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

Alan Watts Lectures Philosophy of the Tao 2



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Like, comment and subscribe. Alan Watts – The Tao of Philosophy (Full Lecture) Slices of Wisdom 0:00 Images Of God 22:35 Coincidence of Opposites 52:05 Seeing Through the Net 1:47:31 Myth.

The Tao Philosophy – Alan Watts.

thank you for joining us for the love of wisdom with Alan Watts in a talk from our radio series on Taoism through his classic books on sin and Taoism and through his many public lectures Alan Watts introduced millions of people in the West to Eastern thought today's program was recorded in the late 1960s on the ferryboat Vallejo this is a session from philosophy of the Dow here's Alan Watts send the same way to try and force the issues wears you out and when you force the lock to usually bend the key so you jiggle gently until you find the right open so in the same way anybody who knows how to conduct business always giggles the key to find the right moment to turn the lock and then it all happens as if it were natural and none of it were forced so therefore the watercourse will give you the sense that your life is a flowing the flowing is equally you and what is not you or called not you it's the process that's happening and when you understand that you'll stop asking questions about it you will see that all asking of questions about it is a Sura shall I say tautological you get explanations but don't explain all explanations of what's happening call for further explanations because big explanations of little explanations upon their backs to buy them little explanations have lesser explanations and so I didn't either this is the analytic process that produces the atomic universe the electronic universe the protonic universe and so on and so on and if you go the other direction you'll find the solar system in a galaxy the galaxy in a system of galaxies and then goodness knows what because obviously the universe as it seeks to know itself must run away from itself as your eyes revolve when you turn to look at them you are the universe you are apertures through which it is aware of itself holes in the wall as it were and so as you look now you see it now you don't it's very simple so you'll find therefore that the deep quest what is it what am I supposed to do what is human destiny why are we here these questions will slowly disappear and that itself will be the answer the answer will be this is why and this is what is going on that cannot be described the Dow the Dow is simultaneously departing and arriving always that's the meaning of the eternal the eternal well then to go with the water cause is called by loud sir wu wei wu means not way as a complex of meanings which can be action striving straining doing or best forcing not forcing his will way so he says Oulu a vowel does nothing but nothing is left undone in other words dow accomplishes all things without forcing them so therefore when you master that's the wrong word because it has the idea of superiority when you come to away by as it were coming down you are working on the same principle as the dow and so this is likened poetically to the difference between a willow and a pine when it snows the tine is a rigid tree and the snow and ice piles up on the branches until they crack the willow is a springy tree and when the weight of the snow is on the branch the branch drops and the snow falls off when the branch comes up again that's will way the drawings are tells a lovely tale that a sage was wandering along the bank of the river near an enormous cataract and suddenly way up beyond the cataract he saw an old man roll off the bank into the cataract and he thought this too bad he must be old and ill and is making an end of himself but a few minutes later the old man jumped out of the screen way down beyond the cataract and went running along the bank so the Satan his disciples went scooting after him and said this is the most amazing thing we ever saw how did you survive well he said there's no special way I just went in with the swirl and came up with the world I made myself like the water so there was no conflict between myself and water so in the same way when a baby is in an automobile accident you'll find often that the baby is uninjured because the baby didn't go rigid to protect itself so when likewise when you learn to fall in judo you learn to curl up limply and yet make your arms very heavy so that they flop with an immense thud on the floor and that making heavy is again like water and this absorbs the shock so it isn't you see you must realize that the water course is not complete limpness because water has weight and therefore strength and that really all energy does the secret of will–we is that all energy is gravity we as a planet would if we encountered some obstacle in space there would be an enormous release of energy the obstacle would feel that it was being hit by some colossal force but the planet is falling around the Sun and the Sun is falling around something else the whole universe is falling but since there's nowhere for it to drop it's sort of not falling at anything just falling around itself sometimes for a collision but everything's got so much space that they're very rare maybe the ones that didn't have space have been eliminated and became all the dust that's in the sky but hey all energy is gravity that's the secret of Judo it's really the secret ultimately of the puzzle of the relationship of gravity to energy it's the real secret of the equals mc-squared so now if you want you see to find an intelligent solution to a problem the work is done with your brain you have all the necessary intelligence inside this bone only most people never use their brains they use their minds instead and think that they use their minds like they use their muscles but you constrain your head and work very hard with it as if it were a muscle to achieve a result but that doesn't work that way when you want to find out the answer for something what you do is you contemplate it you visualize your problem your question as well as you can and then simply look at it because you don't try to find