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

42 Rules For Life ft. Jordan Peterson 🦞🌊 ( JBPWAVE )



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Akira The Don & Jordan Peterson – 42 Rules For Life
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Vocals recorded especially for this song by the good Doctor. Taken from the forthcoming album JBPWAVE: PARADISE
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LYRICS

Tell the truth… or at least don’t lie
Do not do things that you hate.
Act so that you can tell the truth about how you act.
Pursue what is meaningful, not what is expedient.
If you have to choose, be the one who does things, instead of the one who is seen to do things.
Pay attention.
Assume that the person you are listening to might know something you need to know. Listen to them hard enough so that they will share it with you.
Plan and work diligently to maintain the romance in your relationships.
Be careful who you share good news with.
Be careful who you share bad news with.
Make at least one thing better every single place you go.
Imagine who you could be, and then aim single-mindedly at that.
Do not allow yourself to become arrogant or resentful.
Try to make one room in your house as beautiful as possible.
Compare yourself to who you were yesterday, not to who someone else is today.
Work as hard as you possibly can on at least one thing and see what happens.
If old memories still make you cry, write them down carefully and completely.
Maintain your connections with people.
Do not carelessly denigrate social institutions or artistic achievement.
Treat yourself as if you were someone that you are responsible for helping.
Ask someone to do you a small favour, so that he or she can ask you to do one in the future.
Make friends with people who want the best for you.
Do not try to rescue someone who does not want to be rescued, and be very careful about rescuing someone who does.
Nothing well done is insignificant.
Set your house in perfect order before you criticize the world.
Dress like the person you want to be.
Be precise in your speech.
Stand up straight with your shoulders back.
Don’t avoid something frightening if it stands in your way — and don’t do unnecessarily dangerous things.
Do not let your children do anything that makes you dislike them.
Do not transform your wife into a maid.
Do not hide unwanted things in the fog.
Notice that opportunity lurks where responsibility has been abdicated.
Read something written by someone great.
Pet a cat when you encounter one on the street.
Do not bother children when they are skateboarding.
Don’t let bullies get away with it.
Write a letter to the government if you see something that needs fixing — and propose a solution.
Remember that what you do not yet know is more important than what you already know.
Be grateful in spite of your suffering.
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#jordanpeterson
#meaningwave
#lofihiphop
jbpwave

tell the truth or at least don't lie do not do things that you hate fact so that you can tell the truth about how you act pursue what is meaningful not what is expedient if you have to choose either one who does things instead of the one who is seen to do things attention assume that the person you are listening to might know something you need to know listen to them hard enough so that they will share it with plan and work diligently to maintain the romance in your relationships who you shared good news do you share bad news make at least one thing better every single place you going imagine you could be and then aim single-mindedly about cannot allow you to become arrogant coming or resentful compare yourself to who you were yesterday not to who's someone else as you possibly can on at least one thing see what happens if old memories still make you cry rape them down carefully and completely maintain your connections with people do not carelessly denigrate social institutions or artistic achievement treat yourself as if you were someone that you are responsible for helping ask someone to do you a small favor so that he or she can ask you what in the future make friends with people who want the best for you do not try to rescue someone who does not want to be rescued and be very careful about rescuing someone who does nothing check your house are you criticizing like the person you want to be be precise in your speech stand up straight stand up straight is your shoulders back don't avoid something frightening if it stands in your way but don't do unnecessarily dangerous things do not let your children do anything it makes you just like form your wife into a hide unwanted things horn the opportunity we're responsibilities abdicated by someone great linear skateboarding see something that he's sick that what you do not yet know is more important than what you already know be grateful in spite of your suffering you

Can Silence Actually Drive You Crazy?



Views:21371179|Rating:4.87|View Time:10:33Minutes|Likes:307999|Dislikes:8362
*Watch with headphones on!
Is 45 minutes really the longest anyone can stay in a perfectly silent, pitch-black room?
Support Veritasium on Patreon
Check out Audible:
Want to watch the whole hour of silence?

Many stories have circulated claiming the longest anyone has stayed in an ultra-quiet anechoic chamber is 45 minutes, the reason being any longer would drive you insane. To me this sounded like unsubstantiated rubbish, like the claim the Great Wall is the only manmade structure visible from space. So I put my own psyche on the line, subjecting myself to over an hour of the most intense quiet on Earth. No, this was not THE quietest room on Earth (-9dB) but it is one of the quietest, and the truth is once you put a person inside, they are by far the loudest thing in there so the sound rating of the room is irrelevant.

I was not surprised to find that I could stay in there for as long as I liked and feel perfectly fine. What was surprising is that my heartbeat was audible. You can hear it on the sound recording. Now I wasn’t consciously aware of the sound of my heart while in the room, but I was more aware of the feeling of it beating.

Huge thank you to everyone at BYU: Duane Merrell, Spencer Perry, Cameron Vongsawad, Jazz Myers, Ann Clawson, and Robert Willes.

