Onto Part 3 now of The Lorentz Transformation. Let's remind ourselves where we were. Again, the goal here, I brought the props, remember these, Alice and Bob here? So, Alice with her lattice of clocks and measuring system. Bob with his lattice of clocks and measuring system. In our example we set up where we were assuming Alice was stationary, Bob was moving to the right with respect to Alice. But of course from Bob's perspective, his frame of reference, he's stationary and Alice is moving to the left. And so the idea here is that if Bob measures some event, maybe out here that occurs at what clock, one, two, three, four, five, something like that, and on some time in his synchronized lattice of clocks. What is that going to be in Alice's frame of reference? What time and location with respect to her lattice of clocks? So that's the idea here, that's what we're after to make our lives simpler, if we can get a nice formula for this. And it'll also give us some more insights into how the special theory of relativity works. So we've gone through a couple of steps, and it got into this form right here, for our formulas involving X, v, and t sub B, in other words the X location of an event and the time of the event according to Bob's clocks and his measuring system. And then we'll get to t sub A, the time for Alice and X sub A, the location for Alice. And we'd found out that we got sort of the second terms here, but we still need to find out what G and M are. And so now this is where the algebra really comes in, this video clip, maybe when we have the most algebra, but we'll take it step by step. We'll try to get the big picture as well for those of you who like to do a little algebra, you can stop at a certain point and then try to work your way through it, see if you can get the same answer we get at the end, hopefully the correct answer. But here's what we're going to do. We're going to go back to the invariant interval equation. This is one of the big reasons why we derived it a while back. So remind ourselves what it is. It's C squared, for any two frames of reference, this quantity is invariant between those frames of reference. So given t sub B and X sub B, plug those numbers in here. And if another frame of reference, Alice in this case, if the difference between the frames of reference, if they're both inertial and there's some velocity, v, between them, then C squared tA squared- XA squared will all just be equal to the values with Bob's values put in there. So look at what we've got here. We've got these formulas over here, I want to find out what M and G are. And what would happen if I just took my value for t sub A there, my formula, and plugged it in right here. And took the value for X sub A and plugged it in right here, because then what would happen is I'd have XBs and tBs on both sides. The XAs would disappear. And ideally then, I'd also have a G and M in there and ideally, I'd be able to work it down and figure out what G and M are. So let's try that and see where we go with this. So we're going to have a lot of room here, we'll start over here. So I'm going to have C squared. And what's tA? tA is GxB. So GxB + gamma tB, all squared, okay? So I plugged that in there. C squared tA squared- XA squared, and so XA squared is (MxB + gamma vtB) squared =, this side stays the same, C squared tB squared- xB squared. Okay, so now we just sort of work the math. We work the algebra here. We're going to multiply this out, got this squared and got this squared. And let me just remind you about something, and for those of you who maybe haven't done algebra for a while or much of it. Remember when we have something like this, if we have just, we'll call it a + b quantity squared, if we have something plus something quantity squared and we multiply that out, that's equal to (a + b)(a + b). And so you get a times a is a squared, plus ba, and then an ab, so I actually get 2ab, and then plus the b squared. So if you dredge your mathematical memory, you may remember something like that. We're going to use that. Clearly we have sort of an a plus b quantity squared. We've got another a plus b quantity squared, and we'll see where we go with that, so I'll leave that up there for a minute to remind you of that. So let's multiply this out. So we've got a C squared. So from that form I'm going to have a G squared xB squared + the 2ab factor, so 2 times that times that, so + 2G, gamma, xBtB. I just sort of moved the terms around in there, which of course we can do. Plus gamma squared, tB squared. Okay, so that's what I've got for my first term here. Now let's multiply out this second term. And now we will have to make some more room here. Okay, so this is going to be minus, again, this squared, so I've got M squared xB squared + 2 times this times that, + 2M gamma v, XvtB, M gamma v XvtB. Take it step by step, it looks horrible perhaps, but again take it step by step and we'll get there. And plus then this thing squared, gamma squared, v squared tB squared. Gamma squared, vB squared, that's not vB squared, it's v squared, tB squared, Okay, so that's C squared times this, minus this thing squared. And it all equals C squared tB squared- xB squared there. Now, let's note something here. We've got some things that we can sort of pull together. We've got an xB squared term here, got an xB squared