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

Leonard Susskind | Lecture 1: Boltzmann and the Arrow of Time



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First of three Messenger lectures at Cornell University delivered by Leonard Susskind

Theoretical physicist Leonard Susskind delivered the first of his three Messenger Lectures on “The Birth of the Universe and the Origin of Laws of Physics,” April 28, 2014. Susskind is the Felix Bloch Professor of Theoretical Physics at Stanford University, and Director of the Stanford Institute for Theoretical Physics.

it's a pleasure to introduce today's speaker my niece askin who is here giving a series of three lectures this week is one of the messenger lecturers the messenger lectures are described on the university website as one of the most important with Cornell's extracurricular activities I'll save the full description of the series for the public lecture on Wednesday night 7:30 that will be black holes the conservation of information and the holographic principle I'll mention here just that we've had stunningly distinguished messenger lecturers and physics and astrophysics at Cornell including RA Milliken in 1925 Sir Arthur Eddington in 1933 J robert Oppenheimer in 1945 Fred Hoyle in 1960 of course Richard Feynman in 1964 up to and including Steven Weinberg in 2007 Nemo coming event the most recently in physics in 2010 I happened to host Weinberg in 2007 and when I contacted him to ask you about the messenger lectures he told me of course I know about the messenger lectures I frequently referred to fineman's 1964 lectures which later became his book the character physical law what I happened to see Lennie about a year ago and asked him the same question Lenny said of course I know fineman's 1964 messenger lectures I was there so when he was a graduate student here he obtained his PhD from Cornell in 1965 under advisor Peter Carruthers he was it you see the university often from then until 1979 when he moved to Stanford where he's been sent I'll again save the long story for Wednesday night the short story is said since 2009 he's been director of the Stanford Institute for Theoretical Physics and his author two popular science books one about cosmology and the other about quantum mechanics and black holes he's also interested in teaching popular courses he's taught a series of modern courses about celery a series of course is about modern physics entitled the theoretical minimum which can be found online leading the MOOC movement and physics I first sort of met him while I was still an undergraduate he was in 1976 giving a series of lectures on course grade quantum field theory and I didn't attend those lectures I didn't learn about lattice QCD until I came here as a graduate student but I clearly remember that just before the start of one of the lectures this fellow who sort of looked like an Olympic marathon runner he was also taller than came out and asked me a virtual question to which I actually knew the answer down the hall first door clever so as you'll see money is the unique person to be giving this 50 years all along to violence lectures satisfying Forbes specific criteria number one he was there number two he's extremely eminent number three he still sufficiently compos mentis and pull it off we'll see and number four has a crest book to be willing thank you very much Paul oh yes hello am i here ah there we go can you hear me no okay let me say one thing to begin with I really feel sorry and sad for all the young physicists in the world who will never get to meet dick Feynman he was a close friend of mine I think the greatest inspiration that I ever had in my life was listening to the 1964 messenger lectures I never looked at them again I couldn't I said once is all you ever want to do this particular thing in fact I always felt the same thing with though with dick I never wanted to go back over the things I did with him once was enough once was so unique and so exceptional that to see them again not to do them again or to just seeing them something I didn't want to do so I haven't looked at the Fineman lectures again but um I do remember now I could be wrong about where I heard Fineman say this I think if my memory serves me right it was in the first of the messenger lectures I could be wrong it could have been another lecture somewhere else but as I remember Fineman once described theoretical physics as follows first you get an intuitive idea the house of how something works how to explain something then you find equations to quantitatively express your idea from the equations you make calculations and predict something new new number or new relationship after that somebody else usually somebody else goes out and does an experiment to test the prediction and if enough predictions turn out to be right then you've got a theory if the idea covers enough different cases the theory becomes a principle this of course is correct but I think it's not the whole story there are other ways the theoretical physics for pressors or physics in general progresses one I would call the Dirac way I'm specifically interested in theoretical physics one are we call the Dirac way their axons if an equation is beautiful enough that will be right well maybe it did work for Dirac then is the Einstein way sometimes principles these things we call principles collide when two things both of which are so deeply rooted that we cannot conceive of either of them being false but nevertheless when taken together seem to lead the contradiction then we have a conflict of principles it's when such conflicts are resolved the physics makes the greatest conceptual advances my lectures are about three related conflicts of principle I think it's fair to say that all have powerfully changed physics even though not one of them has been completely resolved today I will discuss Boltzmann struggle with the second law of thermodynamics on the one hand entropy always increases on the other hand Newton's laws are reversible we will come to what that means the outcome of the struggle was profound entropy is hidden information Boltzmann's insight deeply affected every area of science but in the end it did not solve the problem that Boltzmann originally set out to conquer why is there an arrow of time why does time go one way what's different about the future than the past I'm going to explain how modern developments in cosmology make this puzzle even more puzzling and how they also suggest a solution but be prepared to be very skeptical will be driven to extremely remote ranges of time and space way beyond anything we can hope to directly access nevertheless the arrow of time is a fact of nature and deserves an explanation so let me begin I will just come and that I think it's an absolute disgrace that a beautiful wonderful room like this in which physics can be presented does not have a permanent blackboard had I known that when Paul asked me I asked him explicitly Paul is there a blackboard in the room and Paul said of cost as a blackboard in the room he was lying to me I don't know if I would have turned down the opportunity to give these lectures I don't think I would have I don't think I would have but nevertheless I have to say I'm deeply saddened and disappointed by the lack of a blackboard in the lecture hall Rockefeller Hall I'm going to use this monstrosity of a machine I don't like using PowerPoint I will use it today all I did there's another interesting fact Paul said give lecture one two and three in that order first the first one in the second one the third one and I assume that this meant that the first lecture should be the first logically order ordered one and that it should be for the general public a broader thing that introduced some concepts at a not exactly a lower level but with fewer equations and fewer technical concepts and so I wrote the first lecture in the first lecture was about Boltzmann and and so forth I then found out just a couple of days ago that the first lecture is the physics department colloquium and the second lecture is the general lecture sorry buddies you are getting the first lecture because it's the first lecture I can't help it so if you find this a little bit trivial then I invite you to – what's that – the same if you're right but you said I was I didn't know what I was going to say next all right the lectures the entire series of lectures had some weird title that Paul made up had nothing to do with me the correct the correct the correct title should have been conflicts of principle I don't know if it's conflicts of principle or conflict of principles but one of the other and basically the three related questions of concept of conflict of principles okay let me start with an experiment this is a real genuine experiment I did this in my laboratory I took pictures of it and I have a hundred I have two hundred and thirty two slides on here incidentally and you'll see but most of my little movies little film clips the film clips are homemade film clips I make them myself and you'll see how they have it we'll see how they work all right we start with a room the room is the purple area it's a sealed room you can't get nothing can get into it or out of it and up in the corner of the room is a little bit of maybe a lot of a lot of molecules gas all stuffed into the corner of the room now I want you to tell me which of the following two little movies makes sense as physics and which does not okay so here we go that's one here's the other one should I do them again No that look pretty good I like that that looks very uncomfortable I don't believe that will ever happen mostly you don't believe that the air in the room will also rush into the corner so what's going on well we usually blame this on the second law of thermodynamics the second law of thermodynamics is that entropy always increases and if you know just a little bit about entropy you will know that the entropy of a room filled uniformly with gas and thermal equilibrium has a high entropy and when all the molecules are stuffed off into the corner it has low entropy entropy and clay increases end of story second law of thermodynamics tells us which film is correct the clash of principles the conflict of principles the second law says that entropy always increases Newton's laws of motion and for practical purposes today I don't want to introduce quantum mechanics but what quantum mechanics also says the same thing it says that the laws of physics and particular Newton's laws are reversible let's remind ourselves what that means what it means in the context of a very simple example incident R here it is the context of a simple example is that if a ball can roll a frictionless ball on a frictionless surface can roll from one point to another in a certain amount of