing in some direction in space with a given length when I teach this I usually make a linguistic distinction between vector spaces and pointers in space which are the things you usually think about as vectors I can never quite figure out how to do this without getting everybody confused including myself I will try as much as possible when I'm talking about a three-dimensional vector in space the kind of vector that could correspond to velocity or position or momentum and that sort of thing I'm going to use the term pointer whenever there is any ambiguity about what I'm talking about a pointer meaning at the direction and ordinary three-dimensional space together with a length I'll use the term pointer in order to not get confused with a completely different mathematical concept or not a different mathematical to a much more general mathematical concept called a vector space a vector space is a space of vectors but of course these vectors are not pointers in ordinary directions or ordinary space point there is an ordinary space or a special case of vector spaces but not the special case we're going to be interested in okay so let me tell you what a vector space is first of all it's a collection of mathematical objects called vectors again emphasized over and over not vectors and or just abstract objects called vectors maybe I should change their name and not the name of the vectors in three-dimensional space we could go with the column vectors or or shmeckler z– but but one way or another we've got to make some distinctions all right it's a collection of objects which we can label let's label them we can also label sets by points and give the points labels like a B and C and so forth but now I'm labeling vectors and to indicate that they are vectors of the right kind namely the kind that come into quantum mechanics we're going to draw a symbol like that this is called a ket vector it was called a ket vector by Dirac there's another kind of vector called a bra vector and if you put them next to each other they make a bra ket or a bracket that's where the name came from or will come to brackets in a little while it's half a bracket a bracket being what happens when you put a bra next to a cat is that clear yeah okay okay so there's a collection of objects collection of objects called vectors okay now what does it mean to say it's a vector space over the complex numbers what it means is that first of all you can multiply vectors by complex numbers now the ordinary vectors in ordinary space are a vector space over the real numbers if I have a vector for example three units of length pointing in the direction North by Northwest that are 45 degrees to the to the horizontal that would define a pointer if you like all right so ordinary three-dimensional pointers are have a length and a direction you can multiply them by numbers if you multiply a certain vector this one here this is sorry a certain point there by 2 it just gets twice as long in the same direction if you multiply it by minus 1 it's the same length but pointing in the opposite direction if you multiply it by 7 and 1/2 it becomes 7 and 1/2 times as long so you can multiply a vector point therefore you're a pointer by an ordinary real number positive or negative mathematicians would say that pointers form a vector space over the real numbers vectors over the real numbers meaning to say that you can multiply them by real numbers and when you multiply them by a real number you get another pointer back we're going to make a generalization of that idea where vectors can be multiplied by complex numbers now eventually we'll see some examples there's nothing like an example to to make it clear but what that means is that given any vector you can multiply it given a you can multiply it by any complex number let's use for complex numbers let's use the Greek alphabet so if alpha is any complex number every I assume everybody here is familiar with complex numbers I'm going to assume that I'm not going to teach you complex numbers here you can multiply any vector by a complex number alpha and it's the and it's a new vector not the same vector it's another vector so the operation of multiplying by a complex number is well defined in a complex vector space we can call this B for example multiplying by alpha maps every vector into another vector alpha being any complex number that's the first property of a vector space over the complex numbers sometimes we can call it a Hilbert spamming sometimes it's just called the Hilbert space vector spaces over the complex number a Hilbert space Hilbert was a mathematician so that's the first property give it a vector you can multiply it by any complex number and it's a new vector the second property is that if you have two vectors any two vectors let's call them a and and B you can add them and you get another vector so adding vectors give new vectors now first of all here's something which is really these both of these things are really new when you go from classical mechanics to quantum mechanics in classical mechanics the notion of a state is a point in a in a space of points in other words in a set it doesn't make any sense to multiply our points in a set by a number for example let's take heads and tails what does it mean to multiply heads by 3 or heads by a complex number doesn't mean anything you just have two points head entails and that's all you have you don't have a thing that you would call three times heads or minus three times heads or two plus four times I times heads so in saying that state vector that states of a system are a vector space this is something new and radical and weird you should not understand this at this point unless of course you've done quantum mechanics before this should not make any sense at all but for the moment we're doing some abstract mathematics to get some definitions and then I'll show you how those definitions apply so first of all you can multiply a vector an abstract vector by any complex number and you get a new vector that's an operation that you can do and the next operation is that you can add any two vectors now of course if you can add any two vectors and you can also multiply them by complex numbers then you can also then you can also multiply the first vector by a complex number the second vector by a complex number and that gives you some new I won't call it C I'll call it C Prime so what you can do with the vector space is you can take any pair of vectors multiply both of them by arbitrary complex numbers add them and you get a new vector so the notion of addition is well-defined of course for ordinary pointers the notion of addition is orden also defined and it's defined by the operation of drawing the parallelogram ordinary vectors satisfy both of these rules that you can multiply a vector by any real number I'm sorry not complex numbers ordinary pointers as I said are vector spaces over the real numbers and that means that you can multiply any of them by a real number and you can add any pair of any pair of pointers to get a new pointer so ordinary pointers in space are a special case of a vector space over the real numbers let me give you a couple of other examples in particular vector spaces over the complex numbers this was the reason Hilbert originally invented the notion of Hilbert space to describe functions functions of X let's say functions of one variable for simplicity that's the simplest example take the class of functions of one variable X let's take X to be the variable but complex functions of one variable functions Phi of X which are the sum of two terms let's call it a real part SCI real of X where SCI real of X always takes on real values plus I I being the square root of -1 times the imaginary part of sorry every complex number is the sum of a real part plus an imaginary part and every complex function now this is a function of only one variable it's not a function of a complex variable it's a function of an ordinary variable but the function itself can be complex alright so this means that for every point X there is a complex numbers I of X or equivalently two real functions now let's take any complex function let's take any complex function of it like this we can multiply it by a complex number any complex number defines a new function you can multiply the complex number sy + sy real plus i sy imaginary by a complex number alpha in other words complex numbers can be multiplied and therefore you can take any complex number and multiply it by any function so first of all functions of X satisfy the first rule the collection of functions the collection of functions of X a collection of complex functions of X form a vector space that's the that's the assertion we're going to check that first of all you can multiply any function by a complex number and you get a new function all right so it satisfies rule number one what about rule number two you can take any pair of functions and add them together and you get a new function so if you have two functions sine Phi let's call them that's another function a perfectly good one so sine and Phi a complex functions this sum is a complex function that's enough to tell you that complex functions are a vector space over the complex numbers real functions are a vector space over the real numbers but they're not a vector space over the complex numbers why not because you can't multiply a real function by a complex number and get back a real function you'll get back a complex function you'll get back a real function so complex functions form a complex vector space and that's the that actually is this with the vector space that the Hilbert was first interested in that's an example of a Hilbert space it's an important example in quantum mechanics maybe the most important example but for now I'm just illustrating the the abstract mathematical definitions complex functions of X form a vector space over the complex numbers let me give you another one it's a very similar let's let's invent a thing called a column vector a column vector is just a collection of numbers which have and we can decide how many we fix the number of numbers 1 2 3 any number we like but let's fix it all right and we simply arrange them in a column let's call those numbers let's give them let's call them a1 a2 a3 and a4 this would be a vector space of dimension 4 we could also think of vector spaces of dimension 2 a1 and a2 there's pairs of numbers pairs of complex numbers is quadruples of complex numbers we could make quintals of complex numbers sex doubles of complex numbers and we can even imagine infinite the columns of complex numbers or just one complex number just one complex number one unit long a one of X all of these each one separately not all together but each one separately for dimensions or two dimensions or three dimensions each of these form is the collection of such objects form a complex vector space vector space over the complex numbers so let's take this particular complex vector space here a 1 a 2 a 3 a 4 where the A's are arbitrary complex numbers if these are arbitrary complex numbers and we display them in a column like this then I assert that there are rules which will allow this to be a vector space so what are the rules I'm going to invent now a rule of addition notice so far there's been no rule of multiplication of vectors only addition of vectors so the rule for adding two vectors here's two vectors what I'll call a the other one I'll call B b1 b2 b3 before just add the entries this is the definition of adding of column vectors a1 plus b1 a2 + b2 a3 plus B 3 a 4 + B 4 obviously with this definition if you take any two vectors and you add them you get another vector the A's and B's are arbitrary complex numbers but each specification of four complex numbers defines a vector here are two vectors one labelled a one labeled B is the third one that you can make out of it alright so I started for some reason I started with the addition rule here but it's also true that we can invent the idea of multiplication by a complex number if we want to take a vector let's call it a1 a2 a3 and a4 and multiply it by the complex number alpha we just do that by multiplying every entry by alpha alpha a1 alpha a 2 alpha 8 whoops alpha a 3 and alpha a 4 we multiply every entry by the same number that defines multiplication by a complex number and all I've done is multiply complex numbers here obviously given any vector in any complex number I can make another vector by multiplication not multiplication of vectors but multiplication of numbers by vectors and given any two vectors you can add them so the collection of objects arranged in a column this way also form a vector space now if I were to take the collection that consists of vectors of length 2 length 3 length 4 and combine them all together that's not a vector space because I haven't given you any rule for adding a vector with with length 2 to a vector of length 3 I haven't told you do that and I don't see any obvious rule for that so it's with this set of rules the collection of two dimensional vectors three for the whole shebang does not form a vector space but for a given length fix the length it forms a vector space and the terminology is that it forms a vector space in this case of dimension 4 now of course this has an analogue for ordinary pointers for pointers we can just think of these as the components of the pointer along the different axes pick some axes XY and z and pointers have components so we can also specify an ordinary pointer by specifying a collection of components or collection of components along different axes and that that also defines a a pointer so it's natural to call these numbers the components of a vector and we'll come back to the uniqueness of components you can if you like think of such a column as a function of the index variable here well let's get that's too abstract let's let's drop that for the moment we'll come back to it can be thought of as a function but we don't need to now of course I've told you nothing about in what possible sense what possible sense can it make to identify vectors with states of systems we're not going to do that tonight tonight we're just going to talk about about the abstract notion of a vector tonight then a bit of next week we'll talk about the abstract notion of a vector and then we'll talk about how vector spaces define the states of a system and then go on and work out some examples of quantum mechanical systems described by such vector spaces at the moment this should be sort of mumbo jumbo right hmm what success in in making mumbo jumbo no I think I've given you rather accurate definitions that you can understand the word vectors are and how to manipulate them have to how to manipulate them how to do the abstract operations with them these abstract operations are in many ways analogous to the abstract operations that you do with set theoretic logic with boolean logic these abstract these abstractions for how you manipulate vector spaces are the generalizations of you like of how boolean set Theory allows you to ask less allows you to combine concepts together and or all these ideas have some meaning in set theory was and and as you take two sets and you combine them and make the union of them or you take the intersection or is it the other way it's the other way isn't it the other way sorry and is the intersection or is the is the union of two sets the analogous logic in quantum mechanics doesn't operate on sets of acts on vector spaces so it's best that we get rid that we go through vector spaces once and for all it's the basic underlying mathematics and once we get familiar with it it won't seem so so much mumbo-jumbo but a moment we're doing abstract mathematics any other question okay so let's continue with the abstractions now there's a notion of every for every such vector space well it's lovely this year there's a notion of a dual vector space the dual vector space is in one-to-one correspondence with the original vector space so let's go back a step let's just talk about complex numbers for complex numbers incidentally complex numbers are themselves a vector space you can multiply complex numbers by a complex number you can add complex numbers they're just the case of one dimension one dimensional vector spaces over the complex numbers are just the complex numbers now there's an operation that you can do on complex numbers which is called complex conjugation it's a very important operation and all it is if you think about the Cartesian plane if you think about the complex plane the real part of a number being on the horizontal axis and the imaginary part being on the vertical axis then the number then is described by a real part the real part is the horizontal component and the imaginary part of the vertical component and we will just write that this number usually called Z is equal to X plus iy the complex conjugate number is just the number reflected about the horizontal if this is a number called Z X plus I Y then X minus iy is called Z star X minus iy is the complex conjugate of X plus I Y or Z star is the complex conjugate of Z let's let's manipulate something let's let's consider the product of Z times Z star Z times Z star what is it that's equal to X plus iy times X minus iy that's x squared now we get I X Y and then we get minus I X Y from the cross terms they cancel and then plus y squared why did I write plus y squared because I x minus I is plus 1 I times I is minus 1 so I times minus I is plus 1 Z star Z is x squared plus y squared what is x squared plus y squared x squared plus y squared is the square of the length of this hypotenuse here so Z star Z is the square of the length or the square of the magnitude of the number or its distance from the origin so complex conjugation is a convenient operation among other things the complex conjugate times a number is the magg the square of the magnitude of the number and if you're not the most of you of course are quite familiar with complex conjugation I want you to think of it as a very basic operation but more than that I want you to think of it as a mapping of the numbers back into the numbers given any number any complex number you can map it to its complex conjugate what about vector spaces I think we should probably quit now I think it's time to quit now next time we'll go through a little more about vector spaces vector spaces operators hermitian operators and eigenvalues and then we'll begin applying it the quantum mechanical problems the preceding program was brought to you by Stanford on iTunes U and is copyrighted by the Board of Trustees of the Leland Stanford junior University please visit us at iTunes dot stanford.edu

This is what a theoretical physics exam looks like at university

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Double pendulum (Q1):

Perturbation theory (Q2):

Ehrenfest theorem (Q3c):

Spin (Q3d):

Special relativity (Q4):

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hey everyone today I'm going to show you inside a physics exam I know that some of you might be interested to know exactly what studying physics is like and therefore what sort of exams and assessment looks like so I have here a theoretical physics exam and I will say as a disclaimer the questions in here I have edited so that they're not exactly the same as an exam I took although it is based on a real exam I've done this for two reasons one so that my uni doesn't arrest me for publishing a real exam and two is so that I can give you guys some solutions as well that I sort of fully worked and available elsewhere online and like textbook solutions having said that it does cover the same ideas that you would expect and that I definitely saw when I took an exam like this for the first time now the exam I've replicated as a theoretical physics exam from a third-year University course and it would take you three hours to do this exam and there are four questions so let me show you what it looks like so this exam is going to cover most areas of theoretical physics that's a bit of classical mechanics and quantum mechanics and special relativity so let's have a look first up we're going to have a classical mechanics problem and this is one that deals with a Lagrangian so we've actually got a double pendulum here and this is probably a bit of a typical problem so you're given a double pendulum and you're asked first of all to find the Lagrangian so that's essentially a way of finding out how to describe the system when the system is quite complicated then you want to find the equations of motion that's probably where you're going to get a lot of marks as well because they might be a bit complicated and there'll be a lot of calculus involved then you're asked to describe the properties of a chaotic system that'll be saying something like a chaotic system which a double pendulum is well small deviations and where it starts or in the initial condition can lead to vastly different outcomes over time and then lastly for this question we've got explain the significance of know–this theorem moving on to part two now we have a quantum mechanics problem so this is actually from Griffiths a really good quantum mechanics textbook and it's about time and dependent perturbation Theory now often when you work with quantum mechanics problems you're dealing with say a particle in a infinite square well but perturbation theory deals with if there is a sort of blip or a bump a perturbation in that square well so it's no longer just a perfect square have a look in the description and I will link the solution here of how to do this our third question is also a quantum mechanics problem it's stealing with some sort of identities commutator relations and really it's a whole bunch of proofs when I was doing sort of this exam getting to this page the page of quantum mechanics proofs and just show things is where my heart sinks because those are to me the hardest questions using mathematical induction or otherwise show that this is true the classic or otherwise statement means really do it using the way they mentioned at the start otherwise you've got some crazy idea up your sleeve I don't know you maybe if you're clever you can think of another way to do it but usually when they give you a hint like this it's a good idea to go with the hint part see I would probably find pretty tricky in this exam and it's showing a relation that's got something to do with Aaron's first theorem I don't know how to pronounce that I barely remember that hyrum but i know this has got something to do with it and then Part D we're finishing off with a question about spin the spin question really comes down to using linear algebra and it's a useful application of some of the maths you learn quite early on so finding a normalization constant and expectation values lastly we have one whole question about special relativity what makes special relativity difficult and say third-year it's not the idea of sort of time dilation and length contraction but it's moving into using tensors and describing things in four space so that's you can see this little notation here with the Greek letters that's called a four vector and it can be pretty tricky and being obstacle to actually describing sort of space-time and special relativity so because of that and because you would have just learned how to use these four vectors most of these questions here are just sort of kind of manipulation of four vectors and once you really master these I think you might be able to go on to being good at things like general relativity and really getting into deeper into theoretical physics concepts that isn't the road I went down but I think they were still pretty cool to learn about I hope no one gets too stressed out looking at this exam that I've showed you because whilst this is pretty realistic of what you can expect if you haven't taken the course and subjects that are relevant to what I'm showing you it can probably look overwhelming even me looking back on this now I'm like geez how did I do any of this I can't remember some of this stuff now but when you're in that mindset of learning and studying for a course and then it's not so alien to you then this something like this would feel I don't know possible if you actually immersed in that environment and you've done the work leading up to it thanks for watching and if you have any questions about this exam or what others might look like please let me know

Demystifying the Higgs Boson with Leonard Susskind

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(July 30, 2012) Professor Susskind presents an explanation of what the Higgs mechanism is, and what it means to “give mass to particles.” He also explains what’s at stake for the future of physics and cosmology.