term over here. We've got an xBtB term here, we've got an xBtB term there. I've got a tB squared and a tB squared term there. So what we're going to do is pull those together because then we're going to look at this side and compare it to the other side and we'll actually get some nice results out of that. So let's pull all the tB squareds together here, so I've got a C squared, gamma squared, tB squared. So I've got a C squared, gamma squared, TB squared. That's one of my TB squared terms, right there. Got another one here. I've got minus gamma squared, v squared, TB squared. Minus gamma squared, v squared, TB squared. And by the way, this is the point where, for those of you who want to practice some algebra here, good place to stop and work through this and see where we end up. So, TB squared, TB squared terms. Now let's look for the xBtB terms, okay? So I've got one here. So I've got a C squared. Don't lose the C squared out here. C squared times 2 times G times gamma. So we'll write that as (2G) gamma C squared xBtB. That's this term right here. I've got another minus times 2M gamma vxBtB. So I've got- 2M gamma vxBtB. So that's my xBtB term here. And then the xB squared terms. I've got one here, and I've got one there, and I think we've got everything correctly here, so we've got C squared times G squared xB squared. So I've got plus C squared times G squared xB squared and then minus M squared xB squared here. Minus M squared xB squared there, and I think we've got all the terms here. We've got one, two, three, four, five, six terms total. One, two three, four, five, six, so it looks like we've got them all there. Hopefully we didn't make any silly mistakes there. And then, still over on this side, I've got c squared tB squared- xB squared. Okay, here's where the real nice part comes in, because I've got all this over here, okay? And note actually, let's simplify it one more step here. And then as I've got a tB squared term here, I've got tB squared there, tB squared there. So let's pull that out. So we're going to write this as (c squared gamma squared- gamma squared v squared)tB squared. And then, same thing here, I've got an xBtB here and an xBtB there so I'll pull that out and write this as 2G gamma c squared- 2M gamma v times xBtB, okay? And then, here I've got an xB squared and xB squared here. So I've got plus c squared, G squared- M squared, xB squared = c squared, tB squared- xB squared. Okay, now here's the nice part. On this side, note we have a tB squared term. Okay, we've got essentially something times tB squared, and then we've got something times xBtB, and then we've got plus something times an xB squared term here. And then on the other side of our equal sign, we've got a c squared tB squared. And then, plus a negative 1 times xB squared or just minus xB squared there. When you have something, a form like this, if you got a tB squared here term here, something times tB squared and over here you have a tB squared. Whatever's in front of this tB squared term has to equal this over here. Okay, so what this is telling me is that for this equality to hold on both sides, this thing right here, that right there, has to equal the c squared. because that's what's in front of the tB squared over there, over here what's in front of tB squared is this thing, so this thing equals c squared, and we're going to, we'll use that in just a minute. Let's look at the xB squared term, okay, so I've got an xB squared term here, I've got an xB squared term there, what that means for the quality to hold is. Whatever's in front here, this thing, this coefficient in front of the xB squared term has to equal the coefficient in front of the xB squared term over there, which actually is just negative 1. So I got a plus this xB squared, and I really have a plus negative 1 xB squared with that minus sign there. So that's another thing I can use. And then finally you say, what about this xBtB stuff in the middle here? I don't have any xBtB term over on the right-hand side. What that means is, again for the quality of hold is this whole thing here has to be 0. Because I have to have plus 0 times xBtB because I have essentially 0 times xBtB over there, I don't have any xBtB term. And therefore whatever this thing is here has to be equal to 0. So, let's write our conclusions down here, we'll keep that last part there. Remember, this is our starting point, a lot of algebra here, fair amount maybe. But the idea is fairly straightforward, we hope. And that is, in variant interval equation, which we derived before, our linear form of our two transformation equations with the G and M in there, because we don't know what those are yet. So we just took that, the tA and xA, plugged it in here, and then just cranked through some algebra. And this is where we've gotten to at the moment. So let's see. We just talk through it. We're going to write down the results there. Keep that last line. Okay, so what we've said here from this, you can say therefore. Therefore, for those of you who like math, you can do a mathematical symbol for therefore. Some of you may know that. So what do we say. Okay, this, the tB squared term has to equal c squared. So we have c squared, gamma squared- gamma squared v squared = c squared. C squared coefficient over there. Let's just work with that a minute, we'll get to the