time then Newton's equations say that there's another solution in which it can roll back to the first position therefore that cannot be a quantity which always increases if it increased going this way it would be creased going that way they can't be Boltzmann's effort to find the quantity and mechanics which always increases was doomed from the beginning and people told him that incidentally he struggled with it ok so that raises the question then what is entropy what is it that seems always to increase despite the what Newton might say about it what's the meaning of the second law and why is there an arrow of time ok let's begin with what entropy is so let's see yeah entropy according to Boltzmann in the end when he finally understood it is hidden information what does it mean that information is hidden well in the practical example that he was thinking about a gas information is hidden because it's contained in a collection of degrees of freedom which are too small to see and too numerous to keep track of when information is contained in to numerous a set of degrees of freedom too small to see that information is called entropy let's take an example here's an example of 64 coins now let the fact that nor the fact that they're on a lattice that's not the important point here I had to draw them some way so here's the 64 coins they might just be in a bag they might even be invisible to you but but there they are sixty-four coins and there each coin has two faces one face is red one face is blue here's a special configuration of the coins in which they're all showing red how many such configurations are there well the number is right over here there's one such configuration it's rather special incidentally if you saw it even if you didn't know where the coins were is those your sure those coins you'd recognize it instantly easily recognized here's a configuration with one blue coin it's less unique there are 64 of them let's just here's another one and there are 64 of them altogether you might not so quickly recognize which was which if it was flashed in front of your eyes 64 configurations with one flipped coin how many were to flip coins well you can work it out at 64 times 63 over two and the answer is 2016 numbers are going up fast rather fast here's another one three coins forty one thousand six hundred and sixty four configurations with three coins I suspect if I flashed that at you pretty fast you would not be able to tell me afterwards which three coins were flipped but maybe you could there's four the number is six better part of a million six hundred and fifty fifth or old or whatever some large number that's how many configurations notice they're going up really fast how about if half the coins are red and half the coins are blue this is a kind of generic situation the numbers are vast two times ten to the eighteenth configurations supposing you were just to pick a configuration randomly what would you be likely to pick you would be very unlikely to pick one you extremely unlikely to pick all Reds picking them at random you probably pick something like this just because there are so many configurations like this this technical definition of entropy for an ensemble a collection of states for a collection of discrete states of a system like the system of coins is this that the particular ensemble it's a technical world ensemble but just means a collection of states that are somehow recognizable in some way recognizable or not is the logarithm of the number of configurations that satisfies a certain criterion okay for example the state with one flip coin with no flip coins there's only one state log one is 0 it has no entropy and you go down the entropy s rise not as fast as these numbers on the left but the entropy goes up and eventually reaches a maximum all right now let's talk about something different let's talk about equations of motion what is an equation of motion an equation of motion is a rule for updating the state given the state the equation of motion tells you what the state is the next instinct think of it that way I'm imagining time is discrete but that's not important these for the authorities six boxes represent six different states how many states are there the 64 coins to to the 64 could with too many to draw I can't draw them all so I've just drawn 6 y 6 is a nice number ok those are six states but think of them as representing all the possible states of some system not the ensembles but the individual states what is an equation of motion an equation of motion is simply a rule which tells you given the state what's the next state 1 goes to 2 2 goes to 5 5 goes to 3 3 goes to 6 6 goes to 4 4 goes back to 1 and what happens next 1 goes to 2 2 goes to 5 and so forth and you cycle through these states this is a typical example of a simple discrete equation of motion it tells how the system moves forward in classical physics it's completely deterministic in quantum physics a similar thing is true it's called unitarity but let's just take the the the classical version of it here's a law of physics or a equation of motion which is also deterministic if you start with 1 you go to 2 if you start with 6 you go to 2 if you start with 5 you go to – that's deterministic I can tell you exactly exactly where you'll be after a certain number of steps something wrong with it what's wrong with it what's wrong with it is it's not reversible if you know you're at 2 you can't tell where you came from you can't retro dict if you know you're at 1 you didn't come from anywhere you couldn't have come from anywhere so this is what's an example of irreversible equation of motion the laws of Newton and the laws of quantum mechanics and the laws of every physical system that we know at the very bottom of its description are reversible all of them and so equations like this are forbidden do we know why well in something in some way we do know why but I'm going to ask you to trust me that this is one of the things that the laws of physics as we know them require what's the rule the rule is that every box has one arrow in in one arrow out an arrow to tell you where you were and the Naro to tell you where you'll be next ok that's the sense in which information in classical physics and in quantum physics if you work out the quantum analog of it that information is never lost you always know where you came from you always know where you go to and there is no contraction into a smaller number of states so that you lose information incidentally in classical physics it's called levels theorem and quantum mechanics it's called unitarity ok let's now ask what an evolution would look like now this is a made-up evolution the I'm going to assume that as you go from one step the next not too many coins are flipped simultaneously one two or three coins are flipped alright here it is the keep track of it because it's important to keep track of the sequence okay I'll go back over it let's do it a little more slowly alright we started with the lowest entropy state why I just decided to start with the lowest entropy state to see what would happen not too many coins get flipped simultaneously so the next step might be one blue coin which blue coin don't know but the law but the point is the equation of motion tells me which blue corn what's going to be next well it can't go back to all red why not because it has to cycle through all the states assumption is it cycles through all the system cycles through all the states it can't just go from red to blue or one red one blue to red one red two one blue so and go to something else let's go back man it might go to a single blue coin someplace else but there are many many fewer of those states than there are states with let's say 2 3 or 4 flip coins much more likely just for that reason that it will go perhaps the 4 flip coins remember though there's about a half a million more than a half a million configurations of that type from here it might go back to one coin it might go back to 3 coins but much more likely that it will go to more coins for the simple reason that there are more configurations like that and thus it goes once it gets to this point where the entropy is maximum where it's almost half-and-half give or take it'll simply rattle around in similar States for a long long time for a very long time much longer than we have to run through this but every so often a fluctuation will happen a Boltzmann fluctuation a rare and unusual configuration will arise here's one with only 3 blue coins what's the next one going to be maybe it's two blue coins now there was just so many more with 4 blue coins that it went to 4 blue coins and so it goes whoo that's getting interesting that's an interesting look I didn't go anywheres oh maybe we'll get back to the beginning not likely and so we go and so we go and so we go whoo whoops there's an interesting one how do we get there there's a star in it not a star not an astronomical star but has a little star in it one star that's a really interesting configuration how did it get there well it got there randomly it just accidentally got there remember you have to cycle through everything not so surprising that you've got something really rare and interesting but it didn't get there for a interesting reason didn't get there for a reason which you could explain by some inevitable law of physics that says that stars will emerge in some very very specific and organized way these the example of the star that I just described is an example of a recurrence you can call it a recurrence a punk a more or less a punk or a recurrence you can call it a Boltzmann fluctuation you can call it a freak unexplained happening they are a general property of finite systems because the system is finite it only has a finite number of states to cycle through and so weird things will happen weird things will happen over and over again and on top of that they will vastly outnumber than normal histories the normal histories are the ones that started with extremely low entropy I will come to why I say that in a moment but the normal history is the ones that we normally understand or think we understand our histories that start with very very low entropy there are many more ways to create that star than to start with low entropy just many many more random statistical ways to how make it happen roughly speaking in the real universe what we're talking about is very very freaked phenomena where a bunch of random dust molecules or gas molecules might simultaneously come together and make a gala see that sounds almost completely impossible but it's not completely impossible and what this tells us is that in a finite system the system that just cycles around itself it is much much more likely than any other way of making that star or that galaxy will go through