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Stanford University the goal tonight is not to tell you why the Higgs boson is the best thing since flush toilets a griddle of that anybody who's here has probably read all of the height the excited reckless superlatives that that have hit the newspapers and so forth things bosons are explained the origin of the universe mr. Mack I'm not going to do that first of all before I say it the the excitement and the enthusiasm is justified it's not that it's not justified it is justified the history is fantastic it's an unbelievable event and so forth but that's not what my thing is it's not what I do well what I do well is explain how things work my goal tonight is going to show be to show you and so far as I can in one hour which is tough which is hard and may not work but as well as I can to explain to you the nuts and bolts of what Higgs physics is about one of my closest friends incidentally is named Francois and Blair and Francois and Blair would be appalled to do that I was sitting here talking about the Higgs that since francois hollande later was the discoverer of it so from time to time and we call it the drought on where kinks effect but I also may slip and slide and just call it a ticks effect because something to get like everybody else oh all right so how it works first of all there's a lot of moving parts a lot of pieces that I would have to lend this I would have to explain to you first to really do it right and I'm going to try to explain those pieces and move a little module shall we start on it's a highly quantum mechanical effect it cannot really be understood without quantum mechanics and so I would begin with a course of quantum mechanics and let's say the course in quantum mechanics consists of just one thing things are quantized quantized means that they come in discrete integer quantities the most famous example of this is angular momentum angular momentum the rotational properties of the rotational momentum of an object are quantized which means that kind of discrete steps and the discrete steps are one Klunk unit 1 unit of Planck's constant we'll take that as all we really need to know about for the most part about quantum mechanics tonight the next concept which is an easier one at least I think it's an easier one of the classical concept is the idea of a field a field is just a condition in space it could be the electric field it could be the magnetic field it could be a gravitational field whatever you think of as being present in space which characterizes the conceived behavior of space at that instant and in that place in in space and time not so steals the space can be filled with fields now ordinarily you imagine in empty space empty space is the thing we call a vacuum from a quantum mechanics point of view the vacuum is just a state of lowest energy nothing there no energy other than what quantum mechanics requires there to be there and so you ordinarily think that the fields that couldn't have its space are zero in empty space the electric field the magnetic field and so forth but there's no important requirement of physics that says that that should be the case let's imagine a world filled with electric field how could that electric field get there well it might be there because there are capacitor plates infinitely far away so far away we can't cover there there we will have a world in which there was an electric field will just be there empty space would have no electric field now when you think about fields of we're beginning a little bit too all right so what I was saying to the audience two things the first was that quantum mechanics we're going to summarize by one simple statement that things are quantized in quantum mechanics quantized means they come in discrete bits the most important example is angular momentum not the most important necessarily the most important example but the most important example for me tonight will be angular momentum and angular momentum is just to do with rotating objects and so forth angular momentum in quantum mechanics unlike classical mechanics comes in discrete units the unit is Planck's constant you can't have a tenth of a unit of angular momentum you can only have angular momentum 0 1 2 3 – 1 – 2 – 3 you can also have 1/2 integers but we're not going to worry about that tonight but no you can't have angular momentum PI only discrete integers that's the first fact that I want you to remember the other fact from quantum mechanics that we'll have to remember also is the uncertainty principle but we'll come to it now the other thing we spoke about was fields fields are things that can fill space electric field magnetic field gravitational field other kinds of fields that exist in physics they are functions of space they can vary from place to place and they affect for example the way things move an example would be an electric field affecting where charged particle moves now the other thing I said was you can imagine a world in which for all practical purposes empty space is filled with a field an example would be if I went out to Alpha Centauri at that end and oh excuse me I don't know what's outfit excuse me yeah whatever's out there and play some capacitor plates big capacitor plate out there big one out there make an electric field in between that's so far apart that we can't see them we would say that the world is a world that exists for the magnetic field and we would say that charged particles move in peculiar ways but that was just a fact of nature um generally speaking fields cost energy the space without an electric field has zero energy with an electric field it has energy and if we were to plot the energy of a field the typical field could be electrical be a magnetic could be something else generally we imagine that the field energy as a function of the field horizontally imagine the value of a field vertically it's energy zero field right here and we imagine exciting the field causing it to vibrate causing it to vibrate by giving it a push at some region of space and nearby the field will vibrate those vibrations are quanta of the field they are particles there are particles the quanta of vibration of the field of particles now you might have a situation where there is more than one field relevant let's call it Phi and Phi Prime or whatever you want to call it doesn't matter then instead of plotting the field is one dimensional we might plot it as two-dimensional now this is not space this is the value of some collection of fields and then the energy would depend on both fields here's an example an energy function which looks like that which simply says that no matter how you displace the field it costs energy now imagine that this upside-down paraboloid here or whatever it is was nice and symmetric nice and rotationally symmetric in the field space exactly like the top of my hat here as which as you can see is symmetric and so the field as a oh yes where's my little deaths that's it right the field as a function of position when most likely if the energy is as low as possible just sit at the bottom of the potential energy of the field just to lower the energy as much as possible so we might think of the field at every point in space a little ball which can be made to oscillate back and forth and do things and those are just oscillations of the behavior of physics in the local region of space as I said they often correspond the quantum particles those oscillations but for the moment they're just oscillations now one of the things we could do if we had a field whose values will like the position in the Hat here would be to start it out displaced from the origin let's say up to here and then start it moving in a circle just in the same way you could take this ball and if it was if the Hat was really nice and smooth and symmetric give it a push in it we go around enough in a circle that circular motion of the field is very very similar in a way to angular momentum it's not angular momentum in space but it's a kind of angular momentum that exists in the field space that angular momentum like all angular momenta are quantized come in integer multiples of Planck's constant what do they correspond to they correspond to something else that is also quantized in nature the value for example of electric charge so in modern physics the way one thinks about the electric charge in a is that in some region of space a particle a charged particle a charged particle is viewed as an excitation of the field in which the field is made to spin around in the internal space of the field not in real space but in the internal space of the field that's one way in fact it's the main way that we think about charge as a kind of rotation in an internal space okay now what I want you to do is imagine taking the Hat and turning it over imagine that the potential energy was not turned it over this way excuse me this way is the way that the potential energy is minimum at the crown of the hat but if the potential energy really looked like that so that it was maximum at the top of the Hat then the top of the Hat would not be a position of equilibrium it would be a position of unstable equilibrium would look like this turning over the Hat the crown of the hat now this is the way that real hat looks like this and it doesn't okay let's just let's make the how's that that look like a hat yeah looks like a what kind of hat does it look like to you looks like a sombrero right looks like a Mexican hat physicists call this kind of potential energy function a Mexican hat believe it or not it's called a Mexican hat it turns back up the top is unstable if a pas ball was put at the top it would roll down and where would it go it would go to the brim of the Hat if for some reason the potential energy of a field was like this then the state of lowest energy would not be at zero field it would be out here now that's kind of interesting it would be a vacuum a world which had a field just like having an electric field except it's not an electric field and what's the value of the field at every point in space was not zero you might notice it how would you notice it well you might notice it because it might affect other things and indeed it does affect other things as we will see but there's now something interesting you can do that you couldn't do here over here if you wanted to set this thing into rotation you would have to displace the field a little bit because it doesn't mean anything to rotate right at the center if you wanted to set up a rotation you displace the field and then give it a flick so making the charge particle costs some energy here you can imagine setting this thing into rotation with just a little flick that costs no energy it costs no energy because you don't have to ride up the side of the hat in other words you could have a motion in which F naught got it you got it you got you understand huh you got have a motion in which that field slowly wound around the top of the potential in fact they could do it everywhere simultaneously not in real space but in this field space that would correspond again to a charge if rotation in this internal space corresponds to some kind of charge but now the whole world if the whole field was moving like that would have a little bit of charge in it a charge density charge filling space and essentially no cost of energy that's the nominal is called a condensate it's called spontaneous symmetry breaking but it's also called a condensate a condensate in space of charge now you might say okay look I want to find the lowest energy that the vacuum can have that empty space can have my best bet is to make the field not move with time just like a ball at the bottom of the sombrero hat here there's also kinetic energy of motion causing the field to move around in a circle like that would cost some energy so you would say the true lowest energy state of the world should be with a field either here or here or here it could be anywhere along the rim of the hat but it should be standing still right the problem with no angular momentum or no charge empty space should not have charged the problem with that is the uncertainty principle let me remind you what the uncertainty principle says it says that if you have a object and you're interested in its position X in ordinary space now and its momentum P velocity if you like the uncertainty principle says that the uncertainty in its position times the uncertainty in the momentum is greater than or equal to what Planck's constant you can't have something both standing still and having zero momentum if it's stand sorry you can't have something standing still namely no momentum and also localized at a point Delta P times Delta X is greater than h-bar same thing here if you know where the field is on this Mexican Hat if you know with great precision then it follows from the uncertainty principle that it must have a very large uncertainty in how fast it's moving around here ah that's interesting now I would say that you can't have empty space with no charge in it can't have empty space with no charge in it because if you lay the field down at this point you know where it is on the rim of the Hat and if you know where it is there's a necessary uncertainty in the charge the charge being like the angular momentum alright so where are we then if this were the case for electric charge for ordinary electric charge we would say that the vacuum empty space not only is filled with charge in a certain sense but a totally uncertain amount of charge totally uncertain and this is a quantum effect a totally uncertain amount of charge there would be equal probability let's take a little volume of space there would be equal probability that the charge was zero or that the charge was 1 or minus 1 or 2 or minus 2 3 minus 3 now this is truly odd this is not something you should try to visualize because you can't visualize an uncertain amount of charge but nevertheless that is what a region of space would look like if you measured its charge it could be anything from minus infinity to plus infinity ok now I want you to imagine that you have an extra charged particle an extra charged particle and you throw it in you don't know initially what the charge is but what does that do it displaces the charge by one unit let's suppose it was a positive charge you've displaced the charge by one unit and so if it was zero to begin with it's now warning if it was one to begin with it's now two if it was two to begin with it's three it was minus one it's 0 1 minus 1 minus 2 and so forth but that's exactly the same as what we started with we started with something which had an uncertain amount of charge equally likely for any value of charge and what did we end up with after we threw the charge in exactly the same thing what if we pluck the charge out of this thing same thing so a condensate is a funny configuration of space where with respect to whatever kind of charge we're talking about it's so uncertain that you wouldn't even realize it if you put an extra one in or pulled one out now the real world is not like that with respect to electric charge we know if we have a charge in space so it's not like that with respect to electric charge however there are materials that behave like this superconductors superconductors are exactly like this so it's not unheard of it's not a totally new thing to have a condensate of charge we're in a region the charge is completely uncertain okay that was module number one if you like condensates or what sometimes called the spontaneous breaking of symmetry modulo number two the standard model now we come the particle physics and I'll give you a short course in particle physics first of all particles have mass and the mass can be anywheres from zero we're talking about small particles now we're not talking about railroad engines or or stars we're talking about small particles we call an elementary particles but there's also a maximum mass that can have if they were bigger than that they would form a black hole if they were more massive than that if a point particle was more massive than something it would form a black hole and it would be something different so up to some maximum and that maximum is called the Planck mass it is not a very large mass it's neither a very large mass nor a very small mass it happens to be about one hundred thousandth of a gram a small dust mote but that is the heaviest of the chart of a.m. that an elementary particle can be without turning into a black hole and if you ask now where on this chart from zero this is called M plunk up to the maximum where are the ordinary particles the electrons of photons the quarks they are way way down here the largest mass of a known elementary particle is about 10 to the minus 17 of the Planck mass why are the particles so light well one answer is in order to detect massive particles you have to have a lot of energy in order to have a lot of energy you need a big accelerator we've only made accelerators up to some size and so for all we know the rest of this is filled with particles and that's probably true that's probably true but what is special about these particles well first of all let me name them and then I'll tell you what's special about them that makes them clump up at zero mass let's name them the particles of the standard model they come in two varieties it is not important that you know the difference well I'll give you a rough idea what the difference is they come in two varieties called fermions and bosons the fermions are all the particles that make up matter in the usual sense the electron which I'll just call e well the neutrino goes along with the electron that's a new electron the neutrino quarks there's a variety of different quarks incidentally there are several different kinds of electrons we call them electron muon tau it doesn't matter but they're very electron like and several kinds of neutrinos the electrons have the electric charge the neutrinos don't and then there are quarks a variety of different kinds of quarks up corpse down quarks this kind of quark that kind of quark and those quarks several different kinds of quarks you know what the role of them are they make up the proton and that's about it for her forum for fermions for bosons on the other hand is first of all the photon gamma gamma for a gamma ray photon there's an object called the gluon G it's very much like a photon it's very much like a photon but it doesn't have anything to do with atoms it has to do with nuclei and protons and neutrons it plays the same role in holding the nucleus or better yet the proton together as the photon plays in creating electrical fields inside an atom so there's the gluon and then there are two others called W bosons and Z bosons for the most part we won't be interested in any of them except the photon me here and there but mostly we'll be interested in the Z boson that's it that's the standard model that's all there is to it with one exception I've left something out it's the thing you came to find out about tonight okay so we'll come to it if there was no expose on then this would be it now what is special about this set of particles what's special about them is for reasons that I'm going to come to reasons that I will come to all of these particles in the standard model as I've laid it out here with nothing else in it would all have mass equal to zero they would be massless and I'll explain why that is in a little while we often hear that's the role of the Higgs boson to create mass for particles or to give the particles their mass that's the expression that I've heard over and over the Higgs gives particles a lot to put why the particles have to be given mass why can't they have mass of their own why do they have to be given mass well as it turns out for reasons we'll explain this set of particles is exactly the set of particles which would have no mass if this was all there was now in part that explains and part it explains why the particles why these particles are so very light it's because the mass lifts they have no mass well not quite we can't live with that because we know that particles really do have mass next question I'm going to draw some figures over here what do these particles do what kind of processes that they are they involved in at the basic process of the standard model this is an oversimplification but it's qualitatively right is that the fermions there's a Fermi on moving along and I will describe a Fermi on by a solid line solid because it's what makes up stuff solid line that's moving from one point in space-time to another point of space-time what the standard model does is it causes the emission of bosons a electron moving along can emit a photon electron moving along can emit a photon and that's connected with the electric charge any electrically charged particle can emit a photon a photon that's the first thing that the standard model does now this of course is just quantum electrodynamics it does not have to be the electron it could be any electrically charged particle next the quark let's see we have room here yeah we'll just do it the quark quark let's just call it Q the quark can emit a glue on precisely the same pattern the quark emits a glue on now the court can also emit a photon if it happens to be electrically charged and quarks are electrically charged but electrons cannot emit gluons gluons are the things that bind quarks together to hold them together into protons and neutrons and then there's one more important process for me tonight there are two more processes but I'll just write down one here and it is either an electron or incidentally a neutrino cannot emit a photon it has no electric charge it cannot emitted lawn it's not a quark okay both electrons and neutrinos and quarks for that matter can emit the Z boson whereas the Z boson here's the Z boson right here and when they do so the Z boson being electrically neutral the electric charge of whatever's here doesn't change so this is another process that the standard model describes now first of all why are the bosons massless well the photon is massless we know that it travels with the speed of light now could we make a theory in which the a photon had some mass yes we could but the more important thing is that we can make a theory in which the photon doesn't have a mass why because the photon doesn't have a mass using the same kind of Theory the Z boson would not have a mass and the gluon would not have a mass everything would be massless these will be the processes that could happen these would be the particles they would all be massless okay now how do fields how do fields give particles mass or better yet more simply I'm a simple example I'm going to show you a simple example now the simple example is how a field can affect the mass of a particle we'll come back in a moment to how it can give something which didn't have mass mass but let's take a more modest question how many fields affect the mass or better yet how might they make different masses for different particles so I'm going to show you an example this example is a little bit contrived but it's a real example a water molecule water molecules have the basic property that they're little dumbbells they have a plus end and a minus end electrically-charged plus and a minus end they're actually not the more likewise you know why were three ends but we can think of them as having a plus end dumbbells and a minus end now the mass of a water molecule water molecules have mass the mass of a more water molecule doesn't depend on its orientation if we turned it over and made a water molecule with its minus end here and the plus end here it would have exactly the same mass why it's the symmetry of space space is the same in every direction and so by symmetry we would say that the water molecule standing up straight has exactly the same mass as the water molecule standing on its head let's not worry for tonight about whether it's lying on the side quantum mechanics tells us we don't have to worry about anything but standing up straight and lying on its head all right so that's that's true about water molecules their mass is the same as their standing up straight and think of water molecules now as particles think of them just as particles we don't know what they are they're just little elementary particles we can't see them and so we have two kinds of particles the upstanding and the standing on his head particle with exactly the same mass now I thought I had a purple no purple I told them to put purple we'll have to use orange we're way over here underneath the board mean under here oh good alright I have my color coding in my notes here and if I blow it out terrible okay so it looks Brown to me it is brown okay let's create a region in which there's an electric field we're going to make a field it could be between two capacitor plates the capacitor plates could be far apart it doesn't matter but let's put them there capacitor plates here and here and inside that region let's create an electric field the electric field in this case pointing up that means it pushes plus charges up and minus charges down if I have my signs right and let's take one of these water molecules and insert it in here once I insert the water molecule in here the energy of the up standing water molecule and the upside down water molecule are different which one has less energy the one with the plus up has less energy and the one turned over has larger energy the water molecule itself is electrically neutral it has no electric charge but it's a little dipole it has a pair of charges and which one has more energy depends on the sign of the electric field okay so there we are we have two water molecules two types of water molecules two different particles we give them different names we can call it water and scotch and what a molecule has one one energy the scotch molecule has another energy and there they are well by e equals mc-squared this also tells us that the two molecules have different mass non practices would be a tiny different mass between them but they would have different mass so the same effect of this field which exerts itself on charged particles does something to neutral water molecules incidentally notice that it doesn't exert any net force on the water molecule the water molecule moves smoothly through it with no force no net force acting on it but there is a difference in the up in the two configurations of the water molecule and so it's as if we had particles of two different mass so this is just an example of how a field creates mass in this case it increases one mass and decreases the other mass incidentally if you read some of the literature and they'll tell you about how the Higgs field gives a mass I've read any number of places that it's something like space being filled with molasses it is not like space being filled with molasses the vacuum is not sticky and one of the things that molasses would do well the idea is that massive particles move slower than massless particles so the idea is that molasses slows them down but fields don't slow particles down if you give the particle a push in this direction it will just continue to move because there's no net force on it it'll just slide right through this thing frictionlessly no no impedance no no friction no molasses there's not the other the other analogy I once heard is that it was like trying to push a snow plow through a heavy snow in the Arctic it's nothing to do with it whatever that's a that's a that's a lazy way to explain it it's a wrong way to explain it okay so there we are but now let's think of this in a slightly different way the electric field in here can also be pictured in terms of photons a field is another way of talking about a collection a condensate of photons an electric field we can replace the electric field by a condensate the same kind of condensate the same kind of condensate of photons let's draw photons by just their little squiggly lines fill this up with photons how does it know which way the electric field is pointing well photons have a polarization they could be up or they could be down so just imagine this thing being filled with photons but not filled in the usual way but filled in a condensate what does the condensate mean a condensate means that if I pull one out it doesn't make any difference if I put an extra one in it doesn't make any difference that's the meaning of a condensate so it's an indefinite number of photons that's what a field is indefinite and if you pull one out nothing happens and now let's reintroduce the water molecule let's just draw the bottom molecule moving through here now I'm going to make the water molecule that I've already blown my my color coding here's a water molecule moving through here and what is it going to do it has charged particles inside it the charged particles can emit and absorb photons they emit and absorb photons we've made the photons green now so it emits photons but when in the emits a photon putting an extra photon in doesn't matter and so we usually draw that by just putting a cross at the end a cross simply means that throwing an extra photon in doesn't affect anything photon is emitted and just is absorbed or is just disappears into the condensate as this object the dumbbell moves through the electric field it's constantly emitting and absorbing these photons which get lost in condensate that is another way of talking about how the field effects the particle and depending on whether the photons are polarized up or down this effect of constantly being absorbing and emitting photons will have the effect of shifting the energy of the two configurations of the of the dumbbell that's simply an example of how a field can affect the mass of a particle and how it can be thought of in terms of particles and condensates that's what I want you to keep in mind that picture okay now let's come to elementary particles not dumbbells not molecules first question is there any reason why a particle or an object just can't have a mass does it need an excuse to have a mass does it need anything called the Higgs phenomenon to have a mass well there are lots of things in nature that have mass and have nothing whatever to do with the Higgs phenomena we give you an example imagine you had a box and let's make that box out of extremely light stuff the lightest stuff you can think of but it's a box with good reflecting walls and fill it with lots of high-energy radiation bouncing off the walls but never getting out it's made out of massless stuff the photons are massless they have no mass the box who are imagining is made out of stuff which is exceedingly light doesn't have much mass but there's plenty of energy in there lots and lots of energy well e equals MC squared and so this box will behave exactly as if it had a mass we didn't need anything to give mass just energy that's all it took are there any particles which are like this which get mass having nothing to do with the Higgs or anything else yes the proton the proton is a particle which is made out of quarks quarks three quarks and a bunch of gluons jeez a bunch of gluons a large number of gluons quarks and gluons in the standard model are massless does that mean that the proton would be massless if the quarks and gluons are massless not at all if the quarks and gluons are massless the effect on the proton would be about a 1% or even less change in its mass not much at all where does its mass come from it comes from a kinetic energy of these massless particles rattling around in a box the Box being created by the proton so mass doesn't have to come from black holes or another example black holes have mass it doesn't come from the Higgs phenomenon doesn't have anything to do with Higgs so what is it about the models of the state of the particles of the standard model which require us to introduce a new ingredient so I'm going to concentrate on the electron let's concentrate on the electron we don't need all of this or I need to tell you about is that their act theory of electrons really we don't have to know very much about the Dirac theory all we have to know is that electrons have spin and furthermore if an electron was moving very fast down the axis here let's say with close to the speed of light we really accelerate that electron then there's two possibilities the spin of the electron can be right handed like that think of my thumb as a direction of motion of the electron it can be going that way like my right hand or could be going that way like my left hand oh I didn't realize it could do that now two kinds of electrons right-handed and left-handed now the right hand that electrons always stay right-handed can they flip and become left-handed in the right hand they'd become a left handed and left handed become a right handed yeah that's exactly what the Dirac theory says but if it was moving with the speed of light it couldn't why not because if a thing is moving with a speed of light time is infinitely slowed down and nothing can happen to the object it just moves along but nothing can happen internally to the object so if it's mass with zero it couldn't flip but in the Dirac theory this flipping back and forth between I tend to do it this way but that's not right this way this way this way this way that is intimately associated with the mass of a particle and in fact the mass of a Dirac particle is simply proportional to the rate at which it flips from left to right that's the Dirac theory in a nutshell mass is the rate for the electron to flip back and forth from left to right okay of course the faster it's going the slower it will flip but that's all right you take that into account so mass is left to right to left to right and we could draw the motion of an electron in the following way here's the electron moving down the axis at first it's right-handed so it's going this way and then it's left-handed it's going this way and then it's writing can you tell the difference maybe not but that's okay and in between it jumps from one to the other the probability or the rate at which it jumps is a measure of the mass of the electron so it jumps back and forth and back and forth now I'm going to ask you to believe something really crazy they remember the Z boson whereas the Z boson the Z boson was associated was emitted it could be emitted from electrons it could be emitted from neutrinos but let's concentrate on electrons it is not the same as the photon and the thing which emits it is not the same as the electric charge it is another kind of charge a completely separate kind of charge it's like charge but it emits Z bosons we need a name for it we don't have a name for it well we do have a name for it's a very awkward name it's called the weak hypercharge I don't like that because it's the thing which emits Z bosons I call it zilch zilch zilch is like electric charge but it's not electric charge when a particle which has zilch accelerates it emits a Z boson it may also emit a photon if it also happens to have electric charge now electrons both right-handed and left-handed have the same electric charge okay but left-handed and right-handed electrons do not have the same zilch in the standard model this is part of the mathematics of the standard model the left-handed and the right-handed electrons have different zilch the left-handed electron has zilch o plus 1 and the right-handed electron has zero zilch I didn't make this up in fact my friend Steve Weinberg didn't make it up if anybody made it up he's up there or down there I don't know where but and it is just the way it is it is the way the mathematics of the standard model works that the left-handed and the right-handed particles have different zilch and now we have a puzzle when the electron moves along and it flips from left to right that means the zilch goes from plus 1 to 0 but zilch is like electric charge it's conserved how can the zilch go from 0 to 1 it can't it can't and that's the reason that the electron in the standard model doesn't have a mass because the left-handed in the right-handed have different value of a conserved quantity so left can't go to right period no mass how do we get around this we get around this by introducing a new ingredient and the new ingredient is called the zigs boson it's not the Higgs boson not yet we haven't gotten to the Higgs boson yet we've gotten to the zigs boson the zigs boson is one new ingredient it is closely connected with this Mexican half the type configuration here it's a kind of particle but it forms a condensate you can't tell how many of there you can put one in you can take one out and so forth without changing the vacuum so we have one more ingredient it's a condensate that space is filled with and the nature of the condensate is it doesn't have electric charge it has zilch and it's a condensate meaning that if you put a zilch in nothing happens if you take one out nothing happens and let's ask now what that means the left-handed electron coming in has a zilch of one let's call it a Z of one the right-handed has Z equals zero back to the left-handed Z equals one is that possible only if you emit something at this point which carries off that Z equals one Ziggs zigs boson gets emitted it carries Z equals one but what happens to it where does it go it goes into the condensate it gets lost in the condensate you put as you put one in and it just gets absorbed into the condensate and so the electron goes on its merry way the condensate absorb the zilch and it goes one to zero but then it can borrow a particle back from the condensate borrow one back it doesn't even have to borrow it if you pull one out nothing changes again and so it goes on its merry way from left-handed to right-hander from left-handed or right-handed everytime it switches it emits a particle carrying this zilch quantum number which then just gets absorbed into the condensate that's the mechanism by which a field and in this case it's a field which forms a condensate by itself it doesn't require capacitor plates it just requires the energy to be such that the field naturally gets shifted and that's the mechanism by which electrons quarks and the various partners of those particles the MU particle the Tau lepton all those ordinary ordinary and extraordinary particles the fermions get there masked by this phenomenon here phenomenon doesn't really have a name it's called the spontaneous breaking of chiral symmetry but it does have a name but this is what it is okay what about the Z boson I told you before the Z boson is like a photon photons are massless how does the Z boson get a mass so I'll just show you something very similar happens to the Z boson let's remind ourselves what a Z boson can do it can take any particle which has a zilch and in particular this green zigs particle it can take the zigs particle and the zigs particle can emit a Z boson it has charge not real charge but zilch and zilch emits Z bosons all right so now let's ask what that means that means that a Z boson moving along can do something a little bit similar to this it can absorb some zilch out of the condensate condensate but now it has zilch originally it was just a Z boson Z bosons don't have zilch it absorbs some zilch and it becomes a zigs Z boson becomes a zigs but then it can emit zigs which gets lost in the condensate again and the Z boson just moves on its merry way constantly going back and forth from being a Z boson to being one of these imaginary not imaginary Ziggs particles that's the nature of the way that particles get mass from fields this phenomenon of the Z boson getting a mass is called the Brout on glair Higgs phenomenon this is the one that's called the Higgs phenomenon the Z boson getting a mass now this could have happened to the photon had there been a condensate of ordinary charged particles the photon would have become massive we would all be dead if that were the case massive photons would not be healthy for us and so we are very lucky that the that this phenomenon here did not apply to ordinary electric charge will we ever discover the zigs particle sure we discovered it long ago it's just part of the Z boson Z boson was discovered means it was postulated 1967 but or even before that by of many people but it was discovered I don't even remember when 1980 I forgot when the when the experiment but slack first discovered the existence experimentally but when it was discovered that there was a Z boson that had had a mass and that when its properties was studied the properties were not only consistent but required that it was a thing which went back and forth and back and forth and back and forth between pure Z boson and the zigs particle so they've existed we're not in doubt about them and we never were at least not for many years so far I have not mentioned the Higgs boson so what is the Higgs boson well the Higgs boson has to do with this condensate it has to do with this condensate but it's a different kind of excitation than sliding around the the edge of the sombrero here does not have to move it's not something which has to do with sliding around here it has to do of putting in two different ways to think about it you have a condensate and you can imagine the condensate has a density a density of these fictitious particles in the condensate imagine something which changes the density of them kind of like a sound wave a compression wave of some kind which squeezes them closer and further and closer apart makes more and more less dense that kind of vibration is what a Higgs volar Higgs boson is another way to think about it is that it doesn't have to do with sliding around the periphery of the sombrero it's you go to a place in space and start the field oscillating this way in and out this way the further away a tie is the stronger the condensate the closer to the center the weaker the condensate so it sloshes back and forth it's kind of a compressional wave in the condensate that mode that phenomena that oscillation is what is called a Higgs boson the Higgs boson is like the sound wave propagating through the through the condensate the reason it has been so important is because it was the one element that had not yet been discovered as I said the zigs was discovered long ago the Z and the W the electrons and all the others were discovered long ago and so the next question which I'll try to answer in a couple five minutes is why it was so hard to discover the Higgs what we discovered about it and very very quickly what the future might or might not bring try to do this in a couple of minutes okay so what kind of thing does the Higgs boson itself through now we're talking about the Higgs boson not the zigs boson not the Z boson the Higgs itself the one them the one that's been so elusive all these years it's called H and what it can do with some probability is for example create we read this from left to right the Higgs boson moving along in time time is now to the left can create an electron and a positron it can create a pair of quarks it can also create other things a new particle or a top quark or a bottom quark all of the different quarks electrons also neutrinos all of Area's fermions can be created in pairs when a Higgs boson decays you say yes if it's like a sound wave why does it decay well believe me sound waves decay if they didn't decay you continue to hear my voice ring forever and ever wouldn't do so sound waves do decay and it is possible to think of sound waves as decaying by creating particles so the Higgs boson decays it the case quickly if it exists if it really exists at the case quickly either into an electron positron or a pair of quarks or maybe some other of the fermions that exist in nature you can read this diagram in two different ways Oh incidentally the probability that the Higgs decays like this is proportional to the mass of the particle that it decays into the heavier the mass the more strongly that particle is coupled to the Higgs boson so heavy particles are favored and life particles are not favored now you can read this diagram in either direction you can say the Higgs boson decays but you can also say an electron and a positron confuse together to make a Higgs boson well if we want to make Higgs bosons and see them in the laboratory we want to read the diagram from right to left and we want to say this is a process whereby a pair of electrons can come together and make a Higgs boson we've been colliding electrons and positrons for a long long time almost as long as I've been a physicist not quite we've been colliding electrons and positrons together and nobody was ever able to discover the Higgs now one reason in the early days is it turns out that the Higgs is a fairly heavy particle I will tell you what its mass is but it's a fairly heavy particle and unless you have enough energy you don't have enough energy to make the Higgs boson but there's a more important reason in fact slack in the later days of slacks or life had plenty of energy to make the Higgs the problem was the weakness of the coupling the smallness of the mass of the electron translated into a very weak improbable cross-section too small in effect too unlikely to make the Higgs and so when you collide electrons together at high energy electrons are just not favourable they're too light and because they're light they tend to not make Higgs with any appreciable probability well how about quirks we can collide quarks together the usual quarks that make up the proton and neutron are also very light and because they are light also unlikely to ever make a Higgs boson well you know I'm sure they were made in slack but never in appreciable numbers that there was possible to to detect them so that was the main difficulty the lightness of these particles was a thing that essentially prohibited us from making Higgs is in abundance at SLAC or in other laboratories where collisions took place what is the most favorable particle most likely particle for the Higgs to decay in the heaviest the heaviest of the fermions and the heaviest of the fermions is called the top quark the top quark is hundreds and hundreds thousands of times heavier than the electron many thousand many many times over many thousands of times heavier than the electron and the Higgs preferentially will decay into top quarks so we'll just call those the quarks they are quarks but they're very heavy 170 times the mass of a proton basically which is heavy top and anti top top quarks and antiquarks so you say well look now it's easy to make the Higgs boson you just oh it actually in fact not possible for the Higgs to the cater to top quarks because the two top quarks are too heavy but if you read it the other way and you take a pair of top quarks and collide them together you can make a Higgs so it's easy we just go in the laboratory take a pair of top quarks collide them together and make a Higgs well the problem is that it's not so easy to find top quarks in nature why not they decay very rapidly to the other quarks they're not sitting or you can't put them into the accelerator and accelerate them they disappear in a tiny fraction of a second there are no top quarks sitting around not even buried inside protons and so forth not even buried inside other kinds of particles there are no top quarks around so we have to make the top quark somehow in the collision how do you make a top quark card so here's a way to make a top quark gluon can come along this is a gluon now and remember what gluons do they coupled two quarks one possibility is that the gluon can make a top quark and an anti top quark well as plenty of gluons around as we'll see in a moment so why don't we just take a gluon and make a top quark and the anti top quark out of it the reason is because gluons are very light they're almost there almost massless they don't weigh very much top quarks are very heavy there's simply not enough energy in the gluon to create a pair of top quarks so what we have to do is we have to take a pair of gluons now here's a process that you can imagine take a pair of gluons with a lot of energy moving toward each other with a huge speed plenty of energy let one of them make a pair of top quarks for a short period of time and then let the other one come and be absorbed by one of the top quarks there we have it a pair of top quarks created by a pair of gluons a pair of high-energy gluons smashed together and make a pair of top quarks once we've created those pair of top quarks the top quarks can come together and make our Higgs boson this is the way we usually draw this is to just draw glue on glue on and then a triangle Higgs is a top quarks going around the loop here that's the most efficient process for making for making Higgs bosons but where do you get gluons from gluons are in floating around well yes they are the proton is filled with gluons the proton mass of the protons maybe 50% energy from gluons or something like that it's filled with gluons and quarks you take two protons and you collide them together and the gluons inside the protons can collide during the collision and do this that was what was detected at LHC LHC is a proton proton Collider it collides protons together and when protons a very indirect way two protons collide together or gluon from each one of them scatter collide create a pair of top quarks and then the top quarks then have plenty of come together and create the Higgs boson that's the process that was discovered at the LHC and it took a long time to get there it was a hard thing to do it was a very very hard thing to do but now it's done we know the mass of the Higgs boson it's 125 GeV about 127 times the mass of the proton and that's I think a finished fact before I quit let's talk about the near future what have we learned we've learned that the standard model is essentially correct we've learned the standard model is essentially correct everything seems to fit together the Higgs boson fitting together nothing it's not the Higgs boson really that gives the particles their mass it's the zigs boson but the Higgs boson is just what's left over when you think of these density oscillations it was the last remaining piece it is now in place it's finished but is everything fitting together exactly right quantitatively right well that we don't know we don't know there's one hint one hint of a discrepancy and I'll tell you what the hint of that discrepancy is let's uh here's I drew this picture let me draw it again over here it's the process of creating a Higgs by two gluons coming together glue on glue on top quark going around the loop and Higgs now this same process once the Higgs is created also allows the Higgs to decay but it's not so easy to see gluons in the laboratory they're difficult to work with that's not the best process for looking for the Higgs boson after you've created it the best process is to replace the gluons by photons I don't have to even change the picture photons it's exactly the same process except with photons out here once the Higgs is created by whatever it can create it it can decay into two photons it's an intricate process it involves a lot of theory and a lot of calculation a fine ling diagram not easy to calculate but you can calculate it and it depends on the properties of the top quark going around here at the moment at the moment and I'm not an expert at this I can only quote what I'm told at a month the moment the Higgs boson that was produced in the laboratory appears to decay into two photons a little too quickly about one and a half times too quickly now everybody agrees that that is not a statistically really significant fact yet but what will it mean if it persists it doesn't seem like a big deal one of the half times too fast but the point is the theorists have the ability to calculate that rate very accurately a one and a half times too big a rate is serious it means something is going on the most likely thing that would be going on is that there's another kind of particle in addition to the top quark that has not discovered yet that can also participate in the same kind of it's called a triangle diagram some other kind of particle that of course would be big news if there's something there that is not described by the standard model that would be big nose it could be a supersymmetric particle it could be anything all kinds of things if this this is something to watch for now the buzz words are the decay of the Higgs into a pair of photons and a excess of about one and a half I think it's a two Sigma effect whatever that means means something the statisticians it means that it's not so robust but it could be right it turns out to be right it means that we've discovered something unexpected well it might be Haven something that's expected but something new beyond the standard model remember the standard model is over 50 years old well over 50 years old and so 1967 am i right season 77 97 97 2007 no I'm not getting on 50 years old so discovering the Higgs boson wasn't really discovering anything was confirming something if this should be off by a factor of one and a half one will have discovered something absolutely new so if you want to watch if you know want to be a spectator in the sport and you want to watch what happens this is the thing to watch for next whether the Higgs decays are consistent with the standard model okay that's a really finish thank you very much and I hope you all one or two questions what would cause do different fermions have different rates of viral oscillation good the answer is going to be an unsatisfying one the answer is that the fermions have dipped what would cause different Fermi onstaff different masses different masses essentially different different also as different rates of oscillation are the same as different masses the coupling strength the coupling constant that couples the relevant particle to the to the Higgs field what happens the particle moves chicken yes each one has a separate coupling constant emits this what did I call it the zigs midst the cigs which gets lost there's a coefficient here which is basically a probability each one and we don't know why they are what they are we know how to parameterize it but we don't know how to explain it for each kind of particle let's say the electron or the mew particle or whatever it happens to be there's a different constant there and that constant is the constant which determines the rate than the mass it's the same constant which comes into telling you how rapidly the Higgs decays into these particles and therefore the heavier the particle the stronger the decay that's a good question I'd like to say something about it I forgot we know the value really that the message yeah we've known that for a long time 240 GeV the value of the expectation value is from one language it's simply the displacement of the field and another language it's the density of the condensate can think of it either way the density of the condensate or as the value of the displacement of the field and that's why oscillation in the magnitude of the field is the same as a density fluctuation what's that apparently not apparently not well yes I think it does but there are many many other things that give it an energy density and for whatever reason they almost all cancel out this is one of the great mysteries of yeah yeah right so that's that's a very good question too which we don't have an answer at the moment right okay I hope you got something out of that I had fun preparing it and figuring out how to try to explain it some of you you probably got something others are just mystified and sitting why are they talking about but that's it for more please visit us at stanford.edu

