Now let's spend a few minutes here exploring Lorentz Transformation. To start, let's remind ourselves where we were and what we ended up with. The goal was, if we have again Alice in this case being stationary, her frame of reference. Bob going by at some velocity v, of course Bob and his frame of reference, he's stationary. He sees Alice moving in the other direction with velocity v. And our goal was to have a simple set of formulas such that give coordinates of an event in Bob's frame of reference Xb, Yb, Zb and Tb we can easily transform those into the equivalent coordinates for that given event analysis frame of reference. And obviously we're not going to redo the derivation here, but remind you where it came from. We had an argument such that the form of the equations had to be linear. And what that meant was that we couldn't have any, given (xB, yB, zB, tB), we couldn't have any xB squares or t squareds or square roots or things like that. The reason being, the argument we put forward, is that if you have, let's consider time okay, so say ticks of Bob's clock, they're equally spaced. One second, two seconds, three seconds, four seconds. Now when you transform time into Alice's frame of reference, as we know there's this time dilation effect. So potentially the distances between ticks can change, but we should have uniformity there, in other words if Bob's ticks, just as example are one apart, one second or whatever unit of time we are using, one, one, one, one. Then Alice could be, between her ticks 1.5, 1.5, 1.5, but we can't have something like where Bob is 1, 1, 1, 1. Alice is 1.5, 1.8 between the next tick, 2.3 between that tick, and so on and so forth. So the distance between ticks when you do the transformation, has to be the same. And you could make a similar argument for the measuring system for lengths, that you can have differences between Bob's length scale and Alice's in terms of a length contraction effect as we've seen. But you can't have non-uniformity where Bob's measuring system is uniform and then it transforms into the non-uniform system. In Alice's, because you get inconsistent results that way. So that was sort of a hand-waving argument, not a rigorous argument. But the idea was that, then we had to have something like this form, where G, H, M and N were unknowns, we would like to find out what those were. And so we took these equations, and plugged them into our invariant interval equation, which we had previously derived for the situation where you have essentially two frames of reference. Or really, multiple frames of reference. And very interval, is invariant. Is constant for all frames of reference. So we knew that was true. Plugged these in here, did a lot of algebra. And got these results here, that t a, given x b and t b, t a was gamma v over c squared. X b + a gamma t b where gamma of course was our Lorenz factor. And x sub a gamma times x sub b plus gamma b sub t b. We wrote those, we just moved things around a little bit. And put them in this form. For t a and xA, it's a little bit easier to memorize them in that form. tA equals gamma t v + v over c squared xB. xA equals gamma xB, so notice this sort of parallelism here between t A, tB, xA, xB. Slight difference in the second term in each. I've got v tB over here in the xA. And v over c squared x(B). Part of the reason for that is the units have to work out. Here we've gotta have units of length and a velocity times the time gives us units of length. Here we have to have units of time throughout the whole thing. So a velocity divided by another velocity squared times a length, if you work that out, will give us units of time. One thing we didn't mention, last time we were doing the derivation, we were concentrating really on XB and XA, tA and tB, was what happened to you, YB, ZA, and ZB. Well, remember last week when we did the analysis and showed that You don't have any width contraction effects or height contraction effects. And therefore in the perpendicular dimensions, if our direction of motion is in the x direction then the y and z directions, there are no length contraction effects. Width contraction, height contraction, radial contraction effects. And so those equations are very simple if we are doing something in three dimensions. And of course normally we're just working in one dimension for simplicity. So those are our Lorentz Transformation equations. What we want to do now is rewrite them slightly, and then investigate them a little bit to see what the math is telling us here. It's also a good idea to not just say okay, there are the equations, let's go use them. But to understand them better, find out what they're telling us. Whether they're consistent, maybe we made a mistake someplace, and so on, and so forth. So to do that, let's rewrite them up here in slightly different form because these are a's and b's and sometimes its easier to forget who's moving here, who's stationary and so on and so forth. And our derivation we assume, this was from the perspective of Alice being stationary. Bob moving with velocity v. We'd like to have some reserve equations for but what about Bob's perspective? If he considers himself stationary then Alice is moving the other direction with velocity v,- v in the essence of the negative direction. And therefore what will the equations look like In that case. First of all, rewrite these, and so then, instead of a for Alice, we're going to have rest for the frame of reference that's considered to be at rest. So, we'll have something like this. We'll say t. Let's get a better pen here. T rest = gamma, gamma still being the Lorentz factor, tb is our moving frame. So we just say t moving plus v over c squared x. Moving okay. So in this case we refer to Bob. And then x rest equals gamma x in the moving frame with respect to Alice. Okay Alice being assumed at rest here. Plus the t moving, not rest. Okay, so we've got that. Now another way we could have labelled these, and you want to choose a way that makes sense to you, often will say we've got the rocket frame and the lab frame. And in many textbooks you'll see it that way as well. So if you prefer that, you can certainly use it. So instead of the rest frame, that would be the lab frame. And then the moving frame is the rocket frame. And sometimes that's even a little easier because you can visualize, you can picture. Okay I got the rocket and I've got the lab observer, Alice in this case in the lab. So you can either use lab and rocket or rest and moving there. Well what we want to do in part one of exploring this is focus on this equation here and just see what happens when we put in some different values here. See if the results make sense based on our previous understanding of what we've been doing with things like length contraction, and so on and so forth. So, to start off, what happens if, let's just say, that v = 0. What if the velocity is 0 here? Well, we know that if velocity is 0, then gamma, let's remind ourselves, gamma. 