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

Nima Arkani-Hamed Lecture: Day 1

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The Salam Lecture Series 2012, with a week-long series of lectures by renowned theoretical physicist Nima Arkani-Hamed. Giving his audience a panoramic view of 400 years of physics in his first lecture, Arkani-Hamed provided insights into the various concepts that have dominated the world of fundamental physics at different points in history. “Everything that we have learned [over the past 400 years] can be subsumed with a basic slogan, and the slogan is that of unification,” he said. “More and more disparate phenomena turn out to be different aspects of the same thing.” “Physics,” he stressed “forces you to remove artificial distinction between disciplines.”

Lecture 1 | String Theory and M-Theory

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(September 20, 2010) Leonard Susskind gives a lecture on the string theory and particle physics. He is a world renown theoretical physicist and uses graphs to help demonstrate the theories he is presenting.

String theory (with its close relative, M-theory) is the basis for the most ambitious theories of the physical world. It has profoundly influenced our understanding of gravity, cosmology, and particle physics. In this course we will develop the basic theoretical and mathematical ideas, including the string-theoretic origin of gravity, the theory of extra dimensions of space, the connection between strings and black holes, the “landscape” of string theory, and the holographic principle.

This course was originally presented in Stanford’s Continuing Studies program.

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Stanford University all right so let me just tell you a little bit about the origins of string theory the origins of string theory really were in hadron physics they did not have to do with quantum gravity they had to do with protons neutrons mesons particularly massan's in fact the theory was put forward at a time when it really wasn't even known that for sure was suspected but it was known for sure that the protons and neutrons massan's and so forth had a quark content to them it was suspected that Amazon was a pair of quarks the idea of gluons did not yet exist the idea of gluons well in fact it actually did but nobody paid too much attention to it Nambu had postulated something like it but gluons were not part of the standard discussion of hadron x' and let's say around 1969 1968-70 what was part of standard hadron physics was one an interesting fact that the number of particle States was large there was the proton and the neutron of course I'm not interested in the difference between proton and neutron just think of them as one thing the nucleon and then there was another particle which was very similar to the proton in the turn had a little more spin and a little bit heavier mass and then there was another one above that with a little bit heavier mass and a little bit larger spin people drew pictures diagrams they were called Chu Frau floods flops not Platz Platz Platz Platz and they were diagrams which that which indicated the spectrum of objects like a proton for example and they plotted horizontally here let's say they plotted vertically angular momentum and horizontally the square of the mass who decided to put the square of the mass there instead of the mass nobody that they the diagrams had a nicer look to them if you plotted mass squared and what was discovered experimentally this is an experimental flat fact getting my keys in my F screwed up today an experimental fact that the spectrum of particles okay let's start with the let's start with the proton that has half a unit of spin and a mass of one in certain units namely the unit in which the mass of the proton is one that's approximately 1 GeV incidentally at that time the GeV didn't exist it was the B ev4 billion now its Giga okay and so the proton and the neutron would are over here a mass of one mass square root of one and an angular momentum of 1/2 then there's another particle with angular momentum three-halves these were fermions and so their angular momentum is quantized in half integers and so there's another one up here with a little bit bigger mass another one another one and another one and rather remarkably these particles all formed a straight line a straight line in the in the plot of L versus M now I should tell you when qu and Frau G first put this forward the logic of drawing a straight line there were only two points on this plot and they thought it was a theorem that through any two points you could draw a straight line well it is a theorem but these guys were not the smartest well they were pretty smart but they they and so they said oh there's two points let's draw a straight line through them and miraculously as experiment went on the additional particles all landed on the mine what an M square yes they actually said that M square they plotted it as a function of M Squared and album and said two points let's draw a straight line and that works out just fine but I mean that explains why M squared is the axis there because they're hypothesizing L is proportional and squared so now they were why would they why did they postulate that instead of L proportional to M they were lucky they were lucky or else they had some deep no it wasn't it and it wasn't entirely luck and it wasn't entirely and it wasn't a stupid guess either there was some interesting reasons for it but the same pattern held true for all hydrants or at least for all hydrants that have been studied in detail for example the pine is on the PI meson also exists on a trajectory these are known as reg a trajectories are eg GE for the for the Italian physicist Tulio Reggie they're called Reggie trajectories and if you plotted the Mazon spectrum the Mazon spectrum neurons are bosons so the angular momenta were integers for example the PI meson has almost zero mass its mass squared is even smaller than its mass it's true in units of GeV all right so the PI meson was almost massless with almost zero angular momentum and then there's a next one up I forget what it's called I don't remember what the next one is called I used to know but I don't remember anymore the next one up and the next one up and five six seven particles along a trajectory like that the Roma's on which was another mess on which starts with angular momentum one also same pattern and what's more all of these trajectories were parallel to each other they were parallel to each other which said whatever this M squared thing is it takes exact Klee the same energy well not exactly but approximately the same energy to increase the N squared by one you to increase M squared when you increase the angular momentum towards the spectrum was quantized of course it was quantized these were particles and this is quantum mechanics the spectrum was quantized but in each case the same relationship between L and M squared and the same quantum jump in M squared when you increased L by one unit one unit now means in units of Planck's constant of course so there's something going on that was giving rise to large numbers of particles of higher and higher angular momentum higher angular momentum is not that uncommon you take a basketball you leave it at rest that basketball has a certain energy and therefore a certain mass and you could plot at some place and now you spin the basketball give it one unit of plunks angular momentum that's not easy to do incidentally but when it has some angular momentum it will be rotating it will have some rotational energy so if you increase its angular momentum by one unit you will have to increase its energy by a little bit tiny tiny bit for a basketball and you can keep increasing the angular momentum of the basketball as you do so the energy will increase in fact it will not look like a straight line it will look like a curve and it will in somewheres why does the curve end the curve ends simply because at some angular velocity the centrifugal forces are so large that the AB basketball will just be torn apart right so in some place at some high angular momentum which represents you know the strength of materials how strong is a whatever basketballs are made out of so trajectories like that were not unusual you can plot them for atoms atoms also have the property that as you increase their angular momentum you increase their energy the mass energy and mass being the same thing but again for an atom there's only so much mass you can give it before you ionize the atom so what was unusual particularly unusual and again incidentally for atoms they would not be straight lines well as unusual here was the simplicity of the formula or the simplicity of the observation rather straight lines I mean they were all straight and parallel to each other and when you say how do you mean I mean that if you were to plot the Mazon or the baryon or the proton or the neutron or the PI meson or the Roma's on or it's excited states would form the same the same line in other words you take a set of part of a family of particles same slope same slope and it was called the universal reg a slope same for bosons and fermions same for different families of of bosons and fermions now this is strictly for those objects which are hydrants those things made up out of quarks and gluons which we now today recognize is being made up of quarks and gluons that was one observation the implication of this observation was fairly clear even though it was misinterpreted at the time in many many quarters it was fairly clear it said that hey drones will composit that you could spin them up this is not something you can do with an electron there is no excited state of an electron with higher angular momentum at least not not though within current experimental bounds so electrons are like points you can't see you can't spin a point spinning a point doesn't mean anything turning a point you can spin a lump so somehow these objects were not simple point particles that was the message that should have been and then in many quarters was taken from this and in fact that they had a stretch ability that from this picture you could deduce if you wanted and we will we will you could deduce the fact that they deform as they spin you wouldn't it's not obvious from here but you can all right but there was something else there was another observation which was a very bizarre up to observation let me describe it to you in terms of mess on mess on scattering let's take mezzo mezz on and then for particular let's take pine there's on primers on scattering here's a PI on coming in applying as on let's call pi doesn't matter whether it's pi plus pi – doesn't matter and it scatters off another prime is on there is a particle called the Roma's on which while it's not a composite of two pi mesons two parmesans can come together at a vertex and a Fineman diagram and make it Roma's on let's make the Roma's on like that we're going to talk when we may or may not talk more about these massan's it's not important that the idea is important but the particular names are not important Roma's on and then that PI meson could then that Roma's on could materialize as a pair of primers ons again just being a Fineman diagram conventional