سوف اقوم بالصراخ باعلا صوت ممكن وسأستمر بالصراخ باعلا صوت بينما ادور حول نفسي, وساستمر بذلك حتى ينقطع نفسي ممتاز هل انت مستعد؟ حسناً شكراً كان الامر ممتعاً شكراً المكان الاكثر هدوءاً على وجه الارض, يبدو انه غرفة معدومة الصدى في مينيابوليس هنا في جامعة بي واي يو لديهم غرفتهم الخاصة معدومة الصدى. وتسمى بالاتينية "انا كويك" اي معدم الصدى ويحققون انعدام الصدى من خلال مادة رغوية ملصقة على الحائط وحتى على الارضيات يمكن ان ترو انني على ارضية من الشبك وضع اسفلها المادة الرغوية انصحكم ان تلبسوا سماعات اذنين اثناء استماعكم لهذه التجربة لانني اعتقد ان التجربة ستكون افضل لكم وسيكون من المهم للغاية. >> انخفاض الصوت تردد سوف تأتي في، و
شكل منه، ترتد، وستبقى بحالة ارتداد الى الابد تقريبا حتى تضيع
في الجدار. ثم الترددات العالية سيتم امتصاصها في الرغوة
بحد ذاتها. >> يمكنك سماع كيف تميت غرفة أصداء
بالتصفيق باستمرار في حين إغلاق الأبواب. >> هناك تقارير أنه من المستحيل
على البقاء في واحدة من هذه الغرف ويبقى عاقل لفترة طويلة من الزمن. سجل
على ما يبدو 45 دقيقة. >> شعرت خانق قليلا. وبعد، فإنه
شعرت أن هناك الكثير من الضغط على رأسي. لماذا يحدث هذا .. خصوصا أن الهواء ليس مختلفا هنا؟ >> ما استنتجه هو أنك حين تستمع في غرفة عادية سيكون هناك الكثير من التردد على أذنك وهذا يعني وجود
الكثير من الفضاء، ولكن هناك شيء من ذلك في >> فهو يبدو وكأنه كنت في غرفة صغيرة، مثل
هل يمكن أن يكون في تابوت. >> نعم، ربما، نعم. >> اعتقد فعلا انه استجابة القلق.
لذلك كنت تستخدم إلى وجود هذه الأصوات في جميع أنحاء أنت ثم أنت لا بعد الآن، وحتى تتمكن
البدء في ذعر، لأنك لم يكن لديك شيء كنت تستخدم ل، وأعتقد يمكن أن القلق
زيادة، يسبب بعض التوتر، وربما هذا هو لماذا يذهب الناس مجنون أو بالهلوسة مع
صوت. انها مجرد لأنهم يحاولون التعويض عن ما يفقدونه وليس هنا >> عندما أحمل الناس في أحذرهم: مهلا،
اذا كان لديك بالدوار قليلا، واسمحوا لي أن أعرف، حتى أستطيع أن تحصل على الخروج من الغرفة قبل أن
جعل حالة من الفوضى في مرافق مكلفة لدينا. اسمعه. ثم سوف تسمع أي السوائل التي
هي في فمك أو حلقك. عليك نسمع كل منهم يتحرك في كل مرة كنت
تحرك فمك. ويعد لكم البقاء هناك وكلما أسمع. لذلك سوف تبدأ
لسماع تدفق الدم عبر الدماغ. >> هل سمعت ذلك؟ >> نعم. هذا يبدو وكأنه نوع من رنين
أو النبض. >> هل أنت نوع من جعل الصوت بالنسبة لنا لذلك
ونحن نعلم ما للاستماع في ل؟ صوت الدم في العروق؟ بعض الناس يقولون أنهم يستمعون لصوت الصدر كذلك >> هل سمعت ذلك؟ لا >> وعازف الكمان توضع في واحدة من هذه الغرف
على ما يبدو كان يقرعون الباب في غضون ثواني يحاول الخروج الآن يقول الناس ستدوخ وسصاب الدوار وتفقد التحكم .. وبعضهم يقول أنك ستستمع لهلوسات ولكن بالنسبة لي فإنه لا صوت الحق. أنت تعلم،
أعتقد أنني يجب أن تكون قادرة على الجلوس في غرفة مع أي صوت مع الأنوار ل
طالما أحب. وهكذا وانا ذاهب الى وضع نفسي للاختبار من خلال البقاء في هذه الغرفة
لطالما أستطيع. تمتع بالوقت الصامت شكرا الآن انها مجرد لي في غرفة كاتمة للصدى.
وربما هذا هو أهدأ مكان أنا لدينا من أي وقت مضى. وفي الثانية سأقول
الرجال، أغلقت الأنوار. يا. أنا مستعد. وقد أطفأت الأنوار سأقوم بإطفاء الكاميرا لكي لا يتواجد أي ضوء وسأبقي معي مسجل الصوت فقط لست متأكدا كم من الوقت قضيت هنا لكنني لا أزال أشعر بأنني مرتاح جدا إنه… إنه لطيف وهادئ.. لكنه يبعث على الاسترخاء إذا ما سمعت أي شيء أشعر وكأنني أسمع يمكنك سماع دقات قلبي لم أكن مدركا لها في البداية، لكن المايكروفون التقطها إذا كان علي أن أخمن كم من الوقت قضيته هنا فسأقول ثمان دقائق أشعر ك يمكنني الشعور بكل نبضة و كان يشع فوق صدري نحو رأسي، الأمور هادئة يبدو بصوت أعلى من ذلك بكثير. مثل الإبهام. نحن في جميع أنحاء الدقيقة 46 أو أكثر، وأنا سعيد
للقفز خارج. لذلك، نعم، عندما كنت على استعداد لك يمكن أن الوجه الأضواء مرة أخرى إذا أردت.
هل أنت هناك؟ لم تغفو، أنا فقط، كما تعلمون،
لقد كان وقتا طيبا. فمن هادئة لطيفة الفضاء. النوع الأول من يشعر مثل هذا من شأنه أن يكون
مادة جيدة لوحة غرفة النوم الخاصة بك في. ليس مشكلة. غرفة كاتمة للصدى، 45 دقيقة، ل
ساعة. سهل. استطيع البقاء هنا لمدة خمس ساعات. أعتقد أن أتمكن من الحصول على الكثير من العمل المنجز
هنا. حتى الآن أنا فقط يجب أن نخرج. وانا ذاهب
لمعرفة ما اذا كان يمكنني الخروج من هذه الغرفة بدونها هناك أناس تجد جميع أنواع الحالات
غير مريحة حقا مثل يجري فقط في الملعب الغرفة السوداء أو كائن في مساحة صغيرة محصورة.
لذلك ربما تلك الأنواع من الناس سيجدون هذا النوع من مخيفة. لقد لاحظت بالتأكيد أن هناك الكثير
من الضوضاء. كما أنه لم يكن مجرد الصمت النقي. مثل أنا فعلا للعمل بجد لجعل
أشياء يشعر الصامتة. خلاف ذلك، كما تعلمون، هناك كان صوت مثلي مجرد فرك حيتي
أو مجرد مثل حفيف ملابسك أو في كل مرة كنت ابتلاع أو الذي نتنفسه. ربما
وكان أغرب شيء لاحظته مثل شعوري من قلبي. أنا فقط شعرت انها تضخ
من الصعب حقا، وشعرت تقريبا مثل دفع الدم يصل من خلالي. لم يكن مثل
كنت أسمع ذلك. وكان مجرد مثل كنت أشعر ذلك. وكنت أشعر أنه كما لو، بطريقة ما،
قلبي كان يرتجف جسدي. لذلك أعتقد أن أسطورة أنه لا يمكنك البقاء
هنا لفترة أطول من 45 دقيقة وضبطت، حتى وإن كان هذا ليس أسطورة منتهكي. مازلت
أعتقد أنه غير صحيح. الآن إذا كنت حقا أكره الصمت، يجب عليك
دائما تحمل كتاب صوتي معك. وأستطيع أن يوصي جيدة واحدة: خطأ
في نجوم لدينا صديقي جون غرين. أنه كتاب ممتاز وأنها كانت في الأساس
الحديث من أكثر الكتب مبيعا منذ أن أصدرت. وقريبا هناك فيلم المقبلة
من وهكذا ربما كنت ترغب في الحصول من خلال الكتاب قبل أن تذهب ومشاهدة الفيلم.
الآن يمكنك تحميل هذا الكتاب مجانا عن طريق الذهاب إلى Audible.com/Veritasium~~V. أو هل يمكن
اختيار أي كتاب آخر من اختيارك ل شهر واحد تجريبية مجانية. مسموع هو مدهش
الموقع مع أكثر من 150،000 عناوين في جميع المجالات الأدب بما في ذلك الخيال، غير الخيالية،
والدوريات. لذلك ينبغي عليك التحقق منها خارج. هذا العنوان هو في دو doobly، وجون
سيقول. لذلك أود أن أشكر مسموع لدعم
لي، وأود أن أشكر لكم لمشاهدة.

Symphony of Science – A Wave of Reason



Views:1805180|Rating:4.90|View Time:3:51Minutes|Likes:18973|Dislikes:398
“A Wave of Reason” is the seventh installment in the Symphony of Science music video series. It is intended to promote scientific reasoning and skepticism in the face of growing amounts of pseudoscientific pursuits, such as Astrology and Homeopathy, and also to promote the scientific worldview as equally enlightening as religion. It features Carl Sagan, Bertrand Russell, Sam Harris, Michael Shermer, Lawrence Krauss, Carolyn Porco, Richard Dawkins, Richard Feynman, Phil Plait, and James Randi.

More science music videos can be found at

Enjoy!

~John
john @ symphonyofscience.com

Lyrics:
Russell:
When you are studying any matter
Or considering any philosophy
Ask yourself only: what are the facts,
And what is the truth that the facts bear out

Sagan:
Science is more than a body of knowledge
It’s a way of thinking
A way of skeptically interrogating the universe

If we are not able to ask skeptical questions
To be skeptical of those in authority
Then we’re up for grabs

Shermer:
In all of science we’re looking for a balance
between data and theory

Harris:
You don’t have to delude yourself
With Iron age fairy tales

Porco:
The same spiritual fulfillment
That people find in religion
Can be found in science
By coming to know, if you will, the mind of God

Krauss:
The real world, as it actually is,
Is not evil, it’s remarkable
And the way to understand the physical world
is to use science

Dawkins:
There is a new wave of reason
Sweeping across America, Britain, Europe, Australia
South America, the Middle East and Africa
There is a new wave of reason
Where superstition had a firm hold

Plait:
Teach a man to reason
And he’ll think for a lifetime

Sagan:
Cosmology brings us face to face with the deepest mysteries
With questions that were once treated only
in religion and myth

The desire to be connected with the cosmos
Reflects a profound reality
But we are connected; not in the trivial ways
That Astrology promises, but in the deepest ways

Feynman:
I can’t believe the special stories that have been made up
About our relationship to the universe at large
Look at what’s out there; it isn’t in proportion

Russell:
Never let yourself be diverted
By what you wish to believe
But look only and surely
At what are the facts

Randi:
Enjoy the fantasy, the fun, the stories
But make sure that there’s a clear sharp line
Drawn on the floor
To do otherwise is to embrace madness