others in a minute, okay? On this side, we have a gamma squared in both terms, so we can do gamma squared times c squared- v squared = c squared. So I just pulled off the gamma squareds there. And then divide both sides here by the c squared- v squared, so I get c squared over c squared- v squared. It should be looking a little familiar to you now. Take the square root of both sides. Everything is positive so we don't have to worry about it. v is less than c, so this is positive down there and that's positive, so we have gamma equals c over, what am I doing here? So square root of c squared- v squared. Got lost there a second. Okay? I just took the square root. The square root of gamma squared is gamma. The square root of c squared is c. The square root of c squared- minus v squared, is square root of c squared minus v squared. You say, what's going on here? Gamma, is that gamma? Really? Well, it is actually. In fact, you can see it a little bit better, if before we took the square root, let's do this, because we did this actually, in an earlier video clip recently. So this is c squared over c squared one minus v squared over c squared like that, okay. And now, that's aligned c squared minus v squared is the same thing as this. because that c squared times one of c squared, c squared times v squared or c squared, is just v squared at the same as that and once I do that, we can cancel the c squareds and I'm left with one over one minus v squared over c squared equals gamma squared. Take the square root and lo and behold, yes it does equal our familiar Lorentz factor gamma. So the unfortunate part of that, is it really didn't tell us anything. Except, it did tell us that we haven't done anything wrong, at least for this term. If we got something, gamma equals something else, where it wasn't in our familiar gamma form, then we know, okay. Some place along the line in the algebra or even before that, we had made a mistake. But that confirms that we're on the right track, at least for this term here. So just confirms gamma is one over square of one minus v squared of c squared. So remember, this here is gamma squared. So gamma equals square root of that. So that's the first term. Now, let's look at one of the other terms here, so I'll erase this, get some room here. And let's take a look at, let's just do the. The x b, t v term here. So, what this tells us, as we've said before, is that this thing inside here, this coefficient has to equal zero because there's no s b t b term on the other side. So, what that is saying is two times g, remember our goal here is to find out what g and m are, two g gamma c squared minus two m gamma v has to be equal to zero for the equality to hold. [INAUDIBLE] Our originally Invariant interval equation we know that's true. At least we derived it. So what do we got here? Well we have a two on each side. We have a gamma on each side. So let's pull that out. So we've got two gamma times g c squared minus m v equals zero. And you may remember, when we have the product of two things. We've got, essentially, this, times this, equals zero. And when you have something like that, and gamma is clearly not zero. Either this has to be zero, or this has to be zero. Since two gamma is never equal to zero, all right, therefore this implies that g c squared minus m v has to equal zero. So what that means is, just do the other step here, g c squared therefore equals m v, pulling m v over there, add it to each side, and then from here, we get g equals m v over c squared. Okay. So, it's maybe a little helpful there. In other words, now we have a little equation involving g and m. If I could find one or the other, then I'd have the other one. So If I had in here I could find g, If I had g, I could find m. So, let's finally look at the last equation here and we have c squared, g squared minus m squared, that's my coefficient for the x p squared term. That has to equal negative one and so let's work with this a minute and we can write this in fact. I'd like to get rid of that negative one. That's the same as minus c squared g squared plus m squared equals one. Or m squared minus squared g squared equals one. Just flip those terms there. And so, what have we got here? Well, we've got m. Let's write it in terms of m here. So actually, let's come back up here, get a little room. So this was our g equation, we found here. Let's actually take g and just plug it in here, is really probably the simplest way to do it. So here's, what we've got. We've got m squared minus c squared, g squared equals one. We know from our middle coefficient that equals zero, that this has to be true. So this is m squared minus c squared. Times m v over c squared, squared, because I got a g squared here, equals one. And now we're getting someplace, because all we've got left is m and v and c in there. So just to do the steps here, we've got m squared minus c squared m squared v squared over c to the fourth, squaring everything inside there equals one. It looks a little messy, but let's see what we've got here. I've got an m squared and m squared there. Let's put in here m squared times one minus c squared squared over c to the fourth equals one. And we'll bring it up here, to get some more room here. Note, I could have done this already. But now, I've got a c squared here and a c to the fourth there. So the