So, we come now to the Lorentz Transformation, so let's set this up. Got a lot on the board here, we're going to go through it as usual, bit by bit, but I want to expedite matters and get some of the key things up here so I'm not just wasting time, as it were, with writing on here, we'll have a lot of, this is probably the most mathematical series of video clips we'll do. Again, it's mostly just algebra. It takes a little while to get through it. So we'll break it up into several video clips and make case slowly as usual. And for those of you who, 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 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 a lattice of his clocks, all synchronized. Remember, in Bob's frame of reference, he is 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 that direction. At the start, when Bob passes right by Alice, we'll assume that they start their clocks at that point, and so Ta T sub a equals zero, Tb equals zero. And we'll also have the origin for their measuring systems also to be 0 at that point. So everything matches up there. We'll get on to this in a little bit, this later at time t sub b where we get a flash of light going on. But here's the basic question we want to try to answer and that is, given the coordinates, spacetime coordinates of an event involves frame of reference. So, XP, YB, ZB and sometime 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 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 to work through examples we've done before but we don't want to have to do that every single time to work through the details. So, it's 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 velocity and 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 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 0 at his X sub B equals 0, we know Alice's measurements is going to be farther on in the positive direction because that's the direction Bob is moving with respect to Alice. And so he'll be whatever position Bob measures something at, plus the essentially the distance traveled by Bob in that given amount of time until a c 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 have ignored them in most of our examples, but to be complete, you can put those in here. And a key thing for the Galilean transformation is time was the same in both references. Now we know, we'll get over to the details of this in a minute, but we know right away that from Einsteins two postulates, we derive the fact that time dilation occurs and length contracts as well. So we know that this thing in particular, 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, 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, we don't notice it, but if you have very precise clocks, it is possible to measure that difference. So we know that the Galilean transformation can't work. We'd like to have something like this that works with special relativity. So that's our goal here. We'll get back to this stuff over here in a minute, but let's set up situation for Bob here and Alice. And what we want to do is later at a time tB, so they pass each other at time t = 0 essentially. And then later at some time tB, according to Bob's clock, he sees a flash of light right at his cockpit. So, in another words, that is occurring at xB=0, because remember he's carrying along his lattice of 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 you see some flash of light right next to his cockpit, the x location for him, xb would be 0 at that point. And we actually could then figure out where Alice sees that flash of light and 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 would be our time dilation equation, tA equals gamma tB. Remember, Alice is observing Bob's clock moving and therefore, time dilation will occur for Alice observing Bob's clock. Alice will see Bob's clock taking more slowly. And running more slowly, right? That's the whole idea of time dilation. Bob does not 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 Lorentz factor gamma involved in there. And of course gamma 1 over square 1- 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. Because remember in 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 light motion going up and down, the light pulse going up and down, is occurring at the same place all the time, Bob's got his clock right next to him there. It's Alice observing it of course, moving along, so that as Alice sees it that moving here, will squeeze us in. We get the familiar triangle effect. So Alice sees at going like this, but to Bob, all this is happening at X of B equals zero. So, our first time dilation equation really assumes that everything is happening at X of B equals zero for the person who is moving. And Lorentz Transformation we want this to be more general than that because again back here at the flashlight example, if the flashlight occurs right at xb equals zero at Bob's clock could there where he is. Then fine we can use our basic time relation equation. But he might see the flashlight out here some place 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 Lorentz transformation will have some equations here that in the case for xb = 0, when whatever is occurring the flash, 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 relationship. Now, we also know, this is a reminder, the invariant interval equation. And this is true in general. And so, this will be very useful for us To derive the Lorentz Transformation equations that want here. And 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, it doesn't appear out of thin air Or anything like that. But the 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. But let's just do the flash of light first. At Bob's location there. So, that's where we're going to head here and 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 some time tB, flash of light and we know therefore, 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 travelling 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 Bon travelled on. And so we can also say, you know if we say just to indicate. This is the location the flash to Bob, the location the flash to Alice is simply going to be the velocity, relative velocity Times whatever time 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 on one side and Alice's coordinates on the other. And so, Bob measures the flash at time TB, we'd like to know if somehow write TA in terms of TB Now 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 = 0 for Bob, sitting right next to him there. And therefore, we can just