it a few more examples to get the idea let's explore the idea further and suppose for a moment that the universe is such a finite system finite system finite number of degrees of freedom number one and number two contained in a finite volume if it's contained in a finite volume and it has a finite energy we describe it but we can describe it in any case by phase space the configuration of space of states is not a series of six boxes but it's a point in the phase space of some number of molecules for example 10 to the 10 to the 80th molecules or whatever whatever the right number is and so it's described by a phase space which is a plot of position and momentum and velocity now to make it a finite system let's assume that the positions of the molecules are bounded they can only go between here and here and that there's a brick wall here that prevents them from going any further furthermore let's suppose that the energy of the system is finite in that case the momentum of the particles cannot exceed a certain amount and so the whole thing is contained in a you know in a box now there's a region of phase space that's my target by my target I mean the thing I want to explain how we got there this is some region of phase space which describes the world let's say as we know it now it's got the certain abundance of Helium certain abundance of hydrogen a certain abundance of lithium it's got planets it's got stars it's got you and me and that's it it's not unique the state is not unique there are many many states that look like that it's an ensemble and it's an ensemble with some pretty significant entropy and entropy of about ten to the hundred in some units or other but still it's a very small region of the whole phase space most states are not like that so the question is then how do we get there and the standard idea of a theory for an explanation forget that we call that explanation inflationary cosmology right now we'll come back to that in a little while but the basic idea is that there's some special family of states not a unique state but some special family of states with a very low entropy very few of them as a starting point but if we follow the phase trajectory the trajectory of the system from any point within that little region there it always evolves into here that's called a explanation of our world if we can find such a thing if we can find a small region of very low entropy such that almost every point within that region evolves into the region where we are then we say we have an explanation the explanation is for reasons unknown we started out in here and for reasons known we evolved to here inflationary cosmology might be one such candidate description okay but once we get into this let's call this the zone of life just to give it a name once we get into the zone of life we don't stop world goes on we eventually will exit the zone of life sadly we will eventually exit the zone of life our universe will expand dilute we will eventually get out of the configuration space where we can exist and we will wind up outside it is that the end no that's not the end the phase point continues to wander around and for long long periods of time it evolved what it evolves around in what's roughly thermal equilibrium in this world in a box it evolved around in thermal equilibrium which is like the series of coins which were in this featureless configuration about half red and half blue and it stays there for a very long time not in any particular state but wandering around through the phase space the wandering phase space point featureless nothing interesting happening but notice over here the phase space manage the phase space point managed to wander back into the zone of life but it didn't wander into the zone of life from the trajectory that started with this low entropy state it wandered in from someplace else the zone of life is much bigger than the starting point here how do we know that we know that because entropy increases the reason entropy increases is because we went from a small region to a bigger region we wander around and then accidentally practically accidentally in a freak event we wander into here the people who live in here look around them and they see a world which is not explainable in this way that we're familiar with explaining phenomena in fact my guess is those people in there would not be too surprised they would say what would be overwhelmingly surprising would be to start from here there's almost no states there we know what's happening our phase point is just wandering around and every once in a while it wanders into the zone of life if we wait even longer we will find out that the phase point wanders in and out and in and out of the of the region of life very very rarely much more rarely does it actually evolve back through this point and wind up behaving in the way that we normally consider explainable what I'm telling you is the thing that we normally consider explainable is the most outrageously unlikely of all possible ways of getting where we are why because it starts with a very very low entropy state and low entropy states are very rare so in this world in a box what one should expect is that the typical example of life the typical example of a world which would support life would be what we would call a freak world we don't live in a freak world we know that inflationary cosmology back to some very low entropy starting point work so there's something wrong with this picture of go back there's something wrong yeah there's something wrong with this picture of living in a box there are ways out maybe we don't live in a finite box that would get us out of it perhaps the problem is that modern cosmology tells us that we do live in a car in a finite box in a finite and in a certain sense on growing box if the box grows then you can get out of this problem okay all evidence that we have about cosmology experimental observational points to the idea that we do live in a finite box the evidence that I'm talking about is the existence of a cosmological constant sometimes called vacuum energy sometimes called accelerated expansion and most often called dark energy they're all the same thing so let me go through in a few minutes what the arguments tell you what the equations tell you a good fraction of you have seen this equation before Murie assualt percentage of people have seen this equation probably about a third I would say something like that does so let me tell you what this equation is this is called the Freedman robertson-walker equation it's the equation for the expansion of the universe a represents the radius of the universe at any time when physicists put a dot on top of a variable it means the time derivative of it and the left-hand side is called the rate of expansion the rate of expansion a dot is the time derivative of the radius universe is expanding a dot over a is finite and this is called the rate of expansion and the equation says that the rate of expansion is proportional to forget this number here this is a number we'll set it equal to 1 in some units or other it really is equal to 1 some Clarkie and units or some sort of units it's equal to 1 this object is the energy density or the mass density in the universe e equals mc-squared same thing and what you can say about this energy density is at least the usual sources of energy the usual types of energy density particles photons all usual things let's call the particles matter those have an energy density which decreases as the universe gets big why because it dilutes energy density dilutes as you say as you make the universe bigger same thing with radiation radiation dilutes even faster for a technical reason that I won't bother you with but there is one form of energy it's often called very mysterious I think it's not so mysterious it's called a cosmological constant or vacuum energy it exists in the world it exists in our equations that exists in quantum field theory it must exist and it does exist what's queer about it is how little of it exists but we're not going to get into that today we're just going to say there is a vacuum energy and the thing about the vacuum energy is it doesn't dilute you take a box of a certain amount of vacuum energy and you'll grow the box what happens is the density of vacuum energy stays the same now you can ask where the energy came from who that's another that's another issue it is understood it's not mysterious but there is a vacuum energy and we know that it really exists it's been measured astronomically here's the number it's even bigger it's very small but it's actually bigger than the density of ordinary material throughout space that tells us that the Friedman robertson-walker equation just has a right-hand side which is just a number it doesn't change you can solve this equation a dot over a equals the square root of lambda and the and the solution is an exponentially growing universe now that doesn't look like a finite box of a certain size that looks like a growing box and with a growing box you can get out of this problem of the Boltzmann fluctuations but this is misleading let me show you let me take you through the arguments about why this is misleading and what the equations really tell us all right first of all the universe the entire universe described by this theory is expanding exponentially that means every point is moving away from every other point moving away according to the Hubble law and the thing that's exceptional about about a cosmological constant is that the Hubble number here the relation between velocity and distance this is velocity distance from you here you are right here I'm also pretty close to there and we're looking out at some distant thing it's moving away from us because the universe is expanding and how fast is it moving out a velocity proportional to the distance away there are the coefficient here is called the Hubble constant and the special thing about vacuum energy is that the Hubble constant is truly constant it doesn't change with time that's special now this is a strange equation if you think about it what it says is if you go far enough away where D times the square root of lambda is equal to the speed of light there will be things moving away from you with the speed of light moreover if you can go further out there moving faster than the speed of light trust me this is okay this is allowed all right things moving away faster than the speed of light but you can't see them when they send signals back to you the signals are also swept up and travel away from you and so the result is that there is a certain radius called the horizon the radius of the horizon and for all practical purposes everything that you can ever see everything that you can ever know about is within that radius that radius or that shell