Special relativity and the Lorentz transformation

52:32right? Our coordinate in this case is let's just think about 1D, right? There's a there's a length I'm like moving in this direction. So there's a length forward and backwards in the direction of my movement. And um there's also time. I've got a I've got a time on my watch. Now time gets dilated. What does that mean? That means that the clock is running slower, which means that a second is getting longer, but my length is getting shorter. So one coordinate is getting shorter and the other coordinate, the time coordinate, is getting longer. That is central to the physics of relativity. And let me

53:12show you by considering this plot. So let's consider this this plot. Um this is a plot that um almost every physics undergrad has seen. On the x-axis is space. On the y-axis is time. So on the y- axis we've got 1 second, 2 second, 3 second, 4 second. On the x-axis is meters, but actually the one is not 1 meter. The one is one light second. >> So it's really like 300,000 km, 600,000 km. So it's it's the speed of light multiplied by 1 second, 2 seconds, 3 seconds, and 4 seconds. Why do we want to do that? Well, if we if we represent the x-axis that way, then the speed of light is a diagonal. >> Yeah. >> Right. Because it's like one one. It's

53:54like, okay, in 1 second it travels one light second. In 2 seconds it travels two light seconds. So you've got a nice the dotted green line is what light would do in this coordinate um system. Okay. So that's that's usually how we represent things in relativity. >> Mhm. >> All right. >> For many folks who math may not be something you do very often, the closest thing you may have seen that touches this concept is interstellar. Yes. >> When they're on the wave planet, the guy is on the ship still and they have that time dilation issue. >> Uh because of exactly what we're talking about. >> Because of exactly what we're talking about. Exactly. And so we've considered that plot. Now let's consider for

54:35example we're in this plot right? It's the two of us. >> Yes. >> Okay. We're both stationary. We're both in the same reference frame. >> May I got a lot of motion. I don't know what you're talking about. >> But as of now I'm I'm going to I'm going to have you move. I'm going to have you move in the next plot. But in this plot >> Yeah. I don't have motion right now, but I will in the future. >> I want to I want to give you a little sense of how relativity is so cool. Okay. >> So let's consider we've got um we're sitting here. This is x equals z this table. Okay. And we set the time. We we have a stopwatch. I've got a stopwatch on my iPhone. I set it to time equals z. And let's say there are three explosions that happen. Okay? Or like three light

55:17switches. The the in the physics textbooks, it's always like light switches. Let's do let's do explosions. Explosions. >> But but we can survive the explosions. Okay. There's an explosion that happens here where we're sitting, but in 2 seconds time. Okay. There's an explosions that happens in front of us um two light seconds away. So 600,000 km away. >> Real close. >> And there's a explosion that happens behind us. Two light seconds behind us. Okay. So 600,000 kilometers behind us. But they all happen at the same time because we're sitting here and we see the we we see the the the explosions happen at the

55:58same time. Okay. So we're like, "Okay, cool. The coordinates of these explosions in spaceime." >> Now we're thinking about coordinates in space time, not just space. The coordinates in spaceime are the explosion that happens here is at space equals 0 cuz it's right here and the time equals 2 seconds. The the explosion in front of us is space equals 2. Yes. >> And the time equals 2 because it happened at the same time. And back there it's -2 time equals 2. >> When you say space time in this example, we're talking about two axes. One is space and one is time. And so you can represent space time >> on a on a two-dimensional xy graph in this example. And so it's not this m

56:40just it's >> no it's literal. Yeah. It's literal on a coordinate axis. This is the genius of Einstein right? >> Right. is he's like putting this stuff on a coordinate plane and he's asking right and this is actually Lorent came up with this stuff even before Einstein he was he was thinking about Lorent transformations and Einstein's the real guy to be like let's just take it seriously guys [laughter] >> right >> okay so now we ask the following we're both we're both sitting here and we agree if we were both sitting here these explosions would happen at the same time now suppose you were moving >> Mhm. that way. Yes. To the front. Yes. At at very close to the speed of light. >> Okay.

57:21The three events that happen simultaneously for me, they are not going to happen simultaneously for you. >> Why? The explosion that's in front of us before you before you Yeah. So, so this is me, right? This is this is my So, so to me, everything's happening at the same time, which is why everyone's in the same time axis. Yes. >> Now let's before we consider the next plot, let's just think about what would happen. You're moving towards >> the explosion in front of me at near >> which is at 22. >> Yep. Which is at 22 for me. >> For you. >> For me. >> But I'm moving towards >> But you're moving towards it. Right. >> At the speed of light. >> At the at near the speed of light. Near the speed of light.

58:02>> Sue me. >> Yeah. Okay. [laughter] But if you're moving out there near the speed of light. Now from Einstein's relativity, we know that length is going to contract for you. Mhm. >> So the distance to that explosion is going to go down >> and the time is going to get a little bit slower, >> but at the same time you're moving towards that explosion. So the light from that explosion is going to come meet you. >> Mhm. >> Because because we're basically it's like two trains coming at each other. >> Yes. >> But because of the speed at which I'm moving towards it >> from your perspective, uh it's going to take time for that light. >> It's going to take two seconds. But because I'm basically meeting the light before it gets to you at that speed,