The True Science of Parallel Universes

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الكل يحب فكرة الاكوان المتوازية , ربما هو إلتماس لعالم مثالي حيث لديك فرص ثانية و الأمور جرت بطريقة مختلفة. واقع بديل دخلت فيه إلى هاغواترس و أفلام حروب النجوم السابقة لم تصنع و توصل فيه سلك اليو اس بي بطريقة صحيحة من أول محاولة لكن هل هناك مكان في العلم للتخمين الحالم هذا ؟ أعني إذا ''الكون'' هو كل شيء موجود , لن تستطيع إمتلاك إصدارين منه, أليس كذلك؟ وإلا سيكون الكونان كل شيء و ما أسميناه الكون لم يكن كل شيء. وإلا سيكون الكونان كل شيء و ما أسميناه الكون لم يكن كل شيء. المشكلة هنا هي المصطلحات,
الفيزيائيون يتكلمون بشكل غير رسمي أحيانا بقول '' الكون '' عندما يقصدون حقاً " الكون المرئي" وهو جزء من الكون الكلي الذي بإستطاعتنا رؤيته. بأمكاننا ايضاً ان نتحدث عن اكثر من كون مرئي مثلا, فضائي بالقرب من حافة كوننا المرئي سيرى اجزاء من الكون الكلي التي لا نستطيع رؤيتها, لأن الضوء لايملك الوقت الكافي للوصول الينا. لكن هذا السؤال مفهوم جيداً وليس هو ما يتحدث عنه الفيزيائيون عندما يتحدثون عن الأكوان المرئية المتعددة او "الأكوان-المتعددة" بإختصار: في الفيزياء , كلمة "الأكوان-المتعددة" عادة تشير الى 3 نماذج فيزيائية للكون لا علاقة لها ببعض لم يتم تأكيد او تجريب اي من هذه النماذج اختبارياً ايضاً. نماذج "الأكوان-المتعددة" الثلاث هي: النوع #1
الكون الفُقاعي او كون الثقوب السوداء الصغيرة هو ابسط نوع من "الأكوان-المتعددة": الفكرة الأساسية هي انه ربما هنالك أجزاء أُخرى من الكون بعيدة لدرجة اننا لن نتمكن من رؤيتها او بداخل ثقوب سوداء ولن نتمكن ايضا من رؤيتها ايضاً. هذا النموذج صُنع لتفسير لماذا كوننا متميز في توليد النجوم و المجرات
و الثقوب السوداء و الحياة. كما تقول الحُجة, اذا كان كل من هذه "الفقاعات" الغير متصلة تماماً في الكون لها قوانين فيزياء مختلفة قليلاً, اذاً حسب التعريف لن يمككنا الوجود إلا في واحدة لها قوانين طبيعة تسمح لنا بالوجود. مثلا يجب علينا ان نكون في كون الارض فيه تشكلت, اذا لم توجد الارض لن نكون هنا. اذا لم تقتنع بهذا المنطق, لاتقلق كثيراً, لايوجد الكثير من الأدلة التجريبية لهذا النموذج من ""الأكوان-المتعددة" حتى الان. نوع #2
الأغشية والأبعاد الإضافية. مستوحاة جزئيا من عجز رياضيات
نظرية الأوتار على التنبؤ العدد الصحيح من أبعاد الكون المرئي لنا ,اقترحت نظرية الأوتار فكرة أن ربما ما نفكر فيه ككوننا
في الواقع مجرد سطح ثلاثي الأبعاد مطمور في كون فائق مكون من 9 ابعاد مكانيه. كمثل أن اوراق الصحيفة هي سطح ثنائي الابعاد مطمورة في عالمنا الثلاثي الأبعاد وبطبيعة الحال، إذا كان للفضاء 9 أبعاد بدلا من الثلاثة، سيكون هناك الكثير من المكان لسطوح ثلاثية الأبعاد الأخرى التي ظهرت، مثل سطحنا ، لتكون أكوان في حد ذاتها،
ولكن، مثل صفحات الصحف، كانت في الواقع جزء من كل أكبر. هذه الأنواع من السطوح
تسمى "أغشية" أو "الأغشية" للاختصار. و للتذكير، وليس هناك حتى الآن أي دليل تجريبي على هذا النوع من الكون المتعدد. نوع #3
صورة العوالم المتعددة لميكانيكا الكم. من المثير للدهشة الفزيائيون مازالوا لا يستوعبون بالكامل كيف تتحطم الدالة الموجية في الفيزياء الكمية و فرضية العوالم المتعددة تفسر ذلك بإفتراض أن كل إطار زمني للكون حقيقي و أن كلهم يحدثون بطريقة متشعبة الى الأبد. كروايات كونية تحدث فيها كل القصص المحتمله. اذا كانت الوضع كذلك, فلن يمكننا ادراك ذلك لأننا سنكون عالقين بالعيش في واحد من العديد من الحيوات الانهائية المتاحة لنا. في بعض الطرق, نظرية العوالم المتعددة تشبه نظرية الاكوان المتعددة الفقاعية بإفتراض ان "ربما كل شيء ممكن أن يحدث, سوف يحدث" وأننا فقط موجودين في سلسلة الاحداث التي نحتاجها للوجود. إذا لم تقتنع بهذا المنطق, لاتقلق, لايوجد اي دليل تجريبي لهذا النموذج من الكون المتعدد. طبعاً, اذا اردت أن تكون تَخيُلياً يمكنك دمج العديد من هذه النماذج معاً لصنع أكوان تعددية متعددة نموذج جديد عالي التوقع مبني على نماذج توقعية و غير مؤكدة تجريبياً. لكن هذا لايعني أنه لا يمكن إجراء التجارب على نظريات الأكوان المتعددة. مثلا, لو أن كوننا المرئي هو بالفعل مجرد واحد من العديد من الفقاعات المتفرقة أو الأغشية و صدف أن إصطدم بفقاعة او غشاء اخر بالماضي فذللك الإصطدام بالتأكيد سيكون له أثر ما على ما نراه في السماء ليلاً. من ناحية اخرى, نظرية العوالم المتعددة قد يتم تجريبها قريباً إذ أن الفيزيائيون التجريبيين تزيد مقدرتهم بالتلاعب والتحكم بأنظمة كمية ضخمة في مختبراتهم أنظمة تقارب الحد بين العالم الكمي و واقعنا اليومي. لذلك كما هو الحال دائما، علينا أن نتذكر أن الفيزياء
علم، وليس فلسفة وفي محاولاتنا لتفسير الكون الذي نراه, يجب علينا ان نجد فرضيات قابلة للتجريب في المبدأ, ثم القيام بتجريبها!

Philosophy: 60-Second Adventures in Thought (combined)

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A look at 6 different ‘Adventures in Thought’ (this is a combination of all 6 parts of the series into one video)

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60-second adventures in thought number one Achilles and the tortoise how could a humble tortoise beat the legendary Greek hero Achilles in a race the Greek philosopher Zeno liked the challenge and came up with this paradox first the tortoise is given a slight headstart anyone fancying a flutter would still rush to put their money on Achilles but Zeno pointed out that to overtake him Achilles would first have to cover the distance to the point where the tortoise began in that time the tortoise would have moved so Achilles would have to cover that distance giving the tortoise time to amble forwards a bit more logically this would carry on forever however smaller gap between them the tortoise would still be able to move forward while Achilles was catching up meaning that Achilles could never overtake taken to an extreme this bizarre paradox suggests that all movement is impossible but it did lead to the realization that something finite can be divided an infinite number of times this concept of an infinite series is used in finance to work out mortgage payments which is why they take an infinite amount of time to pay off number two the grandfather paradox will time-travel ever be possible Renae Bahji val was a french journalist and science fiction writer who spent a lot of his time thinking about time travel in 1943 bar JaVale asked what would happen if a man went back in time to a date before his parents were born and killed his own grandfather with no grandfather one of the man's parent would never have been born and therefore the man himself would never have existed so there would be nobody to go back in time and kill the grandfather in the first place or the last place depending on how you look at it the grandfather paradox has been a mainstay of philosophy physics and the entire back of the Future trilogy some people have tried to defend time-travel with arguments like the parallel universe resolution in which the change is made by the time traveller create a new separate history branching off from the existing one but the grandfather paradox prevails although the paradox only suggests that traveling backwards in time is impossible it doesn't say anything about going the other way number three the Chinese room can a machine ever be truly called intelligent American philosopher and Rhodes scholar John Searle certainly can in 1980 he proposed the Chinese room thought experiment in order to challenge the concept of strong artificial intelligence and not because of some eighties design fad he imagined himself in a room with boxes of Chinese characters he can't understand and a book of instructions which he can if a Chinese speaker outside the room passes him messages under the door Searle can follow instructions from the book to select an appropriate response the person on the other side would think they're chatting with a Chinese speaker just one who doesn't get out much but really it's a confused philosopher now according to Alan Turing the father of computer science if a computer program can convince a human they're communicating with another human then it could be said to think the Chinese room suggests that however well you program a computer it doesn't understand Chinese it only simulates that knowledge which isn't really intelligence but then sometimes humans aren't that intelligent either number four Hilbert's infinite hotel a grand hotel with an infinite number of rooms and an infinite number of guests in those rooms that was the idea of German mathematician David Hilbert friend of Albert Einstein an enemy of chambermaids the world over to challenge our ideas about infinity he asked what happens if someone new comes along looking for a place to stay Hilbert answer is to make each guest shift along one room the guest in room one moves to room two and so on so the new guest would have a space in room one and the guest clique would have an infinite number of complaints but what about when a coach containing an infinite number of new guests pulls up surely he can't accommodate all of them he'll but frees up an infinite number of rooms by asking the guests to move to the room number which has double their current one leaving the infinitely many odd numbers free easy for the guests in room one not so easy for the man in room million six hundred thousand five hundred ninety-seven Hilbert's paradox has fascinated mathematicians physicists and philosophers even theologians and they all agree you should get down early for breakfast number five the twin paradox Albert Einstein didn't have a twin brother but he had some funny ideas of what you could do with one he imagined two identical twins let's call them owl and Bert now al is a couch potato but Bert likes to travel so he hopped into a spaceship and zooms off at close to the speed of light that's when Einstein's special theory of relativity kicks in it says that the faster you travel through space the slower you move through time so from our point of view Bert's time would be moving slower than his own to put it another way time might fly when you're having fun but when clocks fly they run more slowly in relativity after a wild bird decides to head back still at close to the speed of light and return to his brother with his holiday snaps but when Bert arrives home Al will now be older than his twin which makes their double dates a lot more awkward although it seems implausible Einstein just followed his theory to its logical conclusion and it turns out he was right this concept of time dilation provides the basis for our global positioning system which is how your sat-nav knows you need to turn left in 200 yards number six Schrodinger's cat Erwin Schrodinger was a physicist a theoretical biologist and probably more of a dog person in the 1920s scientists discovered quantum mechanics which said that some particles are so tiny you can't even measure them without changing them but the theory only worked if before you measure them the particle is in a superposition of every possible state all at the same time to tackle that Schrodinger imagine the cat in a box with a radioactive particle and a Geiger counter attached to a vial of poison if the particle decays it triggers the Geiger counter releases the poison and bye-bye Tiddles but if the particle is in two states both decayed and not decayed the cat is also in two states both dead and not dead until someone looks in the box in practice it's impossible to put a cat into a superposition you have the animal rights Lobby up in arms but you can isolate atoms and they do seem to be in two states at once quantum mechanics challenges our whole perception of reality so maybe it's understandable that Schrodinger himself decided he didn't like it and was sorry he ever started on about cats

Wormholes Explained – Breaking Spacetime

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Wormholes Explained – Breaking Spacetime

اذا رأيت ثقب دودي في الواقع سوف يبدو دائري ,كروي, مشابه شيئ ما للثقب الاسود الضوء القادم من الجهة الاخرى يعبر ويعطيك نافذة الى مكان بعيد جداً حالما تعبره, الجهة الأخرى (جهة الدخول) ستظهر في مجال النظر مع منزلك القديم ينحسر نحو تلك النافذة الكروية المتلألئة ولكن هل الثقب الدودية حقيقية؟ أم هي عبارة عن سحر (خيال) متنكرة بمعادلات الفيزياء والرياضيات؟ إذا كانت حقيقية, فما هي طريقة عملها؟ وأين نستطيع إيجادها؟ خلال معظم أوقات التاريخ البشري ظننّا أن الفضاء بسيط نوعا ما, عبارة عن خشبة مسرح مُسطّحة كبيرة حيث في تتم فيها أحداث الكون, حتّى ولو قمنا بإنزال جميع الكواكب والنجوم, يبقى شيء ما موجود تلك خشبة المسرح الفارغة هي الفضاء وهي موجودة لا تتغير و خالدة نظرية "آينشتاين" النسبيّة غيّرت ذلك. تقول النظريّة أن الفضاء والزّمن هم الذين يشكّلون ذاك المسرح معاّ, وهما ليسا متطابقان بجميع الأماكن فالأشياء التي على المسرح بمقدورها التأثير على المسرح نفسه, تجعله يتمدّد و يتشوّه. إذا كانت خشبة المسرح القديم غير قابلة للحركة وصلبة, مسرح "آينشتاين" أقرب إلى فراشٍ مائّي هذا النوع من الفضاء المرن قابل للطي, يمكنك حتّى تمزيقه و إعادة لصقه ببعضه مجدّداً, مما يجعل الثقوب الدوديّة معقولة دعونا نشاهد كيف يمكن أن تبدو بمنظور ثنائي الأبعاد.
كوكبنا عبارة عن لوح كبير ومسطّح, منحني بطريقة دقيقة يمكن للثقوب الدودية أن تصل بين نقطتين فضائيتين بعيدتين جداً جداً عن بعضهما بجسر قصير يمكنك عبوره بلحظة مما يمكنك من السفر عبر الكون, بسرعة حتّى أكبر من سرعة الضوء إذن أين نستطيع العثور على ثقب دودي؟ حالياً على الورق فقط. النسبيّة العامّة تقول : من الممكن وجودهم, لكن هذا لايعني أنّه من المؤكّد وجودهم النسبية العامّة هي نظرية رياضيّة. هي عبارة عن مجموعة من المعادلات التي من الممكن أن يكون لها حلول كثيرة, ولكن ليس كل معادلة رياضيّة تصف الواقع لكنهم قابلين للوجود من الناحية النظرية, وهنالك عدّة أنواع : "جسور "آينشتاين-روزين"" النوع الأوّل من الثقب الدودية التي تم وصفها نظريّاً هي(جسور "آينشتاين-روزين") وهي تنص على أنّ جميع الثقوب السوداء هي عبارة عن بوّابة إلى كون موازي ليس له نهاية دعونا نشاهد كيف يمكن أن تبدو بمنظور ثنائي الأبعاد مجدّداً الزمكان (الفضاء) مسطّح لكن يتم ثنيه بالأشياء التي فوقه, إذا ما قمنا بضغط ذلك الشيء, الزمكان (الفضاء) ينثني بشكل أكبر حوله.
في النهاية الزمكان (الفضاء) يصبح مشوّه بشكل كبير لدرجة أنه ليس له خيار إلا بالإنهيار ليشكّل ثقباً أسود ليتشكل حاجز يسمح بالمرور فقط بإتجاه واحد يدعى أفق الحدث, الذي يسمح بدخول جميع الأشياء ولكن لا يسمح بخروجها محبوسة للأبد عند ال "singularity" (مصطلح يعبر عن مركز الثقب الأسود) التي هي النواة لكن لربما لا يوجد نواة في هذه الحالة من المحتمل أن يكون الجهة الأخرى من الأفق هي عبارة عن كون مماثل للكون الذي نعيش فيه لكنه معكوس رأساً على عقب, حيث فيه الزمن معكوس أيضاً.
في الكون الذي نعيش فيه, الأشياء تقوم بالإنجذاب نحو الثقب الأسود, أما في الكون الموازي الذي فيه الزمن يرجع للوراء, الثقب الأسود المعكوس يقوم بنفر الأشياء نحو الخارج, بشكل يشبه الانفجار الكبير (نظرية الانفجار الكبير – أصل الكون) هذا يدعى بالثقب الأبيض لسوء الحظ, "جسور اينشتاين -روزن" لا يمكن في الحقيقة عبورها إنها تتطلب مقدار لا نهائي من الوقت للعبور نحو الكون المقابل, وتتجعّد منغلقة في الوسط إذا قمنا بالدخول في ثقب أسود, لن تكون من الأشياء المنبثقة من الثقب الأبيض, بل ستكون في عداد الموتى لذا للقيام بالسفر عبر المجرّة في لمح البصر, يحتاج البشر إلى نوع مختلف من الثقب الدودي يدعى "traversable wormhole" (قابل للعبور) "نظرية الأوتار القديمة جداً للثقب الدودية" اذا كانت نظرية الأوتار أو واحدة من أشكالها هي الوصف الصحيح للكون الذي نعيش فيه فيمكن أن نكون محظوظين وقد يمتلك عالمنا شبكة متشابكة من الثقوب الدودية التي لا تعد ولا تحصى بعد وقت قصير من الانفجار الكبير (بداية الكون) التقلبات الكمومية في الكون عند أصغر المقاييس أصغر بكثير من الذرّة ربما خلقت الكثيرمن الثقوب الدودية القابلة للعبور (النوع الثاني) مترابطة من خلال أوتار تدعى بالأوتار الكونيّة أول مليار من تريليون من الثانية بعد الإنفجار الكبير, تم تفريق نهايات هذه الثقوب الدودية الصغيرة جداً سنين ضوئية عن بعضها معبثرين عبر الكون. إذا تم تشكيل الثقوب الدودية في بداية حياة الكون سواء عبر الأوتار الكونية أو بطريقة أخرى, يمكن أن يكونوا منتشرين حولننا في انتظار أن يتم اكتشافهم قد يكون أحدها أقرب مما ندرك من الخارج الثقوب السوداء والثقوب الدودية يمكن أن تبدو متشابهة جداً مما قاد بعض الفيزيائيين إلى اقتراح الثقوب السوداء الهائلة في مركز المجرات هي في الحقيقة ثقوب دودية. سيكون من الصعب جداً الذهاب إلى مركز درب التبانة لمعرفة ذلك, لكن لابأس قد تكون هناك طريقة صعبة للغاية لنحصل على ثقب دودي. يمكننا محاولة صنع واحد "ثقب دودي من صنع الإنسان" ليكون قابل للعبور وذو فائدة, هنالك عدّة خصائص نريد من الثقب الدودي أن يملكها. أولا من الواضح أنه يجب أن يصل بين الأجزاء المتباعدة من الزمكان (الفضاء) مثل غرفة نومك والحمام أو الأرض والمشتري ثانياً يجب ألّا تحتوي على أي أفق حدث (نقطة اللاعودة) من شأنه أن يمنع السفر في اتجاهين ثالثاً يجب أن يكون حجمها كافٍ بحيث لا تقتل قوى الجاذبية المسافرين البشر المشكلة الأكبر التي علينا حلّها, هي إبقاء ثقوبنا الدودية مفتوحة, بغض النظر عن الطريقة التي صنعنا فيها الثقب الدودي, الجاذبية تحاول إغلاقها تريد الجاذبية إغلاقها بشكل كامل وقطع الجسر مما يترك في النهاية الثوب السوداء فقط سواء كان ثقبا دوديا قابل للعبور مع كلتا نهاياته في كوننا, أو ثقبا دوديا يصلنا ب كون آخر سيحاول الإنغلاق ما لم يكن لدينا شيء يبقيه مفتوحاً في "نظرية الأوتار القديمة جداً للثقب الدودية",
الأوتار الكونية من شأنها إبقائه مفتوحاً في الثقوب الدودية من صنع البشر, نحتاج إلى عامل معين مادّة غريبة (شاذة) هذا ليس شيئًا يمكن نجده على الأرض أو حتى المادة المضادة "antimatter" إنه شيء جديد تماماً ومختلف ومثير, ذو خواص مجنونة لا تشبه أي شيء رأيناه من قبل المادة الغريبة هي الأشياء التي لديها كتلة سلبية الكتلة الإيجابية مثل الناس والكواكب وكل شيء آخر في الكون, جذّابة بفضل الجاذبية لكن الكتلة السلبية ستكون نافرة أي أنها ستدفعك بعيداً مما يشكل نوعا ما من الجاذبية المضادة التي تبقي ثقبنا الدودي مفتوحاً والمادة الغريبة يجب أن تطبّق ضغطاً هائلاً لدفع الزمكان (الفضاء) حتى يبقى مفتوحاّ, أكبر من الضغط في مراكز النجوم النيوترونية حتّى مع المادة الغريبة يمكننا نسج الفضاء كما يحلو لنا حتى أنّه قد يكون لدينا مرشّح لهذه المادة الغريبة, خواء (إنعدام الضغط) الفضاء بذاته التقلبات الكمية في الفضاء الخالي تقوم بخلق أزواج من الجسيمات والجسيمات المضادة باستمرار لكن سرعان ما يتم إبادتهم بعد لحظة خلوة الفضاء تعم بهم ويممكنا بالفعل التلاعب بهم ليعطو تأثير مشابه لتأثير الكتلة السلبية التي نسعى له يمكننا استخدام هذا التأثير للحفاظ على استقرار ثقوبنا الدودية بمجرد أن نفتحها ، ستبدأ النهايات بنفس المكان, لذلك يجب علينا نقلهم إلى أماكن مثيرة للاهتمام يمكن أن نبدأ عن طريق توصيل النظام الشمسي فنترك نهاية واحدة من كل ثقب دودي بمدار حول الأرض يمكننا أن نقوم بدفع الباقي نحو الفضاء العميق يمكن أن تكون الأرض محور ثقب دودي لحضارة إنسانية هائلة عبر النجوم منتشرة على بعد سنين ضوئية لكن على بعد ثقب دودي فقط لكن الثقوب الدودية لها جانب مظلم حتى فتح ثقب دودي واحد, يقوم بكسر الكون نوعا ما بطرق أساسية, مما قد يشكل متناقضات السفر عبر الزمن و انتهاك البنية السببية للكون يعتقد العديد من العلماء أن هذا لا يعني فقط أنه يجب أن يكون من المستحيل صنعه، بل من المستحيل أن يكونوا موجودين على الإطلاق في الوقت الحالي, نحن نعرف فقط أن الثقوب الدودية موجودة, في قلوبنا وعلى الورق في شكل معادلات نحن نعلم أنك تريد معرفة المزيد عن أشياء المتعلقة بالكون. لذلك نحن نحاول شيئا جديداً نحن "Kurzgesagt" و"Brilliant" نتعاون في سلسلة فيديو من ستة أجزاء حول أشياء العلوم والفضاء المفضلة لدينا بفضل مساعدتهم لنا, سيكون هناك المزيد من مقاطع الفيديو على هذه القناة في الأشهر الستة المقبلة لقد قمنا نحن "Kurzgesagt" بالعمل مع "Brilliant" منذ فترة الآن, ونحن نحب ما يقومون به, في المختصر يساعدك "بريليانت" على المضي قدمًا من خلال إتقان مهارات الرياضيات والعلوم من خلال حل المشكلات الصعبة والمشوّقة بنفس الوقت لدعم تعاوننا معهم, قم بزيارة "brilliant.org/nutshell " وقم بالاشتراك مجاناً اليوم أوّل 688 شخص يستخدمون الرابط سيحصلون على حسم سنوي بمقدار 20% على اشتراك العضوية المميّزة