the solution because any solution that comes in that way is liable to be wrong but when you have watched it for a while the solution comes of itself and that is the way to use your brain it works for you in the same way as your stomach digests your food without your having to supervise it consciously but all our attempts to supervise everything consciously have led to things that aren't too good for our stomachs and so forth you know how it goes the reason for that is quite simple that conscious attention employing words cannot think of very much we ignore almost everything while we're thinking we think along a single track but the world isn't going on along a single track the world is everything all together everywhere and you just can't take that into consideration there isn't time but your brain can because the brain is capable of handling the innumerable variables at once whereas your conscious attention is not or rather more strictly speaking verbal symbols are not capable of handling any more than a one very very crudely simple track that's why we have to trust our brains because you see we are much more intelligent than our understanding of ourselves when a neurologist admits that he's only begun to scratch the surface of understanding the nervous system he is actually saying that his own nervous system is smarter than he is it's up witted him so far and that's remarkable isn't it so you see what you are is necessarily more than anything you can understand for the simple reason that an organism which completely understands itself would be comparable to lifting yourself up by your own bootstraps or kissing your own lips so there's always this element of the nothing or unknown in any process of consciousness or knowledge and if that irritates you remember you are really addressing yourself to a stupid problem because fire doesn't need to burn itself any more than a light needs to shine on itself so for light to ask what am i although it sounds like a sensible question isn't because how could you answer it let me demonstrate this to you in another and perhaps simpler way for many many hundreds of years human beings have wanted to know what is isn't it what is matter substance obviously it seems there must be the basis for all the shapes that we see after all clay is the basis of vessels ion is the basis of tools wood is the basis of furniture but what is the basis of that what is the substance common to everything well we tried to find out and that was the reason at least in the beginning of things why physicists tried to find out what matter energy is well when you consider the problem consider the questions ask yourself what sort of answer are you looking for supposing you found out what matter is in what sort of language would you be able to describe it you realize that you couldn't because all that language can describe whether the language be of words or of numbers is pattern form we can measure form we can outline it we can explain it but we couldn't possibly explain stuff just basic oomph there would be no way of talking about it therefore we are asking a silly question we're asking for a question to be answered in terms which couldn't that kind of question so therefore we might say you don't need really the idea of stuff of matter because what you've got is form and I could say it's the form that matters in anything and you know you take a double take on that when you look at the world as form you will not ask questions about what is it made of what are these shapes made of their because they are not made it says Dao away and Wu Wei means also not making making is different from growing when a plant grows it is not patched or put together all the so called parts of the plant grow out of it whereas when you construct a machine you assemble the parts and put them together this is not the way a plant works it's not the way you work when you eat your food it is ridiculous to suppose that this is the raw material of parts which are going to be manufactured in a factory and distributed to screw on the different parts of your body the silly image so you don't ask therefore what are things made of next thing you don't ask is how do they go together because here becomes a very very fascinating thing that what we are unaware of is that so-called separate things are joined by space we always think of space as separating but that which separates also joins that was why the word teeth is so important it means to stick to answers it divided so we are kept by space when you look at a distant galaxy we call it a nebula because we are seeing it from a long way away when however we get close to it the nebula disappears and we see individual stars we are living in the middle of one from a long way away all the stars in our galaxy appear to be a thing but right close up we don't see that thing what is it that keeps all these stars together all we say gravity gravity is a word for something it means simply we don't know what it is like the ether that was once supposed to be the fluid through which light was transmitted and was discovered not to exist that we could somehow do without it that light went through nothing through space marvelous apparently so here are all sorts of things not joined to each other by strings and yet they constitute a thing and so when we get down to our own atomic structure we are if we were looked at closely enough we would be like galaxies with our atoms vastly distant from each other what holds us together isn't that the wrong question because what is what matters is the form it is your shape now if I saw you some time ago and I meet you again how do I know you it's I recognize your shape if I haven't seen you for some time when I meet you the second time there is nothing of you that I knew before all your so-called substance has changed just as if I visit a waterfall a few seconds later you know it's not the same water it's falling the same way the same style and so in the same way when I meet you today yesterday the day before you are doing your the same style your the same pattern but it's all different so that Pathan can come to an end and then after a while it can happen again just as if you take a