[Bertrand Russell]
Když studujete jakoukoliv věc nebo zvažujete libovolnou filozofii, ptejte se sebe pouze: 'Jaká jsou fakta a jaká je pravda za jejich důsledky?' [Carl Sagan]
Věda je víc než jen soubor poznatků, je to způsob myšlení. Způsob, jak skepticky zkoumat vesmír. Pokud nebudeme moci klást skeptické otázky, skepticky zpochybňovat lidi v pozici autority, pak naletíme každému. [Michael Shermer]
V celé vědě hledáme rovnováhu mezi daty a teorií. [Sam Harris]
Nemusíte se nechat klamat pohádkami z doby železné. [Carolyn Porco]
Stejné duchovní naplnění, které lidé nalézají v náboženství, mohou naleznout i ve vědě. Tím, že poznají, chcete-li, mysl Boha. [Lawrence Krauss]
Skutečný svět, takový, jaký opravdu je, není zlý, ale pozoruhodný. A způsob, jakým porozumět fyzickému světu, je použít vědu. [Richard Dawkins]
Máme tu novou vlnu rozumu, která se žene přes Ameriku, Británii, Evropu, Austrálii, Jižní Ameriku, Střední východ a Afriku. Máme tu novou vlnu rozumu tam, kde dříve pevně vládly pověry. [Phil Plait]
Nauč člověka přemýšlet a bude myslet po celý život. [Carl Sagan]
Kosmologie nás přivádí tváří v tvář těm největším záhadám. Otázkami, kterými se kdysi zabývaly pouze náboženství a mýty. Touha být propojen s kosmem odráží hlubokou skutečnost. Skutečně jsme propojeni; ne tuctovým způsobem, který nabízí astrologie, ale tím nejhlubším způsobem. [Richard Feynman]
Nemohu uvěřit zvláštním příběhům, které si lidé vymysleli, o našem vztahu k celému vesmíru. Jen se podívejte ven, vždyť je to úplně neúměrné. [Carl Sagan]
Věda je víc než jen soubor poznatků, je to způsob myšlení. Způsob, jak skepticky zkoumat vesmír. [Richard Dawkins]
Máme tu novou vlnu rozumu, která se žene přes Ameriku, Británii, Evropu, Austrálii, Jižní Ameriku, Střední východ a Afriku. Máme tu novou vlnu rozumu tam, kde dříve pevně vládly pověry. [Phil Plait]
Nauč člověka přemýšlet a bude myslet po celý život. [Bertrand Russell]
Nikdy se nenechte rozptýlit tím, čemu byste rádi věřili. Hleďte pouze a s jistotou na to, jaká jsou fakta. [James Randi]
Užijte si fantazii, zábavu a příběhy. Ale ujistěte se, že existuje jasná hranice mezi oběma světy. Jednat jinak by znamenalo objímat šílenství. [Richard Dawkins]
Máme tu novou vlnu rozumu, která se žene přes Ameriku, Británii, Evropu, Austrálii, Jižní Ameriku, Střední východ a Afriku. Máme tu novou vlnu rozumu tam, kde dříve pevně vládly pověry. [Phil Plait]
Nauč člověka přemýšlet a bude myslet po celý život. Český překlad: Honza Jíša

The Quantum Theory that Connects the Entire Universe



Views:256333|Rating:4.92|View Time:10:18Minutes|Likes:9950|Dislikes:153
SciShow is supported by Brilliant.org. Go to to get 20% off of an annual Premium subscription.

Quantum mechanics is weird and seems a bit…complicated. But understanding it can help us to understand the universe.

Hosted by: Hank Green

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thanks to brilliant org for supporting scishow I've said it before and I'll say it again quantum mechanics is weird it's also hard to make videos about but we're doing it it is the physics of the super-tiny and it's built around the idea that energy isn't a smooth spectrum it can only come and set amounts and as weird as the implications of that are it's stood up to every experimental test we've thrown at it and as we've learned more about it we've gotten better at using it to solve problems and invent all kinds of useful things like lasers and semiconductors although we understand how to use the math behind the theory understanding what it means has proven to be a difficult challenge there are a number of different interpretations of quantum mechanics out there each of which looks at the theory differently they all make the same predictions about what quantum mechanics looks like on the surface so what will actually happened with experiments in the labs but the math and the meaning of the math can look very different there's a lot going on under the hood one of the more controversial interpretations is called pilot-wave theory at first glance it looks appealing it allows you to get around the uncertainty and randomness that quantum mechanics is famous for but there's a catch getting rid of the randomness involves breaking reality in other ways so the conventional interpretation of quantum mechanics is called the Copenhagen interpretation after the Institute where it was devised in the 1920s it includes a lot of the more well-known ideas around quantum mechanics like that something can be a particle and a wave at the same time it also says that certain things like which way an electron is spinning aren't really set until you observe them and until you do the electron is spinning in both directions at once this concept that particles are in multiple states at the same time is known as superposition and it's what inspired when Schrodinger's famous dead and a live cat Schrodinger was one of the pioneers of quantum mechanics and his cat in the Box thought experiment was actually meant to show that some implications of the theory were just ridiculous but it turns out that that poor cat was actually a good illustrate of superposition and that yes a lot of quantum mechanics makes no sense when you try to apply it to the larger world that we are all much more familiar with in the thought experiment you hide a cat in a box with a flask of deadly poison but the poison will only be released if a radioactive atom decays which has say a 50% chance of happening radioactive decay is a quantum mechanical process whether or not it happens to an individual atom is the kind of random event that the mainstream interpretation of quantum mechanics says is entirely unpredictable you cannot predict it before you open the box you have no idea whether or not the atom decayed and therefore no idea whether the poison was released or not normally if there was a 50-50 chance that a cat was alive you'd say that the cat was either alive or dead you just wouldn't know which here's where the Copenhagen interpretation is different from regular probability before it's observed the atom is in a superposition of decayed and not decayed meaning it's both at the same time so the cat would be both alive and dead once you open the box you turn the superposition into one state you can actually observe let's say alive no kitties were harmed in the making of this episode but opening the box does not tell you that the cat was always alive that whole time before you opened the box it's true state was the superposition of alive and dead now superpositions don't appear in everyday life but according to the Copenhagen interpretation on the tiny scale of particles they're everywhere which is obviously very weird but the idea fits with every experiment we've ever done and over time physicists have come to accept that reality is sometimes kind of blurred that is what the pilot-wave theory tries to fix the theory was first proposed in 1927 by another pioneer of quantum mechanics Louise Dubrow it was shelved until the 1950s when David Baum rediscovered and improved it today it's also known as the Dubrow bomb theory it works by distinguishing between particles and waves instead of treating them as the same thing like the Copenhagen interpretation does in pilot-wave theory there are still particles and waves but they exist separately there's an equation that gives you a particles Vil and that equation depends on the wave the wave interacts with the particle by guiding the way it moves or pilots it in other words that's where the name pilot wave comes from this wave spans the entire system you're looking at whether that's just a few electrons or the whole universe because of the central wave the properties of matter are set before you observe them instead of being superimposed you may not have the information you need to figure out what those properties are in advance but the information is out there it's a little bit like flipping a coin and then covering it with your hand you know it's either heads or tails even before you look at it it was too fast for you to follow the spinning with your eyes but if you had a high-speed camera or something you'd be able to figure it out so Copenhagen interpretation it says that that type of information doesn't exist for something like radioactive decay so until you observe it an atom can be both decayed and not decayed at the same time and Schrodinger's cat can be both alive and dead pilot-wave theory says the information does exist like with the coin you may not have access to the knowledge that would tell you whether the atom has decayed so on a practical level we can't use the math to figure it out so the information is out there and the atom is one or the other not both that's a little closer to how we could like perceive reality and everyday life so that's nice problem is there's a trade-off in a very intrinsic way pilot wave theory breaks a different really big rule in physics locality locality is the idea that everything in the universe can only ever affect its immediate surroundings you can't interact with something far away without sending some kind of signal to it and that signal needs to be transmitted through the space between you and that thing most importantly this means that all signals take time to travel that's why you see lightning before you hear thunder we also know that there should be an upper limit to how fast these signals can move Einstein worked out that the universe's speed limit is the speed of light now the Copenhagen interpretation actually does violate locality in certain situations for example you can generate two electrons in a way that means they must have opposite spins until you actually observe their spins though each electron is in super position spinning in both directions at the same time but if you send the two electrons away from each other wait until they're really far apart and then observe one right after the other you'll always find that they spin in opposite directions that's true even if there's no time for a signal to pass from the first to the second without moving faster than the speed of light Einstein really did not like this but we have tested it and we know that it happens if you're going by the Copenhagen interpretation that means somehow the electrons are affecting each other faster than the speed of light still at least in the Copenhagen interpretation violating locality is the exception rather than the rule pilot-wave theory on the other hand is entirely based on the idea that locality does not need to be a thing and particles can affect each other instantaneously even if they're light-years apart that's the whole point of the pilot wave all particles in a system are tied to each other through that one wave so by extension all particles in the universe affect each other without the time delay you'd get if the signals were traveling at the speed of light in other words if you say the information that would tell you whether the cat is alive or dead is embedded in the rest of the particles in the universe you're also saying those particles somehow effect each other faster than the speed of light and unlike the Copenhagen interpretation it's not just certain situations that have this problem it's everything everywhere always so some physicists take issue with that still pilot wave theory is a good reminder that our theories of physics aren't just guided by what's really happening but also by aesthetic and pragmatic choices if we go with the Copenhagen interpretation then we need to accept some really weird things like that cats are simultaneously dead and alive sometimes we also need to accept that some things are just unknowable but if we subscribe to the pilot wave theory then we need to accept that some effects may be truly non-local both choices have their pros and cons and for the most part physicists have decided that Copenhagen makes more sense pilot-wave theory has seen a bit of a renaissance in recent years though thanks to computer simulations in the world we don't have access to faster than light signals but on a computer we can pretend that we do so physicists are using the math behind pilot waves to do quantum mechanics simulations which could be an improvement on conventional methods for some things it's not just Copenhagen or pilot waves there are other interpretations of quantum mechanics out there to the history of physics is full of people coming up with weird impossible sounding ideas that happen to be right it's also full of ideas that happen to be wrong but are still useful in lots of different ways so even if we never discover whether this theory is actually how the universe works there is a lot we can learn just by exploring it to see what it can do and if you enjoy exploring theories like this I think you'll like the quantum computing course on brilliant org where you'll learn about the laws of quantum mechanics by building your own quantum circuit and racing a classical computer in solving code breaking puzzles you could check it out at brilliant org slash scishow and right now the first 200 people to sign up at that link will get 20% off an annual premium subscription to brilliant so head to brilliant org slash scishow to learn more and check it out and know that when you do you are also helping to support scishow so thanks you