Want to study physics? Read these 10 books

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Books for physics students! Popular science books and textbooks to get you from high school to university. Also easy presents for physics students 😛

Check out my video with Socratica here:

6 Easy Pieces
6 Not So Easy Pieces
Alex’s Adventures in Numberland
Physics of the Impossible
In Search of Schrodinger’s Cat
The Elegant Universe

Mathematical Methods
Fundamentals of Physics
Grad, Div, Curl and All That
Concepts in Thermal Physics

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over the years I built up quite the collection of physics books these are just some of them through my undergraduate and my PhD in physics so in this video I'm going to be recommending 10 books in two categories for those of you who are interested in studying physics or for those of you who are looking for a present for aspiring physicist the two categories are books which introduce concepts in maths and science in a kind of general way popular science books if you like and then also textbooks which you might want to buy to get ahead of the game and read before you go to university and also do exercises in and then also users of reference during the beginning of your course all the books I talked about will have links to their Amazon pages down there in the description and I'd also recommend that you check out the comments section where I'm sure that everyone is posting their favorite physics books to to kick things off I'm gonna recommend five popular science books starting with an absolute classic six Easy Pieces by Richard Fineman so Fineman wrote quite a few famous physics books this was the first one that I read and I also think it's the most appropriate as an introduction over to Fineman and to physics in general so this book is formed of chapters taken from his lectures in physics series which is famous amongst physicists for being really good and it offers an introduction to areas of physics like classical gravitation a little bit of quantum mechanics where physics sits in relation to other sciences and generally it's just a really good introduction to the field and it's for that reason that I was given this about ten years ago at a summer school at the University of Oxford and that's why this copy is particularly battered because I took this everywhere with me when I was in sixth form this thing is really concise it made a big influence on me like it definitely spurred me on to do physics at uni I was pretty set by that point but this was like yeah this there was a particular moment in this actually to do with the uncertainty principle right at the end of the quantum mechanics section when I was I I read something and I had to tell everyone else about it so I I really can't recommend this one enough this is an excellent popular science book and then a follow-up recommendation like the sequel if you like six not so easy pieces which is about relativity which is a subject that isn't in six easy pieces because it's not so easy this also draws on fineman's lecture series but this time talking about special and general relativity these are done in like a depth that's not quite the same as a physics degree but still definitely more advanced than six Easy Pieces so I'd recommend this to someone maybe he was in high school who was particularly advanced or someone who's starting out a degree now this next book isn't about physics it's actually about maths it's Alex's adventures a numberland by Alex bellos the reason I'm recommending this book is because so much of physics is really maths a physics degree is basically a maths degree with applications and if you don't enjoy maths then you are kind of at a disadvantage when studying physics I'm saying that as somebody who didn't enjoy studying maths until very literally until I read this book this book changed the way I thought about maths it made it seem so much less intimidating it made it seemed fun it actually made me interested and it was like a step change it was before about this I just really didn't enjoy it that much after this I have found physics easier I have found working with the maths in my degrees so much easier so honestly this is one of the books I recommend to everyone this this is one that I genuinely think anybody studying physics who like me wasn't particularly keen on the maths part of it everybody like that should read this book next a lighter book but won't covering a broad range of modern physics through the lens of science fiction the physics of the impossible by Michio Kaku I first read this book relatively recently I actually reviewed it on my Goodreads and to be honest I wasn't expecting very much from it I've read a lot of physics books that I I'm a relatively advanced level in physics myself now so I wasn't expecting there to be anything new in it but I did learn a few things and I thought the presentation of it was really engaging it's very American take from that what you will but it's a really interesting way of getting these concepts across particularly seeing as so many people who are interested in physics are obviously kind of sci-fi nerds I'm one of them using kind of sci-fi tropes and background like for example Star Trek Star Wars it's a great way of getting these relatively high level ideas to a younger audience so this one I think is a really good idea for a present for somebody who's a little bit younger maybe kind of sort of 14 to 18 kind of range but I read it as a PhD student and I enjoyed it so generally I think this want to be a good as a as a present then to round out the popular science books we've already had one about relativity so this one is gonna be about the other massive pillar of 20th century physics quantum by Manjit Kumar now I'll be honest with you I don't know if I have found the book on quantum mechanics I don't think one exists really of the ones I've read this is the closest to being the book and I think the reason for that is this introduces quantum mechanics in a historical way so what I mean by that is it starts with Max Planck and his reluctant revolution at the turn of the century and then builds up what happened in the field layer-by-layer von logically now that's just the way my brain works I found that to be the best way to get the ideas into my head I could sort of situate them that way if you are more interested in learning things kind of in a conceptual way then I might recommend in search of Schrodinger's cat by John Gribben or alternatively the elegant universe by Brian Greene though that one is really about string theory rather than in quantum mechanics but it necessarily introduces a lot of quantum stuff too so this is really a matter of personal preference I will definitely say that this is well written is well researched it's very interesting and it kind of it struck my resonant frequency of learning if you like so if you like me maybe would learn things best in a chronological way I'd recommend this if not check the description for the links to those other two books now before moving on to the textbooks if you are interested in learning how to study physics then I just made a video on that very subject with the lovely people at Socratic er so in that video I talk about my experiences studying physics how I got into it how I arrived at my PhD filled about miss Ferrett physics a bit about math star used but about programming that I used and then also a little bit about looks so if that sounds interesting to you then definitely check it out there'll be a card on the screen and there's also a link in the description next up though I'm gonna recommend you for textbooks and then also a bonus book which is kind of halfway between the two categories first the single best most influential math textbook I've ever used mathematical methods for physics and engineering by Riley Hobson and bents this meaty thing was the Bible of maths during both of my degrees and it's quite telling actually that this is the only math textbook I own because in it I have all the maths that are used in my Oxford physics decree so that's everything from high school level maths to integral equations tensors PDEs basically if it's not a graduate level you name it it's in this book the best part about it is that there are loads and loads of exercises at the end of each chapter with the solutions to half of them at the end of this book and then you can buy the other half the solutions to the other half in a separate volume that's just how many questions there are and of course doing those questions is the best way to learn a new topic so I mean for that reason alone if you're going to buy one math textbook to support a physics degree or to get ahead before a physics degree it's it's gonna be this one next up a broad introductory physics textbook fundamentals of physics by Halliday Resnick and Walker this was a textbook which I used a fair bit in my first year studying physics and I don't actually own a copy because of that because my college library in Oxford have enough copies that everyone could have their own so I don't own it but I probably wouldn't used it too much past first maybe second year it's quite an introductory quite broad but shallow textbook about pretty much all the areas that you expect to cover in a physics degree but you wouldn't want to just rely on it it's certainly well made like it's well presented great figures I just remember it being particularly like kind of pleasant to work with but not the most detailed book so perhaps this would be great for somebody who is in high school like a level or IB and they're not challenged enough by the course they maybe want to take it a little bit further so this would be a good way to bridge the gap between high school level and university level physics but like I say I've personally found it a little bit shallow third and flipping back to maths a book which I've not used extensively myself but everybody seems to have read it and loved it and so I basically feel obliged to pay this forward Grant div curl and all that by H M Shea now there is a reason why I am recommending this textbook despite not having used it very much myself and that's because this is a textbook about vector calculus so grab div and curl other three operations the key operations in vector calculus and it's a subject which is super super important in physics like particularly my field atmospheric physics use it all the time but it's not terribly well treated in the literature you don't learn it in high school and it's kind of thrust on you at university level so I'm recommending this because I think it's an important mathematical thing to add to toolkit and this is apparently it was recommended by Oxford University fortunately I didn't need to act on that recommendation because for some reason it just clicked which never happens with me but this is the book that I would recommend to if you want to focus in on like one mathematical technique to leapfrog your game before you go to uni or if you're at uni and you are struggling with vector calculus grad Dave Cohen all that would be an excellent place to start and then lastly another physics textbook and in my opinion an underrated gem concepts in thermal physics by Blundell and blundell now I had the distinct pleasure and privileged to be lectured by both Blundell and blundell they were a husband-and-wife team and this book is based on one of their lecture courses on thermodynamics and statistical mechanics and it's quite a condensed textbook but it's super easy to read it covers a key area in physics a bit similar to grab div and curl this is an area of physics that if you want to get ahead in the game thermodynamics is a fantastic way to do that that's that's a really useful subject area and this particular textbook isn't that intimidating because it's quite small it covers the same material but what it also does is harking back to what I said about quantum it introduces a historical context to the concept because thermodynamics is relatively dry and what it does is at the end of every chapter include a biography of a key player like James Clerk Maxwell or Carnot or Boltzmann and for me that added a lot of interest it helps that the rest of the textbook is really really well-written and engaging and I just found super easy to learn from but the biographies really added something extra so you definitely recommend this as a textbook for thermodynamics if you're taking that course or if you are someone who wants to get ahead for a physics degree this is really more people should read it so that's the textbooks we've already had the popular science books so that just leaves us with the bonus book which doesn't fit neatly into either of those categories and that book is the visual display of quantitative information by Edward Tufte so this book was given to me at the very start of my PhD in the very first week by my supervisor and I had trouble categorizing it because it kind of is a textbook specifically it's a textbook about how to make good figures and it's the kind of thing that you might refer to in a course on I don't know graphic design but it's not something that you'd use in a physics degree so it's something that I read as a popular science book it was something to supplement my learning and I can definitely say without a doubt that this book made me a better scientist this book made me think for the first time really about how I was putting together my figures and it provides you with a bunch of rules that all kind of guidelines that you should follow in putting together a figure and also things that you should avoid doing so it's a genuinely practical book and it also provides you with lots of examples of where things go right and where things go wrong and that is a key part of being a scientist when you have to present your work no one's gonna want to look at it if it's just hard to read or ugly my supervisor game but because he thought that everybody starting a PhD should read it and use its ideas in making their thesis I did I got some great feedback on my thesis and people saying that it was very easy to read I definitely agree with him I think this is an important book to read if you're thinking of becoming a scientist or if you're supposed just interested in graphic design but definitely another underrated book that scientists should be reading to repeat what I said at the start of the video there will be links to the Amazon pages for all of these books in the description so if you found any of them interesting definitely check them out and also recommend that if I missed your favorite physics book then put that down in the comments below of course if you enjoyed the video do give it a like share the video if you think other people should see it and also if you'd like to hear about how I think about learning physics then definitely check out the video I made with Socratic oh thank you for watching I'll see you next time

Lecture 1 | Modern Physics: Classical Mechanics (Stanford)

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Lecture 1 of Leonard Susskind’s Modern Physics course concentrating on Classical Mechanics. Recorded October 15, 2007 at Stanford University.

This Stanford Continuing Studies course is the first of a six-quarter sequence of classes exploring the essential theoretical foundations of modern physics. The topics covered in this course focus on classical mechanics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.

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this program is brought to you by Stanford University please visit us at stanford.edu classical mechanics is the basis for all of physics it's the basis of all of physics not only because it describes the motion of objects like particles and mechanical systems and so forth but because the basic framework the basic structure of all the physics is based on the principles of classical mechanics the conservation of energy the conservation of momentum the principles by which all systems evolve in nature is the same set of rules essentially exactly the same set of rules in a more abstract in the more general setting then the rules which govern how a simple particle moves for example under the influence of gravity but in order to understand that we have to understand the principles in a fairly general context let's begin with the very very simplest kinds of systems that we can think of systems that are so simple that in fact is simple than any real systems in nature laws of nature let's imagine laws of nature which are of the most primitive and simple kind acting on the most primitive and simple systems that we can imagine I want you to first of all suppose that time which evolves continuously under the my watch I see the second hand goes around and around and around that goes around continuously the time can evolve and be any real number t I want you to imagine it only occurs in beats a stroboscopic world in which you only look at the world at intervals of time which we could to be a thousandth of a second or it could be a millionth of a second let's just take it to be one second intervals and ask how in the stroboscopic world systems change with time let's also imagine a very very simple set of systems systems which are so simple that they only have a handful of configurations configurations mean everything we need know about the system to characterize it completely so the simplest system I can think of would be a system that has only two configurations heads or tails a coin lay according down on the table I don't have a coin so we'll take this coffee top here I could put it down heads or I could put it down tails at that if you can tell the difference from where you're sitting but I can tell the difference this is this is heads that's tails all right so we have a system then it's characterized by two states two states of being and we want to add to those two states a law a law of evolution in going from one instant of time to the next instant of time from one beat of the stroboscopic light to the next one what kind of laws are allowable what kind of laws the basic principles of physics allow and what kind of laws don't they allow that's going to be our first kind of question for tonight so our first concept is the space of states in this case just heads in tails it's just two points heads and tails so two points in an abstract space heads and tails it's called the phase space it's called the phase space of a system the space of possible states of a system and what do I mean by a state of a system I mean everything that you need to know in order to predict what happens next everything you need to know in order to be able to say with certainty what the next state of the system will be that's called the phase space of the system in this case just heads in tails what kind of laws can you imagine what kind of laws of nature can you imagine for this extremely simple world it is the simplest well I suppose you could imagine a slightly simpler world you can imagine a world with only one state heads not much can happen in that world there's only one law of physics heads goes to head since heads goes to heads goes to heads it's extremely boring nothing ever happens because there's only one state so how could anything happen but with two states has entails you can have a richer variety of possible laws of physics one law let's take the various varieties of laws of physics we can have one law might just say you stay the same if the state of the system is heads at one instant of time the next instant of time it will be heads if it starts tails it will stay tails that's a very boring law and let's graph that law by drawing an arrow if it starts heads then it stays heads let's just represent that by drawing an arrow from head to head and from tail to tail the meaning of this arrow you start at the tail end of the arrow and you follow it and you see that it comes around to the same point that stands for the rule that says that a heads stays a heads and in this case a tail stays a tail so a dull law of nature this stays that way for endless amounts of time and this stays that way for endless amounts of time another possible law of nature would be slightly more interesting situation if you look at it at one instant of time the Nexen since it's the opposite heads goes to tails because the heads goes to tails these are deterministic laws deterministic laws mean that if you know what is happening at one instant you know forever after you know everything about the system infinitely into the future completely deterministic classical mechanics has that nature to it that is completely deterministic so and in fact it is in a certain sense always reversible but we're coming for that all right so to draw the graph representing heads goes to tails and Tails goes to heads we draw an arrow from heads to tails and from tails – heads and we read that as saying that if you start at heads you go to tail if you start at pals you go to heads what is the evolution of a system under this law of nature if you start with heads it's heads tails heads tails heads tails heads tails forever after these are two consistent laws of physics in a world of only two states whether we can think of more laws yes we can but but for the moment these are two interesting ones now how can we generalize this we can generalize this first of all to systems with more States to States is not very many we could have a die die as in dice die has six states one two three four five and six and we could represent them as points six points now we have a large variety of different laws of physics that we could have for example we could simply have we could label these one two three four five and six which one is one and which one is six is not very important but the there are six of them we could have a law of physics which says 1 goes to 2 2 goes to 3 3 goes to 4 4 goes to 5 5 goes to 6 and 6 goes back to 1 this would be a complicated motion when thought of in terms of a dye you start with a 1 up and then it goes to the 2 up 3 up and so forth but it's a relatively simple picture when drawn in this manner here you can imagine more complicated laws of physics with six states well whether they're more complicated or not is in the eye of the beholder but there first of all we could have a similar law where instead of one going to two one could go to three three could go to to two could go to five one two three something like that that's really not very different than this each one goes to a neighboring one well not on neighboring one each one goes to a next one and they cycle around and one characteristic of such systems is they will just cycle around forever and ever and ever okay now you can have more complicated I don't know if they're more complicated different laws for example you could have a law which goes this way if you start at one you go to two if you go to two then you go to three and then you go back to one if you happen to start at three note this is three this is five right five six six six six one two three four five if you start at three you go to four if you start at four you go to five back to three in this case well that's somewhat similar to this you have two disconnected cycles to do got more complicated things here's another one in fact I think it would take a long time to draw all the possibilities one goes back to 1 2 goes to 3 3 goes to 4 all goes back to 2 five goes to six and six goes back to five right they're all acceptable laws of physics wherever you happen to be you know exactly where to go next so it's deterministic it's deterministic into the future meaning to say wherever you start you know where you will be arbitrarily into the future if you start here and you go a hundred thousand times you'll just wind up writing up some wherever wherever you start here you stay there so these are completely deterministic into the future but they're also completely deterministic into the past which means if you know where you are you know where you were before if you find yourself over here then you know in the previous cycle or the previous instant you were over here and so forth so you can trace your motion either into the future or into the past with complete confidence about where you'll be no matter how far you go that's the character in principle in principle if not in practice and practice things get jumbled up and you can't see them clearly enough and you and you miss detail but if you can see the infinitely small detail in a system and get every single bit of physics absolutely right for any classical system they are in this sense exactly deterministic both into the past and into the future now what kind of laws of physics do we not allow the kind of laws of physics that we don't allow I can best illustrate I think by drawing some some possibilities and explaining to you why they are not allowed here's a law of physics that is not allowed by the principles of classical physics three states three states is perfectly alright nothing wrong with that but let's draw some arrows – 3 2 goes to 3 and 3 goes back to 2 that would be perfectly alright by itself what about 1 well I could have one goes to one but I don't want one to go to one I want one to go to two that's completely deterministic into the future if I start at one I go to two I go to two I go to three I go three back two to two back to three I always know where to go in the next step I just follow the arrow wherever it happens to be but what does it fail it fails to be deterministic into the past supposing I know that I'm at – then where did I come from I could have come from one or I could have come from three so I cannot work my way backward with uniqueness I can work my way forward I cannot work my way backward that's a this is a law of physics which is irreversible it would not allow me to run the laws of physics backward it would lead to an ambiguity every time I were at to another law which is not allowed by the principles of classical mechanics or principles of classical physics would be basically the same thing but with the arrows turned in the opposite direction all arrows reversed here I have a problem not going into the past but I have a problem going into the future let's say yeah supposing now I find myself a – and I want to go into the future I don't know whether to go into the future by following this arrow to one or this arrow to three there are two arrows leading out of two over here one of them goes to three one of them goes to one nothing tells me which I wrote to follow so it's not deterministic into the future I might randomly decide to go from two to three or randomly decide to go from two to one these are rules each one of which is deterministic in one direction but not the other these are the sorts of things which are forbidden by the principles of classical mechanics why are they forbidden by the principal's yeah which one yeah that wouldn't come I wouldn't know where to go from three you get stuck at three yeah no good well you can't have a dangling one you gotta know where to go next now you could let's see we could try to put something like that in yeah okay so I would say we go from 1 to 2 from 2 to 3 2 3 2 3 2 3 now the problem with being going backward I think yeah yeah yeah right so one way or the other you get stuck over these rules all right how do we spot what's allowable and what's not allowable well it's very simple at every configuration 2 or 3 we should have one in and one out we should have a unique one in a unique arrow in and a unique arrow out one arrow in to tell us where we came from and one arrow out to tell us where we're going that's the character of classical physics uniqueness into the future uniqueness into the past when represented in terms of this very simple and analog world analog digital world of our finite number of states then the rules of physics as we know them would say every configuration has one in arrow in one out arrow now with this rule of course life is very boring because if you only have a finite number of states all it can happen to you is you cycle around endlessly among those number of States always in the same order you can have slightly more interesting situations there's no reason why the number of States has to be finite you could have a situation where there's an infinite number of possible state a state corresponding to every integer positive and negative a particle on a position on a line where the position could be any integer value and then a simple law of physics would be you go from one to the next wherever you are you go to the next one each point has one n in one out I can't draw them endlessly it'll take forever but each mark one in and one out this of course would also be a rather boring law of nature you just hop from one to the next to the next and next and next forever and ever and ever but at least you wouldn't be cycling around endlessly again we can have add some more states on to this we could add this on if we start on the line we simply move off and keep going forever and ever and ever but if we start over here we cycle around so we could have mixtures of both kinds of things some states we cycle around in other states we move off to infinity now notice that some of these laws of nature the phase space breaks up into different pieces which are connected among themselves but not connected to each other for example even with just two states we had two possibilities one like this and one like this in this case where it breaks up into more than one more than one piece we have something called a conservation law a conservation law is simply a memory of where we started a conservation law means that something is kept intact for all time some piece of knowledge is kept intact for all time and doesn't change in this case we could label this we could label a configuration over here with a plus one and a configuration over here with a minus one and then we could call we could invent a variable plus one over here and minus one over here it never changes if it's plus one that stays plus one if it's minus one it stays minus one that's a conservation law something which doesn't change with time on the other hand if we hop from plus 1 to minus 1 to plus 1 to minus 1 we don't have a conservation law conservation laws are always associated with these kind of closed families of different trajectories in the phase space which don't mix with each other which remember arm something about the system which might otherwise get mixed up if everything got all mixed together so there's all kinds of possibilities that are inherent in these deterministic laws but always the condition is one in line and one out line for every point that can also be called information conservation its information conservation in exactly the sense that that you never lose memory of where you started either into the past or into the future if you know where you are at any instant you know where you came from and you know where you'll be information about where you are is conserved never changes into the past in the future whereas if you have one of these bad laws laws which are forbidden by the rules of classical mechanics then you do lose information for example if you find yourself over here you don't know whether you came from here or from well no that's not quite right find yourself over here you don't know whether you came from here or over here so you lose information information conservation is perhaps the most fundamental law of basic classical physics that you don't lose information about now why that why is that so it's not written into the laws of physics why they are what they are maybe someday we'll understand Hyper laws physics or metal laws of physics or deeper laws of physics which will tell us why the laws of physics are what they are at the moment it's more or less an experimental fact that all the known laws of physics fit into this class of information conserving laws even those which are quantum mechanical but for our subject this quarter we're only interested in classical physics all right so that's the basic set up if we were interested only in this stroboscopic world of discrete time intervals the real world of course is more continuous than that supposing okay so let's make the law something like this if the last two states were heads heads then it stays heads if the last two states were heads tails then it goes to heads if the last two with heads tails heads then it goes to tails and if the last two with tails tails then it goes to tails I think that's a possible possible law then you would say I can't tell from the fact that it's heads where it goes next and indeed I can't but that would just be another way of saying that a specification of a heads by itself is not what you would call a state where you would call a state would be the specification of the previous last two entries because you need two entries to tell you what happens next now that raises the question that that's very very important in classical mechanics how much and what exactly do you need to know to say what happens next if the phase space is the space of things space of possibilities but always in such a way that they tell you exactly what happens next what is it that you do have to know next so that brings us to continuous physics let's take the motion of a particle let's take the motion of a particle is it enough to know where a particle is to say what happens next let's hypothesize that the generalization the continuous time is that we need to know the exact location of a particle along a line we've run out of ink I'm afraid not there is all right so let's imagine the motion of a particle along a line then you might think that the analog of a state is just a location of a particle where is it but is it enough to know where a particle is in order to say what happens next now what else do you need to know its velocity in order to know where a particle will be next you need to know not only where it is but how fast it's moving you need to know its velocity so that means the state in the same sense that I used it that which you need to know in order to know what happens next does not just consist of the location of a particle but you can say it two ways you need to know not just the location of the particle but you need to know also the previous location or better yet what causes for what is equivalent to knowing the previous location the velocity of velocity that tells you that the phase space the space of states the space of configurations is two-dimensional not one-dimensional it's not just a line it's a line that represents the position of the particle and a second line which represents its velocity either to the left or to the right positive velocity means moving to the right negative velocity means moving to the left supposing you're over here where do you move next you stay the same place why you're at the origin and you have no velocity what if you're over here so we could just we could draw a little loop here to say that you come back to the same place what if you're over here you stay the same place because you're moving with 0 velocity vertical axis is velocity so you come back to the same place what if you're over here where are you one second later let's chop time up into one-second intervals where are you next somewhere to the right huh what if you're over here now what if you're over here move twice as far or roughly twice as far to the right what is it down here you move to the left so we could fill up this space here with little tiny arrows to show where you move next but notice – no way you move next you have to know not only where you are in the sense of what X is but you also have to know the vertical component namely what the velocity is so that means the analog the analog of a point in the phase space is a point in the space not only of positions but also velocities yes why where that take place it means no it means exactly what the gentleman asked me before what if you had a law of nature which tells you in order to know where to move next you have to know your previous two entries all right now one way of saying it is all right in that case I need to know the previous two entries and it just doesn't fall into this class of things or you could say that the space of configurations doesn't just consist of a heads or a tails but it consists of a pair of entries a pair yeah that's right we meant we're not we model that that's right we need two axes that that's right we need an axis for the present configuration in the past one yeah that's exactly right well of course not because in classical physics the position of a particle and so forth is a real number a real number is something that you can never determine exactly ah and so there's always imprecision and that imprecision always represents a degree of uncertainty and where you will be next now it gets worse and worse in general it's likely to get worse and worse as you try to take larger and larger time intervals a given degree of imprecision in what you actually know can get magnified and get the magnified into worse and worse imprecision as time goes on so are in practice in practice classical systems don't really have the property that you can predict endlessly where they're going to be and exactly what they're going to do but you can always say given a time interval I want to be able to predict exactly for the next 30 seconds where every molecule in this room will be let's forget quantum mechanics now I want to be able to predict exactly where every mile kyoool there will be then there's a certain degree of precision in the present information at exactly one instant of time which will permit me to be able to extrapolate for 30 seconds okay it will not permit me to extrapolate for 40 seconds if I try to extrapolate for 40 seconds I will find the errors get magnified out of control if I want to predict correctly for 40 seconds I will have to do even better in my initial conditions and my knowledge of exactly what the state of the system is so in practice this idea of determinism is defective it's defective because in order to determine for a given length of time you'll have to have precision which is so good that it's way way beyond anything anybody can do but in principle given any length of time in classical physics there exist a degree of precision which allows you to extrapolate for that length of time does that answer the question yeah ah so some systems are very predictable some systems are less predictable and get out of control very quickly they're called chaotic systems but the principles are the same that it's just a degree of precision which you need to know in the beginning in order to extrapolate for a given length of time well you could decide I'm not sure what the difference between predictable and detail the way I use the terms I use them interchangeably so I can't say that okay okay yeah all right what's that yeah right the equations are deterministic if you know the initial conditions are deterministic are predictable infinitely predictable if you know the initial conditions with infinite precision you never do and therefore they're never completely predictable now the fact that you need to know both the position and the velocity in classical theory in the physics in order to predict what happens next reflects itself in the structure of the equations of mechanics specifically it tells you that the equations of motion Newton's equations in this case are what are called second-order equations instead of first-order equations let me illustrate it by starting with a first-order equation a first-order equation what a first-order equation means is that it only has quantities of first derivatives with respect to time in other words only contains velocities second-order means it contains not only first derivatives but second derivatives means it contains accelerations the equations of motion acceleration is the second derivative of the motion um we could write we know what the real equations of motion of Newton are F equals MA it contains acceleration let's write a phony equation let's write F equals mass times velocity force is F velocity is velocity and let's suppose that force just depends on where you are we have a particle that moves along a line it's subject to four which vary along the line force may be big here small there and so forth so the force depends on position and let's imagine this fake equation of motion that it's equal to mass times velocity and what is velocity velocity is the time derivative of the position the X by DT and I will use continuously throughout this course the notation that time derivative is just indicated by a dot that means time derivative all right what does this tell me what do I need to know in order to predict what happens next I say for this equation we only need to know the position of a particle if we know the position of a particle I can tell you what the velocity is just from the equation if I know that the position is a particular position then I know the force on it and from the equation that tells me what the velocity is does it tell me the acceleration how about the acceleration let's see if I can compute the acceleration to compute the acceleration from this equation we just differentiate it once more we write that the F by DT is equal to mass mass is just a constant times the second derivative of the position or just the acceleration so we have over here acceleration what about the F DT the time derivative of the force the force varies with time because the position varies with time so the F by DT is just a reflection of the fact that the position of the particle varies with time and we can write the F by DT using standard rules of calculus as just you did the change in F with respect to position times a change in position with respect to time in other words the velocity alright we've already figured out what the velocity is knowing the force knowing the position so we know the velocity and we can read off from this equation what the acceleration is we can figure out all of the derivatives of the motion if we know where the particle is by multiply differentiating this equation so what it tells us to make a short story out of it is it tells us that if we know where the position of the particle at any instant of time then we know where it's going to be in the next instant the next two instance the next three instants it completely predicts the motion but this is not the character of Newton's equations Newton's equations say F is equal to mass times acceleration mass times velocity so let's look at this equation F equals mass times acceleration can I predict from this what the velocity is no this is no equation for the velocity right if I know what the position is I know what the force is that tells me what the acceleration is but there's nothing in these equations which tell me what the velocity is that means I have to add in the velocity as a piece of information to begin with I have no choice I have to tell you in order to predict I have to tell you the position as well as the velocity then if I know the position and the velocity I can then predict the acceleration the next to a third derivative the fourth derivative and all of them so that tells me that in order to know where I am and where I'm going to be I have to know the position and the velocity the phase space is a two dimensional space so we see then that that classical mechanics does have this character of a configurate or a phase space of different configurations with a set of little arrows which tell you where to go next but the phase space itself has a position component to it and a velocity component to it we will go on and study the classical equations of motion and study them in a variety of different formulations but always the connecting link will always be conservation of information the idea that the laws of physics are completely deterministic and described by equations which tell you where you will be next that's the character of classical physics okay bifurcated it's one system but you need two pieces of informations that are warn you I'm not sure what you mean by bifurcated up here here you only need know now here you need one piece of information to say where you'll be next if you here you stay here if you're there you stay there yeah the stuff all there is is heads and tails that's all there is and if you know where you are you know where you'll be next no no we're in oh no no no no this one rule know this this is one rule heads goes two heads tails goes two tails that's a single rule and if you know that you're at heads then you know you'll be next at heads if you know you're at tails you know you but so you only need the piece of information heads or tails that does it that tells you everything now we I suggested a different does well yeah you don't need to know where you were before you only need to know where you are now to know where you'll be next so both of these laws require only knowledge of where you are at one instant to tell you where you'll be next you simply follow the if you start someplace you follow the arrow until you come back or you follow the arrow till you get to the next place and that tells you where you'll be next you don't need any more information than that the only question is what one of these points corresponds to does one of these cut points correspond to how much information does it correspond to a point in this space is it enough to know heads or tails to know what you'll do next or might you need to know the two previous things that's a different that's a different set up weighted stay with the heads and tails for a minute let's suppose that in order to say what happens next you need to know the previous two configurations let's do let's work out that example and let's write it in this form let's make up a law I think I had one down before heads heads goes – heads heads tails goes to watch tails tails heads goes to heads and tails tails goes to tails okay is there something wrong with that oh this only depends on Ed's coffee right you're right you're right right sorry sorry sorry sorry heads yeah we want this one to go to tails uh second one goes to heads good yeah what what what what say it again it's also true yeah yeah yeah yeah it's it also hard to work an example isn't it let's see hmm switch the second one this one make it tails okay I think that's perfectly all right you lose information why well okay here's a here's a set up here's a set up if you know the previous two then you know what happens next in fact let's let's now see if we can make a table out of this not a table but a set of points the set of points then there are four possibilities heads heads heads tails tails heads and tails tails okay let's see has heads heads heads tails tails heads and tails tails now supposing you go from heads heads to heads this is heads heads heads that means you go from heads heads to heads heads right you go from heads heads to heads heads now supposing you go from let's see what happens if you start with heads tails where do you go to so you go from heads head tails – tails tails okay okay what happens if you go from tails tails are we going to run to trouble aren't we tails tails goes to tails so tails tails goes to tails tails bad not allowable by the laws of physics and now tails heads goes where tails tails goes to tails tails great tails heads tails heads goes to tickles – heads tails so that comes here well it seems to me we go we should be able to make a consistent law whoa what uh this one this one Oh tails tails goes to tails tails goes to heads yeah okay so then tails tails goes to heads and that means that tails tails heads so I'm his tails tails goes to Ted – tails heads ah oh okay good good now we have a workable law now we have a workable law the only thing we had to do was to say that what we originally called a configuration namely a single heads or tails was not a complete specification information a complete specification of information involved two pieces of information and once we recognized that we were able to to to write this down as a law of physics which is deterministic and reversible okay so you don't know offhand you don't know to begin with what pieces of information you need to know in order to know what to do next but that's what a stead that's what the configuration space or that's what the phase space is it's the collection of all the things you need to know to know what happens next now of course you could go beyond this you could say I need to know the first three things in order to know what happens next you can do that you'll simply need more states to make a deterministic system what would it mean in classical mechanics to need more information than the positions and velocities suppose you need the positions velocities and accelerations that were in third order differential equations but what would it say about the phase space it would mean you would need positions velocities and accelerations to represent the phase space as it happens that is not the case for classical mechanics that's an experimental fact but it wouldn't stop us if we did if we did need the accelerations we would just write third order equations and we will make our phase space three-dimensional the preceding program is copyrighted by Stanford University please visit us at stanford.edu