newspaper photograph and look at it under a magnifying glass what you saw oh the naked eye as a clean line will appear to be a series of dots and the dots are alike one another and there's no strings joining them change your level of magnification and there's a line you could magnify in time as well as in space only you do it a different way to magnify anything in time you speed up the time so we take a fast motion picture of a pin and we put the beam in on earth and we expose one frame per day and then when we run the movie we suddenly see the beam moving it puts out a little Fela and the Fela Kuti and all the stalks and leaves open from outcomes of being and it eventually crumbles why we saw a Chester we saw happening which we would never notice with the naked eye too slow now that is you see magnifying time now therefore let us suppose we had human faces thousand faces which suddenly began as little fetus cells and we got some turn rapidly into baby face into child adolescent adult old first skull and dust a little all these faces coming enjoying the trip of making their gesture of going through the whole lifespan well we would watch and we would begin to see patterns and continuity which things are moving much too slowly to see these and things are moving to complicatedly to see them but stand off at a distance as it were and you would see the same rhythms occurring again again and again in the game so that they would appear from that distance or from time that speeded up to be a simply contrary continuity like the continuity from here the camera which looked at closely is full of space so you would get what we call in naive language the reincarnation of forms you considered as this kind of a wiggle that is recognizable with me would happen again and again and again with no spook traveling from one happening to the next that wouldn't be necessary anymore than it's necessary to have a line joining one dot in the press photograph with another table so that they will make continuous sense for the different level do you know when you watch television they're actually watching a moving dot it moves so fast that it creates the image now let's supposing you had a different kind of eye altogether that had no memory in it you would look at the television screen and you would see the stop dancing walk all across and leaving no trace is there what kind of a funny thing is that doesn't make no sense we've seen in another ways they all that picture makes sense so you see you don't have to or all these mysteries like our reincarnation you don't need any spooky knowledge to understand them it's all right out in front of you and you will discover that as I talk to you I'm not going to tell you a thing that you didn't already know all this was plain to you all I will do is put it in words in such a way that you will be able to say well that's what I thought but I could never explain [Laughter] you've been listening to Alan Watts from the Alan Watts radio series number nine on Taoism for information on how to obtain the radio series on cassette tape call one eight hundred nine six nine two eight eight seven or you can write to the electronic university vo box to 3:09 San Anselmo California nine four nine seven nine when you call a rye please indicate the name of your local station then you heard a selection from the Ellen watch radio series number nine on Taoism again the phone number is oh you don't it's very simple so you'll find therefore that the big quest what is it what am I supposed to do what is human destiny why are we here these questions will slowly disappear and that itself will be the answer the answer will be this is why and this is what is going on that cannot be described the Dow the Dow is simultaneously departing and arriving always that's the meaning of the eternal the eternal down well then to go with the water cause is called all explanations of what's happening call for further explanations because big explanations have little explanations upon their backs to buy them and little explanations have lesser explanations and so I didn't idle this is the analytic process that produces the atomic universe the electronic universe the protonic universe and so on and so on and if you go the other direction you'll find the solar system in a galaxy the galaxy in a system of galaxies and then goodness knows what because obviously the universe as it seeks to know itself must run away from itself as your eyes revolve when you turn to look at them you are the universe you are apertures through which it is aware of itself holes in the wall as it were and so as you look now you see it not allowed sir wu wei wu means not way as a complex of meanings which can be action striving straining doing our best forcing not forcing his will way so he says thou will a doll does nothing but nothing is left undone in other words thou accomplishes all things without forcing them and so therefore when you master that's the wrong word because it has the idea of superiority when you come to way by as it were coming down you are working on the same printed key so you jiggle gently until you find the right open so in the same way anybody who knows how to conduct business always giggles the key to find the right moment to turn the lock and then it all happens as if it were natural and none of it were forced so therefore the watercourse will give you the sense that your life is a flowing the flowing is equally you and what is not you or called not you it's the process that's happening and when you understand that you'll stop asking questions about it you will see that all asking of questions about it is asura shall I say tautological you get explanations but don't explain thank you for joining us for the love of wisdom with Alan Watts in a talk from our radio series on Taoism through his classic books on sin and Taoism and through his many public lectures Alan Watts introduced millions of people in the West to Eastern thought today's program was recorded in the late 1960s on the ferryboat in Vallejo this is a session from philosophy of the Dow here's Alan Watts send the same way to try and force the issues wears you out and when you force the lock to usually bill

