So, we come now to the Lorentz transformation, so let's set this up. We've got a lot on the board here. We're going to go through it as usual, bit by bit, but I'm going to expedite matters and get some of the key things up here. So, I'm not just wasting time as it were, writing on here we'll have a lot of this is probably the most mathematical series of video clips we'll do. Again, most of it's just algebra, it takes a while to get through it. So, we'll break it up into several video clips and make case slowly as is usual. And for those of you who you know, the algebra is very familiar, to you, then you can obviously speed things up. So, the Lorentz transformation, let's just remind ourselves about, Bob and Alice here. This time we'll have Alice stationary, Bob in the spaceship going at some velocity v. Of course, each of them have their frames of, of reference. Remember the props we've used before in terms of, each of them perhaps a spaceship, the lattice of clocks. So, we can imagine Alice with the lattice of her clocks all synchronized. Bob with the lattice of his clocks all synchronized. Remember in Bob's frame of reference, he's stationary. Alice is moving, in the opposite direction behind him. Alice, in her frame of reference, she's stationary and she sees Bob moving at velocity v in, in that direction. at the start, when Bob passes right by, Alice will assume that they start their clocks at, at that point. And so ta, t sub a equals zero, t sub b equals zero. And we'll also have the origin for their, their measuring systems also to be zero at that at that point so everything matches up there. we get on to this little bit, this later at time t sub B where we get the flash of light going on but here's the, the basic question we want to try to answer. And that is, given the coordinates, space-time coordinates of an event in Bob's frame of reference. So, xB, yB, zB, and some time t sub B. What would the coordinates be for Alice? xA, you, zA, and, and tA. in general, we'd like to go back and

forth between the two coordinate systems. If Bob measures something and says, hey, I saw a flash of light. And it was at this location in my frame of reference, and this time. We'd like to have a, a relatively simple calculation so we can find out the coordinates of that flash of light in Alice's frame of reference. clearly we can sort of work through examples we've done before but we don't want to have to do that every single time to, to work through the details. So, its nice to have a formula and that's what the Lorentz transformation is going to give us. It's going to be similar to the Galilean transformation that we worked on before. So, let's recall that simply that if we have two frames of reference in relative motion with respect to each other. Inertial frames of reference, or constant loss of motion and we'll use again Bob and Alice here. So, Bob moving to the right, Alice stationary if Bob measures something at coordinate x sub B, and time t sub b. Then Alice can find out where that is in her frame of reference by taking the x coordinate of Bob and multiplying by V times, his, his time, to find that out. And note the plus sign here think, it makes sense intuitively, because we know Bob is moving to the right. So, if Alice is standing here as Bob moves on, then if he measures something at zero, at his X would be equal zero. We know Alice, Alice's measurements are going to be farther on in the positive direction, because that 's the direction Bob is moving with respect to, to Alice. So, it'll be whatever position Bob measures at. Plus the essentially the distance traveled by Bob in that given amount of time until AC is the flash or whatever happens happens to be. And then, of course, the y and z dimensions remain the same. They're not changing and we, essentially, ignored them in most of our examples but to be complete you could put those in here. And the key thing for the Geilean transformation is time was the same in both frames of reference. Now we know, we'll get over to the details of this in a minute, but we know right away that from Einstein's two postulates.

We derive the fact that time dilation occurs, and length contraction as well. So, we know that this thing in particular, the, the time relationship here, is not true for the special theory of relativity.When you get certainly when you get higher velocities. But in principle, for any velocity, as long as you have that velocity between frames of reference. There is an effect such that TA and TB are not equal. Again as we've mentioned, it's so small usually in our ordinary everyday world that we don't notice it. But if we have very precise clocks, it is possible to measure that difference. So, we know that that the Galilean transformation can't work, we'd like to have something like this that works with special relatives, that's our goal, here. we'll get back to the, the stuff over here in a minute. But lets set up situation for for Bob here and Alice. And what we want to do is later to time Tb, so that they pass each other at time equals 0, essentially. And then, later at some time Tb, according to Bobs clock, he sees a flash of light right at this clock. So, in other words that is occurring at Xb equals 0 because rememeber he's carrying along his lattice and clocks with him. His measuring system with him. As far as he's concerned he's at rest and Alice is going backwards there and therefore if he sees a flash of light in his cockpit the X location, for him. X sub b would be 0 at that point. And, we actually could then figure out where Alice sees that flash of light. At what time given the X location for Bob and the time, t sub B for Bob. We could figure that out for Alice. In particular one thing we'd use is our time dialation equation. TA equals gamma t b, remember? Alice is observing Bob's clock moving and therefore time dialation will occur for Alice observing Bob's clock. Alice will see Bob's clock ticking more slowly and running more slowly. Right? That's the whole idea of time dialation, Bob does see any difference his clock is working just fine as far as he's concerned. It's Alice observing the moving clock