out at that distance is called your horizon and it's very much like a black hole horizon except looked at from the end not from the inside of the black hole but it's as if there was a black hole on the outside that when things cross over into this Neverland that'd be some beyond here you'll never see them again well that's not quite the way the black holes or cosmological horizons really work the way they really work is that if you with your telescope or following something moving out you would not see it cross the horizon why not you can never see anything cross the horizon first of all you can't see anything beyond the horizon you actually can never see anything cross the horizon what you do see or what your mathematics would tell you is that as time goes on the particles the dust the galaxies asymptotically within your horizon here asymptotically approach the horizon moving slower and slower and slower freezing at the horizon that's what the equations of general relativity say very much like a black hole so here is what either what you would see through a telescope or what or what mathematics would tell you there's all your galaxies how did they form they formed perhaps by starting with some low entropy state and creating galaxies in the standard way and they start to move out they start to move out and all the particles eventually arrive near the boundary near the horizon taking longer and longer and longer to get there let me draw a picture of one way of thinking about this think of all the particles in the universe being contained within this horizon so they're within this region here and there's a potential function a potential function which looks like this you're at the center here you're exactly at the center so you will neither fall off this way or this way but something which is over here will start picking up steam and eventually fall down to here something over here will fall down to here so if this region was filled up with particles they will all fall down to here they will all fall down to here that's what's going on here and from your perspective within your telescope you will see them simply freeze and congregate or pile up pile up at the horizon that's the mathematical description the general relativity tells us is what you would see now if we add some quantum mechanics now we have to add some quantum mechanics to go next if we add some quantum mechanics the main result of quantum mechanics is that a horizon like this cannot tolerate an infinite number of degrees of freedom or an infinite number of particles all piled up like that and in fact the pile up has a finite thickness the finite thickness is proportional to H bar things cannot get that close rather the pile up piles up at a distance which is comparable to a length called plunk length these particles pile up here and because they've fallen down they have some kinetic energy and therefore they're hot this system has temperature that's a well-understood feature of spaces of this type called the sitter spaces the sitter spaces have temperature and all the thermal junk is out near the edge that's your description of the world now of course every once in a while because these particles are hot once in a while one of them will get some extra energy and jump back up to the top one particle who cares about one particle it might jump up to the top two particles might actually make it to the top even rarer so here's what we would expect to see as we follow this system particles a few particles jump up they fall back down they jump up they fall back down they jump up you can see why I have a hundred and eighty slides I think it's two hundred and thirty this keeps happening over and over but every once in a while and except that this when I say every once in a while I mean a long while every once in a while the mathematics in any case tells us that there will be an exceptionally large number of particles that jump out of the horizon and congregate somewheres a dust cloud is formed a dust cloud is formed by no mechanism that you would recognize as a standard astronomical mechanism I random fluctuation that random fluctuation might collapse to form a black hole that's a rather likely thing that it would do and this process is well understood it's been studied for many years the creation of black holes out of out of the sitter space but before it forms a black hole maybe it will not form a black hole in my former galaxy the dust cloud that this cloud may then evolve into a galaxy what were the people who live on that galaxy think they would look out and they would say gee this is interesting we are alone in the whole universe why would they be alone well the probability of making one galaxy is incredibly small by random fluctuation the probability of making two galaxies is vastly vastly smaller so the most likely thing is if you found yourself in a galaxy you would say ooh a fluctuation happened but the likelihood that a double fluctuation happened is negligible so these people would not be all that surprised that they were alone if they understood the theory of fluctuations and if they believed that their birth was due to a random fluctuation they will have a very nice theory of the world so lesson over very long timescales everything happens freak events vastly outnumber the comprehensive Belen quotes histories the ones which start with small entropy and go where we're going and the reason that the comprehensive all of normal histories are so overwhelmed by the other ones is that they be they originate from low entropy configurations which are the most unlikely something is clearly wrong with this picture we did evolve out of a out of a inflating early low entropy state I don't believe that we are simply the one part in 10 to the 10 to the 10 to the 10 to the 10th the civilizations that were just so lucky that they were the first ones to be formed and therefore were not these crazy recurrences that I think is very unlikely so something's wrong there are various possibilities of what could be wrong perhaps this idea that vacuum energy means that we live in a finite box which just recurs and recurs and recurs maybe that's wrong maybe the mathematics the quantum mechanics the combination of ideas of quantum mechanics and gravity don't really fit together and this idea that everything piles up boy – lasers whoo oh wait wait look look one of them is behind the other alright this idea of pileup here maybe that's wrong maybe the use of thermodynamics is wrong on top of everything else the timescales that we're talking about are truly vast exponentially exponentially large can we trust anything about such enormous timescales aren't we being a little bit jumping ahead of the game a little bit by assuming that we know how physics works for 10 to the 10 to the 10 to the 10th years of course we're jumping ahead something unexpected may happen in between but that's just the point that's just the point what this is telling us very likely is that something unexpected that's unaccounted for by the equations as we now know them must come into play over these very long timescales and what kind of thing might it be that was intended but I'm trying to remember what the next one is okay let me let me go through the life that didn't help up there it is here is the standard theory of the evolution of the universe this is inflationary theory inflation is the theory which was confirmed over the last few months by this famous experiment which just took place the bicep2 experiment it didn't lead to this picture it confirms this picture I'm going to tell you what the picture is there is a field the field has a name it's called the inflow time it's a field in space in that field can vary the vacuum energy depends on the value of that field so if the field is at this value here there is a large vacuum energy remember what a large vacuum energy means it means a tremendously rapid growth of expansion but it also means that the horizon is very small because you don't have to go very far until you get to the point where things are moving away with the speed of light the smaller the expansion rate the further you have to go to get to the point where you reach the speed of light and therefore the bigger the horizon this pot this pot of particles here is big when the cosmological constant or when the vacuum energy is small and it's small when the vacuum energy is big so if we start up at the top of this potential where the vacuum energy is large the observable universe within a horizon is very very small and then the standard I'm going to give you the quick nutshell version of all of cosmology the universe rolled down to the bottom where there's a little bit of vacuum energy so the universe is still expanding it's still accelerating it's still in this phase like this but much slower expansion much bigger universe but it's stuck there what happens when it gets down to here incidentally everything interesting that happened in the universe happened on the way down here while it was out of equilibrium including us with somewheres down near the bottom but we're not really at the bottom yet once it gets to the bottom that's when you're in thermal equilibrium that's when the universe has evolved to where every thing has fallen down to the sides and the sitting there in thermal equilibrium and what does it do it jiggles around there with jiggles this is not my nerves this is my purposeful jiggling right now try to stop I can't okay jiggles around here jiggles around here for how long for an immensely long amount of time every so often though it might get a fluctuation which pushes it up the hill a little bit and then it will fall back down a partial incomplete recurrence it'll do it again and again and again over and over again very very rarely will it jump up to the top and when I say rarely I mean it's the most unlikely thing the entropy up at the top here is minimal it's like all the particles having gotten up to the top you don't need all the particles to get up to the top to make a galaxy okay so that's that's the history of the universe in a nutshell a roll from a high value of the vacuum energy down to the bottom sits there for long periods of time and then now and then jumps back what would it look like from the point of view of the pictures we drew well when we're up on the top the universe or at least the horizon the portion that we can see the portion which comes into the mathematical description a quantum description of the universe is very small how small I mean really small this would be really really small up at the top of the potential I can't remember what the number is but microscopic is a lot smaller than a proton and so we sit there with a very very small universe but it rolls down the hill when it gets down to the bottom of the hill the horizon is