Fineman diagram and it would govern the properties of pyon pyon scattering probabilities okay that was the first thing now these are all pylons here any quantum field theories would immediately tell you if you have this diagram here where the tip ions come in this way and make a Roma's on and then go off as to PI ons there will be another diagram which looks like this it's just the same diagram turned on its side where a Roma's on is exchanged between two – ons in this case a Roma's on jumps from here to here and then without the peons ever annihilating but this is just the same diagram turned on its side and if one exists the other has to exist that's a consequence of principles of quantum field theory so that's something that everybody believed but then once it was recognized that this Roma's on was not a unique creature but came along with this whole family this whole Reggie trajectory of excited states it became clear that there was no reason to only have a Roma's on in here you could add together all the various you could add pi mesons come in form not the Roma's on but the next excited state of the Roma's on or the next excited state of the Roma's on and so actually when you draw this diagram you're really committed to adding up the contribution of all the mesons along here likewise here if you can exchange a Roma's on you can also exchange all the excited States well there was something very very suspicious very very peculiar when people numerically went to do this they actually added up from known experimental data the contributions of the Roma's on row Primus on the row double prime measure on and so forth and so on they did that and they did the same thing for the exchange of the Roma's on and they found something rather remarkable they found that to some approximation the sum over all the Romans going this way gave rise to about the same thing as the sum over all the Roma's ons and its partners going in the opposite there any and the other channel it was called in other words it appeared to be over counting all you needed to represent the data and all that you needed to represent the the physics of Pi scattering was summing over Rho Rho prime Rho double Prime in this annihilation process annihilation and recreation process and in fact you didn't need to add this in because this already seemed to contain it numerically numerically on the other hand it was also true that you could ignore this altogether and they are all all this up and again get the right answer get something which looked pretty much like experimental data I was very peculiar any quantum field theories would look at this and say complete how rubbish you give you if you have this you have to add that you have to add it you don't the you don't get to say it's this or this you get to say is this and this okay but peculiarly when you add it up all these contributions it simply gave you for free the effects of another diagram here for that reason I and other people began to draw diagrams which look not so much like Fineman diagrams which look more like this we didn't know what we were doing just saying look something's going on here first of all we have the idea that they were quarks quarks were not a new idea we said what must be going on is there must be a picture which looks something like this replacing the Fineman diagram this would be a quark this would be an anti quark this would be a quark going this way an anti quark going this way and likewise over here we do diagrams like that now it took some time for us to think about filling in what goes on in here we just drew pictures like this and we said look at this if you think about somehow what's going on has a topology let's call it a pathology which looks like this then look if you cut it this way if you imagine slicing it in an instant of time time is running upward of course and all you know pictures finding diagrams time we're allowing to run upward if you slice this right through the middle then you see something that looks like a Fineman diagram in which two particles come together and join and make another dial particle if we think of particles as pairs of quarks then this figure here can represent a picture like this on the other hand if we take that same picture and slice it this way it looks like a picture in which something is jumping across slice it the other way it looks like these two particles produced a thing which jumped from the side to the side so this was kind of the origin of pictures one more ingredient was added it was just added for fun and just just a curia a curious question if there are these quarks what's holding those quarks together well maybe it's something in here these are space-time diagrams of course maybe something is bridging between the quarks and if so then when you slice through these diagrams then what you would see is two quarks with something bridging between them and that's something of course would have the structure of something one-dimensional connecting a quark and an antiquark a string and a pair of quarks if you cut it the other way that same two-dimensional sheet here could be sliced into a picture where a string was exchanged from one side to another this was the very crude origins of the idea of string theory in fact it isn't exactly where it came from but they could have it did to some extent different people thought about different ways um so what's more what's more once a hadron is a string with two quirks connected to it you can spin it something you can't do to an electron you can spin it you can try to calculate with some assumptions about the nature of the material forming the string its elasticity its various properties you can start asking how the energy of it what kind of energy is there incidentally first of all is kinetic energy and second of all is stretching energy so with some kind of assumptions about the nature of these strings or this material in here you can start asking questions about how the energy increases as a function of the angular momentum surprisingly with a relatively simple assumption that we're going to do when we may get to it today I hope we get it to Attila to tour today we'll see that these pictures are not as arbitrary as they might seem l vs. M squared a linear function of L versus M squared is exactly what you need now I yeah why did you suggest that it was something like a string something you normal particle exchange through the two forks well this is not a this is not a particle exchange this is something that's sitting there all the points of it between the between the quarks this is if you were to slice it at an instant let's take a look at it at an instant here it is at an instant that consists of a quark and antiquark and a whole bunch of stuff in between now that bunch of stuff in between might have been a collection of particles that are forming a string you know that are forming a string like thing they might no no prejudice about whether it was truly continuous or whether it was just something which was approximating something continuous in between but there was one thing that made you think there might be something more fundamental about it than just a shoelace and that was that as far as we could tell these trajectories didn't end if you take a shoelace and put a pair of golf balls at the opposite ends of it and spin it around eventually you come to the point where brakes and as far as could be told these rigid trajectories did not break so it seemed maybe that there was something new going on finally to the extent that the standard model gives rise to a string like behavior it can yeah the standard model can describe a great deal about Pi on scattering these days yeah depending on the energy at very low energies the standard model has a very good description of Pi on scattering as the energy goes up where you start getting into the issues of these particles being exchanged back and forth the standard model begins to get a little bit more difficult to deal with and a string like picture becomes becomes more useful did they know that the trajectory does not pass no no no no but didn't give any sign of giving out and yeah so what that meant will it appeared is this thing could be stretched more or less indefinitely without much happening to it will we'll come back to this point well in fact they're not entirely different just glue arms in some sense they may be collections of gluons in fact they have to be collections of gluons the present understanding of it today is something like this that that the gluon field is like the maxwell field the quarks you asked me about bar magnets okay so the quarks are like poles of the North Pole and South Pole of a bar magnet North Pole South Pole you can't have mama poles the only way you can have a north pole or South Pole is to have it as the end of a bar magnet okay or you can say it a different way you can have dipoles you can have magnetic fields which look like this here it looks like is a magnetic monopole lines of flux spreading out here it looks like there are lines of flux spreading out and there might be some reason why the lines of flux form a narrow tube in between the current understanding of the connection between gluons and these strings goes something like this quantum mechanically fields can be described as either particles or fields let's take the field description in the field description the gluon field between two quarks nor quark and an antiquark would be like the field configuration between a particle between a charge and an opposite charge it would look like this that's what the field between a positive charge and a negative charge looks like the energy as you separate them is the energy stored in the field in between now for ordinary electrodynamics those field lines spread out and because they spread out the field diminishes in between them as you separate they feel as you separate the part of the charges the field lines spread out that the field diminishes in between and that's the usual pattern our understanding today is that the nonlinearities we've talked about this before but I'll just mention it again that the nonlinearities in quantum chromodynamics have the effect of causing these field lines to attract in a certain way and the effect of it is that the field lines form strings that look like that as you pull these apart the string doesn't spread this way it just gets longer and longer and longer that's our current understanding and if you like it's permissible to think of the string as being made up out of gluons as you pull it apart it's not like a shoelace or a rubber band as you stretch a rubber band the number of molecules in it doesn't change the number of molecules in it then they just stretch it eventually because the number of molecules doesn't change eventually they get too far from each other and bang the rubber band breaks but imagine a rubber band in which as you stretched it every time there was a gap opening up between atoms a new atom was inserted in between in that case you could imagine that you could stretch this ad infinitum forever without breaking it and that's the nature of the gluon field between a quark and an antiquark as you stretch it the energy of stretching goes into creating more gluons and B Queen and you can just stretch the hell out of that that