So, we made a few videos about relativity, and we’ve talked about how Distances change in relativity we’ve talked, about how. Time changes in relativity So i thought i’d talk about something, which combines those two things together, which is how, speed, which is distance divided? by, time changes in relativity the previous videos, we made we called a Gamma, trilogy, because they all have this gamma factor in it actually one of the interesting things, about the way speed transforms Is that the gammas all disappear they all cancel out so there is no gamma in this gamma video so, we need to think about How you made your speed JosĂŠ’s is distance divided, by time and but particularly how it changes depending on what reference frame you’re in so We need, to think about how Different people measure speed in different reference frames and so the classic kind of thought experiment You, might do is you’ve got somebody on a train Or call, what you call this reference frame s prime and they’re gonna roll a bowling ball, along the train At some speed or other and The train itself is moving so this, whole reference frame is moving if you like in this direction at. Some other speed v so The, way, you’re on the train, you’d measure the speed, is that you’d see how, far the ball in bowling ball Had gone in your reference frame, which, we call this delta x how, far it’s gone prime because it’s in your reference frame And at what time you know it’s um you’ve measured that at, some time Delta t prime after you’d let Go, over the ball and then you could actually, measure the speed in your reference frame of that bowling ball is just how Far it’s gone divided, by the time it took to get there That’s the definition of speed is how far something’s gone divided by the amount of time it took to get there is the definition of? Speed, so that’s in one reference frame so then the question Is what does the person in this reference frame the other reference frame see and let’s deal with a nonrelativistic case first Okay, so what are they, gonna measure there the distance that they’ll measure that that ball has gone it’s going to be how Far the bowling balls rolled, away from you plus, how Far you’ve moved down the line that extra distance how, far the train has moved depends on well that’s basically how Fast the train, was going Exactly that’s all you’re doing So that that distance is v delta t the trains, going at, some speed V and it’s travel for some time delta t then the distance it’s covered is just the speed times the time it’s taken to do It so, well now, we can put all this stuff together That says that delta x is equal to so how far you see the ball that having moved is how Far the trains moved plus, how Far the ball has moved relative to the train just the sum of the two and we can rearrange this in non relativistic case To, say that let’s just divide through delta x divided by delta t is equal to v plus delta x prime Divided by delta t and in this non relativistic case, we don’t have to worry About different people seeing different time so it doesn’t matter whether that’s a delta t or delta t prime We can call it the same thing it’s all the same thing or we can, rewrite this Again, that just says that the speed that you see the balls moving is equal to v so that’s delta x by delta t plus The speed that the person on the train sees the ball moving and that as you said it’s just basically you i just add the Speeds right the speed at, which the bowling ball is moving, is the speed that the ball is moving relative to the train Plus the speed that the train is moving relative to you right that’s the simple galilean transformation, no relativity Everything, works sensibly and the way, we used to be working? The reason, why? People get interested in this is what one of the things that everyone says in relativity is you can’t ever travel faster than light And so one obvious question you could, ask is well why can’t i just get to travel faster than light By, doing these kind of additions in the Supposing instead of rolling a bowling ball the person on the train was actually shining a torch Along, the train then the torch beam would be endued be traveling at the speed of light relative to them and then if you Ask, well how fast would the torch beam then, be traveling relative to you Well it will be this big light plus the speed that the trains moving away, from you, and that’s faster than the speed of light exactly it sorted Unfortunately relativity takes care of that and it’s not really the way that things work, so we need to start again but We need, to now, do the problem properly with relativity Totally, my drawing is not quite up to the job i suspect And then we got the other? chaps in here Moving, along at some speed v okay now. Previously the formula, we had said that the distance the Bowling ball is down the line is equal to the distance relative to the train plus, how Far the trains moved now, what the extra factor that, we haven’t put in yet is special relativity and as? we saw before what special relativity does is introduces these factors of gamma and so the this galilean transformation that We had before it a little bit modified By an extra factor of gamma in front of it gamma is one over the square root of one minus v squared over c squared And we have a whole series of videos you can? Go, and watch about gamma should you wish to do so but then the other thing, we know about relativity? Is that not only does the distance depend on where parents frame You’re, in but actually the time depends on what reference frame you’re in and so the We need, the equivalent transformation for that so there are these things called lorentz Transformations that’s the first of them and then this is the other lorentz transformation We need, turns out these are the two lorentz transformations you need, okay so now, we can say, okay, so what’s the speed as? Measured from the person who’s standing beside the track watching the train, go, by, and we just do the same thing that We did before we just the speed is just the distance divided By the time so we can just divide them we know with delta x? Divided, by, delta t is equal to now the gammas are going to cancel because if we divide that by the other We end up with a camera on the top pounder going or on the dr.