much bigger big enough for us to live in big enough for the galaxies to live in big enough and think of these red dots now as galaxies but then it evolves it continues to evolve and expands and everything goes out toward the edges and sooner or later gets the thermal equilibrium with all the particles down the bottom here they sit there they do things a knock against each other and it's quite boring nevertheless every once in a while a few particles might jump up out and then fall back in still boring but again with a very very low rate a dust cloud may form dust cloud may form make a black hole or may make a galaxy I'm repeating I didn't make these new these are the same ones again and then evaporate back to nothingness again this is the picture that we are now stuck with over and over again the vast majority of observers are these freaks the freaks who occur for no good reason other than random things happen they vastly vastly outweigh the number of times that the universe jumps up to the top where's the top the top is gone this is where we are was stuck there was stuck there with a theory which tells us that the vast overwhelming majority of civilizations they're physicists will correctly have a theory that says that they were born at a random fluctuation they will not be surprised to find themselves alone in the world and they would consider what we see or at least an attempt to describe the universe the way we describe it as deeply deeply misguided we don't expect crazy fluctuations and the craziest fluctuation would be back all the ways up to the top something's wrong now one thing that might be wrong I'd rather think it is what is wrong is that this phase space box is not really closed and sealed imagine had a little hole in it the hole cannot be arbitrarily small for it to do its work but imagine that there was a hole in the side of the box so that the universe phase point could leak out of a box then the history would be you start I mean we still don't understand why we started here which don't understand wheel but let's take it for granted that the cosmologists are right and we did start here what will happen is we will go into the zone of life then eventually we'll leave it we will rattle around a few times maybe many times and then find the way out of the box once you're out of the box finished once you're out of the box assuming the rest is infinite assuming the rest is infinite you will not come back into the box the likelihood of coming back into the box altogether over the infinite range of time beyond that time is negligible and so once you're out you're out you will not recur again if the box was open like this and we started let's imagine the Gedanken experiment where we maybe not we maybe somebody else starts a sequence of universes starts one here and follows it so that's another one follows it the overwhelming majority of them now will pass through the zone of life only once therefore the overwhelming number of observers will see a world which is consistent with being traced back to this very early low entropy state do the equations support this view yes actually they do remember I drew that picture of the potential well I'm not I'm not the world's biggest believer in string theory although I did have something to do with it but I'm not but nevertheless string theory tells us something and the thing might be much broader and a more general in string theory what it says is that there's always a way out of this minimum here there are always vacuums in the technical jargon there are vacuums with zero cosmological constant they're super symmetric doesn't matter what you call them they are configurations where which when you leak out of here you jump around you jump around here and every so often you go over the top and you leak out here when you leak out there you do not come back there's no way back after that why because these things have zero vacuum energy and zero vacuum energy means infinitely big horizon you've leaked out to infinity if this is true it would be a way out it would not explain why we start in a particular configuration but would get us out of this trap of the recurrent nightmare of Boltzmann fluctuations um the world would look something like this then or the evolution you would start small up on the top of the potential you would fall down to the bottom get bigger you would then tunnel through to the other side and simply get big an end of story you get one shot it's a one shot universe where you get one chance to have life in it and after that finished what I am essentially finished that is a possible explanation of what's going on as I said I don't expect you nor do I think that you should be unskipable about everything I said I think you should be skeptical about it we have very little chance of extrapolating that far correctly this is what current theory is pushing us toward will surprises come in between now and the time of recurrences or whatever time it will take for people to get these things straight I rather suspect surprises will happen but nevertheless let me just leave you with a statement that I deeply believe I said it before the arrow of time is a fact of nature and needs and explanation thank so the tradition of relativity started out to the point of view and joke about well we're here about that the horizon was faced at that point you it seems like another point of view would create a different trouble two bubbles will intersect somewhere now have super hot points well okay my next lecture will be about black holes ah I think you are asking the very simple question we could which week a phrase this way imagine somebody not at rest in the center of that bubble here but imagine somebody who was actually traveling within expect with a galaxy that was falling through the where are you yeah that was falling through would they see themselves passing through the superhot region and get themselves scalded no that's the puzzle of horizons which we will come to next time next time is the puzzle the conflict of principle between information conservation which is one side of this and the equivalence principle which is the other side the equivalence principle says that somebody falling down here sees nothing special when they cross that point but the conservation of information at least from the perspective of somebody at the center says in some sense nothing really can fall through so we're gonna this is this is going to be the discussion of next time let me just say again I'll have to say trust me that that the mathematical description that we've evolved that of a kind of largely the work of people like Hawking and bekenstein and other people that that the right way to think of our description at the center of all of this is in terms of a a finite box with all of the degrees of freedom contained within that box each one has a different description and then the region of overlap they have to agree I think that's what you're asking I'm not saying but I wait until the next lecture could won't say that the unit directionality of time is just a statement about the nature of consciousness a small like we see one time but yes I don't think so because I think there's a physical fact there that that's explain now I understand what you're asking I think I don't subscribe to that view of it I do think it's a physical faculties explanation but could consciousness be part of the way we perceive time serious of course is a part of the way we perceive time nevertheless I underlined it is so fat could be well looks like us and in fact I think the way all of the way Boltzmann finally formulated it is not that the entropy always increases whatever configuration you find yourself in the next configuration is most likely to have a larger entropy but it's also true that if you just find yourself at a random point among these configurations the previous configuration also was most likely to have a larger entropy so it does come they're always this term most likely the idea that entropy always increases was misguided as Boltzmann eventually discovered and at the right statement n you might say is that entropy almost always increases so does that it does that address what you were asking yeah close it's better than nothing yeah you didn't you didn't pay much to get in here so what the hell Andy the quantum mechanics and they provide I I don't know as I said I think there's a physical fact that it needs explanation that it's continuous no no I am I think there is always a level at which it could be discrete and we wouldn't notice it I don't think anything I said really depends on on the discreteness of continuity of time the pictures that I drew of evolution were discrete okay the basic set of ideas were a sort of summary of the logic of classical mechanics in classical mechanics it's not discrete upgrading its it's um updating its solving differential equations but the idea of one arrow in and one arrow out becomes what is called levels theorem in classical mechanics what that is you recognize I said not believe it so just so that we don't need people concerned as a final question you suggested yeah yeah jump yeah before I before I lay your concerns before I lay your concerns let me make you more worried jumping to a zero cosmological constant is not an innocent thing it's true if the cosmological constant if that's all that happened and the cosmological constant suddenly became zero I don't think it would affect us very much but you're jumping over this barrier to something new to a place whether we're the parameters of physics are different where everything is different there may or may not be electrons in that world that may or may not be photons in other words physics the properties of a physical world would be very very different than addition to the cosmological constant being different everything would be different in other words it could well be a world in which atoms don't exist so that's the bad news the good news is that such events are exponentially unlikely um the let's see where was it where was the picture let me redraw the picture over here a picture that we dump this and then we do that the time scale for the penetration through here is very very sensitive to the parameters of this bump it's all it's typically exponentially long times but it's ten billion years exponentially long yeah it's exponentially in something namely the log of ten billion so so there's no good reason why it couldn't happen tomorrow but if you take some simple reasonable parameters for the is coming out of various kinds of particle physics frameworks you might expect that the time scales are much much longer than the age of the universe but you know I don't think anybody has a precise theory of this could be come out of it I cannot not tomorrow that next lecture is Wednesday