The Absurdity of Detecting Gravitational Waves



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A head-vaporizing laser with a perfect wavelength detecting sub-proton space-time ripples.
Huge thanks to Prof Rana Adhikari and LIGO:
Here’s how he felt when he learned about the first ever detection:

Thanks to Patreon supporters:
Nathan Hansen, Donal Botkin, Tony Fadell, Saeed Alghamdi, Zach Mueller, Ron Neal
Support Veritasium on Patreon:

A lot of videos have covered the general overview of the discovery of gravitational waves, what they are, the history of the search, when they were found but I wanted to delve into the absurd science that made the detection possible.

When scientists want one megawatt of laser power, it’s not just for fun (though I’m sure it’s that too), it’s because the fluctuations in the number of photons is proportional to their square root, making more powerful beams less noisy (as a fraction of their total). The smoothest mirrors were created not for aesthetic joy but because when you’re trying to measure wiggles that are a fraction the width of a proton, a rough mirror surface simply won’t do.

Filmed by Daniel Joseph Files

Music by Kevin MacLeod, “Black Vortex” (appropriately named)

Music licensed from Epidemic Sound “Observations 2” (also appropriately named)

منذ 1.3 مليار سنة في مجرةٍ ​​بعيدةٍ جداً إندمج ثقبان أسودان أثناء التفافهما العنيف على بعضهما البعض فخلقا تشوهات متنقلة في نسيج الزمكان أو ما تسمى "موجات التجاذبية" في أقل من العشر من الثانية. الطاقة المحررة في هذه الموجات كانت أعظم 50 مرة من الطاقة الصادرة عن كل شيء آخر في الكون المرصود بأكمله مجتمعة إنها مثل نوع مهيب من الطاقة. بعد انتشاره عبر الكون بسرعة الضوء لأكثر من مليار سنة وصلت الموجات للأرض، حيث مددت و قلصت المكان بطريقة تسمح لشعاعين من الضوء الذي يسافر في أنابيب متعامدة بأن تتزحزح قليلاً (عن تحاذيها) مما سماح للبشر للكشف عن وجود موجاتٍ تجاذبيةٍ لأول مرة. هذه قصة بسيطة كافية لنقولها ولكن ما اكتشفته عندما ذهبت لزيارة البروفيسور "رانا أدهيكاري" في معهد كاليفورنيا للتكنولوجيا هو أنه يخفي عدم منطقية ما كان مطلوباً للكشف عن ذلك. هناك الكثير من الأشياء حول موجات التجاذبية و التي هي غريبة * يحاكي صوت موجات الجاذبية * ديريك : هل…هل هذا هو؟
رانا : نعم, هذا هو. المشكلة الرئيسية مع كشف موجات التجاذبية هي أنها ضئيلة. هي مددت وقلصت الفضاء بمقدار جزءٍ واحدٍ في كل 10²¹ وهذا ما يعادل قياس المسافة بين هنا و رجل القنطور
رجل القنطور : أقرب نظام نجمي إلينا يبعد 4.33 سنة ضوئية ومن ثم محاولة لقياس التغيرات في تلك المسافة التي هي بعرض شعرة الإنسان. للكشف عن مثل هذا التلوي الصغير يجب عليك قياس أطول مسافة ممكنة, و هذا يفسر طول الأذرع لمقياس التداخل البالغ 4 كيلو متر
وحتى مع أذرعٍ بهذا الطول فستغير الموجات التجاذبية طول الأذرع بالكثير بمقدار ¹⁸⁻10 متراً لذا يتوجب على الراصد أن يتمكن من قياس المسافات بشكل دقيق بما يساوي 1/10000 لعرض البروتون.
إنه أضأل قياس أُنجِزَ على الإطلاق. إذاً كيف بالإمكان قياس ذلك بالنظر إلى جميع المصادر الأخرى للإهتزازات والضوضاء في البيئة مثل الزلازل، والحركة المرورية، والعواصف الكهربائية (الصواعق والبرق) حسناً , لسبب واحد , إن المرايا هي أنعم شيء تم صناعته من قبل. وهي تزن 40 كجم أو 90 رطل ومعلقةٌ بواسطة خيطٍ من السيلكون ضعف سمك شعره واحدة
لعزلهم من بيئتهم ورغم ذلك, السبيل الوحيد لتكون على يقين من ألَّا تُخْدَعَ من الضوضاء البيئية كان ببناء مرصدين متباعين عن بعضهما البعض في أماكن هادئة بشكل معقول يسمح للتمييز بين الإزعاج المحلي و التي ستظهر في أحد المرصدين و الموجات التجاذبية و التي ستمر عبر كلا الجانبين بشكل آنيٍ تقريباً أنا في المبنى الذي يحتوي على مساحة 1 من 100 من LIGO، للكشف عن الموجة التجاذبية.
LIGO : مرصد الموجات الثقالية بالتداخل الليزري التحدي التالي هو الليزر. مهلاً ! * مصدوم * هذا كم هائل من المعدات ! أنت بحاجة إلى الليزر الذي يمكن أن يوفر طول موجي واحد، وبالضبط طول موجي واحد فقط. يمكنك أن تتخيل، إذا كان الطول الموجي لليزر الخاص بك يتغير وأنت تحاول إستخدام خاصية التداخل في موجات الضوء لتنفيذ هذا القياس فلن تعمل بهذا الشكل أبدًا , لأن هذا مشابه لمحاولة قياس هذه المسافة , ولكن عمود المسطرة يتغير بإستمرار إلى الوراء و إلى الأمام بحيث لا يمكنك أن تقول طول هذه الطاولة. كل هذه المعدات، ما لا يقل عن ثلاثة أرباع منها، كلها ما تحاول القيام به هو جعل الليزر أكثر إستقراراً و في نهاية اليوم ما حققناه هو ما له استقرار وثباتية جزء واحد في 10²⁰. ماذا يعني ذلك ؟ هذا يعني جزء من مئة مليار من ترليون. و هذا آخر ما وصلنا إليه. أفضل اليزرات لهذا الغرض لديها الطول الموجي من 1064 نانومتر. و هذا ضوء الأشعة تحت الحمراء, ولكن هذا يمثل مشكلة. كيف يمكنك قياس ¹⁸⁻10 باستخدام ما طوله الموجي ⁶⁻10 الطول الموجي للضوء؟ نعم، أود أن المزيد من الناس يسألون هذا السؤال. انه لشيء رائع لهذا الرسم المتحركة لإظهار هذا التحول الكبير في الطول الموجي ولكن الحقيقة هي، انه فقط بجزء واحد من تريليون من الطول الموجي هو التغير (الإزاحة) الحاصلة في طول الأذرع. يبدو واضحاً أنه يمكنك قياس نصف الطول الموجي لأن ذلك سيتسبب تداخلاً في الضوء مع نفسه. نعم، ولكن هذا بشكل تام.