system and won't work yeah can you extend the analogy to say that the strain that makes up saying electron is its component well first of all we don't know that string that electrons really are made of strings no no this all of this physics was taking place on a length scale of the size of a proton this is an enormous length scale by comparison with the scales of quantum gravity which take place at the Planck length it's a more or less perhaps accidental fact that that the mathematics of the string theory has described both things they're quite different they occur at completely different length scales but through these considerations I other people began to work out the mathematics of interacting strings can you really quantify this can you really make a theory of interacting strings which will give you all of the physics of the interacting hydrants the answer at the time is you've looked it looked promising it kept it look very promising in fact it kept promising and promising and promising like string theory today a huge promise but never quite did it right okay for reasons that in hindsight are fairly clear the precise mathematics that we were using was not quite the right mathematics for studying hagans it was the right mathematics for studying quantum gravity and so we kept getting the blue as much as we didn't want it we kept getting particles in the theory with zero mass and spin – what is that a graviton nobody wanted a graviton this was a nuisance go away graviton we couldn't make it go away nothing we could do could make it go away and eventually some smart guy named John Schwarz and Joel shirt and said wait a minute wait a minute me well maybe we were being dumb maybe this is a theory of quantum gravity and not a not a theory of hadrons incidentally string theory of hadrons final has been put together with a proper mathematics it does work but it's a little bit different okay let's talk about well okay before talking about strings the mathematics of strings let's talk about relativity versus non relativistic kinematics Oh incidentally just a buzz word this two dimensional structure in here which is what is it it's replacing the idea of a world line a world line is being replaced by a two dimensional sheet such a sheet I think the term actually goes back to me was called a world sheet today it's the standard terminology so strings are world sheets looked at at an instant in the same sense that particles are world lines looked at at an instant so that's the that's the jargon world sheets and world lines sir that diagram okay feed it there's string in there as a stretching that put the gluons in tension no and so how do you get from plus to minus well the lines of flux come out one side and go in the other side same way you go from the North Pole to the South Pole in a magnet I don't do any magnet any magnet has two poles there's always two poles of opposite sign no magnet has two north poles no magnet has two south poles every magnet has one north pole in one South Pole North and South are like plus and minus four what a black wall I don't know what a block wall is despite the fact that I'm the felix bloch professor of physics I don't know yeah – the string be considered either continuous or discrete or does it make any difference well you see now you know you're running into the subtleties of quantum mechanics is the electromagnetic field a continuum well in some ways yes is the electromagnetic field made of discrete quanta yes in some way yes and quantum mechanics tells you that that that distinction between continuum and discrete is a very subtle water and I won't try to answer it right now I think both are true its continuum and it's discrete depending on the way you think about it all right next question arm none of nonrelativistic versus relativistic kinematics or kinematics or simple ideas about the particles energy momentum the symmetries of motion of on the face of it relativistic and non relativistic physics look very very different of course we know that they connected to each other but let's let's quantify or discuss that difference yes yes Michel you can always ask you a question as long as you keep bringing me cookies screaming a point in time that we hit the water no change the angle of that cross-section of a fine you still describe whether something like a spring yes yes the Lorentz transformation of us what you're talking about is Lorentz transformation of course the Lorentz transformation is a moving string but it's still a string absolutely right so if you were to consider a frame in which simultaneity was this line here you would still be seeing a string but you would be seeing a string in motion as opposed to standing still right when I say standing still incidentally I mean the center of mass of it's standing still strings wiggle a lot they've got a lot of tension and they vibrate a lot so they don't stand still but the center of mass can stand still all right let's come now to the issue of relativity versus nominal conductivity how do you describe a relativistic string well that's awfully complicated describing anything relativistically is complicated for example just the let's begin with the energy of a particle the energy of a particle non relativistically point particle of is P squared over 2m momentum squared of the depending on the number of dimensions we would have to add up the various components of momentum divided by twice the mass that's a nice simple algebraic quantity the square of a function is easy to compute and so forth you might add to this a constant and the constant you would think of as the binding energy or just the energy of the particle because it's there so you might put something else there let's call it B the energy that it takes to assemble a particle whatever it is the characteristic of it is that it does not depend on the state of motion it doesn't depend on P in a relativistic the there's a natural candidate for what this bee is it's the energy of the thing when P is equal to zero right I mean with relativity or not relativity it's the energy of the particle at rest what is what are we in the special theory of relativity what do we put there MC squared of course so the natural thing to put there which would be MC squared but let's just think of it as a additive constant and it's constant only in so far is it does not depend on P and I vary from different kind of particle the different kind of particle it could be the binding energy holding together an atom it could be whatever but it doesn't depend on the overall motion and for many purposes you could just drop this because it's always there and doesn't the energy differences don't depend on it so P squared over 2m and P squared over 2m is terribly easy to manipulate it's just the thing that you just quadratic of course if you have many particles in a system then what you do is you add up the energy if they're not interacting and you also add up the internal binding energy internal energy you could call it you weigh them all up again this doesn't matter because energy differences don't are insensitive to it now how do we get this from relativity let's remind ourselves what the formula for the energy of a particle is the energy of a particle this is e I in relativity it's equal to the square root of P squared plus M Squared it's of course this I'm going to correct this in a minute but it's of course equal to the sum of all the particles let's just write it as e equals the sum over all the particles in the system of P squared plus M Squared where P squared is px square plus py squared plus PZ squared or however the many dimensions we wish to take into account but why take into account however many dimensions are appropriate to the problem so this would be P I and M I for the ayth particle and also XY and Z and whatever else okay how do we go from here to here well first of all I left something out I left out the C to the fourth that's the fourth power of the speed of light and M square C squared here as soon as I finish this one little demonstration I'm going to set C equal to one going to us which do I haven't run you're right sorry MC squared yeah C to the fourth very good C squared P squared C squared M Squared C to the fourth good well we're having trouble okay the nonrelativistic limit is appropriate for problems where a particle is moving very slowly which means its momentum is very small and in particular in which P squared C squared is much smaller than M Squared C to the fourth under those circumstances you can take the square root of P squared plus M Squared and write it first in the form M Squared C to the fourth times 1 plus P squared C squared over m squared C to the fourth you can factor out of the square root the M Squared C to the fourth and that gives you MC squared on the outside that's good sign but with a correction and the correction is this over here what do you do with it you expand out the square root do you use the formula that the square root of 1 plus a small quantity is 1 plus the small quantity divided by 2 square root of 1 plus a small quantity is equal to the 1 plus the small quantity over 2 that's an approximation of course it's not exact but as the small quantity gets smaller and smaller it becomes better and better so what do you get you get MC squared plus P squared C squared over MC squared MC to the fourth divided by two all times MC MC squared here okay so let's see what cancel MC squared that's familiar that's the relativistic rest energy but this here has four powers of C in the numerator four powers of C in the denominator C goes away it has one power of M in the numerator two powers of m in the denominator cancel them and you get the good old nonrelativistic formula but it's an approximation it's an approximation and when is it good it's good when all particles are moving slowly it's not just the whole system which has to be moving slowly to use nonrelativistic physics you might for example have a box of particles and the box may be moving slowly but inside the box the particles may be moving with close to the speed of light you cannot use pure nonrelativistic physics for all of these particles because they have relative motions which are up near the speed of light so strictly speaking the nonrelativistic limit is a good thing to do when all of the particles are moving slowly and it is an approximation now there's another sense in which nonrelativistic physics is an exact description of relativistic physics so I'm going to show you this this is something that goes back a long ways in particle physics when I worked on it in 1968 or 67 or sometime it was called the infinite momentum frame now what's called a light-cone frame so if you look up a light cone frame you will see these things described okay but it's easy it's easy if I don't want to do it in great and enormous generality it's easy and here's what the trick is instead of thinking of a system in its rest frame when we said or near the rest frame in other words a frame in which every momentum is slow we're going to do a different trick we're going to look at it from the point of view or a frame where everything the entire system has been boosted up to have huge momentum along one axis in other words boost it up so that it's moving down the z axis let's take that to be the z