that has this Lorenze factor, gamma, involved in there. And of course gamma 1 over the square root of 1 minus v squared over c squared. Remember how we derived this. We derived this using the light clock and didn't make a big deal of it at the time because it wasn't important then. But this was derived for Bob's location at 0. OK? Because, remember, the light clock, we have it here and we have our light bouncing back. Going up and down that's what Bob is seeing on his light clock. The, the light motion going up and down, the light pulse going up and down is occurring at the same place all the time. Bob's just got his clock right next to him there. It's Alice observing it of course moving along so that as Alice sees it. And moving here, we'll squeeze this in, we get the familiar triangle effect. So, Alice sees it going, going like this, but to Bob, all this is happening at x b equals zero. So, our time, our first time dilation equation really assumes that everything is happening at x b equals zero, for the person who is moving. And the Lorentz transformation we want though, needs to be more general than that. Because, again, back here at the flash of light example, if the flash of light occurs right at Xb equals 0, at Bob's clock t here where he is. Then fine we can use our basic time dilation equation but he might see the flash of light out here someplace. And therefore, we'd have a more complicated situation. And this will still be true in a sense but we need to actually get a more general form of it. And we will see that once we get to the end result here. The lens transformation will have some equations here that, in the case for Xb equals 0. When, whatever is occurring, the flash of light, say, is at Bob's location, then this will still be true. But we're going to get a more general form of this type of, of relationship. Now we also know, just as a reminder, the invariant interval equation. And this is true in general.

And so, this would be very useful for us to derive the Lorent's transformation equations that we want here. Again, it's just algebra but it's useful to work through it so you can see where it comes from, it's not just magic, doesn't appear out of thin air or anything like that. It's a sequence of logical steps building on what we've done before. So, let's we're going to start off just by considering actually In a minute we'll get to, in a couple minutes. We'll get to the flash of light out here at a general, location. Let's just do the flash of light first at Bob's location, there. So, that's where we're going to head here, let's see what we get. So that's our general question, that's our goal. We want to again be able to transform back and forth there. And so, we're going to start here with a couple things we know, okay. So, here's the situation again, later at some time TB, flash of light. We know, therefor, that flash of light occurs and Xb equals zero, okay? At Bob's cockpit. what else do we know about that flash of light? Well, where does it occur for Alice? Well, Alice sees Bob traveling along at velocity v. we know we set it up so they started, they measured everything, the origin, from when they're right together. A little while ago and then Bob, Bob travelled on. And so we can also say, you know if we say, just to indicate, this is the location of the flash to Bob. The location of the flash to Alice is simply going to be, the velocity, relative velocity times whatever times she sees on her clock. Just V times TA, okay? Now, here's the key thing. What is TA? Remember, we want to have sort of Bob's coordinates and one side, and Alice's coordinates on, on the other. And so Bob measures the flash at time tb, we'd like to know, if we somehow write ta in terms of tb. Now, in, in actual fact, this is a case where our time dilation applies because we're talking about a situation where the flash is occurring.

Right at xb equals zero for Bob, sitting right next to him there. And therefore, we can just actually put this in here. Ta equals gamma tb, and so this is going to equal just put gamma tb in. I'll put the gamma in front, again, 'cuz we often like to write the gamma, the gammas in front. Gamma v, Tb. OK, so that gives us some good information there. Therefore if the flash of light occurs at Xb equals zero at some time Tb, so Bob records sometime Tb on his clocks. Then I can find out the X coordinate in Alice's frame of reference for that flash gamma times v times t sub B. So, if Bob gives me that answer, or I take a photograph and look at his clock, observe his clock, at that point. I can figure out the location in Alice's frame of reference as well. And then this would give me the time in Alice's frame of reference. Let's also get this in another way using our invariant interval equation. Because we're going to use this more here, in a few minutes, and so just to show you that it does work the same way. Let's just rewrite this so we've got C squared, ta squared minus xa squared equals c squared, tb squared. Minus X b squared, that's our invariant interval equation, that's good for anything, basically. Given ta, Xa, and tb and, and Xb for a given event, or distance between two events. In this case, the distance between the events will be our origin point when they pass each other and then a while later, the flash occurs. And Bob measures it again, T sub b and X sub b, and I want to be able to get the T sub a and X sub a from that. So, what do we know here? We know, plug into the flash of light, right here, that is that the flash of light occurs at 0. So, that meanas this thing right here Is 0, we don't have to worry about that. And so we're left with ta, xa, and tb in here, of course, c being, being the speed of light. And then let's also put in this vta, for the flash of light again. That's Alice's distance for the flash, 'cuz it's just, you know, the velocity of Bob times how much time elapsed on her