هذا سينتقل من مظلمٍ تماماً إلى ساطعٍ تماماً. إذاً.. هل أنت تبحث عن شيء مثل , مظلم بشكل أكثر قليلا وساطع أكثر قليلا؟ نعم نعم, و حدودنا هنا في كمية كفاءتنا لقياس هذا الفرق بين الظلام و السطوع له علاقة مع حقيقة أن الضوء هو منفصل. إنه يأتي في بأجزاء منفصلة والتي تسمى الفوتونات. الاختلاف في عدد الفوتونات التي تضرب المرايا في أي لحظة بسبب هذا الغموض لعدم اليقين الكمي يتناسب مع الجذر التربيعي للعدد الكلي من الفوتونات.
ما يعنيه هذا هو كلما زادت الفوتونات التي تستخدمها… قلّ مقدار "عدم اليقين"، وهذا جزء من المجموع. هذا هو السبب في أن قوة الليزر في الأذرع هو واحد ميجا واط. وهذا يكفي من الطاقة لتشغيل ألف منزل، في شعاع ضوء واحد. وميجاوات، وانت تعرف * فرقعة * بووم ، إنها لن تمزق رأسك حتى… فقط، تبخره و يكون مجرد عمود من الدخان. حتى مع ليزر مثالي و واحد ميجا واط من القدرة ، أي شيء يصدم به الضوء سيتداخل معه… حتى الهواء، لذلك كل الهواء في أذرع الكشاف كان لا بد من استخراجه. ولقد إستغرق الأمر 40 يوماً لضخه ليصبح أقل من تريليون من الضغط الجوي وتم تسخين الأنابيب لدرجة حرارة الفرن لطرد أي غازات متبقية. انها تضخ ما يكفي من الهواء لملء إثنين ونصف مليون كرة قدم (أمِريكية)، مما يجعلها ثاني أكبر مفرغ في العالم بعد مصادم الهايدرونات الكبير. الآن هاك شيئًا معظم الناس لا يفكرون فيه وهو أن موجات الجاذبية يمدد الزمكان ,لذلك, الضوء المسافرعبره يجب أن يمتد في هذا المكان أيضاً. إذا كان كل شيء يمتد كيف لك أن تعرف شيئاً يمتد؟! كيف يمكنك أن تعرف أي شيء يمتد؟ هذا هو اللغز. أنها لا تعطي أي معنى! هذا الأمر كله مزيف, أغلقوه! أود أن أرسل شعاع ليزر أسفل في هذا الأنبوب ثم أنتظره حتى يعود… و من ثم أقول "حسناً لم يحدث أي شيء" لأن الفضاء إمتد و الطول الموجي لليزر إمتد (تبعا له) يبدو…يبدو هو نفسه إذا حصلت عليه بتمدد أو لا. أنها لا تعطي أي معنى في الحقيقة هي مسألة توقيت و كيف يعمل. وبالتالي فإن مقدار الوقت الذي يستغرقه الضوء للسير في هذا الأنبوب و عودته قصير جداً. على أية حال , الموجة… الموجة التجاذبية عندما تمر به إنها تصدر ذلك الشيء البطئ مثل * طنين منخفض * هذا الضجيج الذي أصدرته، والذي يعتبر ضعيفًا. إنه هذا التمتد البطيء…إنه فقط مائة مرةٍ في الثانية الواحدة. وهذا صحيح عندما تأتي الموجة من خلاله, الضوء الذي هو هناك في الواقع يحصل له تمدد. و…ذلك الجزء لا… لا يقوم بإتمام القياس بالنسبة لنا ولكن اممم .. الآن بعد أن امتد المكان ضوء الليزر ذاك قد عبر وانتهى إنه خارج الصورة.
نحن وبشكل مستمر نطلق الليزر مجددا إلى النظام حتى يتسنى للضوء الجديد أن يذهب إلى هناك ويضطر إلى سفر مسافة أكبر من الضوء الذي قبله. لذا, من خلال النظر إلى كيفية تَغَيُّرِ التداخل مع مرور الوقت ومحافظتنا على الطول الموجي لليزر من الليزر نفسه ثابتا، عندها نكون قادرين على القيام بالقياس. إذاً ما المتطلبات للكشف عن موجات الجاذبية؟ حسناً، ميجاوات من طاقة الليزر لتقليل ضوضاء الإطلاق ومن طول موجي واحد بالضبط لأننا نحاول قياس مجرد تريليون من الطول الموجي يطلق بإستمرار لإستبدال الضوء القديم و الذي إنسحق و تمدد… في ثاني أكبر حجرة مفرغة في العالم لحد جزء من تريليون من الضغط الجوي و يضرب أسلس مرايا قد صنعت من قبل معلقاً بخيوط من السيلكون في موقعين بعيدين لاستبعاد عامل الضوضاء مع أذرع بطول أربعة كيلومترات لزيادة مقدار موجات الجاذبية لمجرد جزء من ألف من عرض البروتون. أنت تعرف ما نقوم به بالفعل يومياً هنا هو ما كنت لأقول بأنه مستحيل إذا سألتني عنه قبل 20 عاماً. واحدة من الأشياء التي كانت الأكثر إثارة للإهتمام بالنسبة لي لمعرفته كان ما الذي يحد مقدار حساسية للرصد للكشافات الآن، و أتضح أنه من ميكانيكا الكم وأساساً يمكنك التفكير في الأمر مثل مبدأ الشك لهايزنبرج، حصلنا على شيئين و معاً "لايقينيتهما" يجب أن تكون أكبر من قيمة معينة. لحسن الحظ بالنسبة لنا بأننا نحاول فقط قياس شيء واحد هنا، نحن لا نحاول قياس شيئين في نفس الوقت. كل ما تريد معرفته هو كم مقدار زيادة امتداد هذا الذراع عن ذلك الذراع ؟ وهذه…هذه هي النقطة الرئيسية التي لم يفهمها الناس حتى قبل وقت قليل الطريق لبناء هذه الأنظمة هي أنها جيدة للغاية في قياس شيء واحد و أن كل "عدم اليقين" والتي تأتي من ميكانيكا الكم هي محشورة تماماً في هذا الشيء الآخر و الذي لا نهتم به. أشعر وكأننا نصل إلى هذه المستويات من الطبيعة حيث يبدو أن الطبيعة لا تريدك أن تذهب إلى أبعد من ذلك. ولكن، من خلال براعتنا نحن عرفنا طرق لهندسة ضوضاء الكم، وأعتقد أن هذا مفهوم رائع وإنني أتطلع إلى النتائج التي ستحققها. أعتقد أن الخطوة المنطقية التالية هي الذهاب من إشارتين , إلى الكشف عن كل الثقوب السوداء في الكون طوال الوقت. إنها ليست مثل مستوى حضارة فضائية من التكنولوجيا أنها مجرد…يجب علينا أن نفعل ما هو أفضل كثيراً مما نقوم به الآن ولكن أرى أنها نوعا ما في…في متناولنا.

The Quantum Experiment that Broke Reality | Space Time | PBS Digital Studios



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The double slit experiment radically changed the way we understand reality. To check out any of the lectures available from The Great Courses Plus go to and get ready to learn about everything from cooking to calculus.

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The double slit experiment radically changed the way we understand reality. Find out what the ramifications of this experiment were and how we can use it to better comprehend our universe.

Written and hosted by Matt O’Dowd

Made by Kornhaber Brown (www.kornhaberbrown.com)