Alan Watts Lectures Philosophy of Nature 1



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Like, comment and subscribe. Alan Watts – The Tao of Philosophy (Full Lecture) Slices of Wisdom 0:00 Images Of God 22:35 Coincidence of Opposites 52:05 Seeing Through the Net 1:47:31 Myth.

Essential Lecture Series – Comparative Philosophy – Two of Six.

I really enjoy this lecture. Its definitely a rarity in the Alan Watts library but its worth it. .. Because. WHAT IS REALITY, ANYWAY???

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welcome to another talk from the Alan Watts radio series number four philosophy and society philosopher author lecturer and entertainer Alan Watts is earned a reputation as one of the most underrated philosophers of the twentieth century the Taoist approach to life is to follow the course and current of nature and thus it has become known as the philosophy of nature this talk was recorded during a seminar in 1969 it's called philosophy of nature part 1 here's Alan Watts compare a physical lobe and a political globe the physical globe is a pretty thing with all kinds of green and brown wiggly patterns on it the political globe on the other hand has still got the Wiggly outlines of the land but they are all crossed over with coloured patches many of which have completely straight edges a lot of the boundary between the United States and Canada once you get west of the Great Lakes is simply a straight line what does that got to do with anything with any difference between Canadians on one side of the line or Americans on the other side of the line or what have you it is absolutely a violation of the surface of the territory and look at the fair city of San Francisco it's a lovely place but they planted on the hills of San Francisco a city pattern that was appropriate for the plains of Kansas a gridiron and so you'll get streets that go straight up and that are extremely dangerous where they should have followed the contours of the hills now however I think we should begin by talking a little bit about when we use the word physical reality as distinct from abstraction and what are we talking about because you see there's going to be a fight about this to the sophistry if I say that the the final reality that we are living in is the physical world a lot of people will say that I'm a materialist that are non spiritual and that I think too much of an identification of the man with the body you any any books that you open on yoga or Hindu philosophy will have in it a declaration that you start a meditation practice by saying to yourself I am NOT the body I am NOT my feelings I am NOT my thoughts I am the witness who watches all this and is not really any of it and so if I were to say then that the physical world is the basic reality I would seem to be contradicting what is said in these Hindu texts but it all depends on what you mean by the physical world what is it first of all there must be pointed out that the idea of the material world is itself philosophical it is in its in its own way a symbol and so if I take up something that is generally agreed to be something in the material world and I argue that this is material of course it isn't because nobody has ever been able to put their finger on anything material that is to say if you buy the word material you mean some sort of basic stuff out of which the world is made by say analogy with the art of ceramics pottery we use clay and we form it into various shapes and so a lot of people think that the physical world it's various forms of matter and nobody has ever been able to discover any matter they've been able to discover various forms yeah there is patterns but no no matter you can't even think that what how you would describe matter in some terms other than form because whenever a physicist he talks about the nature of the world he he describes a form he describes the process which can be put into the shape of the mathematical equation and so if you say a plus B equals B plus a everybody knows exactly what you mean it's a perfectly clear statement but nobody needs to ask what do you mean by a or what do you mean by B or if you say one plus two equals three that's perfectly clear but you don't need to know one what to what are three one and all our descriptions of the physical world have the nature of these formally numbers they are simply mathematical patterns because what we're talking about is pattern so one might say then that we are confused through-and-through about what we mean by the material world and what I'm first of all doing is I'm just giving a number of illustrations which show how confused we are and let me repeat this to get it clear because it is rather complicated in the first place we confuse abstract symbols that is to say numbers and words and formulae with physical events as we confuse money with consumable wealth in the second place we confuse physical events the class and category of physical events with matter but matter you see is an idea it's a concept is the concept of stuff of something solid and permanent that you can catch hold of now you just can't catch hold of the physical world the physical world is the most evasive elusive process that there is it will not be pinned down and therefore it fulfills all the requirements in spirit so what I'm saying then is that the the non abstract world which cause it's key called unspeakable which was really a rather good word is the spiritual world and the spiritual