lattice of, of clocks. And so, let's plug this in here for that. So, we're going to get c squared ta squared minus vta squared, just plugging that in, equals c squared tb squared. And the advantage, of course, of these formulas is you just sort of, once you set it up correctly, you just sort of crank through it here. And so note that, well, we'll do every step here. C squared, Ta squared minus, this becomes V squared, ta squared equals C squared tb squared. And one nice thing about that, and I just have the times in there, just ta, and tb, I've got a ta squared in both of these terms here, so let's factor that out. So, that gives me a Ta squared, c squared minus b squared equals c squared t b squared. And now, we're going to bring it up here so we have more room. And so, come up here now what I can do, let's write, let's divide each side by the C squared minus D squared. Right? So, I can write this as ta squared equals C squared over C squared minus V squared TB squared. See what I did there, I just took this and then put it over on the other side of the denominator. Divided each side by this to get one here and C squared over C squared minus D squared over here. Times the TB squared this becomes, for those of you who are up with your algebra, you might want to play around with this a little bit, see what, see what we can get here. Mm-hm. So, pause if you want to do that, and see what our final answer is going to be. It's going to be very familiar. But we'll see what we get here. So, let's do this let's factor out a C squared in the bottom. So, I can write c squared times one minus v squared over c squared times tb squared. Okay? The reason I, I did that, well, first of all, I can do that. I've got c squared minus v squared, so I pull off the c squared, and then I've got v squared over c squared here, so this and this, they're the same things. The reason I did that, as you may see by now is that I can cancel my C squareds

there. And so where does that leave us? It leaves us with an equation like this. We get ta squared equals one over one minus. E squared over C squared. Tb squared. Now we take the square root of each side, all of our Ta's here are soon going to be positive, we're looking at positive time. So, we're going to have to worry about when we're taking square roots a plus or minus sign as some as you may remember with that. So, we've got tA equals 1 over the square root 1 minus b squared over c square, tb. Or, in other words, for ta squared is ta, the square root of tb is tb, and the square root of this is this and this should look familiar, right? This is gamma, so really what we've just shown is ta equals gamma tb. Using, we started with, with this equation, with a invariant interval equation. And the fact that we're looking at a flash of light at x b equals zero, because remember. So, we plug zero in here, and then work through the algebra, and lo and behold, what do we get? We get out time dilation equation, t a equals gamma. So, that's good news, 'cuz if we didn't get that, we would have made a mistake some place along the way. Either here, perhaps in our earlier derivation of the time dilation equation using the light clock. Okay? So, therefore, we say OK we've got this equation. That means Bob measures t sub b. And now Alice can, given that value on Bob's clock, again take out a photo flash, our old photo principal idea. Take a photo of that, could see what's on Bob's clock. And if, if she and her lattice of clocks that was at that location as well, took a photo of her clock, this is what would be the reading on her clock. It'd be whatever was on Bob's clock time the gamma factor times TA. And of course gamma is always greater than or equal to one and so for, so you get the time dilation effect there. So, given T sub B then, we know what T sub B is but what about XA? What the, the X location in terms of, of

Alices lattice? Well, remember we erase them, but remember we had this, we had XA simply equals V times TA. Notice just the velocity of Bob times the time, elapsed time on Alice's clocks. So, she's watching Bob go by with velocity V and therefore, and this just is TA equals gamma, TB. And so we can plug TA into here, and we get this is gamma V, TB. So, that's XA. So, now we have our two, we're going to highlight them here I've got that, and that. That's a form of the Lorentz transformation for the case where the event is occurring at xb equals zero. Okay? and as we set up the origins like that, which is standard procedure, to set up the origins like that. You know, if they're not set up like that, you can always, start your clocks at a proper time. Or you can adjust the clock, you can also adjust the clocks back and forth if you want, to get a zero point at that instant. To have everything match up and both Bob and Alice have their clocks at zero. Later on, of course, if Bob is moving then they're not going to be synchronizing more as we've talked about with the relativity of simult, simultaneity. But here's the Lorentz transformation equations. For this case, with a flash of light is occuring at xb equals zero, in other words, Bob says, "hey, I measured a flash of light". It was at xb equals zero and tb equals something or other and then Alice could immediatly say," OK, I know where that location is in my frame of reference". It's going to code at time ta equals gamma tb, so Bob tells me t sub b. Alice then calculates T sub A, got that? And also given T sub B, Alice calculates X sub B as gamma VT sub B. And, so again, that's, sort of a limited form of the Lorrentz transformation for this special case. What we're going to do in the next video then, is consider the case of, what happens if the flash of light is not here as Bob measures it. But it's out here some place, some general value of x sub b.

So, he'll have some value of xb for the event, is flashed, or whatever it happens to be. And the value of t sub b, and then we want formulae sort of like this, slightly more complicated, that Alice can use to figure out the time, T sub A, of that flash. Or event out here, and the location X sub A so that's coming up in the next video clip.