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An Imposter

[音乐] 本短篇由 Great Courses Plus 提供 一个非常奇怪的测试结果 那就是单个粒子 的双缝实验 它是一个非常令人惊讶的实验, 它演示了量子世界非常不同于 我们直观感知的宏观世界。 我们的物理直觉。 事实上,它蕴含着我们真实世界里最本质的自然现象 也可以不是物理在
总之,至少在任何意义 我们所熟悉的。 [主题音乐] 让我们我们从我们熟悉的认知开始 有一个橡胶小鸭 它在水池里上上下下,导致周期性的波纹 往外传播 在一定距离之外,这些波纹遇到了一个障碍物 这个障碍物有两条细缝 大部分波纹被堵住了,但有部分波纹通过了细缝。 新的波纹会相互重叠 他们产生这种靓丽的纹理 它被称作为 “干扰纹理” 那是因为在某些地方, 从一条缝里出来波浪的波峰与从另外一条缝里出来的波浪的波峰重叠 产生了更高的波峰。 同理,也产生了更深的波谷 我们称之为 ”构造干扰“ 当一个波峰与另一个波谷相遇, 它们相互抵消,水面平坦。 这是 “湮灭干扰” 因此我们获得了波浪与平坦交替的水纹。 任何类型的波浪都会产生与此类似的条纹。 例如,水波、声波 包括光波 这个光的双缝干扰现象首先由 汤玛斯.杨 在 1801 观察到。 一束光线经过两条非常细的细缝 产生了明暗交替的条纹 也就是“构造干扰” 与 ”湮灭干扰“ 当然,我们都知道光其实就是 一种电磁波 对光的这种认识要归功于100年后的麦斯维尔。 因此光是电磁波是不是完美的解释了 这种干扰条纹? 但是,我们也都知道光 是由一组不可分割的 电磁能组成,也就是所谓的 “光子”。 爱因斯坦通过光电效应证明了这个。 但是他的思路来源于 麦克斯布朗可 的 黑体辐射理论的 的量化能量 观看我们的其他短片可以了解这部分的细节。 好。 既然每个光子是一小块光波 也就是电磁波 而且每个光子不能再被分割 也就意味着每个光子 都应该要么经过这条细缝, 要么经过另一条细缝。 它肯定不能分成两半,然后再在另一边合并。 本来这事情很好解释 有两个光子 一个光子经过一个细缝 然后两个光子经过细缝后 互相干扰 因此产生了这种干扰条纹 但是到这里,我们得到了一个非常令人惊讶的测试结果。 这种干扰条纹始终都存在 即使我们每次只发射一粒光子。 我们重新在梳理一下。 我们首先发射一粒光子,第一个光子被精准的检测到 落在了屏幕上的某个位置上。 如此,发射第二个、第三个、第四个光子 都一样落在了某个位置上 因此这表现出它们的颗粒特性, 并且有精准的位置。 但是持续这样下去 一颗一颗的发射出去 你会逐渐观察到这种干扰条纹又出现了 顺便提一下 维尔塔什 实际上 在他的 一系列的双缝实验的视频里进行了这项实验 值得大家一看。 这实在是太奇怪了。 这个干扰条纹 与每个光子能量传播方式无关 这不同于水波的传递。 每个光子一次性把自己的能量全部释放到屏幕的一个点上。 然后,最后产生的干扰条纹却由 好多个完全无关联的光子导致。 怎么会这样呢? 每个光子并不知道上一个光子落在屏幕上的位置 也不知道下一个光子会落在哪里。 光子落在的区域分 最有可能被光子射到的区域 与最没可能被光子射到的区域 也就是我们知道的干扰条纹, 类似一个波浪通过两条细缝。 这些光子就会按照这种样式落在屏幕上。 其实并不仅仅 光子是这样的 一粒一粒的把电子射向这种细缝 它也是一个点一个点的落在屏幕上 不断的逐个逐个发射电子 也会最终出现同样类型的干扰条纹。 这种疯狂现象已经在原子甚至 分子上发现了。 C60 是由 60 个碳原子 组成 的大型分子 也被观察到了这种 双缝实验里的干扰条纹 我们不得不得出结论,每个光子、电子 以及C60分子 像波一样 同时 穿越 两条细缝. 然后他们自己与自己干扰 从而形成这种干扰条纹。 条纹的亮处就是 这种粒子出现的最多概率的地方 发现自己。 就好象是概率波 出现在某个点,因为某个理由 让自己出现在某处 量子隧道效应里也有类似的现象 事实上有些量子属性,如量子的动量 量子的能量、旋转都呈现这种 波的特性。 我们用数学的方式描述这种 波一样的特性,称为 波函数。 波函数的描述 是量子力学的核心。 那么这些波函数到底意义着什么? 这些波到底是什么? 我们再次回到双缝实验的结果。 我们可以知道例子从哪里发出并落在哪里。 它从我们的激光枪或电子枪里发出 它落在屏幕的某个点上 因此这些粒子在起点与终点都呈现粒子特性 在起点与终点之间却更像波。 这个波里包含了这个粒子 可能落在屏幕上某处的所有信息。 也包含了它在整个途中的 任何一个阶段的某个位置。 事实上,波必须包括这个粒子 所有可能通过的路径。 这些所有 从起点到终点的路径, 到达屏幕后,落在最终的终点上 那意味着选择
从这些可能的路径。 那么是什么原因导致这
波之间的过渡 的多种可能性
和一个明确的东西 在一个特定的点? 在那神秘的起点 与终点之间 是这个粒子在这段空间 里的任何一处的概率? 好。 我们加入了更多的问题。 我们仍然无法弄清楚什么波是什么。 事实上,
答案是不知道 但是量子力学正在尝试的各种各样的解释。 让我们来谈谈最受青睐的 由沃纳·海森堡 和 尼尔斯·玻尔 提出的 他们是20世纪20年代。哥本哈根大学量子力学的先驱。 哥本哈根解释
说,波函数 没有一个物理性质。 相反,它是 一个 纯 概率 事件。 它表明,一个双缝实验中 的粒子的所有可能的路线构成一个波函数 一个概率波,这个概率波涵盖了 所有可能的路径。 只有当粒子最终被检测到 该位置和路径才最终被确定。 哥本哈根解释
称这种转变 从概率事件变为既定的事件 为波函数坍塌 它告诉我们,在概率波函数坍塌之前, 尝试讨论一个粒子的属性是没意义的。 这就像宇宙中存在着任何的可能性 而且这些可能性同时存在 直到最后一刻、才知道过程是哪个。 怪异,这些不同的可能路径, 这些不同的可能现实, 互相交流。 这种互动增加了某些路径的机会 同时又减少了某些路径的可能性。 也就是可能的现实之间的互相影响。 这是出现在
最终位置的分布 即干涉图样。 这种模式是真实的,尽管
绝大多数 的产生路径从来没有 成为现实。 在哥本哈根解释, 这个最后的选择 是随机的 且在这个最终波函数限制内。 量子力学理论可以 精准的预言现实 和它是完全
与哥本哈根一致 但这哥本哈根并不是唯一的解释 还有些解释赋予了这个波函数 物理意义。 请记住,我们知道
光是电磁场里的波 量子场论告诉我们 所有的基本粒子都是他们领域的波。 这可能给我们一个
更多的物理介质 驱动这些概率波。 如果你认为
哥本哈根解释是 非常怪异,可以浏览我们制作的“多重宇宙” 短篇。 感谢 Create Courses Plus 伟大的课程
Plus是一种服务 这可以让你了解一个
从教育的主题范围, 包括常青藤
教授和其他老师 来自世界各地的。 转到thegreatcoursesp
lus.com/spacetime并获得 访问不同的库
关于科学的视频讲座, 数学,历史,文学,或
甚至如何做饭,下棋, 或者成为一名摄影师。 新的课题,讲座,
教授 每月添加。 我最近一直在探索
伟大的课程加 量子力学的内容。 这是一个全面审查
非常复杂的主题。 与伟大
课程此外,您可以 看着许多不同的讲座
你想要的任何时间,任何地点, 没有测试或考试。 帮助支持展会,并开始
转到您的试用一个月 到thegreatcoursesp
lus.com/spacetime。 好。 让我们来看看一些
的评论 从我们的小插曲
木星的作用 在形成
我们的太阳系。 杰森·布兰克问:“是不是
木星几乎是明星吗?” 嗯,最低质量恒星
有7.5%左右的质量 太阳,而木星
是太阳能质量的1 / 10,000。 所以它不是真的
所有密切。 你需要大约75木星
堆积在彼此的顶部 点燃持续
融合的核心。 你有几个不知道为什么
我们认为,即使木星 需要有一个岩石核心。 那么,太阳和其他
明星不需要岩石内核 因为他们是巨大的
足以让所有的气 本身崩溃。 有一个最小质量
这是能够这样做的。 这就是所谓的“牛仔裤的质量。” 这取决于云的大小,
温度,转速, 和组合物。 对于典型的星际云,
牛仔裤质量颇有几分 比太阳的小
质量,但仍然太多, 比木星大得多。 对于木星形成
它的气体巨球, 它需要一个岩石核心
启动进程。 该核心可能已经溶解
由于木星首先形成。 朱诺将人物
说出来仔细 映射木星的引力
和磁场。 自行车杰克会
像我多聊 关于谐振频率。 我pleasure–一
谐振频率 是当两个轨道机构
有形成轨道周期 小整数的一个比整齐。 例如,对于每一个
木星的卫星木卫一的轨道, 它的卫星木卫二轨道两次
木卫三和四次。 每八个地球
轨道,金星做13。 这些整数比
最大限度的时间量 该行星花
在最接近。 当这些机构
最接近在一起, 他们拥有最强的
引力 在相互
那拉他们停止 从偏离出来的
该谐振频率。 冒名顶替者抱怨说,
木星情节是这样 也可以理解的。 别担心。 我们已经有了一些
难以理解的内容 真正很快降临到你的方式。 [主题音乐]

