Lorentz transformation for change in coordinates | Physics | Khan Academy

We spent several videos now getting familiar with the Laurence Transformations. What I want to do now, instead of thinking of what X Prime and CT Prime is in terms of X and CT, I'm going to think about what is the change in X Prime and the change in CT Prime going to be in terms of change in X and change in CT.

We'll see it's just going to involve some fairly straightforward algebraic manipulation. So let's think about it. Change in X Prime is going to be X Prime final minus X Prime initial. Well, X Prime final, let me just pick a suitable color for that. X Prime final is going to be gamma times X final minus beta times CT final.

All I did is I used this formula up here. If I want to figure out my final X Prime, well, I'm just going to think of my final X and my final CT, so that's that. And from that, I am going to subtract the initial X Prime. Well, X Prime initial is just going to be—get another color here—X Prime initial is just going to be Laurence Factor gamma times X initial minus beta times CT initial.

So now let's see, we can factor out the gamma. So this is going to be equal to, and I'll do it in my color for gamma. If we factor out the gamma, we're going to get gamma times, we're going to have X final—let me do this in a. So we have—let me do that in white, actually—we're going to have X final; and then if we distribute this negative sign, minus X initial.

And then let's see, if we distribute this negative sign, well I don't want to skip too many steps here, so that's that. And then we're going to have negative, we're going to have negative beta CT final negative beta CT final, and then we have plus; we distribute this negative plus beta CT initial plus beta CT initial.

And so what can we do here? Well, that's just going to be change in X. So this piece right here is just changing X, so I can rewrite this as being equal to gamma times change in X.

Now let me subtract, let me just subtract. Let me take, let me take out factor out a negative beta. So I'll say minus beta times—well, then you're going to have CT final minus CT initial.

And well, what's CT final minus CT initial? Actually, I'm not going to skip any. I think I'm skipping too many steps already. Well, that's just going to be change in CT. So we get this is all going to be equal to gamma our Loren Factor times change in X minus beta times change in CT.

And since C isn't changing, you could also use C times change in T either way. So there you have it. Notice it takes almost the exact same form. X Prime is equal to gamma times X minus beta CT, and change in X Prime is going to be gamma times change in X minus beta times change in CT.

And I'm not going to do it in this video, but you can make the exact same algebraic argument for your change in CT Prime, as you'll see. I encourage you to do this on your own. Change in CT Prime, which you could also view as—since C isn't changing—C times delta T Prime, these are equivalent, is going to be equal to exactly what you would imagine.

It's going to be gamma times change in CT minus beta times change in X. I encourage you, right after this video, actually do this one too. Just hey, delta X Prime is going to be X Prime final minus X Prime initial, and then do what I just did here—just a little bit of algebraic manipulation.

And you can make the same exact argument over here to get to the result that I just wrote down. You say, well, our change in CT Prime is going to be CT Prime final minus CT Prime initial, and then you can substitute with this, do a little bit of algebraic manipulation, and you'll get that right over there.

And the whole reason I'm doing this is, well, now we can think in terms of change in the coordinates, which will allow us to think about what velocities would be in the different frames of reference, which is going to be pretty neat.