axis so that it has a mungus lee large momentum along the z axis there's no loss of generality there we can take any system and just boost it so that it's along the z axis and then rewrite what this formula looks like ok so I'm going to you know I'm going to set C equal to 1 now I'm not going to bother keeping C the energy is the sum of all the particles of again square root of P squared plus M squared which is equal to square root of PZ squared plus px squared plus py squared plus M squared but now we're boosting the hell out of the system along the z axis until every single particle has a huge momentum along the z axis every single one of them if there's any particle which is going backward on the z axis you just haven't boosted it enough just boost it more until it's going forward with a large momentum in that case all of the PZ s are very large what happens to px py and M when you boost something nothing that's the rest mass we don't even speak about moving mass anymore the rest mass and the components of momentum perpendicular to the boost don't change when you boost something ok so now the big quantity is PZ and P X py and M are kept fixed and much smaller than P Z so the appropriate thing to do here in taking the limit is expand it for large P Z expand it for this being small a way to do that is to write this in the form square root of 1 plus let's just call it P Square P X let's write it out P X square plus py squared plus M squared divided by P Z squared all times PZ on the outside right PZ on the outside if I brought the PZ inside the square root it would have to be squared it would be PZ squared and then it would cancel this PZ squared here okay what's the next step expand use the binomial expansion and binomial approximation to do exactly the same thing we did over here this is now the small quantity and so this becomes P Z times 1 plus the X square plus py squared plus M Squared over twice P Z twice P Z squared excuse me P Z squared or to summarize it all the energy is the sum of all the particles of PZ of the iPart achill plus the sum of let's call it P P will now stand for px and py let's use little P little P stands for px and py little P squared over twice big PZ plus M Squared over twice big PZ no no those are PZ up here and a PZ squared down here okay good I don't need to put brackets in okay first observation if we believe in momentum conservation which we do in this class if we believe in momentum conservation then first of all this is just the total momentum of the system the first term here is the total Z component of momentum going down the z axis it's huge very large but it's a constant it's a constant that as various things go on in this system the total momentum never changes if you have a constant term in the energy which doesn't change in any way during the course of a a constant additive thing adding it to the energy or subtracting it from the energy doesn't do anything for example if you added the electric charge to the energy since electric charge is conserved and the only thing that's ever important in physics is differences of energy you could just drop it or keep it that doesn't matter the same is true here you have a conserved quantity which is conserved for other reasons than energy conservation energy of course is also conserved but P Z is conserved for other reasons here's something which never changes you can just drop it if you were to think of the energy as being the Hamiltonian of a system it would make no difference whether you whether you drop it or don't drop it because it's a conserved quantity which never changes so you can drop this will make no difference the rest of the energy here is this thing here now first of all notice that PZ is in the denominator what does that mean why is there why is the energy so small in particular energy differences for example differences depending on the state of motion in the XY plane they will be tiny why are they tiny anybody know why are the energy difficult I will tell you for this it's useful to remember a bit of quantum mechanics even though we don't need to be doing quantum mechanics what is the meaning of the energy the energy of course in quantum mechanics is the same as the Hamiltonian it is also in classical mechanics but what's the meaning of the Hamiltonian in quantum mechanics do you remember your quantum mechanics it is an operator it's a hermitian operator but it's also the Opera mm-hmm it's an eigenvalue of the energy but it's also associated with something else it's also associated with time evolution all right remember that this is the same as I D by DT namely IH bar probably D by DT this is its action as an operator on a state that what it means to say energies are very small is that systems are changing very slowly this is also true incidentally in classical mechanics or just to point this out there quantum mechanics if the energies of a system are very very small it means changes take place very very slowly the smaller the energy if the energy scales with someone over P Z here it means that the larger the P Z is the slower things take place in the system what's going on here very simple it's time dilation the more you boost the system up to higher and higher momentum in your reference frame the slower things take place okay that's interesting but of course we have all the time in the world are a system moving we can wait as long as we like to see things take place if we're trying to make a theory of radioactive decay sure boosting it up will make the radioactive decay go slower but we can rescale that out we can say instead of working on a scale of microseconds we'll work on a scale of millions of years and we'll also see the we'll also see the the nucleus decay everything just has to be rescaled so this one over P Z there the total this is the one over P Z this is incidentally for the ithe article and we add them all up so the fact that all the PCs get large incidentally in fixed proportion they all get large and fixed proportion that said the energy got smaller that's a completely expected phenomenon apart from that if we rescale only pzs ignore the fact that they get big or just rescale the evolution of the system this Hamiltonian or this expression for energy really does look like the nonrelativistic nonrelativistic expression with respect to the motion in the XY plane for the motion in the XY plane the energy is proportional to the square of the X Y momentum just as it is for the non relativistic particle but notice that the role of the mass of the particle in this nonrelativistic analogy is not the rest mass it's the momentum along the z-axis what this means what is mass mass is inertia right that got to do with the difficulty of deflecting something well this is saying is that the momentum along the z axis is functioning as a kind of inertia with respect to forces in the perpendicular direction and the whole thing is looking very very much like if we think of PZ is a constant then all this is is the nonrelativistic formula for the energy of a two dimensional particle now notice we now have only two dimensions of motion and what is it what about this term over here well how will should we interpret that again remember that we think of PZ as being independent of the state of motion at least a two dimensional motion so with respect to this two dimensional analogy analogy between relativistic and nonrelativistic physics it's an analogy between relativistic physics and two dimensional motion in which PZ plays the role of the mass and how about this object over here it plays the role of a binding energy does it have the right properties to be a binding energy the only thing about a binding energy is that it should be independent of the state of motion it should not depend on this does not depend on the two dimensional motion so this is kind of interesting and it's not only interesting it's incredibly useful in studying particle dynamics and absolutely central to studying strings is that in a very precise and exact way the motion of a relativistic system when it's boosted up to enormous ly large momentum behaves completely non relativistically with respect to the motion in the plane perpendicular to the to the boost okay it's for this reason that string theory is also often described in terms of mathematics which is the mathematics of a nonrelativistic string a non relativistic string a non relativistic string is a collection of point particles in some limit than which to let the point particles get more and more continuous all moving non relativistically why by what the hoods bar do we do we use non relativistic physics to describe anything is complicated as a relativistic string well the answer is that in the infinite momentum frame which these days is called the light-cone frame mostly because there's nothing to do with cones nothing whatever to do with cones I'll tell you another time where it's called a light cone frame not important but in the infinite momentum frame motion is nonrelativistic and you have a chance that perhaps the motion of a string when it's boosted up may be described by not by a kind of nonrelativistic quantum mechanics and this seems to be borne out not seems to be this has been the techniques that have been used for since the very beginning of string theory to analyze relativistic strings let me show you the simplest fact yet ugh no okay let me show you one of the very simple connections that that follow from thinking this way let's now hypothesize or postulate that we can think of particles as strings using the two dimensions using the two dimensional analogy with nonrelativistic physics to explore those strings as if they were conventional nonrelativistic math shoelaces but something closer to rubber bands stretchable they can move they can flap they can do all the things that a rubber band an ideal rubber band can do what what's the mathematical description of a two-dimensional rubberband which is moving around in two dimensions let's take our rubber band to be an open rubberband that means somebody took a scissor cut it and opened it up let's begin with open strings open strings mean strings with two ends that may or may not be something interesting attached to the ends but we're interested more in the strings let's write the physics of a of a string what is the where z energy what is the energy stored in a string we can think of the string as a collection of points point particles which later on we will take limits one of the things we will do when we take a limit as well let the mass of each point go to zero that's because we're going to have an the whole string has a finite mass we're going to think of it as being a collection of a virtual infinity of point masses it had better be that in taking the limit we let the mass of each point go to zero all right but what's our what's the energy of this the energy is going to be proportional the kinetic energy it'll be the sum of all the points of X I dot squared these are two dimensional now so we can write this as X plus y dot squared that's the ithe point divided by two and we might put here in a mass of the I part achill which later on we're going to let go to zero but let's say let's not there be two I'll just tell you how to do to contain the continuum limit I'll show you how to do it I'll just