actually put this in here. Ta = gamma tb so this is going to equal Just put the gamma tb and I'll put the gamma in front because we often like right, the gamma was on front, gamma vi tb, okay? So That gives us some good information there. Therefore, if the flash of light occurs at xB equals 0 at some time tB, so Bob records some time tB on his clocks, then I can find out the x coordinate in Alice's frame of reference for that flash. It's just gamma times v times t sub B. So if Bob gives me that answer or I take a photograph and look at his clock or absorb 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 got c squared, TA squared- XA squared equals c squared. TB squared Minus xb squared, that's our invariant interval equation. That's good for anything, basically, given, ta, xa, and tb, and xb, for a given event, or a distance between two events. In this case, the distance between the events will be our origin point when they pass each other, and then, awhile 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 applying it to the flash of light right here, that flash of light occurs at 0. So, that means this thing right here Is zero. We don't have to worry about that. And so we're left ta xa and tb, of course c being 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. because it's just the velocity above times how much time elapsed on her lattice of clocks. And let's plug this in here for that. We're going to get c squared, ta squared minus v Ta squared, just spot you that in, equals c squared tb squared and the advantage of this formulas is that if you set it up correctly, you sort of cling though it here and note we will do every step here, c squared ta squared minus just becomes V squared, tA squared equals tB squared. And the 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 v squared equals c squared tB squared. And now we're going to bring it up here so we have some 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 V squared. So, I can write this as tA squared equals C squared Over c squared minus v squared tB squared. You see what I did there? I just took this and going to put over on other side of the denominator if I need to side by this. So, I got 1 here and c squared over, c squared minus v 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 we get here in a minute. Pause if you want to do that. And see what our final answer's going to be here. It's going to be very familiar. Let's do this. Let's factor out a c squared. In the bottom, so I can write c squared times 1- v squared over c squared time tB squared. Okay, the reason I did that, well, first of all, I can do that. I've got c squared- v squared so I pull off the c squared. And I've got b squared over c squared here. So this and this, they're the same things. The reason I did that is you may see by now is I can cancel my c squareds there. And so where does that leave us? It leaves us with equation like this, we get tA squared = 1/1- v squared/c squared, tB squared. Now we take the square root of each side. All our tAs here are soon to be positive. We're looking at positive time, so we don't have to worry about taking square roots plus or minus signs, as some of you may remember with that. So we get tA = 1/the square root 1- v squared/c squared tB. In other words, square of tA squared is tA, square root of tB squared is tB, square root of this is this. And this should look familiar, I hope, right. This is gamma, so really what we've just shown is tA = gamma tB. Using, we started with this equation, with their invariant interval equation and the fact that we're looking at flashlight at xB = 0. Because remember, so we plug 0 in here and then work through the algebra and lo and behold, what do we get? We get our time dilation equation, tA = gamma tB. So that's good news because if we didn't get that, we would have made a mistake some place along the way, either here, or perhaps in our earlier derivation of the time dilation equation using the light clock. Okay, so therefore, he said, okay, 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 a photo flash, our old photo principle idea, take a photo of that, could see what's on Bob's clock. And on her lattice of the clocks, that was at that location as well, took a photograph of her clock, this is what would be the reading on her clock. It'd be whatever was on Bob's clock times the gamma factor times tA. And of course, gamma is always greater and equal to 1, and so you get the time dilation effect there. So given t sub B then, we know what t sub A is. But what about xA? What's the x location in terms of Alice's lattice? Well, I just erased them, and then I go, but remember we had this. We had xA = v(tA). Now this is just the velocity of Bob times the elapsed time on Alice's clocks as she's watching Bob go by with velocity v. And therefore, this is just tA = gamma tB, and so we can plug tA into here, and we get, this is gamma vtB, so that's xA. So now we have our two, highlight them here, we've got that and that. That's a form of the Lorentz transformation for the case where the event is occurring and xB equals 0, okay. And as we set up with the origins like that, which is standard procedure to set up the origins like that. If they're not set up like that, you can start your clock to the proper time, or you can adjust the clock. You can also adjust the clock back and forth if you want to get a 0 point at that instant, to have everything match up, and both Bob and Alice have their clocks at 0. Later on, of course, if Bob is moving, then they are not going to be synchronizing more as we talked about with the relativity of simultaneity. But here is the Lorentz transformation equations for this case, where the flash of light is occurring xB = 0. In other words, Bob says hey, I measured a flash of light, it was at xB = 0 and tB equals something or other. And then Alice could immediately say, okay, I know where that location is in my frame of reference. It's going to occur at time tA = 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 A as gamma v, t sub B. And so again, that's a limited form of Lorentz transformation for the special case. What we are going to do in the next video then, is consider the case of what happens if a flash of light is not here as Bob measures it, but it's out here someplace, at some general value of x sub B. He'll have some value of xB for the event, this flash or whatever it happens to be, and the value of t sub B. And then we want formulas sort of like this, it'll be 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.