1 over the square root of 1 minus v squared over c squared. So if velocity is 0, gamma is 1. So we know gamma equals 1 in that case. So what does this equation become? Well we get x rest, Alice's perspective Gamma is 1 here equals x moving and v is 0, so this is 0 as well. So we just get x rest equals x moving. In other words If we were thinking about again it lies in clocks here, as the underlying foundation of our observations. Alice has her lattice of clocks and that's her measuring system for length and time. Bob would have his as well. If via 0, they're not moving with respect to each other. So any measurement that Bob makes here. Here Bob says okay. And his moving frame, I see something at xb or in this case we'll call it x moving frame at 23.8. Well, since our space ships are together, Alice would also see that on her system at 23.8. So this is what that's telling us and we hope we get this answer or else something went wrong along the way. So that's one simple check we can do. What happens if, let's just get some room here. So that was our first test case, if that made sense. Now let's say, let's not let vb equal to 0, but let's make it really small compared to the speed of light. Sort of everyday type of velocity. So if v is much less than c. Then gamma is approximately equal to 1, right? If v is much less than c here. This is essentially 1- pretty close to 0. 1 divided by the square root of pretty close to 1, is going to equal 1. So gamma is approximately equal to 1. And so what do we get then? So we get Xrest = gamma,this is approximately equal to 1 here and so it becomes Xmoving in the moving frame. And by the way this again the terminology here this does not mean x is moving. This belongs an x coordinate, an x measurement in the moving frame. So just fix that in your mind, don't get confused about that. So x rest = x moving here, plus a velocity. Okay it's going to be non 0 it's going to be much less than c but it's still non 0 times t in the moving frame. In other words, on one of Bob's clock in the moving frame. Well what does this remind ourselves of, hopefully it reminds us of the Galilean Transformation especially because if we go back up here for a minute. Look at the t equation, if v is much less than c, and gamma = 1. Look what we have here. Gamma, of course, being 1. If v is much less than c, this is a very small number. V divided by c squared. Not just v divided by c, it's v divided by c squared. That's a very small number and essentially we just have time in the rest frame equals time in the moving frame. And therefore, this is just t. And so that would make this a Galilean Transformation, where t is the same for all observers, all frames of reference. And you just have your basic x in the rest frame equals the x value in the moving frame plus v times t. Essentially, this shows how. For Bob has moved along. And so if he measures something again, say at 23 out here on his lattice of clocks, is measuring system. But he's also moved a certain amount, then the distance to Alice, the x in the rest frame, is the amount Bob has moved, plus the amount he measures out to 0.23 on his measuring system. So basic Galilean Transformation there. So that checks as well because we know that works in our everyday experience when velocities are much less than the speed of light. Okay, so now though what happens when we have velocity being large? Let v be large obviously not equal to c but up there. Some place, so let v b large, then what do we have, well we essentially just have this equation. We can't do any simplifications of it, but let's split it apart here and take a look at it, so we've got x in the rest frame. X in the rest frame = gamma times x in the moving frame plus gamma times v times time, as measured in the moving frame. In Bob's frame. There all I did was multiply that by gamma and then that frame by gamma, to split it apart here. So essentially this is very similar to the Galilean Transformation which we were just talking about. In other words for Alice here Bob is going to measure some event at some x value in his frame. Maybe it's at 23 again on his lattice of clocks. And his measuring system, out there. So, where is that going to occur, on Alice's lattice of clocks? What's going to occur, again, the general idea is, how far has Bob moved, what time is is the measurement being made. And then plus, whatever he's measuring out here. We'll assume it's a positive value. So we're assuming he measures it at 23 out here, but in the meantime he's also moved over here. And so the distance for Alice would be that distance plus that distance. So this looks similar to the Galilean Transformation. Same concept, except we've got these gamma factors in here. And in fact if Bob is measuring this, at his time t =0 and we also assume that they're right together at t = 0. Bob is just passing by Alice at t = 0. So if we're measuring this at t = 0 for Bob. T =0 then this term goes to 0 and look what we're left with here. That's our length contraction equation. In other words what that's saying is if Bob says, okay right at T = 0 on my clock. I measure out and I see something at 23 out here. Alice, remember length in the moving frame is going to be contracted. So Alice here will see x moving as less than her value for whatever the distance is 23 or whatever. So another way of saying this, the way we wrote the equation before, was x in a moving frame equals 1 over gamma. X in the rest frame is stationary, friend. Just to emphasize because gamma is always less than, is always greater than one, greater than equal one, of course. That means the moving value as measure by Alice in this case is less than, you take x rest, multiply by maybe gamma 2, 1/2 and then you get a smaller value. So this is the length effect. That's built into the Lorentz Transformation. As it better be, if it isn't then we have got something really wrong here. Either our original conclusion is incorrect or our Lorentz Transformation derivation was incorrect. So what we see here is that, first of all we can get the Galilean Transformation out of this; that equation for sort of every day velocities much less than the speed of light. We also see that when v is large that we get the length contracting effect coming out of it, where again length contracting in its simple case is simply when Bob measures something at t =0 so he is there, Alice is there, Bob measures a certain length. And then Alice gets that length as well and you get the length transfraction going on there. So that's in there a well. And so that's exploring the x equation here. In our next video clip, part two, we want to do a little bit of exploration of the time equation.