tell you how to do it all right what are we missing out of this formula interactions yeah are these points are attracting each other if they weren't attracting each other they would just fly apart they're forming a string they are in addition to the points we have to put in the little Springs that connect them so think of it as a chain of little balls and little Springs can you see the Springs little balls and little Springs let's just call this X sub I squared X sub I squared now stands for X x squared X dot square plus y dot squared okay what is the potential energy between the points the potential energy is a sum also over all neighboring pairs so there's another sum of I here there's a spring constant let's just call it K all the mass points have the same mass there's a spring constant there and what is the potential energy between a pair of points it'll be proportional to the distance between them X I minus X I'm plus 1 squared probably a 2 there this is Hookes law this is Hookes law the energy stored in a stretched spring is proportional to the distance of stretching squared that's the Hookes law formula for they now what happens when you go to the continuum limit in other words you let the points get denser and denser and denser more and more of them you have to do two things you have to let the mass of each one go to 0 and you'll have to also let the spring constant what it is you want the spring constant to get bigger small big big can you earth you you know why supposing you take a rubber band and you take a rubber band a big long piece of rubber band and you stretch it it's easy to stretch now take two points very close to each other and try to stretch them that same distance much harder okay so the spring constant gets big and the mass gets small but at the end what you get just take it from me what you get is of course an integral represent an integral replacing the sum the integral is over a parameter along the string you have to introduce a mathematical parameter along the string we can call that parameter we'll give it a name Sigma Sigma goes from one end of the string where we can arbitrarily say it's zero so Sigma is zero at this end and at the other end we can arbitrarily say Sigma is equal to PI I could have taken it to be one I could have taken it to be seven that doesn't matter it will be useful to think that to call it pi the reason is later on we're going to study closed strings which go all the ways around in a loop and it's nice to say they go from zero to two pi that's all but they go from zero to pi so this sum over the mass points is going to be an integral from zero to PI D Sigma this is adding them all up and we're going to have the kinetic energy of the little element of string at point Sigma now continue a string now we take a little element a point Sigma we take this velocity squared and divide by two and what about this term over here I've chosen the mass to go in the appropriate way of dropping the mass here by the time you finished you can absorb the mass into something else doesn't matter what it's just X dot square it's clearly kinetic energy what about this term here what's that going to look like how about X I minus X I plus one which you replace that with derivative derivative this is like the derivative of X with respect to Sigma squared so the other term here will be derivative of X with respect to Sigma this is derivative of X with respect to time this is derivative of X with respect to Sigma squared this is the energy of the string if I wanted to write the Lagrangian you all remember what a Lagrangian is energy is kinetic energy plus potential energy Lagrangian is kinetic energy minus so if I wanted the Lagrangian it would be this if I wanted the energy it would be with a plus sign okay I'll write the energy will write the energy Hamiltonian plus let's focus for a little bit I'm going to stop at a few minutes and we'll take a rest but let's focus a little bit on a string which happens to have no overall center of mass motion in the two dimensions in the XY plane we're coming back now to here what we're going to do is use a model for a relativistic string which is simply based on this kind of infinite momentum thinking but in which there are only two X's the two X is moving in the in the direction perpendicular to the motion so this could be called X&Y but I'll just call it X dot squared it really consists of X dot square plus y dot squared this one consists of the X by the Sigma squared plus DX plus dy by D Sigma squared is that clear yeah okay oh I just absorbed it I just chose oh sorry there isn't two two is important I chose K in such a way to make sure that when I got to the final continuum limit the coefficient was 1 remember it's something that has to that has to vary as you vary the spacing and it can be chosen in such a way as to make this and this is the conventional energy of a vibrating string it has two terms kinetic and potential potential proportional to the stretching this is this DX T Sigma is the stretching of the string okay I want to point out one interesting fact this Hamiltonian here or this expression for energy is the generalization of this expression here for a system of particles which also has a interaction between them but the whole thing the whole object it may be vibrating and doing things but the whole object is an object we can call it a particle who's to say it's not a particle protons and neutrons have spin they rotate there's all sorts of internal motions in particles we know there are internal motions of particles internal motions of atoms internal motions of quarks inside protons and neutrons the best bet would be there are all sorts of internal motions in every particle so this stringy vibrations and internal motions and so forth that perhaps not perhaps but would add up to all the internal motion in the particle all the internal energy in the particle the internal energy would be the contributions to the energy from the potential stretching and from the relative motion of the different parts the overall motion will separate that out soon enough the overall motion of the string the center of mass of it that would just be treated as the ball as the position of the particle but the relative stretching and the relative vibration that's internal energy so when we calculate the internal energy of this particle what should we relate it to we should relate it not to the mass but to the mass squared in this correspondence it's not an analogy it's an exact statement about the properties of relativity is a very precise mathematical statement which I won't make now but there is a there was an exact sense in which fast moving systems are completely relative and non relativistic in the two dimensional sense what would the internal energy now respond to it would correspond not to the mass of the particle but to the mass squared so for a string at rest think of a string which has no motion in the in the XY plane all is doing is vibrating and it has internal energy that internal energy has to be identified with the square of the mass of the entire assembly of constituents of the string if the constituents of the string are adding up to something that we want to call a particle then that particle has a mass squared which is the sum of all of the internal energies inside the particle now this is an interesting fact we get mass squared for the energy of a particle in this framework hmm there are some seas around which I haven't tried to keep track of yeah there's seas in the yellow season right yeah I haven't tried to keep track of them but this connection was something I knew that nobody else knew at the time I'd worked on both these things and so this is interesting this is exciting another fact another fact is that a string is not so different than a spring if you look at the spectrum of energies of a string it's quantized in pretty much the same way we'll come we're going to do the quantization of it carefully but the basic fact about the quantization of it is that the string is a collection of Springs and springs have quantized energy and what's the formula for the energy of a quantum mechanical oscillator and integer multiple of something each time you increase the energy of a spring or a string the internal energy by one unit it could increases the mass squared by one unit increases the mass squared by one yeah else energy zem trip at the end is this work the simple confusion is M the mass of the strength yeah it's a mess of the whole screen the whole script and the energy is socially doubled with the size of mastering four times because M squared would be is that correct yeah yeah you've stolen my thunder okay but now look well okay let's let's look at this formula a little bit carefully now and see if I see anything interesting you like sir you have them squared there that would kind of kind of Connaught the rest mass of the system today it is in fact the rest mass of the string in a frame of reference where it's not zipping along but where it was really stationary by the equation seems to be simply not oh but I'm describing it put on it will that's right because the photon is massless well that just corresponds to M equals zero on the right hand side and you're going to ask me how can this thing be zero well we're going to come to that that is a very significant and interesting point we're going to come to it for the moment it's just the mass of the entire string the entire including its internal energy including its stretching energy all of the energies that you would normally add up to find MC squared to find the rest mass that's what this is here okay it's just a this is just a classical mechanics and relative relativity applied so far we haven't been it any other we haven't introduced London currency and things like that okay but if we did introduce quantum mechanics we would know that this string would be could have quantized energy levels and therefore quantized math squares in fact if we increase the angular momentum by one quantum then the quantized energy the quantum of energy that would be introduced would be a quantum of squared not a quantum of in this was an immediate piece of evidence that that more over yeah okay this was this was one of the hints that one of the hints there's another interesting fact here um supposing you took a string which was not moving but what you stretched out but you stretched out to a certain length okay how much energy would have have well in all of its energy would be potential energy not kinetic energy let's calculate what it would be how big is the xt sigma well if you stretch that out uniformly then the change in x along the length of it would just be the length of the string l would just be we're stretching it out to a physical distance l we stretch it out to a physical distance L over a distance from 0 to PI right all right so the derivative of X with respect to signal would be something like L divided by PI I don't care about the PI's right now they're not what CL what's interesting the X by D Sigma would just be proportional to the length of the string if you stretched it out by distance L and divided it by the range of Sigma from 0 to PI that would give you the x by d sigma and so we can say that the X by D Sigma is proportional to the length of the string