world isn't something fun gaseous abstract form in that sense of shakers its formulas in another sense the formless world is the wiggly world see it's when we say something is shapeless like a cloud what shape has this cloud you say well it's so vague it says it's shapeless that's the real formless world the formal world is the one that human beings try to construct all the time see wherever human beings have been around you see rectangles and straight lines as we're always trying to straighten things out and so that's the the very mark of our presence I don't know why we do it it's always been a puzzle for me why architects are always using rectangles but the thing is little bit they make us feel very uncomfortable if they don't I have an architect friend who built somebody a house like a snail shell and it was it spiraled in and in and in and in and the John was right at the center but everybody rebels against this house they just feel very uncomfortable these are the furniture doesn't fit because all furniture is made to fit in a rectilinear scene and we we're always putting things in boxes see all thoughts or words are labels on boxes therefore we feel we've got to get everything boxed and so we put ourselves in boxes everything is put in boxes but actually everything else in nature doesn't go that way as for example the snail doesn't put itself in a box the crab doesn't put itself in a box it has these fascinating gorgeous objects what is for example more beautiful than a conch shell or a lovely scallop shell these are gorgeous things we could make the most delicious shells out of concrete or plastics there they could be very beautiful and we could distribute ourselves over the landscape like shellfish along the seashore but instead we have to live in boxes there's nothing you can't fight it it's the system so you know then you have to you begin to build your furniture and chairs everything accordingly to these shapes because they are easy to store away in a place that is a box in the first place but you see then that is this rectilinear world this is unspiritual this is the world of what we will call the artificial has distinct from the natural and when we live in a world like that we begin to have ourselves bamboozled by it you think you begin to think that reality is this sort of straightened out situation that we all have to live in and you don't remember that reality is precisely the wiglet world you see we don't realize that we are all Wiggly the problem is that we we go in rather the same way we have head two arms two legs etc but notice how we do all sorts of things to ourselves to sort of evade our weakness the way we dress especially men but this world this physical world is Wiggly and this is the most important thing to realize about it as I have sometimes said we're living in the middle of a Rorschach blot and there really is no way that the physical world is in other words that the nature of truth I said in the beginning somebody had said that thoughts were made to conceal truth this is this is the fact because there is no such thing as the truth that can be stated in other words ask the question what is the true position of the stars in the Big Dipper well it depends where you're looking at them from and there is no absolute position so in the same way accountants a good accountant will tell you that any balance sheet is simply a matter of opinion there's no such thing as the true state of affairs others of a business but we're all hooked on the idea that there is you see an external objective world which is a certain way and that there it really is that way history for example is a matter of opinion the history is an art not a science it's something constructed which is accepted as a more or less satisfactory explanation of events which in as a matter of fact don't have an explanation at all most of what happens in history is completely irrational but people always have to feel that they're they've got to find a meaning for example you get sick and you've lived a very good life and you've been helpful to other people have done all sorts of nice things and you get cancer and you say to the part to the clergyman why did this have to happen to me and you're looking for an explanation and there isn't one it just happened that way but people feel if they can't find an explanation they feel very very insecure why because they haven't been able to straighten things out the world is not that way so the truth in other words what is going on is of course a lot of Wiggles but the way it is is always in relation to the way you are in other words however hard I hit a skinless drum it will make no noise because noise is a relationship between a fist and a skin so in exactly the same way light is a relationship between electrical energy and eyeballs it is you in other words who evoke the world and you evoke the world in accordance with what kind of a you you are what kind of an organism one organism evokes one world another organism evokes another world and so everything is reality is a kind of relationship so once one gets rid of the idea of the truth as some way the world is in a fixed sense say it is that way see then you get to another idea of the truth altogether the idea the truth that cannot be stated the truth that cannot be pinned down I might say that I'm interested in Japanese materialism because contrary to popular belief Americans are not materialists we are not people who love material but our culture is