1. Course Introduction and Newtonian Mechanics

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For more information about Professor Shankar’s book based on the lectures from this course, Fundamentals of Physics: Mechanics, Relativity, and Thermodynamics, visit 

Fundamentals of Physics (PHYS 200)

Professor Shankar introduces the course and answers student questions about the material and the requirements. He gives an overview of Newtonian mechanics and explains its two components: kinematics and dynamics. He then reviews basic concepts in calculus through two key equations: x = x0 + v0t + ½ at2 and v2 = v02+ 2 a (x-x0), tracing the fate of a particle in one dimension along the x-axis.

00:00 – Chapter 1. Introduction and Course Organization
21:25 – Chapter 2. Newtonian Mechanics: Dynamics and Kinematics
28:20 – Chapter 3. Average and Instantaneous Rate of Motion
37:56 – Chapter 4. Motion at Constant Acceleration
52:37 – Chapter 5. Example Problem: Physical Meaning of Equations
01:08:42 – Chapter 6. Derive New Relations Using Calculus Laws of Limits

Complete course materials are available at the Yale Online website: online.yale.edu

This course was recorded in Fall 2006.

this is a first part of a year-long course introducing you to all the major ideas in physics starting from Galileo and Newton right up to the big revolutions of the last century which was on relativity and quantum mechanics the target audience for this course is really very broad in fact have always been surprised at how broad the representation is so I don't know what your major is I don't know what you're going to do later so I picked the topics that all of us in physics find fascinating some may or may not be useful but you just don't know some of you're probably going to be doctors and you don't know why I'm going to do with special relativity or quantum mechanics but you don't know when it'll come in handy if you're a doctor and you've got a patient who's running away from you at the speed of light you will know what to do or if you're a pediatrician with a really small patient who will who will not sit still it's because a loss of quantum mechanics don't allow an object to have a definite cannot have a definite position and momentum so these are all things you just don't know when they will come in handy and I teach them because these are the things that turn me on and got me going into physics and whether or not you go into physics you should certainly learn about the biggest and most interesting revolutions right up to present day physics all right so that's what the subject matter is going to be and I'm going to tell you a little bit about how the course is organized first thing is this year it's going to be taped you can see some people in the back cameras as part of an experimental pilot program funded by the Hewlett Foundation and at some point they will decide what they will do with these lectures most probably they'll post them somewhere so people for elsewhere can have the benefit of what you have sitting in the Yale classroom so I've been told that from now on we just ignore the camera and do business as usual nothing's going to be changed I try to negotiate a laugh track so that the jokes don't work super focusin laughter I was told no I just got to deal with it as it happens so it's going to be it's like one of the reality shows where things are going to be as they are and hopefully after a while we'll we'll learn to act and behave normally and not worry about his presence then coming to the rest of the details of course by the way there are more details on the website that I posted that was given to me by the University if you want to know more about what all this is about the course organization is fairly simple we're going to meet Monday and Wednesday in this room 11:30 to 12:45 I will give you some problems to do on Wednesday and I'll post them on the website you guys should get used to going to the classes website I'm really really dependent on that now finally learn how to use it I will use that to post information maybe once in a while send email to the whole class if you want to get those emails you got to sign up for the course because I push a button and it goes to anybody who signed up there the homework will be given on Wednesday and it's due before class the following Wednesday and I let me introduce you to our head ta Mara Daniel who's recently Mara Barabbas right she will come a little okay so Mara is going to be the person who will see you after class and she will take the problem sets that you have submitted before class and she will give you the graded ones from after class okay that would be sorted out will be up there so you should drop the homework before you come into class rather than furiously working on it during class and the solutions will be posted the same afternoon so there is not much point in giving a homework that's late but once in a while you know you come up with a reason that I just cannot argue with you got married getting a transplant whatever it is that's fine you've got a transplant I want to see the old body part you got married I want to see your spouse if something happened to a grandparent I'm counting and up to four I don't get suspicious go five six seven eight I will have to look at a family tree but you know look any reasonable excuse will be entertain and the relative importance given to these different things is 20% for your homework 30% for the midterm which will be sometime in October and 50% for the final that'll be the weighted average but I have another plan called the amnesty plan in which I also compare just your final grade what your in the final exam and whichever is higher of the two is what I will take to determine your overall course grade this is something I used to announce near the end but then some people felt that it's not fair not to know this from the beginning so I'm telling you from the beginning but don't don't dream and think that somehow the final is going to be so much different from your regular day to day performance but give you some reason to live after the midterm you feel that is hope I can change everything overnight it does happen I put that in for a reason because sometimes some of you have not taken a physics course and you don't know how to do well in physics and slowly you catch on and by the time it's final exam you crack the code you know how to do well as far as I'm concerned that's just fine if at the end of the semester you take a three-hour exam in a closed environment and you answer everything I don't care what you did in your homework or your midterm that's not relevant so that's how the grading will be done we have a maras group of TAS she is the head TA and she's the one you should write to whenever you have a problem then we also have two faculty members one is a postdoctoral fellow mark Caprio mark okay so he will have a discussion section on Tuesdays between 1:00 and 2:00 in Sloan lab and Steve for Lynette oh I don't know if Steve is here or not there's Steve our new assistant professor he will have a section on Tuesday night in Dunham's lab a room 220 Tuesday night is the night when you people realize homework is due on Wednesday so we know that so he will be there to comfort you and give used whatever help you need all right my own office hours I've not determined yet I'll have to find out when it is good for you I used to you know I live and work out of sloan lab up on the hill and it was easy to have office hours before or after class but now you have to make a special trip so just give me a little bit of time to find out maybe by soliciting email responses from you what would be a good time for my office hours but for any procedural things like you know this problem set was not greater properly and so on there's no point emailing me because I'm going to send it to Mara anyway so directly deal with the powers that be okay finally I want to give you some tips on how to do well in this course and what attitude you should have first likewise this you should come to the lectures it's not self-serving it's not so much for my benefit I think there's something useful about hearing the subject presented once orally secondly the book you can see one of you had a book here it's part it's part of 1100 pages and when I learned physics it was like 300 pages now I look around this room I don't see anybody whose head is three times bigger than mine so I know that you cannot digest everything the books have so I have to take out what I think is a really essential part and cover that in the lecture so you come to class to find out what's in and what's not in if you don't do that it's a danger you will learn something you don't have to and we don't want that okay so that's where you come to class second thing most important thing for doing well in physics is to do the homework the 20% given to the homework is not a real measure of how important it is homework is when you really figure out how much you know and don't know if you watch me do the thing on the blackboard it looks very reasonable it looks like you can do it but only we are going to find out is when you actually deal with the problem that's the only time you're going to find out so I asked you to do the problems as and when they're posted so if you post it on Wednesday is to cover the material for that week then you should attempt it as quickly as possible because I'm going to assume you have done the problems when you come for the next few lectures and in doing the homework it is perfectly okay to work in groups you don't have to do it for yourself that's not how physics is done I'm now writing a paper with two other people that are my experimental colleagues who write papers with 400 other people maybe even a thousand other people when they do the big collider experiments in Geneva and Fermilab collaborations can run into hundreds so it's perfectly okay to be a part of a collaboration but you got to make sure that you're pulling your weight you got to make sure that if you explain to others how to do this problem then somebody else contribute or something else but you know what everybody contributed in the end so the game is not just to somehow rather get the solution the problem set but to fully understand how it's done and the TAS will be there to help you there's every day that's going to be TA in the undergraduate lounge I would urge you to use that there's a beautiful new lounge that the Provost's office allowed us to build for physicists and chemists whoever happens to be in the building if you go there on the third floor of Sloan you may run into other people like you we're trying to work on problems you may run into upper-class students students or more advanced you run into your TA so that's a good climate that our coffee machine isn't that allowing sofas and everything else that our computers that our printers so it's a good lounge and I think if you go there one day a week to do your problem sets or more often that's a good meeting place I recommend that the final piece of advice this is very important so please pay attention to this which is I ask you not to talk to your neighbors during lecture now this looks like a very innocuous thing but you will find out it's the only thing that really gets my back up most of the time I don't really care I'm very liberal but this disturbs me because I'm looking at you I'm trying to see from your reaction how much of your my lecture you're following and then it's very distracting when people are talking so please don't do that if you talk I'm going to assume you're talking about me if you laugh I'm going to assume you're laughing at me let's make that that's not really what I think but that's how disturbing it is when people talk and very nice students who do not realize this often disrupt my straight my line of thinking so I ask you to keep that to minimal once in a while you'll have to talk to your neighbor and say can you please pass me my pacemaker that fell down that's fine then you go back to your business but don't don't do too much of that finally that is this ancient issue about sleeping in the class now my view is it's just fine okay I know you guys need the rest and interestingly the best sleepers are in the first couple of rows I haven't met you guys it's not person I have found some people really have to come to the first a second row because they claim that if they don't hear me they cannot really go to sleep now that was true in Sloan but I think loose has got very good acoustics so you can stretch out in the back but my only of only criterion is if you talk in your sleep that's not alarming it's talking it's not what next if you're going to sleep I ask you to sit between two non sleepers sometimes what happens the whole row will topple over we don't want the domino effect now it's going to be captured on tape and that's going to be really bad for my reputation so spread yourself thread yourself around the other people all right so that's it in terms of class you know logistics and everything I'm going to start going into the physics proper I will try to finish every lecture on time but sometimes I'm in the middle of a sentence or in the middle of a derivation and I have to go over a couple of minutes there's no need to shuffle your feet and move stuff around I know what time it is I also want to get out like you guys but let me finish something other days I may finish a few minutes before time that's because the ideas of physics don't fall into seventy five minute segments and sometimes they spill over a little bit also I'm used to teaching this course three times a week and now it's certainly twice a week and so things that fell into a nice 50-minute units are now being snipped up different ways so it's pretty difficult so even for me some of it will be new and the timing may not be just right okay anybody have any I should tell you first of all that in this class the taping is not going to affect you because the camera is going to be behind your head I mentioned to you in the website that this is not the big opportunity you've been looking for to be a star only the back of here it will be seen in some cases the back of the head could be more expressive than the front in which case this is your opportunity and I wish you luck but otherwise just don't worry about it because you will be only heard you may not even be heard so I've been asked that if a question is not very clear I should repeat it so that people listening to this later will know what the question was but I would ask you let me make one thing very clear that is I'm not in favor of you are talking to each other because it's distracting you're stopping me anytime is just fine I welcome that because I've seen the subject for god knows how many years the only thing that makes it different for me is the questions that you people have so you can stop me anytime and you should not feel somehow you're stopping the progress of the class there is no fixed syllabus we can move things around it is far more exciting for me to answer your questions than to have a monologue so don't don't worry about that so stop me anytime you don't follow something and don't assume that you're not following something because there's something wrong with your level of comprehension quite often you guys come up with questions that never crossed my mind very interesting and things we've been repeating year after year after year because they sound so reasonable suddenly sound unreasonable when some of you point out some aspect of it that you didn't follow so it could be very interesting for all of us to have issues discuss in class and quite often some presents are very common and your classmates will be grateful to you that you brought it up otherwise you know you get 10 TAS get 10 emails all with the same question okay so I'm going to start now anybody have any questions about class the format the midterm exams all right yes oh you mean my office hours no the discussion sections are Tuesday afternoon from 1 to 2 and Tuesday night from 8 to 10 and the the website has got all the details on when and where yes yeah there are many many lab times and you have to go to the website for the lab and by the way that reminds me I got here lots of flyers given to me by the director of the laboratories which will tell you which lab is the right lab for you they're offered many times a week yes I think it's a good idea to take the lab because if particularly in this particular class because I don't have any demonstrations or they're all in the other building so if this will remind you that physic is after all an experimental science and you will be able to see where all the laws of physics come from so if you're going to take it you should take it at the same time yes ah very good this is a calculus based class and I expect everyone to know at least the rudiments of differential calculus what's the function what's the derivative what second derivative how do you take derivatives of elementary functions how to do elementary integrals sometime later I will deal with functions are more than one variable which I will briefly introduce to you because that is may not be prerequisite but certainly something you will learn and you may use on and off but there are different ways of doing physics – to demonstrate over and over how little mathematics you need to get the job done there are others who would like to show you how much mathematics you could somehow insinuate into the process okay there are different ways of playing the game and some of us find great pride in finding the most simple way to understand something that's certainly my trademark that's how I do my research also so if you feel that's not enough math used I guarantee you that I certainly know enough eventually to snore the whole class but that's not the point I will use it in moderation and use it to use it to the best effect possible rather than use it because it is there okay so I don't know your mathematical background but the textbook has an appendix which is reasonable measure of how much math you should know you got to know your trigonometry you got to know what's the sine and watch the cosine you cannot say I will look it up your birthday and social security number is what you look up trigonometric functions you know all the time okay I will assume you do all right and of course trigonometric identities you know from high school pages and pages of them so no one expects you to know all those identities but there are a few popular ones we will use all right anything else yes yeah midterm will be sometime around 20th of October I have to find out exactly the right time we have 24 lectures for this class and the first 12 roughly will be part of the midterm but after the 12th lecture I may wait a week so that you had time to do the problems and get the solutions then I will give you the midterm yes yeah this tells you in detail this flier answers exactly that okay there's one more question somebody yes I don't have a strong view I think you should take the lab sometime but I don't know how many semesters semesters that you have to take but I would say the advice of your predecessors is very important if they tell you this is what works that's better than what somebody like me can tell also you should talk to Stephen ions who is the director of the labs he has seen every possible situation he will give you good advice I start now okay so we are going to be studying in the beginning what's called Newtonian mechanics it's pretty remarkable that the whole edifice was set up by just one person Newton and he set us on the road to understanding all the natural phenomena till the year 1800 and something when Max will invented the laws of electromagnetism and wrote down the famous Maxwell equation but except for electromagnetism the basics of mechanics which is the motion of billiard balls and trucks and marbles and whatnot was set up by Newton so that's what we are going to focus on and you will find out that the laws of physics for this entire semester certainly can be written on one of those blackboards or even half of those blackboards and the purpose of this course is to show you over and over and over again that starting with those one or two laws you can deduce everything and I would encourage you to think the same way in fact I would encourage you to think the way physicists do even if you don't plan to be a physicist because that's the easiest way to do the subject and that is to follow the reasoning behind everything I give you and my purpose will be not to say something as a postulate but to show you where everything comes from and it's best for you if you try to follow the logic that way you don't have to store too many things in your head in the early days when there are four or five formulas you can memorize all of them and you can try each one of them till something works but after a couple of weeks you will have one formulas cannot memorize all of them you cannot resort to trial and error so you have to know the logic so the logical way is not just the way the physicists do it it's the easier way to do it if there is another way that will work for non physicists I won't hesitate to teach it to you that way that turns out to be the best way to try to follow the logic of everything okay so Newtonian mechanics is our first topic so Newtonian mechanics has two parts or all of physics consists of tube is a two-part program the plan every time is to predict the future given the present that's what we always do when we do that right we are satisfied the question is what do you mean by predict the future what do you mean by the future what do you mean by the present by present we mean you will pick some part of the universe we want to study and we will ask what information do I need to know about that system at the initial time like right now in order to be able to predict the future so for example if you were trying to study the motion of some object here's one example here see that's an example of Newtonian mechanics I'll give you one more demonstration let's see we can catch this one that's a good example so that was Newtonian mechanics at work because what did I do so I released a piece of candy threw it from my hand and the initial condition had to do with where did I release it and with what velocity that's what he sees with his eyes then that's all you really need to know then he knows it's going to go up it's going to curve follow some kind of parabola then his hands go there to receive it that is verification of a prediction his prediction was the candy is going to land here and he put his hand there he also knew whether candy land but he couldn't get his hand there in time but we can always make predictions but this is a good example of what you need to know what is it he had to know about this object that was thrown I claim is the initial location of the object the initial velocity the fact that was blue or red not relevant and if I threw a gorilla at him it doesn't matter you know what the color of the gorilla is what mood it is in they said things we don't deal with in physics or if there is a tall building its standard physics problem and object falls off a tall building object could be a person so we don't ask why is this guy ending it all today that's not we don't know and we cannot deal with that so we don't answer everything we just want to know when is going to hit the pavement and with what speed so we ask very limited questions which is why we brag about how accurately we can predict the future so we only asked limited goals and we are very successful in satisfying them so we are basically dealing with inanimate objects so the project of Newtonian mechanics of predicting the future given the present has got two parts and now one is called kinematics and the other is called dynamics so kinematics is a complete description of the present it's a list of what you have to know about a system right now for example if you're talking about the chalk if I throw the chalk you will have to know where it is and how fast it's moving the dynamics doesn't dynamics then tells you why the object goes up why the object goes down why is it pulled down and so on that's dynamics the reason it comes down is gravity is pulling it in kinematics you don't ask the reason behind anything you simply want to describe things the way they are and then dynamics tells you how they change and why they change so I'm going to illustrate the idea of kinematics by taking simplest possible example that's going to be the way I'm going to do everything in this course I'm going to start with the simplest example and slowly add on bells and whistles and make it more and more complicated so some of you might say well I've seen this before so maybe there is nothing new here that may well be I don't know how much you've seen but quite often the way you learn physics earlier on in high school is probably different from the way professional physicists think about it the sense of values we have the things that we get excited about it different problems maybe more difficult but I want to start in every example in every situation that explain to you the simplest example then slowly add on things so what we are going to study now is a non living object and we are going to pick it to be a mathematical point so the object is a mathematical point has no size if you rotate it you won't know it's not like a potato you take a potato you turn it around it looks different so it's not enough to say the potato is here you got to say which way the notice is pointing and so on so we don't want to deal with that now that comes later when we study what are called rigid bodies right now we want to study an entity which has known spatial extent so it just a dot and the dot can move around all over space so we're going to simplify that too we're going to take an entity that lives along the x axis it moves along a line so you can imagine a bead with a wire going through it and the bead can only slide back and forth so this is about the simplest thing I cannot reduce the number of dimensions one is the lowest dimension I cannot make the object simpler than being just a mathematical point then you got to say what do I have to know about this object at the initial time what constitutes the present or what constitutes maximal information about the present so what we do is we pick an origin call it zero we put some markers there to measure distance and we say this guy is sitting at one two three four or five this is sitting at X equal to five now of course we got have units and the units for length is are going to be meters units for time will be second at time will be measured in seconds then we'll come to other units right now in kinematics this is all you need now there are some tricky problems in the book sometimes they give you the speed in miles per hour kilometers per year pound dollars per square foot whatever it is you gotta learn to transform them that I won't do that I think that's pretty Elementary stuff but sometimes I may not write the units but I weren't the right to do that and you guys have it so you still have to keep track of your units everything's got to be in the right units if you don't have the unit's then if you say the answer is 19 then we don't know what it means okay so here's an object at a given instant it's got a location so what we would like to do is to describe what the object does by drawing a graph of time versus space and the graph could be something like this you gotta learn how to read this graph I'm assuming everyone knows how to read it this doesn't mean the object is bobbing up and down I hope you realize that even though the graph is going up and down the object is moving from left to right so for example when it does this it's cross the origin and it's going to the left of the origin now which is the left with the on it and it turns around to start coming to the origin and going to the right that is X versus T so the language of calculus X is a function of time and this is a particular function this function doesn't have a name that other functions which have a name for example this is x equals T X equal to T Square you can have X equal to sine T and cosine T and log T so some functions have a name some functions don't have a name what a particle tries to do generally is some crazy things doesn't have a name but it's a function except tea so you should know when you look at a graph like this what it's doing so the two most elementary ideas you learn are what is the average velocity of an object it is denoted by the symbol V bar so the average is found by taking two instance in time say T 1 and later T 2 and you find out where it was at T 2 where as where is was a T 1 divided by time so the average velocity may not tell you the whole story for example if you started here and it did all this and you came back here the average velocity will be 0 because you start and end at the same value of x you get something 0 divided by time will still be 0 so you cannot tell from the average everything that happened because another way to get the same 0 is to just not move at all the average is what it is it's an average it does not give you enough detail so it's useful to have the average velocity is useful to have the average acceleration which you can find by taking similar differences of velocities but before you even do that I want to define for you an important concept which is the velocity at a given time so this is the central idea of calculus right I'm hoping that if you learned your calculus you learned about derivatives and so on by looking at X versus T so I will remind you again this is not a course in calculus I don't have to do in any detail I will draw the famous picture of some particle moving and it's here at time T at some value of x a little later which is time T plus delta T the delta T is going to stand always for a small finite interval of time infinitesimal interval of time not yet 0 so during that time the particles gone from here to there that is X plus Delta X and the average velocity in that interval is Delta X divided by delta T graphically this guy is Delta X and this guy is delta T and Delta X or Delta t is a ratio so in calculus what you want to do is to get the notion of the velocity right now we all have a intuitive notion of velocity right now when you're driving in your car there's a needle and the needle says 60 that's a velocity at this instant it's very interesting because velocity seems to require two different times to define it the initial time in the final time and yet you want to talk about the velocity right now that is the whole triumph of calculus know that by looking at the position now the position slightly later and take the ratio and bringing later as close as possible right now we define a quantity that we can say is the velocity at this instant so V of T is the limit delta T goes to 0 of Delta x over delta T and we use the symbol DX DT for velocity so technically if you ask what is the velocity is stand for it stands for let me drop general situation if a particle goes from here to here Delta x over delta T I don't know how well you can see it in this figure here is the slope of a straight line connecting these two points and as the points come closer and closer straight line will become tangent to the curve the velocity at any part of the curve is a tangent to the curve at that point it's tangent of that theta this angle is Theta then Delta X and or delta T by trigonometry stand data okay once you can take one derivative you can take any number of derivatives and the derivative of the velocity is called the acceleration and we write it as the second derivative of position so I'm hoping you guys are comfortable with the notion of taking one or two or a number of derivatives interestingly the only the first two derivatives have a name first one is velocity second was acceleration the third derivative unfortunately was never given a name and I don't know why I think the main reason is that there are no equations that involve the third derivative explicitly F equals MA the a is this fellow here and nothing else is given an independent name of course you can take a function and did it take derivatives any number of times so you are supposed to know for example if X of T is T to the N you're supposed to know DX DT is n to the T to the N minus 1 then you're supposed to know derivatives of simple functions like sines and cosines so if you don't know that then of course you have to work harder than other people if you know that that may be enough for quite some time okay so what I've said so far is a particle moving in time from point to point can be represented by a graph X versus T or any point on the graph you can take the derivative which will be tangent to the curve at each point and this numerical value will be what you can call the instantaneous velocity at that point and you can take the derivative of the derivative and call it the acceleration so we are going to specialize to a very limited class of problems in the rest of this class the limited class of problems is one in which the acceleration is just a constant now that is not the most general thing but I'm sure you guys have some idea why we are interested in that so anybody know why that's such a big so much time is spent on that yes for me right the most famous example is that when things fall near the surface of the earth they all have the same acceleration and the acceleration that's most common is called G and that's 9.8 meters per second squared so that's a very typical problem when you fall in the surface of the earth you are describing the problem of constant acceleration that's why there's a lot of emphasis on sharpening your teeth by doing this class of problems the question we are going to ask is the following if I tell you that the particle is a constant acceleration a can you tell me what the position X is normally I will give you a function and tell you to take any number of derivatives that's very easy this is the backwards problem here you're only given the particle as acceleration a and you're asked to find out what is X in other words your job is to guess a function whose second derivative is a and this is called integration which the opposite of differentiation an integration is just guessing integration is not a an algorithmic process like differentiation if I give you a function you know how to take the derivative change the independent variable find the change in the function take the ratio and that's the derivative the opposite is being asked here I tell you something about the second derivative of a function and ask you what is the function the way we do that is we guess and the guessing has been going on for 300 years so we sort of know how to guess so let me think aloud and ask how I will guess in this problem I would say okay this guy wants me to find a function which ready is to the number a when I take two derivatives and I wrote somewhere here this result which says that when I take a derivative I lose a power of T in the end I don't want any powers of T very clear I got to start with a function that looks like T Square this way when I take two derivatives there will be no T left unfortunately we know this is not the right answer because if I take the first derivative I get 2t I take the second derivative I get two but I want to get a and not two then it's very clear the way you find pat's it up is you multiply it by this constant and now we are all set this function will have the right second derivative so this certainly describes a particle whose acceleration is a but a is not dependent on time but the question is is this the most general answer or is it just one answer and I think you all know that this is not the most general answer it is 1 answer but I can add to this some number like 96 they'll still have the property that if you take two derivatives you're going to get the same acceleration so 96 now we said typical constants I'm going to give the name see the consulate everyone knows from calculus that if you if you're trying to find a function about which you know only the derivative you can always add a constant to one person answer without changing anything but I think here you know you can do more right you can add something else to the answer without invalidating it and that is anything with one power of T in it because if you take one derivative it'll survive but if you take two derivatives it will get wiped out now it's not obvious but it is true that you cannot add to this anymore basic idea in solving these equations in integrating is you find one answer so that when it derivatives the function does what it is supposed to do but then having found one answer you can add to it anything that gets killed by the act of taking derivatives if you take only one derivative you can add a constant if you're taking two derivatives you can add a constant and something linear in T if you knew only the third derivative of the function you can add something quadratic in T without changing the outcome so this is the most general position for a particle of constant acceleration a now you must remember that this describes a particle going side to side I can also describe a particle going up and down if I do that I will I would like to call that coordinate Y then I will write the same thing you got to realize that in calculus the symbols that you call x and y are completely arbitrary if you know the second derivative of Y to be a then the answer looks like this if you knew the second derivative of X the answer looks like that now we have asked what are these numbers B and C so let me go back now to this expression X of T equals 8 e squared over 2 plus C plus BT it is true mathematically you can add two numbers but you can't ask yourself what am i doing as a physicist when I add these two numbers what am I supposed to do with a and B I mean with this B and C what value should I pick the answer is that simply knowing the particle has an acceleration is not enough to tell you where the particle will be for example let's take the case where the particle is falling under gravity then you guys know you just told me acceleration is minus 9.8 but G's minus 9.8 we call it – because it's accelerating down and up was taken to be the positive direction in that case Y of T will be minus 1/2 G t square plus c plus bt so the point is every object falling under gravity is given by the same formula but there are many many objects that can have many histories all falling under gravity and what's different from one object the other object is when was it dropped from what height and with what initial speed that is that's what these numbers are going to tell us and we can verify that as follows if you want to know what the number C is you say let's put time T equal to zero in fact let me go back through this equation here if you put time T equal to zero X at zero it doesn't have this term doesn't have this term and it is C so I realize that the constant C is the initial location of the object and it's very common to denote that by X not so the meaning of the constant is where was the object at the initial time it could have been anywhere simply knowing the acceleration is not enough to tell you where it was at initial time you get to pick where it was at the initial time then to find the meaning of B we take one derivative of this DX DT that's velocity as a function of time and if you took the derivative of this guy you will find it's a t plus B that's the velocity of the object then you can then understand that V of zero is what B's which we write as B zero okay so the final answer is that X of T looks like X naught plus V not T plus one-half a t-square okay so what I'm saying here is we are specializing to limited class of motion where the particle has a definite acceleration a then in every situation where the body has an acceleration a the location has to have this form where this number is where it was initially this was the initial velocity of the object so when I threw that thing up and you caught it what you're doing mentally was immediately figuring out where it started and at what speed that was your initial data then in your mind without realizing it you found the trajectory at all future times now that is one other celebrated formula that goes with this I'm going to find that then will I'll give you an example now I'm fully aware that this is not the flashiest example in physics but I'm not worried about that right now you will have you'll see enough things that will confound you but right now I want to demonstrate a simple paradigm of what it means to know the present and what it means to say this is what the future behavior will be and we want to do that in the simplest context then we can make the example more and more complicated but the phenomenon will be the same so what we have found out so far I'm purposely going from X to Y because I want you to know that the unknown variable can be called an X or can be called a Y it doesn't matter as long as the second derivative is a that's the answer now that's the second formula one derives from this you guys probably know that too from your days of the day care but I want to derive that formula and put it up then we will see how we use them second formula tries to relate the final velocity at some time T to the initial velocity and the distance traveled with no reference to time so the trick is to eliminate time from this equation so let's see how we can eliminate time you know that if you took a derivative of this you will find V of T is v-0 plus 80 what that means is if you know the velocity at a given time and you know the initial velocity you know what time it is the time in fact is V minus v-0 over a if I don't show you any argument for V it means V at time T and the subscript zero means V of zero so what this says is you can measure time by having your own clock the clock tells you what time is this but you can also say what time it is by seeing how fast the particle is moving because you know it started with some speed it gaining speed at some rate a so if the speed was so and so now then the time had to be this time can be indirectly inferred from these quantities then you take that formula here and you put it here wherever you see a time T you put this expression so what will you get we get an expression in which there is no T T has been banished in favourites so I'm not going to waste your time by asking what happens if you put it in I will just tell you what happens what happens is you will find the v square is equal to V 0 square plus 2 a times X minus x0 how many people have seen this thing before ok that's a lot look I know you've seen this at the moment I have to go through some of the more standard material before we go to the more non-standard material if this part is very easy for you there's not much I can do right now so let me draw a box just drawing a box you guys means important so these are the two important things I claim now remember I want you to understand one thing how much of this should you memorize suppose you've never seen this in high school how much are you supposed to memorize I would say keep that to a minimum because what the first formula tells you should be so intuitive that you don't have to cram this we are talking about particles of constant acceleration that means when I take two derivatives I want to get a then you should know enough calculus you know it has to be something like a t-square and the half comes from taking two derivatives the other two you know are stuff you can add and you know what what you're adding those things because the particle has a headstart it's got an initial position even at t equal to zero it has an initial velocity so even without any acceleration it will be moving from why not – why not plus VT the acceleration gives you an extra stuff quadratic in time once you got that one derivative will give you the velocity then in a crunch you can eliminate T and put it into this formula but most people end up memorizing these two because you use it so many times eventually it sticks in you but you should try to memorize everything so we are now going to do one standard problem where we will convince ourselves we can do we can apply this formulas and predict the future given the present so the problem I want to do there are many things you could do but I'd espect one this is the one with their own numbers so I can do it without a calculator here's a problem that is this building and it's going to be 15 meters high and I'm going to throw something and it's going to go up and come down something I throw up has an initial speed of 10 meters per second so we have to ask now now that my claim is you can ask me any question you want about this particle and I can answer you you can ask me where it would be nine seconds from now eight seconds from now how fast will it be moving I can answer anything at all but what I needed to do it to do this problem was to find these two unknowns so you got to get used the notion of what will be given in general and what is tailor-made to the occasion so we know in this example the initial height should be 15 meters and the initial velocity should be 10 and for acceleration I'm going to use minus G and to keep life simple I'm going to call it minus 10 as you know the correct answer is 9.8 but we don't want to use the calculator now so we call it minus 10 consequently for this object the position Y at any time T is known to be 15 plus 10 t minus 5 T squared that is the full story of this object of course you got belittle when you use it for example let's put T equal to 10,000 years what are you going to get if you put t equal to 10,000 years of 10,000 seconds you're going to find Y is some huge negative number now said right there what's wrong with that reasoning so you cannot use a formula once it hits the ground because once it hits the ground the fundamental premise that a was a constant of minus 9.8 or minus 10 is wrong so that's another thing to remember once you get a formula you got to always remember the terms under which the formula was derived if you blindly use it beyond the bella D you will get results which don't make any sense conversely if you get an answer and it doesn't seem to make sense then you got to go back and ask am i violating some of the assumptions and here you will find the assumption that the particle had that acceleration is true as long as it's freely falling under gravity but not when you hit the ground now if you dug a hole here till there then of course it may work till that happens okay but you thought you got into every time when you this is so obvious in this problem but when you see more complicated formula you may not know all the assumptions that went into the duration and quite often you'll be using it when you should all right see this you agree is a complete solution to this miniature tiny Mickey Mouse problem you give me the time and I'll tell you where it is if you want to know how fast it's moving at a given time if you want another velocity I just take the derivative of this answer which is 10 minus 10 T so let me pick a couple of trivial questions one can ask one can ask the following question how high does it go how high will it rise to what height will it rise so we know it's going to go up and turn around and come down we're trying to see how high that is so that is a tricky problem to begin with because if you take this formula here it tells you why if you know T but now we are not saying that we don't know the time and we don't know how high it's rising so you can ask how am I supposed to deal with this problem then you put something else that you know in your mind which is that the highest point is the point when it's neither going up and are coming down if it's going up that's not the highest point which is coming down that's not the highest point so at the highest point it cannot go up and it cannot go down that's the point where velocity is zero if you do that let's call the particular time T star then 10 T star minus 10 is equal to 0 or T star is 1 second so we know that it will go up for one second then it will turn it on and come back now we are done because now you can ask how high does it go then you go back to your Y of 1 and Y of 1 is 15 plus 10 minus 5 which is what 20 meters by the way you will find that I make quite a lot of mistakes on the blackboard you're going to find out you know one of these years when you start teaching that when you get really close to blackboard I you just cannot think it's definitely some inverse correlation between your level of thinking and the proximity to the blackboard so if you find me making mistake you got to stop me why do you stop me for two reasons first of all I am very pleased when this happens because I'm pretty confident that I can do this under duress but I may not do it right every time but if my students can catch me making a mistake it means they're following it and they're not hesitating to tell me secondly as we go to the more advanced part of the course we'll take a result from this part of the blackboard let's take it into the second part and keep manipulating if I screw it up in the beginning and you guys keep quiet we'll have to do the whole thing again I would I would ask you when you follow this thing to do it actively try to be one step ahead of me for example if I'm struck by lightning can you do anything can you guess what I'm going to say a next you have any idea where this is going you should have a clue if I die and you stop that's not a good sign okay you got to keep going a little further because you should follow the logic so for example you know I'm going to calculate next when it hits the ground you should have some idea how I'll do it because this is not a spectator sport if you just watch me you're going to learn nothing like watching the US Open and thinking you're some kind of player you will have to do you will have to shed the tears and you got to bang your head on the wall go through your own private struggle I cannot do that for you I cannot even make it look hard because I raised this problem from childhood so there is no way I can make this look difficult that's your job all right so we know this point at one second is 20 meters so let's just ask one other question we'll stop one other question maybe when does it hit the ground and with what speed at typical physics question so when does it hit the ground well I think you must know now how to formulate that question when does it hit the ground is when is y equal to zero by the way I didn't tell you this but I think you knew that I pick my origin to be here and measured Y positively to be upwards and I call that 15 meters you can call that your origin if you call that your origin your y 0 will be 0 but ground will be called -15 so in the end the physics is the same but the number is describing it can be different but you have to interpret the data differently but the standard origin for everybody is the foot of the building you can pick your origin here some crazy spot it doesn't matter but some origins are more equal than others because there are some natural landmark there here foot of the building is what I call the origin so in that notation I want to ask when is y equal to 0 so ask when Y is equal to 0 then I say 0 is 15 plus 10 t minus 5 T Square or I canceling the 5 everywhere and changing the sign here I get T Square minus 2 t minus 3 equal to 0 that's when it hits ground so let's find out what the time is so T is then 2 plus or minus 4 plus 12 over 2 which is 2 plus or minus 4 over 2 which is minus 1 R 3 okay so you get two answers when it hits the ground so it's clear that we should pick three but you can ask why is it giving me a second solution anybody have an idea wives yes that's correct so her answer was if it had if it was the full parabola then we know it would have been at the ground before I set my clock to zero first of all negative time should not bother anybody T equals zero is when I set the clock a measured time forward but yesterday would be T below minus one day right so we don't have any trouble with negative x so the point is this equation let's not know about the building doesn't know the whole song and dance that you went to building any threw up a rock it knows nothing what does the mathematics know it knows that this particle happened to have a height of 15 at time zero and a velocity of 10 at time zero and it is falling under gravity with an acceleration of minus ten that's all it knows if that's all it knows then in that scenario if there is no building at anything else it continues a trajectory both forward in time and backward in time and it says that whatever seconds one second before you set your clock to zero it would have been in the ground what it means is if you release the rock at that location one second before with a certain speed that we can calculate it would have ended up here with precisely the position and velocity it had at the beginning of our experiment so sometimes the extra solution is very interesting and you should always listen to the mathematics when you get extra solutions in fact when very famous physicist called Dirac was looking for the energy of a particle in relativistic quantum mechanics he found the energy of a particle is connected to its momentum this P is what we call momentum and its mass by this relation so particle of mass m and momentum P has this energy so you solve for the energy you get two answers now your temptation is to keep the first answer because you know energy is not going to be negative particles moving it's got some energy and that's it but the mathematicians told Dirac you cannot ignore the negative energy solution because it tells you there's a second solution and you cannot throw them out and turns out the second solution with negative energy was when the theory is telling you hey there are particles and there are anti particles and the negative energy even properly interpreted will describe anti particles so the equations are very smart the way the physics works is you find some loss of motion in mathematical form you put in the initial conditions or whatever you solve the equations and the answer that comes you have no choice you have to accept the answer if there are new answers besides the one you were looking for you got to think about what they mean and that's one of the best things about physics because here is a person who is not looking for anti particles he was trying to describe electrons but the theory said there are two routes in the quadratic equation and the second route is mathematically as interesting as the first one it has to be part of the theory and in trying to adjust it so it can be incorporated you discovered anti particles so always amazing to us how we go into the problem our eye or mind can see one class of solutions but the math will tell you sometimes there are new solutions and you got respected and understand and interpret the unwanted solutions and this is a simple example where you can follow what the meaning of the second solution is it means that the problem you post there is more than the answers that you could imagine here it meant particle that was released from the ground earlier that it meant something much more interesting namely anti particles accompanying particles they are going to accompany particle surely as every squad ratting equation has two solutions all right so now in this problem we can do something slightly different and that's use this expression here and I will do that then I will stop for today if you are asking questions like how high does it go but you don't ask when go to the highest point then you don't have to go through the whole process of finding the time at which it turned around I don't know where that is that disappear in the blackboard then putting the time equal to one second into this formula if the question of time is not explicitly brought up then you should know that you have to use this formula so how do we get it here well we say at the top of the loop when the course of it comes down the velocity is zero therefore you say zero square equals initial velocity squared plus two times minus G that's my acceleration times y minus y zero if you solve for that you find y minus y 0 equals V zero squared over 2g and if you put in the V zero I gave you which is what 10 100 over 20 which is 5 meters so y equal to y 0 plus 5 meters and that is the height to which it rises I think we got it somewhere else we found the maximum height to be 20 meters another thing you can do is you can find the speed here if you want to find the speed there you put the equation V square equals V zero squared plus 2 times minus G what is y minus y 0 the final Y is 0 the initial Y is minus is 15 you solve for that equation then you will find the final velocity so if time is not involved you can do it that way I want to derive the last result in another way then I will stop and that's pretty interesting because it tells you use and abuse of calculus I'm going to find for you this result using calculus in a different way so from the calculus we know DV DT is equal to a so multiply both sides by V now you have to know from elementary calculus that V times DV DT is really D by DT of B squared over 2 now I hope you guys know that much calculus that when you take a derivative of a function of a function namely V squared over 2 is a function of V and V itself as a function of T then the rule for taking the derivative is first take the V derivative of this object then take the D by DT of V which is this one the right hand side I'm going to write this a DX DT this much is standard but now here is what I'm I'm going to do something which somehow we are told never ever to do which is to just cancel the DTS you all know that when you do the Y DX you're not supposed to cancel that D that's actually correct okay you don't want to cancel the D in the derivative but this happens to be completely legitimate so I'm going to assume it's true and I'll maybe take a second explain why it's legitimate what this really means is in a given time delta T the change in this quantity is 8 times the change in this quantity therefore you can multiply both sides by the delta T but the only thing you should understand is delta T as long as they're small and finite will lead to some small and finite errors in the formula because the formula is really the limit in which Delta X and delta T both code is 0 so what you have to do is multiply both sides by delta T but remember it's got to be in the end made vanishingly small as long as we understand that we can do this cancellation and this says on the left hand side the change in the quantity V squared over 2 is the change a times the change in the quantity X so add up all the changes or what mean by integral same thing add up all the changes the change in V squared over two will be the final V squared over two minus the initial V squared over two and the other side will be eight times the change in X X minus X naught and that's the formula I wrote for you u squared is V zero squared plus two a X minus X naught so the point is whenever you have derivatives with something over DT do not hesitate to cancel the DTS and think of them as a Delta V squared over two is equal to a times Delta X this will be actually true as long as both quantities or vanishingly small they will become more and more true as Delta X and Delta B squared become vanishingly small in the limit in which they're approaching zero the two will be in fact equal if Delta X is a finite amount like one second this will not be true because in the starting equation Delta X and delta T and Delta V Square or all assumed to be infinite decimal so don't hesitate to do manipulations of this type and I will do them quite often so you got to understand when it's okay and when it's not okay what this means is in a time delta T if this quantity changes by some amount in the same time delta T that quantity changes by some amount then keeping the delta T equal to some number we may equate the changes in the two quantities provided it is understood the delta v square over two is a change in V squared over two in the same time in which the particle moved a distance Delta X then by adding the differences we eliminate time and we get this final result alright so if you go to your website today you will find I have assigned some problems and you should try to do them they apply to this chapter then next week we'll do more complicated problems that involve motion in higher dimension how to go to two dimensions or three dimensions