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Roger Melko Public Lecture: Artificial Intelligence and the Complexity Frontier



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Christianity and Stoicism



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Braess's Paradox – Equilibria Gone Wild



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Mythstory #9 – Aztec Mythology



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The Aztec were a flourishing Mesoamerican culture from the 1300-1521. The capital of the Aztec empire was Tenochtitlan. During the empire, the city was built on a raised island in Lake Texcoco. Modern-day Mexico City was constructed on the ruins of Tenochtitlan. The Spanish colonisation of the Americas reached the mainland during the reign of Montezuma II. In 1521, Hernán Cortés, along with an allied army of other Native Americans, conquered the Aztecs through germ warfare, siege warfare, psychological warfare, and direct combat. Lucky for us, much of their mythology still survives to this day. So let’s begin from the beginning

hello everyone welcome to myths tree the shore we talk about myths from around the globe for today's show we're going to be discussing the central religion of Mesoamerica Aztec mythology the Aztek were a flourishing Mesoamerican culture from 1300 to 15 21 the capital of the Aztec empire was Tenochtitlan during the empire the city was built on a raised island in Lake Texcoco modern-day Mexico City was constructed on the ruins of Tenochtitlan the Spanish colonization of the Americas reached the mainland during the reign of Montezuma ii in 1521 Hernan Cortes along with an allied army of other Native Americans conquered the Aztecs through germ warfare siege warfare psychological warfare and direct combat lucky for us much of their mythology still survives to this day so let's begin from the beginning creation there are two variations of the Aztec creation myth that are very popular in discussions I picked the one that seemed to have the most support if you would like to learn about the other creation myth feel free to check out the big myth site for an overview in the beginning there was only chaos and the god of duality it was both male and female the Lord of duality no meant acutally and the Lady of duality oh mecha waddle this God was good and bad male and female and gave birth to four other gods who each preside over one of the four cardinal directions over the West resides Quetzalcoatl the god of light mercy and wind over the South presides witzy lapaki the God of War over the east presides she pathetic the god of gold farming and springtime and over the North presides tests cat lopaka the god of judgment night deceit sorcery and the earth these gods set out to create the world they created many other gods and the first humans who are Giants one of these gods had to play the part of the Sun and they would end up switching their roles as time went on this creation model is called five sons each son had to be replaced by a new God because the world always came to end Tezcatlipoca was chosen as the first son but either because he had lost the leg or because he was god of night he only managed to become half a son the world continued on in this way for some time but a sibling rivalry grew between Quetzalcoatl and his brother the mighty son who Quetzalcoatl knocked from the sky with a stone club with no son the world was totally black and in his anger Tezcatlipoca commanded his Jaguars to eat all the people the gods created a new group of people to inhabit the earth this time they were normal-sized Quetzalcoatl became the new son and as the years passed the people of the earth grew less and less civilized and stopped showing proper honor to the gods as a result Tezcatlipoca demonstrated his power and authority as a God of sorcery and judgment by turning the animalistic people into monkeys Quetzalcoatl who had loved the flawed people as they were became upset and blew all the monkeys from the face of the earth with a mighty hurricane he then stepped down as the Sun to create a new people tolik the god of rain became the next Sun but Tezcatlipoca seduced and stole his wife Chuckie Quetzal the goddess of sex flowers and corn Pollock then refused to do anything other than wallow in his own grief so a great drought swept the world the people's prayers for rain annoyed the grieving son and he refused to allow it to rain but the people continued to beg him then in a fit of rage he answered their prayers with a great downpour of fire it continued to rain fire until the entire earth had burned away the gods then had to construct a whole new earth from the ashes the next Sun and also Pollux wife was Shashi Whitley qu she was very loving toward the people but Tezcatlipoca was not both the people and sha Chi wheatley ku felt his judgement when he told the water goddess that she was not truly loving and only faked kindness out of selfishness to gain the people's praise sha Chi weakly ku was so crushed by these words that she cried blood for the next 52 years causing a horrific flood that drowned everyone on earth during the fifth son the current one humans would once again walk the earth Quetzalcoatl rescues humanity Quetzalcoatl would not simply stand by as the gods continued to kill people he needed to save them so he asked for the help of his brother Shilla tell the God with the face of a hound she little said I am shuttle the evening star every night I lead the Sun down to machlin the Aztec hell to die I know the way to the land of the dead and will guide us there Quetzalcoatl with his wise old face and brilliant feathers said I am Quetzalcoatl the morning star every morning I lead the Sun back out of Midland to be reborn with the dawn I know the way out of the land of the dead and I will guide us back home to the sweet paradise to mountain the Aztec heaven they retrace the path that the Sun took every night down to the depths of the underworld all the way to the Palace of the Lord of the Dead we must be careful Quetzalcoatl said I know Lord McClintock wittily will not be pleased by our request he is a wily God and may try to trap us Shalako agreed and they cautiously proceeded to the throne of the Lord and Lady of the Dead Quetzalcoatl approached the Lord of mifflin who sat on his throne surrounded by spiders and owls as well as the bones of humans piled up like treasure I've come for the bones the precious bones the Jade bones said Quetzalcoatl can I have them in order to populate the earth MacLeod too quickly replied and how do i benefit from this no I don't think I'll give up my splendid bones if I give them to you I'll never get them back and I'll be poorer for it no you can't have my bones Quetzalcoatl had anticipated this oh no you misunderstand me we don't intend to keep the bones we just want to borrow them the humans would be mortal and would eventually return to you just like how everything else is born and eventually dies even the Sun itself only we the gods live forever you wouldn't really lose anything in the end and in the meantime your fame would grow lady McClintock wattle looked pleased by these words Lord McClintock oddly considered them then spoke hmm an interesting idea all right you can have the bones if continued McClintock Whitley and shelah Tov Rose if you can play my conch shell trumpet and circle my kingdom four times in honor of me he handed Quetzalcoatl a conch shell that did not look like any kind of instrument and they left the chamber she Lovell looked at the trumpet and dismay the conch shell couldn't make a sound he's trying to trick us I've got a plan said Quetzalcoatl and he called the worms and other gnawing insects and ordered them to chew holes into the conch shell then he took the shell and held it up and summoned the bees to climb inside through the holes and buzz loudly the sound echoed through the shadowy realm like a trumpet blast Quetzalcoatl and shelah rule came back into the room and asked for the bones very well then said McClintock wittily you can have them for now but the humans will not be immortal they must die again someday and return to me just as you had said earlier the gods agreed gathered up the bones and left lady McClintock wattle looked terrified our treasure we can't let them carry it off of course we won't I may have said they could have the bones I never said they could leave my kingdom with them and then he ordered some of his servants to dig a pit along the path that the two gods must take to escape and others to chase after them the gods were able to escape but lost some of the bones along the way all the gods got together and sprinkled the bones with their blood restoring them to life and thus humankind was born from the penance of the gods of themselves journey of a princess when the Aztecs settled down on a magical island that had appeared they did not go to war as they usually did they still had to feed their hungry gods but they used their own people instead of captives that's what their God had told them to do after a while the Aztec emperor sent a message to a nearby tribe inviting the Chiefs daughter to visit the Aztec capital and meet his son his invitation was accepted the princess of the nearby tribe arrived at the capital city with many servants and many presents for the royal family she was delighted to meet the emperor's son they had a lovely dinner together by the end of the evening she was more than willing to become his bride a few days later her father arrived in Tenochtitlan the Aztec capital city he expected to discuss what goods his daughter would bring to her marriage with the Emperor's son that's when he heard that his daughter and her many attendants had been sacrificed to feed the many hungry Aztec gods in fact he probably would have been sacrificed himself only fortunately for him he had traveled to the city with many armed guards the Emperor tried to explain that it was an honor to be sacrificed but the chief would have none of it he hurried home to his people and the very next day sent his army to destroy the awful Aztec people as their God had told him to do the Aztecs had taken time to grow in strength before they had contacted this nearby King their young men had become capable warriors they had many weapons and they won easily the Aztecs demanded tribute in the form of jewels food clothing and of course captives to feed their hungry gods that made the Aztecs very happy the Aztecs went on to conquer other tribes in the area and that made them very rich there aren't many Aztec tales that aren't related to creation most of the mythology is just the story of the five sons or expansions on it I was lucky to find some that had nothing to do with it Aztec mythology is very in-depth and contains a very rich story thanks for watching this episode of myths Treon Aztec mythology make sure to LIKE and subscribe and tell us what mythology you want us to cover next in the comments below and if you really love us consider supporting us on patreon we'll see you next time

SUNSCREEN in UV



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“The Food and Drug Administration (FDA) is proposing to limit the maximum labeled SPF value for over-the-counter (OTC) sunscreen drug products to ‘‘50+.’

NYT Article on SPF 50

FDA: Understanding over the counter sunscreens

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“Titanium dioxide and zinc oxide provide UV protection primarily via absorption of UV radiation and not through significant reflection or scattering.”

From the FDA’s website:
“Although the protective action of sunscreen products takes place on the surface of the skin, there is evidence that at least some sunscreen active ingredients may be absorbed through the skin and enter the body”