and the X by D Sigma squared would just be the square of the length of the string right this is Hookes law this is Hookes law for a string if you stretch it out the distance L the energy stored in it none relativistically will be l squared but that's what has to equal the mass squared now we know something interesting about how about the energetics of the string if we were to study it in the rest frame in the rest frame of the string the energy of the string is the mass we got from mass is squared by boosting the string but if we went back now and we said look wait a minute we know a equals MC squared that's rest mass what is the rest mass of the string and the answer is that the rest mass is proportional to is a proportionality factor proportional to its length in other words this string has the property that the energy if you think about it in its rest frame if you stretch it out the distance L it will have an energy which will grow with L and be proportional to L in the rest frame it doesn't look like a Hookes law string at all it looks like a different kind of string whose energy is proportional to its length well that's very interesting because it fits with another picture it fits with the picture which I described before of lines of flux connecting quarks and antiquarks lines of flux would produce I slice something like this they would produce a patch of electric or magnetic field here it doesn't matter whether it's electric or magnetic lines of flux in a tube like this would produce a magnetic or an electric field in here electric fields have energy and the energy density and them depends on the field the energy density along this long tube of flux would be uniform if the number of flux lines passing through this little area is the same as number passing through this little area and so forth the field strength would be uniform along this tube of flux here this is in fact a property of tubes of magnetic flux and superconductors and so forth and superconductors are superconductors you don't have of course monopoles and superconductors but you can have long lines of magnetic flux and they have the property that the magnetic field is uniform along them and therefore the energy density is uniform along them that means that the energy is proportional to their length this is a common thing in in field theory and condensed matter physics in a variety of different context with with field energy in a field forming a long string is proportional to the length of the string not the length squares a different kind of strings another way to think about it is that the string is made up of a lot of little particles but as you pull on it and separate the distances here new particles form in between so as to keep the number of particles per unit length fixed that's another way to think about these long flux lines that they're uniform along their length and as you pull them apart more particles form to fill the gaps then in that situation it would also be true that the energy per unit length would be fixed and the energy would be proportional to the length this is by now actually an experimental fact about about hydrants that you can spin them up you can stretch them and they have the property that the energy per unit length is fixed they have a stretch called the string tension the string tension is a constant that would not be the case of an ordinary Hookes law if you strict them but that's but this is the picture in the rest frame and the rest frame the energy of the string is proportional to its length in the infinite momentum frame where the physics is all nonrelativistic the energy is proportional to the square of the length like Hookes law so these two kinds of strings Hookes law and flux tube are related to each other some sense they're just the same object being described in two different reference frames one at rest and one now father if you had an ordinary world rubber band and I was vibrating it with that most relativistic speeds in one there no way order your rubber bands if they vibrate with relativistic speed we haven't got the vaguest idea how to describe them we don't want to do that no uh we've we've made an indirect deduction we made the indirect deduction first but first half of the deduction was in the infinite momentum frame everything is nonrelativistic at least in two dimensions we use that to discover the fact that the stretched energy of a string which is l squared because it's described non relativistically like a hawk's law like a Hookes law spring is l squared that is to be related to M Squared indirectly from that we conclude that if we were in the rest frame the energy of the string would be proportional to its length and that's interesting because there's a wide variety of interesting string like objects that occur in field theory not made out of atoms but made out of field field configurations which have exactly the same property all right let's say let's come back in a few minutes I think I think I've probably exhausted your attention for today but let me well let me summarize let me summarize let me summarize experimental properties of of hadron indicated this kind of excitations along a line had we've been smart at the time we probably would have realized that this pattern here is the appropriate pattern for Strings whose potential energy is proportional to their length that's something we actually could have deduced directly from here but in fact why am I saying we could have I did that that the energy grew as the length of such a string that was the consequence of this relationship here that was one fact next fact non-relative are relativistic physics in a frame in which everything is moving fast is the same as nonrelativistic physics except in one less dimension or in one less dimension and so we can try to build a simple theory of relativistic strings by going to such a frame and just using non relativistic physics but one less dimension here it is here's the non relativistic string in two-dimensional space the only thing we have to remember is that wherever we saw our energy we have to think of it or internal energy in particular internal energy should be really identified with the square of the mass not the mass okay that if you remember came from the two different expansions if you like one of them was an expansion in which this was the big term and then the whole thing is approximately of water MC squared square root of this thing in here the other expansion was determined which this was big and then the excess energy was proportional to M Squared not M all right so when you do that and you go through this little exercise your conclusion is that the Hookes law energy of the effective nonrelativistic string should be identified with a mass squared which indirectly tells you that that the rest mass of the string is proportional to its length and finally there are lots and lots of field theory in condensed matter systems which have the same property so that was something encouraging if you like ok let's take a rest and then come back and either ask some questions or I don't think I'll discuss the quantization of the string today the quantization is easy we've done most of the things that are necessary to figure it out it's just a collection of harmonic oscillators um but I think we can take a rest first to help my engineer it seems you start with a classic strings dan you you lose the hell out of it without to the z-direction Jenny basically do a Lorentz transform houses Janu you've got the same physics when you go back should you still you get back to the physics that you started with you do with me but you do it's just easier this way no but you don't have the same physics yeah I first started with a spring that had a length proportional to its mass square in both cases in both cases the mass is proportional to the length and the mass squared is proportional to the length squared in both cases but in one case you call the energy the mass in the other case you call the energy the mass squared let's go through that low I'll come back to your question in a minute let me just go through it again because there was some slight of hand there was some tricky business there we wrote that energy is equal to square root of P squared plus M Squared okay let's so for the moment forget the motion in the XY plane it's just concentrate on the Z Direction in the time direction energy is related to time P is the related to space there's two ways I could expand this one of them is good when P is small and M is large all right in other words when I'm enough and when the particle is moving slowly in my frame of reference in that case let's see what we do then we write that this is equal to P times the square root of 1 plus M Squared over P squared right sorry I'm sorry I want to start them the situation where P is small and M is large good P is small and M is large so I write this then as M squared times 1 plus P squared over m squared P is small and M is large so P squared over m squared is small quantity okay not good okay the M can come out and this becomes M times the square root of 1 plus P squared plus M Squared but that's approximately 1 plus P squared over 2m ok squared which is equal to M which really means MC squared plus P squared over 2m so in that context the internal energy when the particle is at rest in space is proportional to the mass that's your good ol D equals MC squared ok and that tells you that for a particle at rest its inertia a particle at rest its inertia in other words its usual resistance to to acceleration is the same as its mass and its energy that's mass meaning inertia and the general energy are proportional to each other ok now let's do the other expansion the other expansion we boost like hell so that the momentum is very large okay and then we expand it and the other way let's see P is very large now so this is the square root of P squared times 1 plus M squared over P squared which is p+ did I do that right yeah P plus M squared I think over to P all right now this is the momentum along the direction that we did the boost there are two other directions of momentum and we can put them in here but notice that the energy apart from this fact this piece which is just total momentum which we can drop because it drops out of all interesting things is proportional to the square of the mass so in this form the energy is proportional to the square of the mass and what that says is that if a particle or a system system of particles is moving down the axis with an enormous momentum that its inertia that its inertia relative to this direction here well let me let me go back let me go back let me put in here the other terms plus px squared plus py squared they just went together with M Squared this was PZ squared plus M squared plus px square plus py squared we're taking this to be small and PZ to be large okay one of the things that this says is that the inertia is now not the mass of the particle it's the momentum along the z axis and that actually makes a lot of sense though it's not true none relativistically but relativistically it is true that a given force perpendicular to the direction of motion will produce a smaller acceleration the larger the larger the momentum so this that's the first thing this P here is the inertia and the M Squared