Bottom so, we can just cancel them out and i’ll do a little bit of tidying up i’ll divide top and bottom through by Delta t prime so i can just write this as delta x prime over delta t prime this Is kind of the answer that We wanted because delta x over delta t is the speed of the bowling ball or whatever it is that’s being thrown? Along, the train as seen, from the person for signing the track and Delta x prime over delta t prime is the speed as seen, by the person on the train so i can Write that in terms of the user new. Primes that, we had before that’s just says that u is equal to u prime plus v? divided, by 1 plus that’s au prime, again v Over c squared which is the final answer in the non relativistic limit so that what We were looking at before the galilean transformation so both u prime and v are small compared to the speed of light that Means this term is small so this thing’s just 1 basically, and that just says that the speed you see is just the sum of the speed of the ball relative to the train and the train relative to u Which is the answer, we have before so that’s right it kind of comes out right if things aren’t closest be alike but Ok, but now let’s do the more interesting case of actually where the relativity really matters and in particular let’s go to the really extreme case of actually instead of Rolling a bowling ball on train let’s shine A light beam, along the train so in that gauge you Prime the speed of that whatever it is relative to the trainee is equal to the speed of light And let’s see what happens when, we put that in there so in that case, we end up with if u prime is equal to the speed of light then u Is equal to u prime is just a speed of light plus v divided by 1. Plus u prime is the speed of light c v over c squared Now, let me just do a little bit of playing around with this so i’ll rewrite the top here as c Into 1 pull that factor out plus v over c Because if i multiply this out i understand that with c. Plus c b over c. Which is just v so c plus b Divided, by, and then all i’m going to do here is cancel that c over c. Squared so i end up with 1 plus v over c Which you notice is just the same as the term there so this is the same, as that, which just basically, means that this Whole thing, is just equal just the speed of light still So there’s the bizarre thing that we’ve taken we’ve remember what we’ve done here is said, ok so there’s a light Beam that’s moving relative to the train at the speed of light now if you’re watching from beside the train what speed You, see that light beam, going at the Answer still the speed of light you haven’t actually added to its speed at all And i guess the physics behind it is that actually you really have to worry, about both space and time Being changed, by, what reference frame you’re in which Means that not surprisingly the speed that comes out at the end is going to be changed in a slightly strange way as Well and it turns out that relativity takes care of this invariance of the speed of light that No, matter what reference frame you’re watching of light beam traveled from you’ll always see it traveling at the speed of light is the universe Wanting to keep things at the speed of light and everything Changes to cater for that or does everything change all the time and the speed of light falls out of that Is it chicken or egg It’s a very good question and i actually really liked Your first explanation, that actually the universe arranges things in such a way that the speed of light always comes out As the speed of light and no matter how. Much you, mess around with things By trying to run, away from a light beam or run towards it when you come to measure its speed You’ll always find that its speed comes out As the speed of light and everything else kind of adjusts the distances and the times just in just such a Way so that when you come to combine those distances in times to measure a speed for a light beam it will always come out as a speed of light, is there something that is Important about the speed of light being constant and unchanging So what motivated iearn stein to come up with all this in the first place is he had this idea that The laws of physics should, be the same whatever reference frame you’re in so if you were in a sealed Box there it should be no experiment you can, do that will tell you whether that sealed, box is moving at A constant speed or is stationary and in fact it’s even in his view of things that are kind of a meaningless question and What do you, also knew? Is that the speed of light comes out from the laws of electromagnetism and? so Then the question is okay so but in what in what reference frame to those laws of physics work and his argument Is that those laws of physics should work in whatever reference frame you’re in which Means that actually the speed of light Has to be invariant if you believe that the laws of physics are the same in all the different reference frames What if it hadn’t been that way, what if the universe is so now, that can be different for different reference frames like Would like when you and i be ripped to pieces in? Some cosmic rapport with the universe just be a bit different and quirky like does the universe benefit From the fact this is what happens i? Get i mean it’s very hard to construct. A universe in which this, wasn’t true in retrospect it’s hard to construct A universe in which this isn’t true bear in mind that for example you know all the other things that come from Relativity kind of flow, from that so i think famous things, like e equals mc-squared All flow, from this invariance of the speed of light, and that means that The equivalence of, mass to energy is actually a natural consequence of relativity and so for example what, powers the sun fusion, converting Mass into energy wouldn’t work, and if that We weren’t living in a relativistic universe now you could imagine that The universe the laws of physics in such a universe would come up with Some other way of generating you know energy from fusion, but the picture we currently have it really, would be completely different And the quantity of the speed of light the the speed that it actually is Does that matter like would the universe Be the same if the speed of light had been halved Or it was quite you know if it was something close to the speed, we walk out like, does it need to be as Fast as it is to our human, brains so it comes out of the laws of electromagnetism So it’s to do with how. Strong Electric fields are how, strong magnetic fields are which are really just arbitrary constants of nature as far as we know? Which i guess, means that actually the speed of light Which is some combination of these things is also an arbitrary law Of nature the universe would be a very strange place if the speed of light were very much slower because of course all these Relativistic effects that lead to all the weird things that come out of relativity, we don’t usually have to worry About them in everyday life, but if it actually turned out that the speed of light you know Was walking pace then all sorts of strange relativistic effects? Would happen every time you walk to the post box you know or being you would have to worry About all the relativistic time dilation and length contraction Effects so it would be like if i go and post this letter is it gonna i’m not gonna die before i get oh yeah? We’ll have all your relatives have died of old age before you get home again so Yes indeed it would be a very very strange universe so the light instead of going straight up and down follows a path like that So it has further to go From our perspective that light’s gone further but, also the speed of light, is the same in every reference frame