Imagining the Fourth Dimension

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imagining the fourth dimension here's where we start getting into some possible confusion because the same word can have many different meanings when people say that time is the fourth dimension what does that mean the fourth dimension adds a way for the third dimension to change this is obvious when we say the third dimension is space without time but the entropy-driven arrow of time that people associate with this concept is obviously not spatial because it behaves in ways that are different from the first three dimensions this is why some people prefer to say that the fourth dimension is a temporal dimension while the first three are spatial but the more we learn about space-time and general relativity the more we realize that time is not just an arrow the fourth dimension stretches it bends and quantum entanglement shows that it's possible for particles to make instantaneous connections within it even for there to be causality in times reverse direction and as mind-blowing as this may be to fathom the accepted definition for antimatter is that it's matter which is moving backwards in time this is why with this project I prefer to call the fourth dimension duration I ask people to accept that time is a direction not a dimension in the same way that up or forward are directions rather than dimensions two opposing directions can be used to describe a spatial dimension and time and anti-time are two words we can use to describe the fourth dimension but they're not the only words and this is important because all we're really trying to do here is come up with words that describe the dimension which is at right angles to the third dimension here's something important to remember none of these dimensions exist in isolation you can't make a 1d line without using points you can't make a 2d plane without lines you can't make a 3d space without planes and you can't have a 4d duration without multiple plank frames of space saying the fourth dimension is duration makes no more sense than saying the third dimension is depth if when we say those phrases we're thinking you can have duration without space or depth without length and width saying the fourth dimension is space-time then at least acknowledges that the fourth dimension encompasses the dimensions from which it is constructed and doesn't exist in isolation from the other dimensions let me say this again it doesn't matter what label you put on the fourth dimension or any additional dimension as long as you're thinking about how the new dimension is somehow at right angles to the ones before a rose by any other name still smells a sweet to paraphrase mr. Shakespeare so time is not really a dimension but no matter what to mention you're examining the direction of time is a word we can use for tracking change from state to state in art there really eleven dimensions I insist that it makes no sense to say that the first three dimensions are spatial and the fifth dimension and above our spatial or at very least space like but then to say that the fourth dimension isn't spatial if that were the case then the mental castle we're building here has a very rickety layer at the fourth dimension and the whole structure is prone to crashing down last entry we talked about how it's really impossible for us to see the third dimension because it takes a certain amount of time for the light from anything in the third dimension to breach our eye and that's just as true for our hand in front of our face as it is for a star ten light years away saying that a third dimensional object has length width and depth is a phrase we casually say but we have to keep in mind that discussing a third dimensional object like a cube is the same as discussing dragons or Flatlanders a 3d cube is an idea which we can freely discuss but without using the fourth dimension to view such an object it's only a concept likewise persons who talked about tesseracts as being four dimensional objects say that this is what the real fourth dimension is like but what we're really talking about with a tesseract or any other n-dimensional shape is the same as a cube it's an idea in order for a tesseract to really exist it has to have a duration within its dimension and when we watch an animation of a rotating tesseract we are visualizing how that structure could rotate and change from state to state over time when viewed from the third dimension likewise just as a cube represents a simple and idealized shape within the third dimension but there are the limitless range of other shapes that can exist within the third dimension the additional degree of freedom afforded by the fourth spatial dimension allows for an even larger number of other shapes which can exist within that dimension one word physicists used to describe the path an object takes within space-time is a world line another word for a fourth dimensional shape coined by author and futurist Bruce sterling is a spine with my imagining the tenth dimension project I asked people to visualize themselves in the fourth dimension as a long undulating snake which is a way to think about the data set that represents a person's length or duration within the fourth dimension from conception to death do you see how that snake is a spine depending upon your point of view though that snake could be much blurrier than what we show in the original animation every day our bodies are exchanging atoms with the outside world through the air we breathe the food we eat and the water we drink a constant cycle of repairs and replacement means the spine representing a person from conception to death is a much more wide-ranging and interconnected shape than what we might first imagined one of the 26 songs attached to this project called change and renewal is about this idea let's finish off by thinking about the point line plane postulate again which can be used to visualize any number of spatial dimensions the trick I've suggested you start with each time is to think of a point that encompasses the entire dimension then find a point that is outside of what that first point encompasses so a one-dimensional point in the largest version of its indeterminate state occupies the entire length of a line and some new point not found anywhere on that line allows us to visualize the second dimension a two-dimensional point in its largest version fills an entire plane and a point not within that plane gets us to the third dimension a third dimensional point at its largest version is like a single Planck unit sized slice of the entire universe and allows us to think about the possibility that Julian Barbour is pointed out that each of those 3d frames allows for the instantaneous quantum connections often deemed as supremely mysterious and unfathomable having said that though we still have to decode the mystery of how we can have a physical world made out of objects that are not infinitely large within the third dimension and this is why I say those quantum connections are at right angles to space-time so let's continue the point line plane postulates logic into the fourth dimension a 4d point at its largest version would encompass the universe not just in space but in space-time the point would reach from the beginning to the end of the universe in the same way that a photon traveling at the speed of light would perceive itself to be simultaneously emitted from a distance are and arriving at an observers retina this is an important concept we looked at in light has no speed it also ties nicely to something Einstein said a number of times there is a way of thinking about reality in which the separation between past present and future is only a stubbornly persistent illusion what's outside that largest possible 4d point we've just imagined well if you are a person who has been trained to believe that free will is also nothing more than a stubbornly persistent illusion you might well say that's as far as we need to go after all if the universe was set in motion at the Big Bang and anything we do is an inevitable outcome based on what has come before then the largest 4d point we can imagine accounts for all of that from the beginning to the end including the now that each of us is observing at this very instant but what if you believe in free will with this project that's where we start to think whoa the fifth dimension

English Novel – A brief History of Time by Stephen Hawking Part 1 Complete analysis in Hindi

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Einstein's General Theory of Relativity | Lecture 1

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Lecture 1 of Leonard Susskind’s Modern Physics concentrating on General Relativity. Recorded September 22, 2008 at Stanford University.

This Stanford Continuing Studies course is the fourth of a six-quarter sequence of classes exploring the essential theoretical foundations of modern physics. The topics covered in this course focus on classical mechanics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.