is playing the role of an internal energy or M squared over 2 P is playing the role of an internal energy so internal energy becomes mass squared in this frame and inertia just becomes PZ oops that doesn't look good but all right now somebody asked me about the connection between these kind of strings which have this property of having a energy per unit length and superconductors I think I mentioned it as we were talking I will spell it out superconductors have the property of repelling a magnetic field they repel they don't want to accept magnetic field penetrating through them they actually repel the magnetic field ah because they repelled a magnetic field it's kind of a pressure that was pushing magnetic field out of the way but if you somehow push a magnetic field I'm going to tell you how to do that minute if you push a magnetic field into the into the superconductor in such a way that the lines of force are passing from one side of the conducts super here's a big piece of superconductor lines of force are passing through it like that what it will do would be to squeeze will push those lines of force out of the way and push them into a into a narrow string like thing like that that's a call to flux I'd it's called a flux ID or a superconducting flux line superconducting yeah superconducting flux line magnetic flux not electric flux magnetic flux and how can you make one in principle in a in a in a superconductor here's what you might do I doubt very much well this is probably not the way it's done in the way in really but the you take a peep a piece of superconductor you drill an incredibly narrow hole through it this is a Gedanken experiment this is not something that I want you to go away with as a practical experiment you drill through it an incredibly small let's see my going to get this right ah before you pull it before you cool down the superconductor before you cool down the soup winter conductor you take a long bar magnet and you stick it through the hole like that flux lines come out this way from the bar magnet and this way now you cool it down so that becomes a superconductor and you draw out the magnet these lines of flux here will get pulled into the magnet even after you've pulled out the magnet lines of flux will go through that magnet and they will form a thin tube through that magnet the field along the tube will be uniform and because the field is uniform that means the energy per unit length is fixed okay now let's go a little bit further imagine that we have magnetic monopoles we can actually simulate magnetic monopoles but let's suppose we really did have magnetic monopoles that they really were magnetic monopoles in the world and they may well be but we haven't discovered them yet but let's suppose we have discovered them and we can manipulate them okay then we could take a monopole and an anti monopole put them right on top of each other don't let them annihilate keep them a little bit apart take them and put them into the superconductor monopole an anti monopole I'm not going to tell you which one is which I'm just going to draw two of them and of course the monopole and the anti monopole have some flux lines between them now cool down the superconductor and take the monopole and the anti monopole and start to separate them what happens exactly the same thing that we think happens between a pair of quarks except that this is just an ordinary magnetic field between the monopole and my anti monopole so inside a superconductor a monopole and anti monopole would have an energy which would be proportional to the distance between them while the distance between them because the string between them has a an energy proportional to its length okay the monopole and the anti monopole could not separate from each other they would be confined because as you start to separate them the energy goes up with the length and and that's the that's the character of what happens to two hydrants or quarks and a hadron you separate them and their energy goes up so okay so this is a system for the energy is proportional to the length and it has the characteristics of the same kind of string does that answer your question whoever asked me and who asked me how well I guess he's gone the question was what's what's taking a role of a superfluid for the hey groans what is this super superconducting tool virtual monopoles that's what it's thought to be virtual monopoles but not these with different monopoles or chromodynamic mana poles in the superconductor here the condensate the superconducting condensate is made out of electric charge and it causes confinement of magnetic charge in quantum chromodynamics quarks are confined and they are kind of the electric charges of of quantum chromodynamics the things which condense our magnetic charges you say where are those magnetic charges why don't we see them because they're always condensed in the vacuum so but this is this is beyond the what I had intended to talk about today okay what other questions come up oh it's almost nine o'clock okay yeah so is the rule or highly relativistic sir well a superconductor is not a highly relativist it's it's a it's not no no but it's string like so if you took a a regular Hookes law stream made of Springs and had a going relativistically inside the circuit market so assuming you could you have to make up some theory every after this in theory but that that construction would then have an energy to work into its life an order would do depends in the details no there's something special about energy that there grows at length as I said what it really means in particle language is that as you separate the constituents of the string as you pull it apart the energy of pulling it apart instead of just separating and making larger distances makes more make small particles so that the so that the density of them along the string always stays the same that's the character of a these kind of strings and the other way to think about it is they're strings made up at a field where because field lines are not allowed to end field lines are not allowed to end as long as the field doesn't spread out this way it has to remain uniform in the other direction so it's a characteristic of flux lines flux lines which are confined to a tube if you had any kind of situation where an electric field will confined and prevent it from spreading out in the perpendicular direction the energy per unit length of it would be constant along with yeah that's a very good question yeah remarkably the way string theory works is there are not independent degrees of freedom for motion along the other direction this is a remarkable and strange fact that in string theory you do not include directly degrees of freedom for the motion of the string along the direction of the the boost here it is thought that that's connected with something called the holographic principle it really in a gravitating system you need one less direct dimension to describe it but it's one of the very remarkable features of strength eery that you don't describe the string in let's say in three-dimensional space you in this limit you describe it only by the two-dimensional motion and yet as we will see it's consistent with with Lorentz invariants yeah how many how many data points do you have on the Reggie plots in five or ten or I don't know what the maximum number is now uh seven something like that so it's not endless but it doesn't give any evidence of giving out I'll call I'll talk about it another time but there the the evidence for that but remember that we're not really interested in hadron so I was just giving you a historical perspective of where the whole thing came from but now we want to study this mathematical theory and not insist that it looks like hadrian's but just ask what it does look like and what we'll find out is it looks more like gravity than it does hydrants well the idea that we had had sometimes spatial element of the part of the composite Ness of the particle for a particle it doesn't seem to be composit well okay let's let's be get so the question is can be expressed in two different ways let's say um it can be expressed though the answer to your question can be expressed in terms of the smallness of the particle or in terms of the energy that it takes to increase the particle by one unit all right so let's let's remember the formula for the energy of a rotating system as a function of its angular momentum l squared where that's the angular momentum divided by two times something number one of this moment of inertia and what's the moment of inertia related to the mass the mass and the square of the size of the system right the sky mr squared not M squared R squared M R squared but the point is that for a given mass the moment of inertia gets smaller and smaller as assistant gets sorry yes the moment of inertia gets very small when the system is very small now what is L L is angular momentum and it comes in quanta so this is proportional to N squared H bar squared or something like that and now you can ask how much energy does it take to go from N equals zero to N equals one N equals zero the energy it might be a little bit of energy but the energy this piece of the energy is zero the first excited state will have an energy H bar squared over 2 times 1 divided by the moment of inertia and if the object is very very small the moment of inertia is very large no 1 over the moment of inertia is very large so the excitation energy the energy that it takes to increase the angular momentum by one unit becomes very very big if the object is small well hadrian's are big objects they have small moments of inertia on the gun on the scale of quantum gravity hey Jones are just enormous I mean they're they're you know they're big blobs they have large moments of inertia the energy taken to excite one is not very big electrons are known to be much smaller the expectation is that they're exceedingly small maybe 10 to the 16th times smaller than a proton you square their radius to get the moment of inertia so that could mean 10 to the 32 times a smaller moment of inertia and that means that the energy of excitation would be 10 to the 32 times bigger well maybe that's an that's that's probably too much but they're so if electrons have excited states because they're so small if they have rotational excitations those rotational excitations well I think I lost my true for our cheap lot but the mass necessary to increase the angular momentum by one unit would be way off the know that end of the of the plot just too much it just takes too much energy to spin it up when I say too much I mean too much compared to the energies that are available in particle collisions okay so the energy needed to excite a hadron by one unit is less is roughly of order a GeV you collide particles at a GeV or whatever it is you see these excitations the energy needed to looks to spin up an electron is very much higher you don't see them how about the energy needed to spin up a a basketball by one unit of angular momentum very small very small in fact you can't even see that it's quantized so atom would be okay whatever it is first there are a couple of electron volts need to spin up an atom by one you a couple of electron volts good okay for more please visit us at stanford.edu