Albert Einstein – Mark Steel Lectures



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most people today have only a limited interest in science and that it shouldn't really be surprising from the way that it's taught in school for example when I was at school a spaceship landed outside the classroom and the teacher would have gone what are you staring at you're supposed to be learning about science not looking at aliens of where do you think you're being sucked out to Timothy the overpowering beam of light is a signal for me it is not a signal for your now what comes after lithium scientists can appear part of a conspiracy not to be trusted responsible for all sorts of modern abominations but for some reason the one scientist who survived as a popular figure into our celebrity age is Albert Einstein which may be because he was the school failure who went on to work out how night and time can bend or maybe because he was a peace activist whose discoveries were crucial to building the atom bomb or maybe because he was a proper scientist meaning he had sticking out here did equations no one can understand poked his tongue out and sometimes forgot where he lived Einstein was born in 1879 in the south of Germany and was deemed a failure at school the dyslexic society claims he was dyslexic but in some ways he had a different sort of intelligence to other kids for example when he was first shown his newborn sister he said yes but words wheels until Einstein was 9 he hardly spoke his parents consulted a doctor and then hired a private tutor but Einstein threw a tantrum and it was reported that he grabbed a chair and struck of the teacher who was so terrified that she ran away and was never seen again when you think about it once he was grown RP to 184 tonight cuz if Einstein said he was gonna knock you into the middle of next week you'd crap yourself but once he was 13 he developed two obsessions the violin and a compass that his dad gave him when he was ill Einstein was always careful to remember the importance of youthful inspiration he said the most beautiful experience we can have is the mysterious whoever does not know it and can no longer wonder no longer marvel is as good as dead the tragedy of the adult mind he said was it's in a continuous flight from wonder so he was frustrated by his typical formal education and it was this that led him to be expelled from school for being a disruptive influence when Einstein objected he'd done nothing wrong his teacher replied yes but you sit in the back row and smile and that violates the feeling of respect which a teacher needs from his class what a comment on formal education that they managed to expel Einstein even as a young boy when Einstein was taken to a military parade the automated movement of the soldiers had made him cry silence Einstein had to start national service at 17 so instead he left the country and renounced his German citizenship he spent his time wandering around art galleries in Italy and then went even further and renounced his Jewish faith altogether and the enormity of this can easily be lost see I'm probably like many people in the West in that religions played no big power life so I'm not that bothered one way or the other though when Joe Strummer Nina Simone and Johnny Cash died in one year while Cliff Richard lives that's gotta be a sign there is no God eventually he moved to where he could best pursue his scientific interests and came to Switzerland Einstein went to college in Zurich by which time his rebellious spirit his intellect and his violin playing was making him a hit with the women he applied to loads of scientists asking if he could be their assistant but didn't get a single reply we didn't seem to bother in much as he could spend all day studying and playing the violin by this time Einstein had fallen in love with another physicist called me live–ah who his mother detested to the extent she said no decent family will ever by the time you're 30 she'll be an old witch in 1902 he started working over there at the Patent Office an Einstein and maleva got married soon after there was no honeymoon and after the wedding they came home to this apartment except that they couldn't get in because Einstein had lost the key once he found a way in he spent the next three years of his life writing the theories that would redefine the way the human race thought about space and time and throughout this time he barely thought about anything else even after they'd had a child one professor who visited him said he was rocking his baby while smoking a very bad cigar and in his other hand was an open book while the stove was smoking horribly when his wife returned from a holiday in Serbia she announced that she'd become a Catholic and he took no notice whatsoever in every bit of his spare time he kept working which didn't help his marriage as his wife then announced that she'd lost all interest in physics so his obsessive methods helped to wreck his marriage but the good news was it did mean he was able to write a groundbreaking paper on the random movement of pollen and then he turned to light it was already known that light travels at a remarkable speed and that the light we see from stars has taken years to reach us we're seeing the Stars as they were many years ago when that light started on its way towards us just as someone from same 20 light-years away would look at earth now and see us watching little and large applause and greetings people it had been established that light travelled at the same speed as radio waves x-rays and all other electromagnetic waves 186,000 miles per second and this was true no matter where it was measured and this puzzled scientists because surely if light was shone from a moving object the speed it was traveling at would be 186,000 miles per second plus the speed of the object but it was always the same the one thing they could have tried to slow it down was to privatize it Einstein tried to make sense of the unchanging speed by analyzing speed itself every speed he said its relative we might say a train travels at 40 miles an hour but we mean it travels at 40 miles an hour relative to the station we don't take into account other things such as the speed of the earth around the Sun this business of the speed of light always measuring the same created a few problems if it's traveling at that speed relative to someone moving on a train and relative to a person in bed that doesn't add up or if you were in a spaceship traveling at 180 thousand miles a second how could light still be traveling at 186,000 miles a second faster than you who would be measuring the speed of light correctly the person in the spaceship or the person in bed the temptation was to try and prove that the speed of light did change in certain circumstances maybe there was a cosmic local council putting in speed bumps and cameras all over the place but Einstein approach the problem from the opposite direction the speed of light was constantly said so there must be something else undiscovered the answer Einstein decided was the person in the spaceship and the person in bed were both right from their perspective and are still got three alive lines you could have in defending light could be traveling at 186,000 miles per second faster than both of them if for each of them a second lasts a different length it's all for Chris the main objection to this argument is the one that you're probably thinking now which is they can't do that that's mental it challenges all the themed acceptable wing common sense but how it works is that for the person traveling close to the speed of light a second to them feels exactly the same as it always has done but if the person on earth could see the person on the spaceship moving they would appear to be going incredibly slowly because on the spaceship one second would last longer relative to one second on earth spaceship minicab controllers it's oh yeah it'll be five minutes and when they turn up six years later they can go work on more clock than early Einstein had literally rewritten the laws of physics for 200 years there'd been no scientific dispute against Newton's belief that time and space were in movable but Einstein replaced that by saying there is no absolute meaning to the statement of the time of an event this could mean he had the outlook of a thirteen year old so if you asked him what time he was coming over just go huh whoever but the spell had been broken time wasn't fixed as well as redefining our view of space and time Einstein was determined to sit out his ideas in such a way that they could be understood by anyone who wished to grasp them according to Newton the movement of every body can be exactly predicted from the way that its bang bang something with twice as much force and it will accelerate twice as much but why if you're traveling at 98% of the speed of light and then somebody doubles the force that's moving you according to Newton you'd accelerate past the speed of light but Einstein said it's impossible to travel faster than the speed of light far from anything else you'd look back see yourself arriving now this next bit is so complicated I'm not actually allowed to explain it without wearing this Einstein realized that no matter how much energy you used to push something it could never go faster than the speed of light and to test Einstein's theories in Geneva they built this huge machine that could with microscopic particles close to the speed of light but what they've proved is that the closer something is to the speed of light the more energy is needed to make it go even a tiny bit faster but then what happens to all that energy where does it go if it doesn't manage to make the thing it's pushing go any faster the answer said Einstein is in effects the extra energy goes into increasing the object's mass give it more energy and instead of going faster it will get heavier the implications of this are even more extraordinary than the stuff about time bending for start it means never travel near the speed of light with an anorexic she'll get on the scales and go for megatons greengrocers at love it which won't have free pads nil be over there done in 85 tons be forty three million quid all right precious the idea that energy could change mass involved as bigger philosophical jump as the notion of time slowing down previously they've been thought of as entirely separate it was as if someone discovered that if you do writing to oranges they become Sugar Puffs again this wasn't just a trick the mass of the particles traveling near the speed of light has been measured and does agree with Einstein's calculations and his calculations show that energy is associated to mass in a precise way energy equals the amount of mass times the constant speed of light times itself or e equals mc-squared even the simplicity of this equation seems to fit on Stein it's as if Einstein had an advertising agency going we need something simple sexy and direct but the consequences of its discovery were earth-shaking if a huge amount of energy could become an amount of mass then even a tiny amount of mass could release a huge amount of energy when he finished writing with special theory of relativity Einstein went to bed for two weeks along with his work on molecules and lights this meant he published three amazing papers in 1905 and he was awarded a Doctorate by the University of Zurich it was a scientific version of a band that's been used to play in tiny venues and then suddenly attracts the attention of everyone in the music business because of an unexpectedly brilliant album except he'd done three brilliant albums at once the top positions at the universities in Bern and Zurich and only completely resigned from this patent office in 1909 so what an amazing idea that as he was writing the theory of relativity he was still going through his normal daily office rigmarole and if it made him get in late I wonder if his colleagues did that thing office workers do where he'd say morning and they'd all go afternoon and then if they knew what he was writing they probably added unless of course you're traveling