by and large devoted to the transformation of material into junk as rapidly as possible God's own junkyard in science we really work in two different ends of the spectrum of reality we can deal with problems in which there are a very few variables or we can deal with problems in which there are almost infinitely many variables but in between we are pretty helpless in other words the average person cannot think through a problem involving more than three variables without a pencil in his hand that's why for example it's difficult to learn complex music think you're of an organist who has two keyboards or three keyboards for work with his hands and each hand is doing a different rhythm and then he's feet on the pedals he could be doing a different rhythm with each foot now that's a different difficult thing for people to learn to do just like to rub your stomach into a circle and pat your head at the same time takes a little skill now most problems with which we deal in everyday life involve far more than three variables and we are really incapable of thinking about them actually the way we think about most of our problems is simply going through the motions of thinking we don't really think about we do most of our decision-making by hunch you can collect data about a decision that you have to make but the data that you collect has the same sort of relation to the actual processes involved in this decision as a skeleton to a living body it's just the bones and there are all sorts of entirely unpredictable possibilities involved in every decision and you you don't really think about it at all the truth of the matter is that we are as successful as we are which is surprising the degree to which we are successful in conducting our everyday practical ride because our brains do the thinking for us in an entirely unconscious way the brain is far more complex than any computer the brain is in fact the most complex known object in the universe because our neurologists don't understand it they have a very primitive conception of the brain and admit it and therefore if we do not understand our own brains that simply shows that our brains are a great deal more intelligent than we are meaning by we the thing that we have identified ourselves with instead of being sensible and identifying ourselves with our brains we identify ourselves with a very small operation of the brain which is the Faculty of conscious attention which is a sort of radar that we have that scans the environment for unusual features and we think we are that and we are nothing of the kind that's just a little little trick we do so actually we our brain is analyzing all sensory input all the time analyzing all the things you don't notice don't think about don't have even names for and so it is this marvelous complex goings-on which is responsible for our being able to adapt ourselves intelligently to the rest of the physical world the brain is furthermore an operation of the physical world but now you see though we get back to this question physical world this is a concept this is simply an idea and if you want to ask me to differentiate between the physical and the spiritual I will not put the spiritual in the same class as the abstract but most people do they think that one plus two equals three is a proposition of a more spiritual nature than say for example a tomato but I think the tomato is a lot more spiritual than one plus two equals three this is where we really get to the point that's why in Zen Buddhism when people asked what is the fundamental principle of Buddhism you could very well answer it tomato because look how when you examine the material world how diaphanous it is it really isn't very solid the tomato doesn't last very long nor for that matter do the things that we consider most exemplary of physical reality such as mountains the poet says the hills are shadows and they flow from form to form and nothing stands because the physical world is diathermic it's like music when you play music it simply disappears there's nothing left and that for that very reason it is one of the highest and most spiritual of the arts because it is the most transient and so in a way you might say that transiency is a mark of spirituality a lot of people think the opposite that the spiritual things are the everlasting things but you see the more a thing tends to be permanent the more it tends to be lifeless so then the physical world we can't even find any stuff out of which it's made we can only recognize each other and I say well I realize that I met you before and that I see you again but the thing that I recognize is not anything really except a consistent pattern let's suppose I have a rope and this rope begins by being Manila rope then it goes on by being cotton rope then it goes on with being nylon then it goes on with being silk so I tie a knot in the rope and I moved or not down along the rope now is it as it moves along the same knot or a different knot we would say it was the same because he recognized the pattern of the knot but at one point it's Manila but another point is cotton another point it's nylon and another itself and that's just like us we are recognized by the fact that one day you face the same way as you did the day before and people recognize you're facing so they say that's John Doe or Mary Smith but actually the contents of your face whatever they may be the water the carbons the chemicals are changing all the time you are like a whirlpool