c squared disappears and that becomes just a c squared down there. So let's write that. We've got m squared times one, and just to be clear, and an m squared here and an m squared here, so I pulled out the m squareds and then I'm left with one, m squared times one is m squared, M squared times this term was that term, equals one still over here. So what do we here? I've got m squared one minus v squared over c squared. Well that looks nice because I recognize, anytime you get something with that form when you're doing when you're doing the theory of relativity. You say that looks good. Because that looks very close to gamma, right. Form, I can write that in terms of gamma, if I wanted to. Equals one. Okay. Looks even better because now, I can write this as I divide each side by this thing. This equals m squared over one, one minus v squared over c squared. Looking better all the time because now we take the square root of each side and that means we get m Equals one over square root of one minus v squared over c squared. And that is gamma, right. One over square root of [INAUDIBLE] equals c squared. It's gamma. Very nice news there. So now we have everything. M is gamma. So right there, that's gamma. G is just M v over c squared. M is gamma so G is now gamma v over c squared. G is gamma v over c squared over there. So let's write all of that down. So we worked through all this. Remember how we got this again, M variant to interval. We took our two equations that we developed so far, plugged them into the M variant interval equation, worked through the algebra here and set things equal to each other, worked through some more algebra and found out that M was simply gamma after all that work. And G was gamma v over c squared. So here's what we have. Oops, well erase that. M is this, just to remind ourselves there. So we have tA, the final result here or we gotta rewrite it in slightly different form. G was gamma v over c squared. So we got gamma v over c squared times XB plus gamma tB. And XA was M, which is just gamma, times XB plus gamma VtB. And those are the Lorentz transformation equations, named after Hendrik Antoon Lorentz, who later on in their careers became sort of a scientific father figure to Einstein. Einstein was younger than Lorentz, as we've mentioned before. Lorentz really extended Maxwell's theory of electromagnetism during the 1890s and into beginning of the 1900s. And working with Henri Poincare as well, very sophisticated theory. And he had come up with these equations in the context of that theory. Einstein came up with them, as we've seen, just from his two basic postulates involving, essentially, clocks and links, and so it was coming to them from a very different perspective. Although, many people at first sight, Einstein, you're just re-deriving what we already know here from Lorentz and Poincare as well. These Lorentz transformation equations. But because Einstein was coming from such a different perspective, they were renamed certainly for awhile, the Lorentz-Einstein transformation equations. And then as time went on, people realized that Einstein's approach really was the more fundamental approach to the matter. And yet still, we call them the Lorentz Transformations to honor Lorentz's contributions to this. And Einstein himself looked up to Lorentz a great deal. So here are the two equations. We want to re-write them in slightly different form just to make it easier to remember and also to bring out the difference and similarities to the Galilean transformation. I'm going to rewrite this as tA equals, knowing I've got a gamma in each term there. So I'm going to pull out the gamma. I'm going to put the tB term first, tB plus v over c squared x of B, okay. So I just sort of flipped the terms, I put the tB term first and I pulled out the gamma factor. And then XA, is again I can pull out the gamma here. So, gamma xB plus vtB there. So, first of all, especially the XA term here, first of all, these two equations, if you look at them, they both have gamma in front of them. Okay. That indicates special theory of relativity is going on here. And the form is similar. I've got an XB term plus a tB term, and for the t, I wrote the tB term first, because you've got the tB and XB and the second terms have the extra coefficient in front of them. For the XA equation here, it's v times tB because the units have to match. V is meters per second, for example. Length divided by time. This is time. So this gives us units of length, for example, meters. Here, it's the equation for time. And velocity is meters per second. We'll use the meters per second system, MKS system as it's known. C squared is a velocity squared, okay. So, that's going to give me, if I velocity here and a velocity squared down here, I get a velocity in the denominator and I've got meters here a length and so, if you do the cancellation there you get, essentially, a time factor of seconds in terms of the units there, and it all works out. Gamma, of course, being dimensionless. All right. So, the units check out. If they didn't check out, we would have made a mistake someplace along the way. So, that's the general form, and this is probably a nice form to memorize them in if you're going to be using them fairly frequently. And we're certainly will be referring to them as we go on throughout the course.