near the speed of light he had another child with maleva Edward and travelled around Europe to lecture but his living conditions were worse than ever according to one account Einstein was crawling with so many fleas he had to take a bath despite this while in Berlin he met his cousin Ilsa she was fascinated by physics and gave poetry readings and the two fell in love in the way that cousins seemed – back then the situation puts a certain strain on his marriage Einstein drew up a set of conditions that the laborer had to sign for the two of them to carry on living together you'll make sure that my clothes and laundry are kept in good order and repair you do without one by sitting at home with you two by going out or traveling with you in your relations with me you are neither to expect intimacy from me or reproach me in any way you must leave my bedroom or office immediately if I request it now throughout this Paul me labor lost for stone in white so suppose she's lucky I so I didn't say to her you want to get yourself traveling near the speed of light now without saw dead air eventually he took a job in Berlin leaving Malaya and his children the deal being that if he won the Nobel Prize which he expected to he would give her the money exactly Einstein's moved to Germany came in 1914 just in time for the First World War and then almost every organization across Germany supported it a group of leading intellectuals issued a manifesto to the civilized world declaring that were it not for German militarism German culture would have been wiped off the face of the earth one of Einstein's heroes Max Planck signed it and ninety-one other prominent academics including vilhelm appointed the man who discovered x-rays Einstein was asked to sign it but instead he drew up his own manifesto denouncing the war he circulated it with a friend but they only got two others to sign he even joined an organization that became illegal for campaigning against the war but throughout he remained above all a physicist he was indeed awarded the Nobel Prize and gave the money to me Laver as far as his physics was concerned this was the most exciting period of his life back in 1907 he'd been sitting in his house in Bern when he suddenly considered how a person plummeting in a lift would feel their own right not only that but if the lift was falling fast enough and a beam of light was shone from one side to the other the beam would appear faint because the lift would have dropped a bit in the time it took for the night to get across so if you're ever in a lift that's hurtling to the ground you'll know which one's the physicist because while everybody else is screaming and wringing their loved ones they'll be the one going does anybody have a torch this is marvelous opportunity the reason he was so excited by this was it alerted him to the possibility of light being affected by gravity according to Newton gravity was a force in which the mass of one object for example the earth drew the mass of other objects towards it such as people or the moon but light has no mess it doesn't weigh anything so it can't be attracted through gravity unless said Einstein Newton was wrong again it is quite comforting to know that no one is ever completely right isn't it Newton's theory of gravity fitted his belief that space was nothing and objects exercised gravity as if by magic across this empty space whereas Einstein was working towards something different he suggested that space is connected to stars and planets and to the rest of space now this next bit is so complicated that I'm not actually allowed to explain it without the assistance of a co-presenter Joe well mark the wave gravity actually works is this imagine you take an ordinary trampoline and you place an object in the middle now all the trampoline is bent so anything traveling near the object has to negotiate the curves that have been made by the object well that was a success now imagine this trampoline is the universe and I am a body such as Saturn a body in three-dimensional space works in much the same way a star or a planet distorts the surrounding space changing its shape so anything traveling through that space has to negotiate the bends and curves that have emerged including light so now you know all about Einstein's general theory of relativity you miss the opportunity to say the car went round your anus so the key to grasping this is to imagine space as a three-dimensional sheet of rubber but when it clicks and you actually get it it's amazing you just want to tell random strangers like on the day after you lose your virginity I was sat in this cafe and suddenly thought I see it I just wanted to grab the way year ago don't wonder about delivering tea can you sing spicy spanking Einstein clearly felt like this a dinner party with his friend Freund Lee he suddenly pushed all the plates away and started doing equations with the food on a posh tablecloth this new theory made Einstein an international celebrity thousands bought his photo but he gave the money to beggars and one journalist wrote women lost sight of domestic worries and discuss the principle of relativity and negatively charged electrons Einstein was offered a three-week run for whatever money he wanted at the London Palladium an advertising company asked him to promote a new product called relativity cigars hi penny from Sigmund I'm phoning about the offer for our new line cheese and Einstein flavored crisps so many agencies asked him to advertise things he said I feel something like a whore hi it's Dave from channel 4 make your program called I love 1905 Albert and I wondered if you could and it's probably for the best that he has such contempt for his celebrity status or his image now could be very different crazy hey but that's how it go time for dinner they're all in days this house is a madhouse parvo annoyed Einstein is the media always asked for sound buy explanations for his theories Albert Einstein in layman's terms how would you describe the theory of relativity well it's when a competition was set up for who could give the best explanation of relativity in 3,000 words Einstein said I shan't enter for it isn't possible we had no conception of well oh I'm sorry we are out of time so a researcher at the University very good reading for joining us he is the source of light on a moving object from a moving motorcar even though you might think that the motive he wore the cheapest clothes and would turn up to a prestigious function to sit next to someone like the Archbishop of Canterbury wearing no socks when he went on tour it took hardly any clothes at all and he took no washing stuff which he described as unnecessary ballast he did have two other pursuits sailing and the violin but as playing led to possibly the greatest sentence spoken in the whole of the 20th century during one recital Einstein got hopelessly out of time with the rest of the orchestra and the whole thing had to stop and one of the other band players with no sense of irony shouted Einstein can't you count Einstein had another problem an anti-semitic group called the German natural philosophers was fog obsessed with discrediting him and at one of their rallies Einstein hired a box and sat there going along with the whole thing clapping at the most absurd comments the German natural philosophers complained about the Jewish nature of relativity a series of prominent German Jews were assassinated so without telling Einstein Elser ensured whenever he toured there was secret policemen protecting him he was informed that he was on an assassination list with a fascist group offering a $5,000 reward for his head to which he said I had no idea I was worth so much as the Nazi Party grew they came up with all sorts of ways to discredit him professor thomas check offered through physics is the creation of the German spirit statistical laws in physics must be racially understood by the time Hitler came to power in 1933 it was becoming impossible for Einstein you get an idea of the atmosphere from the greeting sent by the rector of Frankfurt University to Hitler to Kaiser Wilhelm Institute for the Advancement of science pigs leave to tender reverential greetings to the Chancellor so 164 leading Jewish professors were sacked Einstein had his savings confiscated and he had no illusions that the Nazis were a passing fad so he accepted an offer to work in America shortly after Einstein took up his position at Princeton University there was a phone call to his office and a man asked where Albert Einstein lived so the secretary said well I can't give out that information and asked who was calling and the man said his Albert Einstein I've forgotten me address he toured the world addressing audiences on science and peace but usually gave away his fee and then on the issue of peace he changed his mind the Nazis were such a threat to civilization he decided that a war against them could be justified in particular when he realized that in accordance with his theories the energy of an atom could be released as explosive energy it became terrified that the Nazis might be able to make an atom bond so he wrote to President Roosevelt but Roosevelt brushed Einstein aside just writing him a short letter saying that he found this most interesting eventually the Americans put together the Manhattan Project hundreds of scientists were locked in a huge compound in Los Alamos with orders to build an atom bomb and Einstein was told nothing about the project and despite all the fears that the Nazis would use e equals mc-squared to build a bomb first the Nazis never got near to producing one but the open Homer team did and in an interesting twist in the race with Germany to build an atom bomb having won the race they then decided to drop the bomb on a different country altogether so about ten past eight one morning an American pilot dropped a package about ten feet long 2,000 feet above the ground one extra Neutron destabilized the nucleus in one atom this set up a chain reaction splitting the atom apart and dislodging thousands of other nuclei the mass lost by all these atoms times the speed of light and squared was the amount of energy that spiraled out of the uranium Einstein was proved right as a e equals mc-squared wiping out 100,000 people in an instant but I knew nothing about it until he heard it on the radio when he said oh how horrible some people have suggested that Einstein was partly responsible as he uncovered the science that made it possible be it might as well blame Marconi for Chris Moyles from then on he did all he could to promote the peace movement as a result the FBI compiled a one thousand five hundred page document detailing his political activities at one point the CIA accused him of working with former Nazis on a beam of light that could melt and destroy cities but in 1952 he was offered the job of President of Israel I suppose they saw read written about something that could occupy space at 186,000 miles a second and they thought he's a man but he thought that he would not be popular as president and by 1955 he was getting increasingly ill when he became aware that he was dying he refused surgery and he was told that his eye auto could burst at any moment but responded by saying let it burst one night in the hospital he had trouble breathing and as the nurses tried to help him he spluttered a few sentences in German that no one present could understand and then died the cancer itself Tim Tisch part of his genius must have been due to an attitude he summed up by saying man is too obsessed with the trivialities of living Einstein was dealing in such vast quantities of time and space that it's easy to imagine he didn't have any time for Humanity at all but I don't think that's what he meant by trivialities perhaps he thought about the Manhattan Project and wondered what might be achieved if the same resources were put into aiding rather than into ending it or perhaps he was trying to get humanity to see what he saw that if you do strive to understand time and space the universe may still not be any more comprehensible but it does become far more beautiful