in the stream the stream is doing this consistent well pulling and we always recognize like at the Niagara there the whirlpool is one of the sites but the water is always moving on and this is why it's so spiritual to be non spiritual is not to see that in other words it is to impose upon the physical world the idea of thinness of substantiality that is to be involved in matter to identify with the body to believe in other words that the body is something consonant something tangible so therefore if you cling to the body you will be frustrated so the whole point is that the material world the world of nature is marvelous so long as you don't try to lean on it now when you look at jade you see it has this wonderful mottled markings in it and you know somehow and you can't explain why those Mullings are not chaotic when you look at the patterns of clouds or the patterns of foam on the water isn't it astounding they never never make an aesthetic mistake look at the way the Stars are arranged why they're not arranged there despite they seem to be scattered through the sky like spray but would you ever criticize the stars for being in poor taste when you look at a mountain range it's perfect but somehow this spontaneous Wiggly arrangement of nature is quite different from anything that we would call a mess look at an ashtray full of cigarette butts and screwed up bits of paper look at some modern painting where people have gone out of their way to create expensive messes you see they're different and this is the whole joke that we can't put our finger on what the difference is although we've jolly well know it we can't define it if we could define it in other words if we could define aesthetic beauty is it cease to be interesting in other words if we could have a method which would automatically produce great artists anybody could go to school have become a great artist their work would be the most boring kind of kitsch but just because you don't know how it's done that gives us an excitement and so it is for this there is no formula that is to say note sir no rule according to which all this happens and yet it's not a mess so this idea of re you can translate the word read as organic pattern and this 3 is the word that they use for the order of nature instead of our idea of law you've been listening to the philosophy of nature with Alan Watts from the Ellen Watts radio series number four philosophy and Society for information how to obtain the radio series on cassette tape call one eight hundred nine sixty nine to eight eighty seven or you can write to the electronic University P o box two three zero nine San Anselmo California nine four nine seven nine when you call a right please indicate the name of your local station you heard the program philosophy of nature from the Alan Watts radio series number four philosophy and society again the phone number is welcome to another talk from the Alan Watts radio series number four philosophy and society philosopher author lecturer and entertainer Alan Watts is earned a reputation as one of the most unrooted philosophers of the twentieth century the Taoist approach to life is to follow the course and current of nature and thus it has become known as the philosophy of nature this talk was recorded during a seminar in 1969 it's called philosophy of nature part 1 circle reality as distinct from abstraction and what are we talking about because you see there's gonna be a fight about this to the sophistry if I say that the the final reality that we are living in is the physical world a lot of people will say that I'm a materialist that are none spiritual and that I think too much of an identification of the man with the body you'll any any books that you open on yoga or Hindu philosophy will have in it a declaration that you start a meditation practice by saying to yourself I am NOT the body I am NOT my feelings I am NOT my thoughts I am the witness who watches all this and is not really any of it and so if I were to say then that the physical world is the basic reality I would seem to be contradicting what is said in these Hindu texts but it all depends on what you mean by the physical world what is it first of all there must be pointed out that the idea of the material world is itself philosophical it is in its in its own way a symbol and so if I take up something that is generally agreed to be something in the material world here's L&Y compared a physical lobe and a political globe the physical globe is a pretty thing with all kinds of green and brown wiggly patterns on it the political globe on the other hand has still got the wiggly outlines of the land but they are all crossed over with colored patches many of which have completely straight edges a lot of the boundary between the United States and Canada once you get west of the Great Lakes is simply a straight line what does that got to do with anything with any difference between Canadians on one side of the line or Americans on the other side of the line or what have you it is absolutely a violation of the surface of the territory and look at the fair city of San Francisco it's a lovely place but they planted on the hills of San Francisco a city pattern that was appropriate for the plains of Kansas a gridiron and so you'll get streets that go straight up and that are extremely dangerous where they should have followed the contours of the hills now however I think we should begin by talking a little bit about when we use the word