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this program is brought to you by Stanford University please visit us at stanford.edu gravity gravity is a rather special force it's unusual it has different than electrical forces magnetic forces and it's connected in some way with geometric properties of space space and time but before and that connection is of course the general theory of relativity before we start tonight for the most part we will not be dealing with the general theory of relativity we will be dealing with gravity in its oldest and simplest mathematical form well perhaps not the oldest and simplest but Newtonian gravity and going a little beyond what Newton certainly nothing that Newton would not have recognized or couldn't have grasped Newton could grasp anything but some ways of thinking about it which will not be found in Newton's actual work but still lutonium gravity the Toney and gravity set up in a way that that is useful for going on to the general theory ok let's begin with Newton's equations the first equation of course is F equals MA force is equal to mass times acceleration let's assume that we have a frame of reference a frame of reference that it means a set of coordinates and as a collection of clocks and those frame and that frame of reference is what is called an inertial frame of reference an inertial frame of reference simply means 1 which if there are no objects around to exert forces on a particular let's call it a test object a test object is just some object a small particle or anything else that we use to test out the various fields force fields that might be acting on it the inertial frame is one which when there are no objects around to exert forces that object will move with you for motion with no acceleration that's the idea of an inertial frame of reference and so if you're an inertial frame of reference and you have a pen and you just let it go it stays there it doesn't move if you give it a push it will move off with uniform velocity that's the idea of an inertial frame of reference and in an inertial frame of reference the basic Newtonian equation number one I always forget which law is which there's Newton's first law second law and third law I never can remember which is which but they're all pretty much summarized by f equals mass times acceleration this is a vector equation I expect people know what a vector is a three vector equation will come later to four vectors where when space and time are united into space time but for the moment space is space and time is time and a vector means a thing which is like a pointer in a direction in space as a magnitude and that has components so component by component the X component of the force is equal to the mass of the object times the X component of acceleration Y component Z component and so forth in order to indicate that something is a vector equation I'll try to remember to put an arrow over vectors the mass is not a vector the mass is simply a number every particle has a mass every object has a mass and in Newtonian physics the mass is conserved that does not change now of course the mass of this cup of coffee here can change it's lighter now but it only changes because mass has been transported from one place to another so you can change the mass of an object by whacking off a piece of it and but if you don't change the number of particles change the number of molecules and so forth then the mass is a conserved unchanging quantity so that's first equation now let me write that in another form the other form we imagine we have a coordinate system an X a Y and a Z I don't have enough directions on the blackboard to draw Z I won't bother there's x y and z sometimes we just call them x1 x2 and x3 I guess I can draw it in x3 is over here someplace XY and Z and a particle has a position which means it has a set of three coordinates sometimes we will summarize the collection of the three coordinates x1 x2 and x3 incidentally x1 and x2 and x3 are components of a vector the components they are components of the position vector of the particle position vector of the particle I will often call either small R or large are depending on on the particular context R stands for radius but the radius simply means the distance between a point and the origin for example we're really talking now about a thing with three components XY and Z and it's the radial vector the radial vector this is the same thing as the components of the vector R alright the acceleration is a vector that's made up out of the time derivatives of XY and Z or X 1 X 2 and X 3 so for each component the compose for each component one two or three the acceleration which let me indicate well let's just call it a the acceleration is just equal the components of it are equal to the second derivatives of the coordinates with respect to time that's what acceleration is the first derivative of position is called velocity we can take this to be component by component x1 x2 and x3 the first derivatives velocity the second derivative is acceleration we can write this in vector notation I won't bother but we all know what we mean I hope we all know we mean buddies by acceleration and velocity and so Newton's equations are then summarized and summarized but rewritten as the force on an object whatever it is component by component is equal to the mass times the second derivative of the component of position so that's the summary of I think it's Newton's first and second law I can never remember which they are Newton's first law of course is simply the statement that if there are no forces then there's no acceleration that's Newton's first law equal and opposite right so this summarizes both the first and second law I never understood why there was a first and second law it seems to me there was just one F equals MA all right now let's begin even even previous to Newton with Galilean gravity gravity as Galileo understood it actually I'm not sure how much of this mathematics Galileo did or didn't understand he certainly knew what acceleration was he measured it I don't know that he had thee but he certainly didn't have calculus but he knew what acceleration was so what Galileo studied was the motion of objects in the gravitational field of the earth in the approximation that the earth is flat now Galileo knew the earth wasn't flat but he studied gravity in the approximation where you never moved very far from the surface of the earth and if you don't move very far from the surface of the earth you might as well take the surface of the earth to be flat and the significance of that is to twofold first of all the direction of gravitational forces is the same everywhere as this is not true of course if the earth is curved then gravity will point toward the center but in the flat space approximation gravity points down down everywhere is always in the same direction and second of all perhaps a little bit less obvious but nevertheless true then the approximation where the earth is Infinite and flat goes on and on forever infinite and flat the gravitational force doesn't depend on how high you are same gravitational force here as here the implication of that is that the acceleration of gravity since force apart from the mass of an object the acceleration on an object is independent the way you put it and so Galileo either did or didn't realize well he again I don't know exactly what Galileo did or didn't know but what he said was equivalent to saying that the force on an object in the flat space approximation is very simple its first of all has only one component pointing downward if we take the upward sense of things to be positive then we would say that the force is let's just say the component of the force in the X 2 direction the vertical Direction is equal to minus the minus simply means that the force is downward and it's proportional to the mass of the object times a constant called the gravitational acceleration now the fact that it's constant Everywhere's in other words mass times G doesn't vary from place to place that's this fact that gravity doesn't depend on where you are in the flat space approximation but the fact that the force is proportional to the mass of an object that is not obvious in fact for most forces it's not true for electric forces the force is proportional to the electric charge not to the mass and so gravitational forces are rather special the strength of the gravitational force on an object is proportional to its mass that characterizes gravity almost completely that's the special thing about gravity the force is proportional itself to the mass well if we combine F equals MA with the force law this is the law force then what we find is that mass times acceleration the second X now this is the vertical component by DT squared is equal to minus that's the minus M G period that's it now the interesting thing that happens in gravity is that the mass cancels out from both sides that is what's special about gravity the mass cancels out from both sides and the consequence of that is that the motion of an object its acceleration doesn't depend on the mass it doesn't depend on anything about the particle a particle object I'll use the word particle I don't necessarily mean the point the small particle or baseb as a particle an eraser is a particle a piece of chalk is a particle that the motion of the object doesn't depend on the mass of the object or anything else the result of that is that if you take two objects of quite different mass and you drop them they fall exactly the same way our Galileo did that experiment I don't know if they're whether he really threw something off the Leaning Tower of Pisa or not it's not important he yeah he did balls down an inclined plane I don't know whether he actually did or didn't I know the the the myth is that he didn't die I find it very difficult to believe that he didn't I've been in Pisa last week I was in Pisa and I took a look at the Leaning Tower of Pisa galileo was born and lived in Pisa he was interested in gravity how it would be possible that he wouldn't think of dropping something off the Leaning power tower is beyond my comprehension you look at that tower and you say that I was good for one thing dropping things off now I don't know maybe the Doge or whoever they call the guy at the time said no no Galileo you can't drop things from the tower you'll kill somebody so maybe he didn't but he must have surely thought of it all right so the result had he done it and had he not had to worry about such spurious effects as air resistance would be that a cannonball and a feather would fall in exactly the same way independent of the mass and the equation would just say the acceleration would first of all be downward that's the minus sign and equal to this constant G excuse me that mean yes now G is a number it's 10 meters per second per second at the surface of the earth at the surface of the Moon it's something smaller and the surface of Jupiter it's something larger so it does depend on the mass of the planet but the acceleration doesn't depend on the mass of the object you're dropping it depends on the mass of the object you're dropping it onto but not the mass of the object that's dropping that fact that gravitational motion is completely independent the mass is called or it's the simplest version of something that's called the equivalence principle why it's called the equivalence principle we'll come to later what's equivalent to what at this stage we could just say gravity is equivalent between all different objects independent of their mass but that is not exact were the equivalents an equivalence principle was about that has a consequence an interesting consequence supposing they take some object which is made up out of something which is very unwritten just a collection of point masses maybe maybe let's even say that not even they're not even exerting any forces on each other it's a cloud a varied a few diffuse cloud of particles and we watch it fall let's suppose we start each particle from rest not all at the same height and we let them all fall some particles are heavy some particles are light some of them may be big some of them may be small how does the whole thing fall the answer is all of the particles fall at exactly the same rate the consequence of it is that the shape of this object doesn't deform as it falls it stays absolutely unchanged the relationship between the neighboring parts are unchanged there are no stresses or strains which tend to deform the object so even if the object were held together by some sort of struts or whatever there would be no forces on those struts because everything falls together the consequence of that is the falling in a gravitational field is undetectable you can't tell that you're falling in a gravitational field by you when I say you can't tell certainly you can tell the difference between freefall and standing on the earth that's not the point the point is that you can't tell by looking at your neighbors or anything else that there's a force being exerted on you and that that force that's being exerted on you is pulling you down word you might as well for all practical purposes be infinitely far from the earth with no gravity at all and just sitting there because as far as you can tell there's no tendency for the gravitational field to deform this object or anything else you cannot tell the difference between being in free space infinitely far from anything with no forces and falling freely in a gravitational field that's another statement of the equivalence principle for example these particles could be equipped with lasers lasers and optical detectives of some sort what's that oh you could certainly tell if you was standing on the floor here you could tell that something was falling toward you but the question is from within this object by itself without looking at the floor without knowing the floor was it well you can't tell whether you're falling and it's yeah yeah if there was something that was calm that was not falling it would only be because there was some other force on it like a beam or a tower of some sort of holding it up why because this object if there are no other forces on and only the gravitational forces it will fall at the same rate as this all right so that's another expression of the equivalence principle that you cannot tell the difference between being in free space far from any gravitating object versus being in a gravitational field that we're going to modify this this is of course it's not quite true in a real gravitational field but in this flat space approximation where everything moves together you cannot tell that there's a gravitational field or at least you cannot tell the difference I will not without seeing the floor in any case the self-contained object here does not experience anything different than it would experience far from any gravitating hating object standing still or uniform in uniform motion no you're accelerating if you go up to the top of a high building and you close your eyes and you step off and go into freefall you will feel exactly the same you feel weird I mean that's not the way you usually feel because your stomach will come up and you know do some funny things you know you might you might lose it but uh but the point is you would feel exactly the same discomfort in outer space far from any gravitating object just standing still you feel exactly the same peculiar feelings one of those peculiar feelings due to they're not due to falling they do to not fall well they do to the fact that when you stand on the earth here there are forces on the bottoms of your feet which keep you from falling and if the earth suddenly disappeared from under my feet sure enough my feet would feel funny because they used to having those forces exerted on their bottoms you get it I hope so the fact that you feel funny in freefall is because you're not used to freefall and it doesn't matter whether you're infinitely far from any gravitating object standing still or freely falling in the presence of a gravitational field now as I said this will have to be modified in a little bit there are such things as tidal forces those tidal forces are due to the fact that the earth is curved and that the gravitational field is not the same in every same direction in every point and that it varies with height that's due to the finiteness of the earth but in the flat space surprise and the Flat Earth approximation where the earth is infinitely big pulling uniformly there is no other effective gravity that is any different than being in free space okay again that's known as the equivalence principle now let's go on beyond the flat space or the Flat Earth approximation and move on to Newton's theory of gravity Newton's theory of gravity says every object in the universe exerts a gravitational force on every other object in the universe let's start with just two of them equal and opposite attractive attractive means that the direction of the force on one object is toward the other one equal and opposite forces and the magnitude of the force the magnitude of the force of one object on another let's let's characterize them by a mass let's call this one little m think of it as a lighter mass and this one which we can imagine as a heavier object will call it begin all right Newton's law of force is that the force is proportional to the product of the masses making either mass heavier will increase the force or the product of the masses begin tons of little m inversely proportional to the square of the distance between them let's call that R squared let's call the distance between them are and there's a numerical constant this for this law by itself could not possibly be right it's not dimensionally consistent the if you work out the dimensions of force mass mass and R will not dimensionally consistent there has to be a numeric constant in there and that numerical constant is called capital G Newton's constant and it's very small it's a very small constant I'll write down what it is G it is equal to six or six point seven roughly times 10 to the minus 11th which is a small number so in the face of it it seems that gravity is a very weak force you might not think that gravity is such a weak force but to convince yourself it's a weak force there's a simple experiment that you can do week week by comparison with other forces I've done this for car classes and you can do it yourself just take an object hanging by a string and two experiments the first experiment take a little object here and electrically charge it electrically charge it by rubbing it on your sweater that doesn't put very much electric charge on it but it charges it up enough to feel some electrostatic force and then take another object of exactly the same kind rub it on your shirt and put it over here what happens they repel and the fact that they repel means that this string will shift and you'll see a shift take another example take your little ball there to be iron and put a magnet next to it again you'll see quite an easily detectable deflection of the of the string holding it next take a 10,000 pound weight and put it over here guess what happens undetectable you cannot see anything happen the gravitational force is much much weaker than most other kinds of forces and that's due to the or not due to but the not due to that the fact that it's so weak is encapsulated in this small number here another way to say it is if you take two masses each of one kilometer not one kilometer one kilogram kilogram is a good healthy mass right nice chunk of iron mm and you separate them by one meter then the force between them is just G and it's six point seven times ten to the minus eleven the you know the units being Newtons so it's very very weak force but weak as it is we feel that rather strenuously we feel it strongly because the earth is so darn heavy so the heaviness of the earth makes up for the smallness of G and so we wake up in the morning feeling like we don't want to get out of bed because gravity is holding us down Oh Oh the equal and opposite equal and opposite that's the that's the rule that's Newton's third law the forces are equal and opposite so the force on the large one due to the small one is the same as the force of the small one on the large one and but it is proportional to the product of the masses so the meaning of that is I'm not heavier than I like to be but but I'm not very heavy I'm certainly not heavy enough to deflect the hanging weight significantly but I do exert a force on the earth which is exactly equal and opposite to the force that they're very heavy earth exerts on me why does the earth excel if I dropped from a certain height I accelerate down the earth hardly accelerates at all even though the forces are equal why is it that the earth if the forces are equal my force on the earth and the Earth's force on me of equal why is it that the earth accelerates so little and I accelerate so much yeah because the acceleration involves two things it involves the force and the mass the bigger the mass the less the acceleration for a given force so the earth doesn't accelerate quickly I think it was largely a guess but there was certain was an educated guess and what was the key ah no no it was from Kepler's it was from Kepler's laws it was from Kepler's laws he worked out roughly speaking I don't know exactly what he did he was rather secretive and he didn't really tell people what he did but the piece of knowledge that he had was Kepler's laws of motion planetary motion and my guess is that he just wrote down a general force realized that he would get Kepler's laws of motion for the inverse-square law I don't believe he had any underlying theoretical reason to believe in the inverse-square law that's correct he asked a question for inverse square laws no no it wasn't the ellipse which was the the the orbits might have been circular it was the fact that the period varies is the three halves power of the radius all right the period of motion is circular motion has an acceleration toward the center any motion in the circle is accelerated to the center if you know the period in the radius then you know the acceleration toward the center okay or we could write let's let's do it anybody know what if I know the angular frequency the angular frequency of going around in an orbit that's called Omega you know a–they and it's basically just the inverse period okay Omega is roughly the inverse period number of cycles per second what's the what is the acceleration of a thing moving in a circular orbit anybody remember Omega squared R Omega squared R that's the acceleration now supposing he sets that equal to some unknown force law f of r and then divides by r then he finds Omega as a function of the radius of the orbit okay well let's do it for the real case for the real case inverse square law f of r is 1 over r squared so this would be 1 over r cubed and in that form it is Kepler's second law remember which one it is it's the law that says that the frequency or the period the square of the period is proportional to the cube of the radius that was the law of Kepler so from Kepler's laws he easily could have that that one law he could easily deduce that the force was proportional to 1 over R squared I think that's probably historically what what he did then on top of that he realized if you didn't have a perfectly circular orbit then the inverse square law was the unique law which would give which would give elliptical orbits so who's to say well then of course there are the forces on them for two objects are actually touching each other there are all sorts of forces between them that I'm not just gravitational electrostatic forces atomic forces nuclear forces so you'll have to my breaks down yeah then it breaks down when they get so close that other important forces come into play the other important forces for example are the forces that are holding this object and preventing it from falling these we usually call them contact forces but in fact what they really are is various kinds of electrostatic for electrostatic forces between the atoms and molecules in the table in the atoms and molecules in here so other kinds of forces all right incidentally let me just point out if we're talking about other kinds of force laws for example electrostatic force laws then the force we still have F equals MA but the force law the force law will not be that the force is somehow proportional to the mass times something else but it could be the electric charge if it's the electric charge then electrically uncharged objects will have no forces on them and they won't accelerate electrically charged objects will accelerate in an electric field so electrical forces don't have this Universal property that everything falls or everything moves in the same way uncharged particles move differently than charged particles with respect to electrostatic forces they move the same way with respect to gravitational forces and as a repulsion and attraction whereas gravitational forces are always attractive where where's my gravitational force I lost it yeah here is all right so that's that's Newtonian gravity between two objects for simplicity let's just put one of them the heavy one at the origin of coordinates and study the motion of the light one then Oh incidentally one usually puts let me let me refine this a little bit as I've written it here I haven't really expressed it as a vector equation this is the magnitude of the force between two objects thought of as a vector equation we have to provide a direction for the force vectors have directions what direction is the force on this particle well the answer is its along the radial direction itself so let's call the radial distance R or the radial vector R then the force on little m here is along the direction R but it's also opposite to the direction of R the radial vector relative to the origin over here points this way on the other hand the force points in the opposite direction if we want to make a real vector equation out of this we first of all have to put a minus sign that indicates that the force is opposite to the direction of the radial distance here but we have to also put something in which tells us what direction the force is in it's along the radial direction but wait a minute if I multiply it by r up here I had better divide it by another factor of R downstairs to keep the magnitude unchanged the magnitude of the force is 1 over R squared if I were to just randomly come and multiply it by r that would make the magnitude bigger by a factor of our so I have to divide it by the magnitude of our this is Newton's force law expressed in vector form now let's imagine that we have a whole assembly of particles a whole bunch of them they're all exerting forces on one another in pairs they exert exactly the force that Newton wrote down but what's the total force on a particle let's label these particles this is the first one the second one the third one the fourth one that I thought that thought this is the ithe one over here so I is running index which labels which particle we're talking about the force on the eigth article let's call F sub I and let's remember that it's a vector it's equal to the sum now this is not an obvious fact that when you have two objects exerting a force on the third that the force is necessarily equal to the sum of the two forces of the two are of the two objects you know what I mean but it is a fact anyway obvious on how obvious it is a fact that gravity does work that way at least in the Newtonian approximation with Einstein it breaks down a little bit but in Newtonian physics the force is the sum and so it's a sum over all the other particles let's write that J not equal to I that means it's a sum over all not equal to I so the force on the first particle doesn't come from the first particle it comes from the second particle third particle fourth particle and so forth each individual force involves M sub I the force of the ice particle times the four times the mass of the Jade particle product of the masses divided by the square of the distance between them let's call that R IJ squared the distance between the eigth article his I and J the distance between the earth particle and the J particle is RI J but then just as we did before we have to give it a direction but a minus sign here that indicates that it's attractive another R IJ upstairs but that's a vector R IJ and make this cube downstairs alright so that says that the force on the I've particle is the sum of all the forces due to all the other ones of the product of their masses inverse square in the denominator and the direction of each individual force on this particle is toward the other all right this is a vector sum yeah hmm the minus indicates that it's attractive excellent but you've got the vector going from like a J oh let's see that's a vector going from the J yes there is a question of the sine of this vector over here so yeah you know absolutely let's yeah I actually think it's yeah you're right you're absolutely right the way I've written that there should not be a minus sign here all right but if I put our ji there then there would be a minus sign right so you're right but in any case every one every one of the forces is attractive and what we have to do is to add them up we have to add them up as vectors and so there's some resulting vector some resultant vector which doesn't point toward any one of them in particular but points in some direction which is determined by the vector sum of all the others all right but the interesting fact is if we combine this this is the force on the earth particle if we combine it with Newton's equations let's combine it with Newton's equipped with Newton's F equals MA equations then this is F this on the ice particle this is equal to the mass of the I particle times the acceleration of the ice particle again vector equations now the sum here is over all the other particles we're focusing on number I I the mass of the ice particle will cancel out of this equation I don't want to throw it away but let's just circle it and now put it over on the side we notice that the acceleration of the ice particle does not depend on its mass again once again because the mass occurs in both sides of the equation it can be cancelled out and the motion of the ayth particle does not depend on the mass of the earth particle it depends on the masses of all the other ones all the other ones come in but the mass of the iPart achill cancels out of the equation so what that means is if we had a whole bunch of particles here and we added one more over here its motion would not depend on the mass of that particle it depends on the mass of all the other ones but it doesn't depend on the mass of the i particle here okay that's again the equivalence principle that the motion of a particle doesn't depend on its mass and again if we had a whole bunch of particles here if they were close enough together they were all moving the same way before before i discuss lumo mathematics let's just discuss tidal forces what tidal forces are once you set this whole thing into motion dynamic young we have all different masses and each part what's going to be affected by each one is every particle in there is going to experience a uniform acceleration oh no no no no no acceleration is not uniform the acceleration will get larger when it gets closer to one of the particles it won't be uniform anymore it won't be uniform now because the force is not independent of where you are now the force depends on where you are relative to the objects that are exerting the force it was only in the Flat Earth approximation where the force didn't depend on where you were okay now the force varies so it's larger when you're far away it's sorry it's smaller when you're far away it's larger when you're in close it changes in a vector form with each individual particles each one of them is changing position yeah and and so is the dynamics that every one of them is going towards the center of gravity of the fire not necessarily I mean they could be flying apart from each other but they will be accelerating toward each other okay if I throw this eraser into the air with greater than the escape velocity it's not going to turn around and fall back changing with what with respect to what time oh it changes with respect to time because the object moves moves further and further away it's not uniformly the radius is changing and it's yeah let's take the earth here's the earth and we drop a small mass from far away as that mass moves in its acceleration increases why does its acceleration increase the deceleration increases because the radial distance gets smaller so in that sense it's not the alright now once the gravitational force depends on distance then it's not really quite true that you don't feel anything in a gravitational field you feel something which is to some extent it different than you would feel in free space without any gravitational field the reason is more or less obvious here you are his is the earth now you're you or me or whoever it is happens to be extremely tall a couple of thousand miles tall well this person's feet are being pulled by the gravitational field more than his head or another way of saying the same thing is if let's imagine that the person is very loosely held together he's just more or less a gas of we are pretty loosely held together at least I am right all right the acceleration on the lower portions of his body are larger than the accelerations on the upper portions of his body so it's quite clear what happens to her he gets stretched he doesn't get a sense of falling as such he gets a sense of stretching being stretched feet being pulled away from his head at the same time let's uh let's all right so let's change the shape a little bit I just spend the week two weeks in Italy and my shape changes whenever I go to Italy and it tends to get more horizontal my head is here my feet are here and now I'm this way still loosely put together right now what well not only does the force depend on the distance but it also depends on the direction the force arm my left end over here is this way the force on my right end over here is this way the force on the top of my head is down but it's weaker than the force on my feet so there are two effects one effect is to stretch me vertically it's because my head is not being pulled as hard as my feet but the other effect is to be squished horizontally by the fact that the forces on the left end of me are pointing slightly to