In 2014, the FDA updated sunscreen ingredient proposal rules

thanks to curiosity stream for supporting PBS Digital Studios whoa it looks like I'm wearing like really bad tanning oil hey I'm Diana you're watching physics girl just relaxing here in the shade on this beautiful summer day and that right there yeah that's right okay but you're not needed for this video yet so you just chill so a couple months ago a few guys got in touch with me and told me that they had access to a UV camera do you know what that is yes basically it's a camera that allows you to see the world in ultraviolet which meant a few things for me it meant that I could see what this looks like in UV I could see what your face looks like without any lotion or makeup on in UV and I could see how sunscreen shows up on your face in ultraviolet and so I said yes please do come visit me in San Diego and bring your UV camera and so the guys from the very well made YouTube channel how to make everything showed up at my house this is what we did on your pants your pants are so Tigers right now you have some weird tan lines after this I was just thinking that this is so crazy showing up oh yeah yeah that was the fun day but I should explain why I'm here with Derek from veritasium who is also making a video about the world in UV I've been making this video for like a year and I'm here to make him finish it but for this video we're gonna cut in a few times to discuss the sunscreen controversies there are a lot of misconceptions about sunscreens and we're gonna clear them up yeah but for now I'm gonna turn it over to Diana in the studio back to you Diana the first thing I wanted to look into was whether different SPF s that is sun protection factor which actually is supposed to indicate how strong the sunscreen is but anyways I wanted to see whether different SPF of sunscreen look different under UV light and then I wanted to compare whether high SPF like SPF 100 are more effective than just regular high SPF like SPF 50 here was my thinking with a UV camera things that look lighter are emitting or reflecting UV light like the wall behind us and things that look darker like the shirt that I'm wearing are absorbing UV light which is something that I think you'd want to know when it comes to sunscreen because UV light ultraviolet light of course is part of the spectrum of light or electromagnetic waves from the Sun and UV is just on the far side of the violet part of the visible light rainbow which means it's a wave that has a shorter wavelength which means it has a higher frequency than any visible light and in the end it can do more damage to molecules like DNA or to cells and as we all know it's been linked to skin cancer so my thinking was that if you look at your skin with sunscreen on you should see that UV light is absorbed so it should look darker and yeah like when we put this on screen on it looked like face paint we were even able to graffiti this out of the truck with sunscreen but of course you can only see it with the UV camera so my hypothesis was that the sunscreen with the highest SPF should look darker with the UV camera and so we tested a bunch of different sunscreens with different SPF on a grid on my arm drawn with permanent marker here's what we saw since they can't see anything it's just like a sharpie it looks like pink yeah looks like metallic black thing initially we saw that yeah the low SPF areas do look less dark you can tell they don't seem to be absorbing as much ultraviolet light but interestingly there's not a ton of difference between the 30 to 110 SPF and I've heard this rumor that SPF s above 50 anything really hide doesn't actually do anything more than SPF 50 so I looked into the scientific literature on this and I actually found a study that came out just this year in May 2018 that tested 199 people with SPF 50 and SPF 100 sunscreens over the course of about six hours of sun exposure the subjects got more sunburned with the SPF 50 but they do put a nice little disclaimer at the bottom of the paper that leads you to conclude that more research needs to be done on whether there are more benefits over the long term of using SPF 100 versus 50 hey guys I just want to pause on this for a second because I still think it's weird that I have this conception that above SPF 50 you don't get any more benefits and and Derek let me know what you think about this but I I think I figured out where that conception came from and I think that it's from a 2011 proposed rule from the FDA the American Federal Drug Administration yeah they proposed a rule that would limit the maximum allowed SPF labeling to 50 plus thoughts I also don't think it's strange the conception that the high number sunscreens are all basically equivalent you know when I lived in Australia people were limited to putting only 30-plus on the sunscreen bottles and now they've moved that up to 50 plus but it's it's still I think a good idea because as those numbers increase you're getting kind of diminishing returns so what's the difference between 50 and 100 well it's not really that much because a lot of how effective they are comes down to how much you apply and how well you apply it so you know I think it's just a marketing gimmick to say that like this is a 75 or this is an 80th a 90 so obviously SPF and the efficacy of different sun protection factors is still an active area of research anyways back to the fun day at hand when we were looking at the sunscreens with a different SPF we tried a different sunscreen SPF 25 and something weird happened it was actually lighter like really light this means that it's not absorbing UV light but could it still be Sun screaming yes if it reflects UV light which is most likely what's happening here with this sunscreen that has zinc oxide in it which is a substance that's known to reflect some types of UV light so oh this is where I got really confused I started looking up some sunscreen ingredients and what they do and why some are reflective and others are absorb and I ended up going down the rabbit hole of sunscreen controversies I'll get to the discussion on health effects of different sunscreen ingredients in a minute but I did find that Hawaii recently banned some sunscreens from the state the reason why is that the ingredients oxybenzone and octi NOx 8 were found to be contributing to coral reef bleaching a process whereby coral polyps expel the little algae living inside of them actually kind of sounds like an exorcism but it is in fact a bad thing so starting in 2021 Hawaii is banning any sunscreens with those ingredients this ban was based off of recent research which begs the question are the health effects and environmental effects of sunscreen ingredients not well known okay I'm just gonna pause here again because I feel like this is a good discussion for us there's a lot to unpack here but I want to start out with that SPF 25 sunscreen that was really reflective yeah I mean there's two major ways that sunscreens can protect your skin from UV light one of the ways is by absorbing that light and turning that energy into heat another way is by reflecting the light and what you saw was clearly more reflection now in the media they're often called physical versus chemical sunscreens which is maybe not the best designation I don't think it's the best because you know they are all chemicals maybe it's a better distinction to say that the chemical sunscreens are organic because they are organic molecules that have these chains of carbon and the so called physical sunscreens are inorganic molecules things like titanium dioxide and zinc oxide so they do a bit more reflection but they also do scatter the light they absorb the light this chemical versus physical distinction bothers me as you know because I think it's you know leading way on a lot of health blogs and in a lot of media indicating you should avoid these chemical sunscreens because they're chemicals you don't like it leather like it because they're all chemicals right there everything's are chemicals right I think there is sort of a deeper reason why people would say chemical versus physical because chemical could describe like by virtue of its chemistry that's how it interacts with the light whereas you know a physical sunscreen by virtue of its structure is really how it interacts with the light you're right when it comes to the marketing people do try to use the chemical label as sort of a negative term and I think that it actually hides an important discussion about the potential safety issues of what they call chemical sunscreens so we're discovering things about sunscreen ingredients all the time for example recent research has shown that some sunscreen ingredients and in particular some of the organic sunscreen ingredients can get into the bloodstream and into breast milk through your skin which led the FDA to recently state that this is a significant discovery that needs to be considered as we continue to evaluate the health and safety of sunscreen ingredients in fact the FDA just recently rejected eight new proposed sunscreen ingredients because they felt there wasn't enough science to show that these ingredients were safe but my question is that now that we know sunscreen ingredients can enter the body through the skin should we be looking back at those 16 ingredients should we be looking at them through the lens of the new research and and judging them with the same thoroughness that we're judging these eight new ingredients against for example what's surprising to me is that there is still so much to be learned about sunscreen I think it's a little weird because on the one hand you're saying these health blogs are saying chemicals bad but on the other you're saying well maybe chemicals are bad because they haven't been really tested yet that's not why people are saying that they're bad and in fact some of them may be perfectly fine and they get lumped in with the ones that do have that do cause skin allergies like to say because they're chemicals is the reason like you might lose out on potentially a ton of great sunscreen ingredients that are gonna be really effective for protecting the populace against skin cancer but people have lumped them in into this chemical category which really just means organic molecule category maybe we're gonna find ones that don't get absorbed into the bloodstream as readily you mean yeah there is so much to be learned but at the end of the day I think put on the sunscreen at the end of the day do you want skin cancer because it is the most common cancer in the u.s. you heard it from the man wear sunscreen kids oh well yeah also thank you for watching this video and check out Derek's video which is gonna be about the world in UV something about sunscreen but about everything else two really cool experiments that we filmed yesterday check out the video by the how to make everything guys they made sunscreen in the end yeah awesome yeah watch that I will link to both of those videos in the description and at the end of this video or whatever thanks so much for watching and happy physics thing thanks to curiosity stream for supporting PBS Digital Studios curiosity stream is a subscription streaming service that offers documentaries and nonfiction titles from a variety of filmmakers including curiosity stream originals for example you could check out vitae mania a documentary hosted by Derek Muller himself the world streaming premiere of vitamin E a– is on curiosity stream on august 2nd and vitae mania is all about the 100 billion dollar vitamin and supplement industry to learn more you can go to curiosity stream comm slash physics girl and use promo code physics girl during the signup process that's cool you have like a tagline I don't have a tagline I know yeah but it's happy physics thing it's still pretty good like I no one's gonna steal that that's but my god you know someone tried to steal it once and I was like you can have 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How to understand BLACKHOLES using a cricket ball? | EXPLAINED | Dumb science tamil



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How to understand BLACKHOLES using a cricket ball? | EXPLAINED | Dumb science tamil



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How to understand BLACKHOLES using a cricket ball? | EXPLAINED | Dumb science tamil



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Civilization Part 6.2 – BBC Series by Niall Ferguson



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Civilization Pt3 – BBC Series – Niall Ferguson

The most informative history lesson in 12 hours.