UCL Economics MSc & Economic Policy MSc

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Economics MSc

Modern economics is an increasingly rigorous discipline and advanced degrees are now essential for careers in international institutions, government and industry. The Economics MSc at UCL will equip the professional economist with the powerful tools required to understand the rapidly changing, complex and uncertain modern world economy.

Prospectus entry:

Economic Policy MSc

The Economic Policy MSc is a unique programme which takes advantage of UCL’s role as a global leader in policy-orientated research. The core teaching in microeconomics, macroeconomics and econometrics focuses on understanding the policy implications of economic models and their applications to real-world examples.

Prospectus entry:

the aim of this course is simple we want our graduates to be top economies people with an in-depth knowledge of theoretical and applied economics the two courses share common elements do like the core theoretical material macro micro and econometrics or do prefer taking that material and applying it to practical questions of economic policy if the former your MSC in economics if the latter do our embassy in economics the teaching staff is great they are well prepared there are all 12 researchers and this helps a lot especially because they have a lot of connections and huge networks they do the kind of research that is at the state-of-the-art of the economic environment the Department has close links with a number of different research centers the Institute for Fiscal Studies for example as well as other research centers that are housed within the department for example the creme Research Centre which has a strong emphasis on the economics of migration so the strong ethos within the departments the students are exposed to work in seminars and in some of these research centers it's very complementary to what they'll be learning in their core adoption courses you see how is this physically placed in the centre of London and so this clearly makes it an ideal platform for interacting and also and being made aware of all job opportunities the Environment Agency's so I think the location itself and also a less let's not forget to mention all the networking opportunity that happens is because you are under the current prospects of our graduates are excellent you can find our graduates around the world in academic departments economic consultancies management consultancies central banks investment banks and international organizations everyone this program itself is very unique in the way it's and design for it being more applied and I the reason I picked apply this because I don't want to go ahead with the PhD I want to apply is normos in the day to day life I'm going to work with a consultancy so this course is perfect for that we want people fascinated by economics and up for a challenge you need to be good at math that's the basis of everything we do but intellectual curiosity is important as well you're going to spend a whole year of your life doing these sources and if you're not gripped by economics it's a waste of your time everybody is extremely smart extremely driven which is something that you don't always find especially in undergrad so it's it's really really motivational group is for me it's one of the first parts of course and I've been able to speak to get perspectives from fearful of varying backgrounds and experiences our masters programs among the best in the world tough quantitative and rigorous will turn you into better accomplices you