Lectures | Entanglement and Complexity Gravity and Quantum Mechanics



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Professor Leonard Susskind describes how gravity and quantum information theory have come together to create a new way of thinking about physical systems.

Professor Mark van Raamsdonk of the University of British Columbia gives the Stanford Physics and Applied Physics Colloquium. The AdS/CFT correspondence .

Professor Leonard Susskind describes how gravity and quantum information theory have come together to create a new way of thinking about physical systems.

Until now theories about quantum physics were negated by gravity, but can entanglement actually help explain gravity? Are We Living In A Holographic .

Chemistry / रसायन विज्ञान for SSC CGL and CHSL ( हिंदी में )



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Why Read: The Importance of the Liberal Arts Literature Against Philosophy (2004)



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The liberal arts (Latin: artes liberales) are those subjects or skills that in classical antiquity were considered essential for a free person (a citizen) to know in order …

What is Abstract Algebra? (Modern Algebra)



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Abstract Algebra is very different than the algebra most people study in high school. This math subject focuses on abstract structures with names like groups, …

Want to Understand Drought? Follow the Water! – Science Nation



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Already parts of the world suffer from lack of water, and with increasing demand it’s expected to get worse. To better understand and predict drought, 30 universities are collaborating in a multidisciplinary effort called the Shale Hills Project. Among the studies, is field research following the life cycle of water along the Susquehanna River Basin, the main tributary to the Chesapeake Bay. With support from the National Science Foundation, civil engineer Chris Duffy and his team at Penn State are tracking rainfall, where it flows, how much is absorbed by plants and soil, how water turns rock into soil, where soil builds up and where it erodes, and how the water flows into rivers, streams and underground aquifers. He says understanding the flow of water will help us better understand areas prone to drought and how urban populations will be affected.

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water a precious resource for sure and one many take for granted until there's too little or too much scientists and engineers have positioned instruments at the Susquehanna shale Hills observatory at Penn State to find out more about the water cycle it's one of six criticals own observatories what we're trying to do is build an experimental test beds across the United States and actually we are also working with the Europeans to understand the cycle of water in all the details environmental engineer Chris Duffy and his team are focused on learning everything they can about the shale Hills watershed we use things like laser precipitation monitors or infrared lasers that measure droplets in fact and it tells us the type of rainfall whether it's rain or snow or sleet some of it evaporates any water vapor that's leaving the watershed and going into the atmosphere is captured by those sensors when you're working with trees it's hard to measure all the water being transpired this SAP flow sensor measures the rate water moves through trees it's no surprise that plants are huge water guzzlers we're climbing trees to collect branch samples they measure how deep roots of plants and trees go to quench their thirst so we climb up and we get these french samples and we put them in the vials we can take them back to the lab take the water out of them and get an idea of where the water in the tree actually came from geology plays a big part the type of soil and rock under the observatory determines how much of the water will seep into an underground basin does it have a chance to clean itself as it moves is it picking up material as it goes so some understanding of what water does in the subsurface is important to all of us and Duffy says from a big-picture standpoint that's more critical than ever because of climate change global change and global warming is accelerating climate effects and accelerating rainfall in some areas and accelerating drought impacts and other he says a key goal is to help planners better predict the impact of floods and droughts on water supplies what they're finding out at shale Hills has really whet their appetite to know more for science nation I'm Miles O'Brien

Greatest Trick Plays in Football History



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best trick plays in football of all time

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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