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Classical and Quantum statistics in hindi || Raj Physics tutorials #physics #science



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UNDERSTAND TIME TRAVEL WITH RUBBER BAND! | PART 1 | Dumb science tamil



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Note for skeptics: Before confusing the concept with one of your own theories, please read the book Special relativity and interpret with sense. This video is based on observer who is static in outer space. For, an observer who is traveling near speed of light, the theory works opposite to what I said.

XII-1-14- Charging A Body By Induction Method



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Charging A Body By Electrostatic Induction video Lecture of Physics for JEE IIT, NEET,ACADEMIC by FAIZ AHMAD FAIZ Sir. FAIZ AHMAD FAIZ Sir is known for
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Physics – Mechanics: The Inclined Plane (1 of 2) Frictionless



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This lecture series will cover Newton’s Second Law of Physics: F=ma.

In this video, I will demonstrate how this applies to a frictionless incline.

Problem Text:
A 50kg mass is placed on a frictionless incline that makes an angle of 30 degrees with respect to the horizontal. Find its acceleration.

An Introduction to Quantum Biology – with Philip Ball



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What is quantum biology? Philip Ball explains how strange quantum effects take place in the messy world of biology, and how these are behind familiar biological phenomena such as smell, enzymes and bird’s migration.
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In this guest curated event on quantum biology, Jim Al-Khalili invited Philip Ball to introduce how the mysteries of quantum theory might manifest themselves at the biological level. Here he explains how the baffling yet powerful theory of the baffling yet powerful theory of the subatomic world might play an important role in biological processes.

Philip Ball is a science writer, writing regularly for Nature and having contributed to publications ranging from New Scientist to the New York Times. He is the author of many popular books on science, including works on the nature of water, pattern formation in the natural world, colour in art, and the cognition of music, and he has also broadcast on many occasions on radio and TV.

Jim Al-Khalili is Professor of Theoretical Physics and Professor of Public Engagement in Science at University of Surrey. He is author of several popular science books and appears regularly on radio and television. In 2007, he was awarded the Royal Society Michael Faraday Prize for Science Communication.

This event took place at the Royal Institution on 28 January 2015.

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Philip Ball is a science writer, writing regularly for Nature and having contributed to publications ranging from New Scientist to the New York Times.

He is the author of many popular books on science, including works on the nature of water, pattern formation in the natural world, colour in art, and the cognition of music.

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How Hard Can You Hit a Golf Ball? (at 100,000 FPS) – Smarter Every Day 216



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TEDxCaltech – Tony Hey – Feynman and Computation



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Tony Hey is corporate vice president in Microsoft Research, and responsible for its multidisciplinary eScience Research Group and research collaborations between Microsoft and university researchers worldwide. Previously, he directed the U.K.’s e-Science Initiative, helping to build a new scientific infrastructure for collaborative, multidisciplinary, data-intensive research.Before that he was Head of the School of Electronics and Computer Science, and Dean of Engineering and Applied Science at the University of Southampton, and led a group researching parallel computing. Tony is a fellow of the U.K.’s Royal Academy of Engineering, the British Computer Society, the Institute of Engineering and Technology, the Institute of Physics, and the American Association for the Advancement of Science.  He was awarded a CBE for his services to science in 2005. Tony is passionate about conveying the excitement of science and technology to young people.  He has co-authored “popular” books on quantum mechanics and relativity, and written technical books on particle physics and computing.

About TEDx, x = independently organized event: In the spirit of ideas worth spreading, TEDx is a program of local, self-organized events that bring people together to share a TED-like experience. At a TEDx event, TEDTalks video and live speakers combine to spark deep discussion and connection in a small group. These local, self-organized events are branded TEDx, where x = independently organized TED event. The TED Conference provides general guidance for the TEDx program, but individual TEDx events are self-organized. (Subject to certain rules and regulations.)

On January 14, 2011, Caltech hosted TEDxCaltech, an exciting one-day event to honor Richard Feynman, Nobel Laureate, Caltech physics professor, iconoclast, visionary, and all-around “curious character.” Visit TEDxCaltech.com for more details.

The Real Meaning of E=mc² | Space Time | PBS Digital Studios



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You’ve probably known OF E=mc² since you were born, and were also probably told that it meant that it proved Mass equaled Energy, or something along those lines. BUT WAIT. Was E=mc² explained to you properly? Mass equalling energy is mostly true, but E=mc² actually describes a much more interesting, and frankly mind-blowing aspect of reality that likely wasn’t covered in your high school physics class. Join Gabe on this week’s episode of PBS Space Time he discusses THE TRUE MEANING OF E=mc²

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Veritasium: Your Mass is NOT From the Higgs Boson

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What Happens At The Edge Of The Universe?



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What Happens At The Edge Of The Universe?
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What Would Happen If The Sun Suddenly Disappeared?

Have you ever wondered what might lie at the end of the universe? It’s amazing to imagine what it could be like at the very edge of the universe, what it looks like, what is out there. If it were possible to see this part of space, we would see the beginning of the universe as it began. But once something slips past the event horizon, it becomes lost to our sight, no longer giving off light signals.

Spacetime is a fascinating subject. One that needs to be explored to fully understand just how big the known universe is. So let’s take a journey to the outside of the known universe and answer: What Happens at The Edge of the Universe.

Reality Lab Lectures: Thomas Furness – "My Attempts to Save The World"



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The Reality Lab Lectures – Tuesday, April 9, 2019
Talk Title: My Attempts to Save The World
Speaker: Thomas Furness (UW Professor, HITLab Founder)

Talk Abstract: Over a career spanning 53 years, Prof. Furness has been exploring and developing technologies for facilitating bandwidth between humans and computing machines. His work has encompassed fighter cockpits, virtual reality, retinal displays, educational tools, medical simulators, pain, phobias, molecular modeling, scanning fiber endoscopes and entertainment systems. This quest has been punctuated with side trips and ‘aha’ experiences that have led to unanticipated destinations. Dr. Furness plans to talk about lessons learned on his journey including unexpected delights…with an aim to inspire, entertain and challenge.

Event held on the UW-Seattle Campus and recorded by UW CSE Production Team

© UW Reality Lab, 2019

How Engines Work – (See Through Engine in Slow Motion) – Smarter Every Day 166



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Mapping Particle Physics – with Jon Butterworth



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What is the universe made of? Join pioneering physicist Jon Butterworth as he sets out explore the standard model and the exciting world of particle physics.
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Come on a journey into the world of the unseen in search of atoms and quarks, electrons and neutrinos, the forces that shape the universe and the mysterious territory currently being explored at the energy frontier.

Jon Butterworth is the head of Physics and Astronomy at UCL. He works on the ATLAS experiment at the CERN Large Hadron Collider and has written several books on particle physics.

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We Need to Talk About Physics – with Helen Czerski



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Helen Czerski explores the everyday oddities that shed light on some of the most important science of our time.
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Helen’s book “Storm in a Teacup: The Physics of Everyday Life” is available now –

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When we hear about physics, we often hear about the weirdness of the tiny quantum world or the bewildering vastness of the cosmos. But there’s a lot in the middle, and it’s time someone starting talking about it. Helen Czerski will take us through some of the little everyday oddities that shed light on some of the most important science and technology of our time. Once the patterns are visible, a new perspective on the world beckons.

Helen Czerski is a Lecturer in the department of Mechanical Engineering at UCL. She is the author of “Storm in a Teacup”, a new popular science book exploring the physics of everyday life.

This lecture was recorded at the Royal Institution on 15 February 2017.

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