the right and the forces on the right end of me are pointing slightly to the left so a loosely knit person like this falling in freefall near a real planet or real gravitational object which has a real Newtonian gravitational field around it will experience a distortion will experience a degree of distortion and a degree of being stretched vertically being compressed horizontally but if the object is small enough or small enough mean let's suppose the object that's falling is small enough if it's small enough then the gradient of the gravitational field across the size of the object will be negligible and so all parts of it will experience the same gravitational acceleration all right so tidal for these are tidal forces these forces which tend to tear things apart vertically and squish them this way tidal forces tidal forces are forces which are real you feel them I mean certainly new the car the cause of the tides yeah I don't know to what extent he calculated what do you mean calculated the well I doubt that he was capable I'm not sure whether he estimated the height of the of the deformation of the oceans or not but I think you did understand this much about tides okay so that's the that's what's called tidal force and then under the tidal force has this effect of stretching and in particular if we take the earth just to tell you just to tell you why it's called tidal forces of course it's because it has to do with tides I'm sure you all know the story but if this is the moon down here then the moon exerting forces on the earth exerts tidal forces on the earth which means to some extent that tends to stretch it this way and squash it this way well the earth is pretty rigid so it doesn't it doesn't deform very much due to due to these two the moon but what's not rigid is the layer of water around it and so the layer of water tends to get stretched and squeezed and so it gets deformed into the a the form shell of water with a bump on this side and the bump on that side alright I'm not gonna go any more deeply into that that I'm sure you've all seen okay but let's define now what we mean by the gravitational field the gravitational field is abstracted from this formula we have a bunch of particles don't you have need some some coordinate geometry so that would you have the four kind of middle is being pulled by all the other guys on the side I'm not explaining it right it's always negative is that what you're saying doesn't know I'm saying so she's attractive all right so you have but what about the other guys that are pulling upon him a different direction here and we're talking about the force on this person over here obviously there's one force pressing this pushing this way and another force pushing that way okay no the cone no they're all opposite to the direction of the object which is pulling on that's what this – sorry instead well you kind of retracted the minus sign at the front and reverse the ji yeah so it's the trend we can get rid of a – like a RI j and our ji are opposite to each other one of them is the vector between I and J I and J and the other one is the vector from J to I so they're equal and opposite to each other the minus sign there look as far as the minus sign goes all it means is that every one of these particles is pulling on this particle toward it as opposed to pushing away from it it's just a convention which keeps track of attraction instead of repulsion yeah for the for the ice master that's my word you want to make sense but if you can look at it as a kind of an in Samba wasn't about a linear conic component to it because the ice guy affects the Jade guy and then put you compute the Jade guy when you take it yeah now what this what this formula is for is supposing you know the positions or all the others you know that then what is the force on the one additional one but you're perfectly right once you let the system evolve then each one will cause a change in motion and the other one and so it becomes a complicated as you say nonlinear mess but this formula is a formula for if you knew the position and location of every particle this would be the force something you need to solve some equations to know how the particles move but if you know where they are then this is the force on the particle alright let's come to the idea of the gravitational field the gravitational field is in some ways similar to the electric field of our of an electric charge it's the combined effect of all the masses Everywhere's and the way you define it is as follows you imagine an one more particle one more particle amount you can take it to be a very light particle so it doesn't influence the motion of the others and one more particle in your imagination you don't really have to add it in your imagination and ask what the force on it is the force is the sum of the forces due to all the others it is proportional each term is proportional to the mass of the sec strip article this extra particle which may be imaginary is called a test particle it's the thing that you're imagining testing out the gravitational field with you take a light little particle and you put it here and you see how it accelerates knowing how it accelerates tells you how much force is on it in fact it just tells you how it accelerates and you can go around and imagine putting it in different places and mapping out the force field that's on that particle or the acceleration field since we already know that the force is proportional to the mass then we can just concentrate on the acceleration the acceleration all particles will have the same acceleration independent of the mass so we don't even have to know what the mass of the particle is we put something over there a little bit of dust and we see how it accelerates acceleration is a vector and so we map out in space the acceleration of a particle at every point in space either imaginary or real particle and that gives us a vector field at every point in space every point in space there is a gravitational field of acceleration it can be thought of as the acceleration you don't have to think of it as force acceleration the acceleration of a point mass located at that position it's a vector it has a direction it has a magnitude and it's a function of position so we just give it a name the acceleration due to all the gravitating objects it's a vector and it depends on position here X means location it means all of the position components of position XY and Z and it depends on all the other masters in the problem that is what's called the gravitational field it's very similar to the electric field except the electric field and the electric field is force per unit charge it's the force on an object divided by the charge on the object the gravitational field is the force of their on the object divided by the mass on the object since the force is proportional to the mass the the acceleration field doesn't depend on which kind of particle we're talking about all right so that's the idea of a gravitational field it's a vector field and it varies from place to place and of course if the particles are moving it also varies in time if everything is in motion the gravitational field will also depend on time we can even work out what it is we know what the force on the earth particle is all right the force on a particle is the mass times the acceleration so if we want to find the acceleration let's take the ayth particle to be the test particle little eye represents the test particle over here let's erase the intermediate step over here and write that this is in AI times AI but let me call it now capital a the acceleration of a particle at position X is given by the right hand side and we can cross out BMI because it cancels from both sides so here's a formula for the gravitational field at an arbitrary point due to a whole bunch of massive objects a whole bunch of massive objects an arbitrary particle put over here will accelerate in some direction that's determined by all the others and that acceleration is the gravitation the definition is the definition of the gravitational field ok let's um let's take a little break we usually take a break in about this time and I recover my breath to go on we need a little bit of fancy mathematics we need a piece of mathematics called Gauss's theorem and Gauss's theorem involves integrals derivatives divergences and we need to spell those things out there a central part of the theory of gravity and much of these things we've done in the context of a lot of electrical forces in particular the concept of divergence divergence of a vector field so I'm not going to spend a lot of time on it if you need to fill in then I suggest you just find any little book on vector calculus and find out what a divergence and a gradient and a curl we don't do curl today what those concepts are and look up Gauss's theorem and they're not terribly hard but we're gonna go through them fairly quickly here since they we've done them several times in the past right imagine that we have a vector field let's call that vector field a it could be the field of acceleration and that's the way I'm gonna use it well for the moment it's just an arbitrary vector field a it depends on position when I say it's a field the implication is that it depends on position now I probably made it completely unreadable a of X varies from point to point and I want to define a concept called the divergence of the field now it's called the divergence because one has to do is the way the field is spreading out away from a point for example a characteristic situation where we would have a strong divergence for a field is if the field was spreading out from a point like that the field is diverging away from the point incidentally if the field is pointing inward then one might say the field has a convergence but we simply say it has a negative divergence all right so divergence can be positive or negative and there's a mathematical expression which represents the degree to which the field is spreading out like that it is called the divergence I'm going to write it down and it's a good thing to get familiar with certainly if you're going to follow this course it's a good thing to get familiar with but are they going to follow any kind of physics course past freshman physics the idea of divergence is very important all right supposing the field a has a set of components the one two and three component but we could call them the x y&z component now I'll use x y&z are X Y & Z which I previously called X 1 X 2 and X 3 it has components X a X a Y and a Z those are the three components of the field well the divergence has to do among other things with the way the field varies in space if the field is the same everywhere as in space what does that mean that would mean the field that has both not only the same magnitude but the same direction everywhere is in space then it just points in the same direction everywhere else with the same magnitude it certainly has no tendency to spread out when does a field have a tendency to spread out when the field varies for example it could be small over here growing bigger growing bigger growing bigger and we might even go in the opposite direction and discover that it's in the opposite direction and getting bigger in that direction then clearly there's a tendency for the field to spread out away from the center here the same thing could be true if it were varying in the vertical direction or who are varying in the other horizontal direction and so the divergence whatever it is has to do with derivatives of the components of the field I'll just tell you exactly what it is it is equal to the divergence of a field is written this way upside down triangle and the meaning of this symbol the meaning of an upside down triangle is always that it has to do with the derivatives the three derivatives derivative whether it's the three partial derivatives derivative with respect to XY and Z and this is by definition the derivative with respect to X of the X component of a plus the derivative with respect to Y of the Y component of a plus the derivative with respect to Z of the Z component of it that's definition what's not a definition is a theorem and it's called Gauss's theorem no that's a scalar quantity that's a scalar quantity yeah it's a scalar quantity so it's let me write it it's the derivative of a sub X with respect to X that's what this means plus the derivative of a sub Y with respect to Y plus the derivative of a sub Z with respect to Z yes so the arrows you were drawn over there those were just a on the other board you drew some arrows on the other board that are now hidden yeah those were just a and a has a divergence when it's spreading out away from a point but that there vergence is itself a scalar quantity oh let me try to give you some idea of what divergence means in a context where you can visualize it imagine that we have a flat lake alright just the water thin a a shallow lake and water is coming up from underneath it's being pumped in from somewheres underneath what happens that the water is being pumped in of course it tends to spread out let's assume that the height let's assume the depth can't change we put a lid over the whole thing so it can't change its depth we pump some water in from underneath and it spreads out okay we suck some water out from underneath and it spreads in it anti spreads it has so the spreading water has a divergence water coming in toward the towards the place where it's being sucked out it has a convergence or a negative divergence now we can be more precise about that we look down at the lake from above and we see all the water is moving of course it's moving if it's being pumped in the world it's moving and there is a velocity vector at every point there is a velocity vector so at every point in this lake there's a velocity vector vector and in particular if there's water being pumped in from the center here right underneath the bottom of the lake there's some water being pumped in the water will spread out away from that point okay and there'll be a divergence where the water is being pumped in okay if the water is being pumped out then exactly the opposite the the arrows point inward and there's a negative divergence the if there's no divergence then for example a simple situation with no divergence that doesn't mean the water is not moving but a simple example with no divergence is the waters all moving together you know the river is simultaneous the lake is all simultaneously moving in the same direction with the same velocity it can do that without any water being pumped in but if you found that the water was moving to the right on this side and the left on that side you'd be pretty sure that somebody is in between water had to be pumped in right if you found the water was spreading out away from a line this way here and this way here then you'd be pretty sure that some water was being pumped in from underneath along this line here well you would see it another way you would discover that the X component of the velocity has a derivative it's different over here than it is over here the X component of the velocity varies along the x direction so the fact that the X component of the velocity is varying along the direction there's an indication that there's some water being pumped in here likewise if you discovered that the water was flowing up over here and down over here you would expect that in here somewhere as some water was being pumped in so derivatives of the velocity are often an indication that the some water being pumped in from underneath that pumping in of the water is the divergence of the velocity vector now the the the the water of course is being pumped in from underneath so there's a direction of flow but it's coming from from underneath there's no sense of direction well okay that's that's what diverges just the diagrams you already have on the other board behind there you take say the rightmost arrow and you draw a circle between the head and tail in between then you can see the in and out the in arrow and the arrow of a circle right in between those two and let's say that's the bigger arrow is created by a steeper slope of the street it's just faster it's going fast it's going okay and because of that there's a divergence there that's basically it's sort of the difference between that's right that's right if we drew a circle around here or we would see that more since the water was moving faster over here than it is over here more water is flowing out over here then it's coming in over here where is it coming from it must be pumped in the fact that there's more water flowing out on one side then it's coming in from the other side must indicate that there's a net inflow from somewheres else and the somewheres else would be from the pump in water from underneath so that's that's the idea of oops could it also be because it's thinning out with that be a crazy example like the late guy young well okay I took all right so let's be very specific now I kept the lake having an absolutely uniform height and let's also suppose that the density of water water is an incompressible fluid it can't be squeezed it can't be stretched then the velocity vector would be the right thing to think about them yeah but you could have no you're right you could have a velocity vector having a divergence because the water is not because water is flowing in but because it's thinning out yeah that's that's also possible okay but let's keep it simple all right and you can have the idea of a divergence makes sense in three dimensions just as well as two dimensions you simply have to imagine that all of space is filled with water and there are some hidden pipes coming in depositing water in different places so that it's spreading out away from points in three-dimensional space in three-dimensional space this is the expression for the divergence if this were the velocity vector at every point you would calculate this quantity and that would tell you how much new water is coming in at each point of space so that's the divergence now there's a theorem which the hint of the theorem was just given by Michael there it's called Gauss's theorem and it says something intuitive very intuitively obvious you take a surface any surface take any surface or any curve in two dimensions and now suppose there's a vector field that the field points now think of it as the flow of water and now let's take the total amount of water that's flowing out of the surface obviously there's some water flowing out over here and of course we want to subtract the water that's flowing in let's calculate the total amount of water that's flowing out of the surface that's an integral over the surface why is it an integral because we have to add up the flows of water outward where the water is coming inward that's just negative negative flow negative outward flow we add up the total outward flow by breaking up the surface into little pieces and asking how much flow is coming out from each little piece yeah how much water is passing out through the surface if the water is incompressible incompressible means density is fixed and furthermore the depth of the water is being kept fixed there's only one way that water can come out of the surface and that's if it's being pumped in if there's a divergence the divergence could be over here could be over here could be over here could be over here in fact any ways where there's a divergence will cause an effect in which water will flow out of this region yeah so there's a connection there's a connection between what's going on on the boundary of this region how much water is flowing through the boundary on the one hand and what the divergence is in the interior the connection between the two and that connection is called Gauss's theorem what it says is that the integral of the divergence in the interior that's the total amount of flow coming in from outside from underneath the bottom of the lake the total integrated and now by integrated I mean in the sense of an integral the integrated amount of flow in that's the integral of the divergence the integral over the interior in the three-dimensional case it would be integral DX dy DZ over the interior of this region of the divergence of a if you like to think of a is the velocity field that's fine is equal to the total amount of flow that's going out through the boundary and how do we write that the total amount of flow that's flowing outward through the boundary we break up let's take the three-dimensional case we break up the boundary into little cells each little cell is a little area let's call each one of those little areas D Sigma these Sigma Sigma stands for surface area Sigma is the Greek letter Sigma it stands for surface area this three-dimensional integral over the interior here is equal to a two-dimensional integral the Sigma over the surface and it is just the component of a perpendicular to the surface let's call a perpendicular to the surface D Sigma a perpendicular to the surface is the amount of flow that's coming out of each one of these little boxes notice incidentally that if there's a flow along the surface it doesn't give rise to any fluid coming out it's only the flow perpendicular to the surface the component of the flow perpendicular to the surface which carries fluid from the inside to the outside so we integrate the perpendicular component of the flow over the surface that's through the Sigma here that gives us the total amount of fluid coming out per unit time for example and that has to be the amount of fluid that's being generated in the interior by the divergence this is Gauss's theorem the relationship between the integral of the divergence on the interior of some region and the integral over the boundary where where it's measuring the flux the amount of stuff that's coming out through the boundary fundamental theorem and let's let's see what it says now any questions about that Gauss's theorem here you'll see how it works I'll show you how it works yeah yeah you could have sure if you had a compressible fluid you could discover that all the fluid out boundary here is all moving inwards in every direction without any new fluid being formed in fact what's happening is just the fluid is getting squeezed but if the fluid can't squeeze if you cannot compress it then the only way that the fluid could be flowing in is if it's being removed somehow from the center if it's being removed by by invisible pipes that are carrying it all so that means the divergence in the case of water would be zero there was no water coming it would be if there was a source of the water divergence is the same as a source source of water is the source of new water coming in from elsewhere is right so in the example with the 2-dimensional lake the source is water flowing in from underneath the sink which is the negative of a source is the water flowing out and in the 2-dimensional example this wouldn't be a 2-dimensional surface integral it would be the integral in here equal to a one dimensional surface and to go coming out okay all right let me show you how you use this let me show you how you use this and what it has to do with what we set up till now about gravity I think hope a lifetime let's imagine that we have a source it could be water but let's take three dimensional case there's a divergence of a vector field let's say a there's a divergence of a vector field del dot a and it's concentrated in some region of space that's a little sphere in some region of space that has spherical symmetry in other words doesn't mean it doesn't mean that the that the divergence is uniform over here but it means that it has the symmetry of a sphere everything is symmetrical with respect to rotations let's suppose that there's a divergence of the fluid okay now let's take and it's restricted completely to be within here it does it could be strong near the center and weak near the outside or it could be weakened near the center and strong near the outside but a certain total amount of fluid or certain total divergence and integrated there vergence is occurring with nice Oracle shape okay let's see if we can use that to figure out what the field what the a field is there's a Dell dot a in here and now let's see can we figure out what the field is elsewhere outside of here so what we do is we draw a surface around there we draw a surface around there and now we're going to use Gauss's theorem first of all let's look at the left side the left side has the integral of the divergence of the vector field all right the vector field or the divergence is completely restricted to some finite sphere in here what is incidentally for the flow case for the fluid flow case what would be the integral of the divergence does anybody know if it really was a flue or a flow of a fluid it'll be the total amount of fluid that was flowing in per unit time it would be the flow per unit time that's coming through the system but whatever it is this integral doesn't depend on the radius of the sphere as long as the sphere this outer sphere here is bigger than this region why because the integral over that there vergence of a is entirely concentrated in this region here and there's zero divergence on the outside so first of all the left hand side is independent of the radius of this outer sphere as long as the radius of the outer sphere is bigger than this concentration of divergence iya so it's a number altogether it's a number let's call that number M I'm not Evan let's just Q Q that's the left hand side and it doesn't depend on the radius on the other hand what is the right hand side well there's a flow going out and if everything is nice and spherically symmetric then the flow is going to go radially outward it's going to be a pure radially outward directed flow if the flow is spherically symmetric radially outward direct directed flow means that the flow is perpendicular to the surface of the sphere so the perpendicular component of a is just a magnitude of AE that's it it's just a magnitude of AE and it's the same everywhere is on the sphere why is it the same because everything has spherical symmetry a spherical symmetry the a that appears here is constant over this whole sphere so this integral is nothing but the magnitude of a times the area of the total sphere if I take an integral over a surface a spherical surface like this of something which doesn't depend on where I am on the sphere then it's just I can take this on the outside the magnitude of the the magnitude of the field and the integral D Sigma is just the total surface area of the sphere what's the total surface area of the sphere just 4 PI 4 PI R squared oh yeah 4 PI R squared times the magnitude of the field is equal to Q so look what we have we have that the magnitude of the field is equal to the total integrated divergence divided by 4 pi the 4 pi is the number times R squared does that look familiar it's a vector field it's pointed radially outward well it's point the radially outward if the divergence is positive if the divergence is positive its pointed radially outward and it's magnitude is one over R squared it's exactly the gravitational field of a point particle at the center here that's why we have to put a direction in here you know this R hat this art will this R over R is it's a unit vector pointing in the radial direction it's a vector of unit length pointing in the radial direction right so it's quite clear from the picture that the a field is pointing radially outward that's what this says over here in any case the magnitude of the field that points radially outward it has magnitude Q and it falls off like 1 over R squared exactly like the Newtonian field of a point mass so a point mass can be thought of as a concentrated divergence of the gravitational field right at the center point mass the literal point mass can be thought of as a concentrated concentrated divergence of the gravitational field concentrated in some very very small little volume think of it if you like you can think of the gravitational field as the flow field or the velocity field of a fluid that's spreading out Oh incidentally of course I've got the sign wrong here the real gravitational acceleration points inward which is an indication that this divergence is negative the divergence is more like a convergence sucking fluid in so the Newtonian gravitational field is isomorphic is mathematically equivalent or mathematically similar to a flow field to a flow of water or whatever other fluid where it's all being sucked out from a single point and as you can see the velocity field itself or in this case the the field the gravitational field but the velocity field would go like one over R squared that's a useful analogy that is not to say that space is a flow of anything it's a mathematical analogy that's useful to understand the one over R squared force law that it is mathematically similar to a field of velocity flow from a flow that's being generated right at the center at a point okay that's that's a useful observation but notice something else supposing now instead of having the flow concentrated at the center here supposing the flow was concentrated over a sphere which was bigger but the same total amount of flow it would not change the answer as long as the total amount of flow is fixed the way that it flows out through here is also fixed this is Newton's theorem Newton's theorem in the gravitational context says that the gravitational field of an object outside the object is independent of whether the object is a point mass at the center or whether it's a spread out mass or there it's a spread out mass this big as long as you're outside the object and as long as the object is spherically symmetric in other words as long as the object is shaped like a sphere and you're outside of it on the outside of it outside of where the mass distribution is then the gravitational field of it doesn't depend on whether it's a point it's a spread out object whether it's denser at the center and less dense at the outside less dense in the inside more dense on the outside all it depends on is the total amount of mass the total amount of mass is like the total amount of flow through coming into the that theorem is very fundamental and important to thinking about gravity for example supposing we are interested in the motion of an object near the surface of the earth but not so near that we can make the flat space approximation let's say at a distance two or three or one and a half times the radius of the earth well that object is attracted by this point that's attracted by this point that's attracted by that point it's close to this point that's far from this point that sounds like a hellish problem to figure out what the gravitational effect on this point is but know this tells you the gravitational field is exactly the same as if the same total mass was concentrated right at the center okay that's Newton's theorem then it's marvelous theorem it's a great piece of luck for him because without it he couldn't have couldn't have solved his equations he knew he meant but it may have been essentially this argument I'm not sure exactly what argument he made but he knew that with the 1 over R squared force law and only the one over R squared force law wouldn't have been truth was one of our cubes 1 over R to the fourth 1 over R to the 7th with the 1 over R squared force law a spherical distribution of mass behaves exactly as if all the mass was concentrated right at the center as long as you're outside the mass so that's what made it possible for Newton to to easily solve his own equations that every object as long as it's spherical shape behaves as if it were appoint appointments so if you're down in a mine shaft that doesn't hold that's right but that doesn't mean you can't figure out what's going on you can't figure out what's going on I don't think we'll do it tonight it's a little too late but yes we can work out what would happen in the mine shaft but that's right it doesn't hold it a mine shaft for example supposing you dig a mine shaft right down through the center of the earth okay and now you get very close to the center of the earth how much force do you expect that we have pulling you toward the center not much certainly much less than if you were than if all the mass will concentrate a right at the center you got the it's not even obvious which way the force is but it is toward the center but it's very small you displace away from the centre of the earth a little bit there's a tiny tiny little force much much less than as if all the mass was squashed toward the centre so right you it doesn't work for that case another interesting case is supposing you have a shell of material to have a shell of material think about a shell of source fluid flowing in fluid is flowing in from the outside onto this blackboard and all the little pipes are arranged on a circle like this what does the fluid flow look like in different places well the answer is on the outside it looks exactly the same as if everything were concentrated on the point but what about in the interior what would you guess nothing nothing everything is just flowing out away from here and there's no flow in here at all how could there be which direction would it be in so there's no flow in here so the distance argument like if you're closer to the surface of the inner shell yeah wouldn't that be more force towards that no you see you use Gauss's theorem let's do count system Gauss's theorem says okay let's take a shell the field the integrated field coming out of that shell is equal to the integrated divergence in here but there is no divergence in here so the net integrated field coming out of zero no field on the interior of the shell field on the exterior of the show so the consequence is that if you made a spherical shell of material like that the interior would be absolutely identical to what it what it would be if there was no gravitating material there at all on the other hand on the outside you would have a field which would be absolutely identical to what happens at the center now there is an analogue of this in the general theory of relativity we'll get to it basically what it says is the field of anything as long as it's fairly symmetric on the outside looks identical to the field of a black hole I think we're finished for tonight go over divergence and all those Gauss's theorem Gauss's theorem is central there would be no gravity without Gauss's theorem the preceding program is copyrighted by Stanford University please visit us at stanford.edu