58:43it's it's going to happen. >> Yeah. And so for you, that event is going to happen sooner than 2 seconds, right? And it's going to happen closer than two light seconds because the length is contracted. Similarly, the event that happens here is going to take a little bit longer than 2 seconds because this thing happens for me at 2 seconds, but then the light has to go to wherever you are. [clears throat] >> So, it's going to happen a little bit longer than 2 seconds. and the guy and the explosion that was way back there. That's going to take even longer. >> So, what we should see when we look at your perspective, right, and if we look at your perspective, the event that's in front of you should go down on the y- axis and down on the x- axis, right? For your x

59:26and y. >> Yes. >> My x and y is is is stationary. >> And and when we say down on the yaxis, cuz our y >> So, it's sooner in time. >> Sooner in time, >> and it's sooner in space. >> Space, right? and and getting closer to the origin. Yep. Yep. Yep. Okay. Got it. So now let let's let's see how let's see how we did right with that logic. >> So we go to the next plot. >> In this plot nothing has happened so far. I'm g giving you some analysis of of what we're talking about. The as I said the diagonals are light >> right? [clears throat] So the diagonals are what light would travel like the path that light would travel in my spaceime >> because again our our x-axis is light uh

1:00:08light seconds >> and our yaxis >> is just time >> time >> seconds right and so that's why it's diagonal and all of the dotted lines are diagonal because if a light beam started at t= 1 then it would travel like at the it would be the dotted line that's just above the t equals z. This is why we want to decrement the x-axis in this way so that it creates >> Yes. >> this simplicity to understand the >> Exactly. Exactly. Now, one of the one of the and the other big thing is um there's there's two points and there's actually a line a black line right on the right on the um y-axis here which says I'm not moving, >> right? Because you're the one who's moving. So to me, I'm just stationary at

1:00:49x equals z. >> Yes. >> For all of time, right? So no matter what time it is, I'm always at xals 0. Okay. So now what would what would you see for me? Well, you would see me if you're coming from back there, you'd see me with a positive x-axis and then and then you'd see me go backwards, right? Because to you, I'm coming from up there and then going backwards. Okay. Now, let's go ahead and do the Laurent transformation to see what you would see. >> Okay. >> Okay. What would you see? What you would see is the following. you'd see the coordinate axes change. >> Yeah. >> Okay. And notice, first thing we want to notice is the three red dots >> are doing [clears throat] exactly what

1:01:29we had predicted. >> The event that's in front is getting closer to you. So the time is decreasing and the x is decreasing. The event that's happening right here >> is going forward in time because it's not 2 seconds, it's a little bit more because the the light has to catch up to you for you to be like, "Ah, there was an explosion." And the event that happened way back there is happening way farther at 2 seconds. Right? >> But notice two things. And and and for one, the line that is me, >> which is usually time tals 0, xals uh time time equals whatever x equals 0. That thing is shifting to where at negative time. So before time t equals 0, I'm I'm ahead of you and then I go

1:02:11behind you. Right? That's the black line. Yep. >> Here's the key thing about this coordinate transformation, which is the genius of the Lorren transformation. All of the diagonals are still diagonal. >> Mhm. >> What does that mean? That means even in your reference frame, light is still moving at the same speed. >> It is still moving at the speed of light. >> Right? What's really happening is the length is contracting and the time is contra is is getting is getting faster, but they're happening at the same rate such that the diagonals are always diagonal. >> Yeah. They're they're just shifting on the same ratio of proportionality to each other. >> Yes. Yes. By the lance factor actually

1:02:53is is what is what is what it is. And and so it preserves relativity because one of the tenants of relativity is that the speed of light is the same in any reference frame. It's constant. >> So, so even when we go into your reference frame, the diagonals are still diagonal but >> the space has changed and the time has changed. >> The other cool thing about this is the volume of space-time is always the same. The the area of the square becomes a rectangle, but the area of each rectangle is exactly the same. >> Yes. Right. >> Again, because all everything is proport it's a shift that's happening. It goes back to this mapping point that you're bringing up earlier. We are making this

1:03:33transformation in our map but fundamentally our Jacobian remains as one the entire time. >> Yes, that's exactly right. And what that means is if like the volume of spaceime is always preserved. >> Yes. Yes. >> In the Lorren transformation. Okay. That's a central tenant of relativity. >> Okay. That ensures that the speed of light is always the same for the both of us. >> And really what it's kind of crazy when you think about it. The universe is doing all this >> just to make sure that the speed of light is the same for me as it is for you >> because neither of us are right. >> Correct. >> Right. Your >> perspective is the same as my perspective. So the speed of light better be the same. And the universe is

1:04:15doing all this nonsense of contracting length and making time go slower just so that the diagonals remain diagonal. I'm not saying it's a simulation, but if you wanted to make it a simulation, >> yeah, this would be a rule. This would be it would make sense that that's and I'm being facitious here, but I think the point you're bringing up is important, which is that the constant of the speed of light uh requires the universe to do all this other stuff. >> Yeah. >> So that regardless of your speed or your position in the universe, it's it remain that constant remains the same. Exactly. >> Which requires it to do all this other stuff. all the other things. >> So, so it's almost like a it's like the

1:04:55the speed of light is a golden rule. >> It always must be followed. >> Yes. No matter where you are, no matter how fast you're going, because I don't know if you're moving or I'm moving, right? >> We'll figure out the other stuff to make it happen. >> Yeah. Exactly. And so that's a c central tenant of Einstein special relativity. Okay. And that's the Lorren transformation that gives rise to all of the rich behavior that is a consequence of special relativity. So that again is just a that's a linear transformation as I said right it's um >> it's not too complicated >> I was able to get it very quickly so it's not it's not >> special relativity is like actually not