Einstein velocity addition formula derivation | Special relativity | Physics | Khan Academy

Let's say this is me and I am floating in space. My coordinate system, my frame of reference. We've seen it before; we'll call it the S frame of reference. Any space in any point in space-time, we give it X and Y coordinates.

And let's say that we have my friend. We've involved her in many other videos before. She is traveling with a relative velocity to me of V, so V right over there. Her frame of reference we call it the S Prime frame of reference. You can denote any event in space-time by the coordinates X Prime and Y Prime and CT Prime. Now, you could also do it as T Prime if you like, but we've been doing it as CT Prime so that we have similar units.

And let's say that we have a third character. Now this is going to get interesting. Let's say this third character is traveling with a velocity U in my frame of reference. So U is equal to change in X over change in time. If we know all of this information, let’s see if we can formulate a way. If we know what the change in X and change in time is, and we know what V is, if we can figure out what this velocity is going to be in the S Prime frame of reference—in this purple friend's frame of reference.

So what we want to figure out is what is—let me do it in that purple color—we want to figure out what is going to be the change in X Prime over change in T Prime. If we figure this out, then we know what will this velocity look like in the S Prime frame of reference.

Well, let’s just go back to the Lorentz Transformations. Let’s first think about what our change in X Prime is going to be. Well, our change in X Prime is going to be the Lorentz Factor times our change in X—change this in white—times our change in X minus beta times change in CT.

Let me close those parentheses. And then we want to divide that by our change in time. We want to divide it by our change in time, so change in time—or change in T Prime, I should say. Well, let me just go back to the Lorentz transformation. Let me write it the way that I'm used to writing it. I'm used to writing it as... I'm used to writing it as change in CT Prime, which is the same thing as C times the change in T Prime, is equal to the Lorentz Factor times... This is going to be C change in CT or I could write it as C change in T minus beta times change in X.

Once again, I could have written this as change in CT or I could write this as C times change in T because the C isn't changing, so we know that already. So if we wanted to solve for just change in T Prime, we could just divide both sides by C.

So let's do that. If we divide all of this by C, what do we get? And this is a form we've seen in other videos that you might recognize, and you might see this in some textbooks. Our change in T Prime is going to be equal to gamma times... Well, C times delta T over C is just going to be delta T. Delta T.

And then if you take so minus beta divided by C—well, let's just remind ourselves that beta is equal to V over C, so beta over C is V over C squared. So I could write this as minus V over C squared change in X. Change in X.

So change in T Prime, let’s write that now. That's going to be equal to... you have your gamma, your gamma times change in time in my coordinate system. So not the T Prime; just change in T. Change in T minus V over C squared change in X.

So immediately there's at least one simplification we can do. We can divide both the numerator and the denominator by gamma. I keep wanting to change, so we can divide the numerator and denominator by gamma. That simplifies a little bit.

And so we can write this as our change in X Prime over change in T Prime is equal to—in the numerator we have change in X—and actually let me just write out what beta is. So beta... actually let me write it over—I'll do it over here. Minus V over C. That's what beta is.

And actually, let me take the C out. If change in CT is the same thing as C times change in T, so times C delta T. Well, V over C times C is just going to be V, so let me write it that way. So this part, these cancel out. This is the same thing as change in X minus V change in T.

So let me write it just that way, simplify it. So minus V change in T—that's our numerator. And then our denominator, our denominator is change in T minus V over C squared change in X. So it looks like we're getting close here, but we don't have the change in X over change in T.

We have how much change in... how much of a change in X we have for a given change in T. So what we could do is we can divide the numerator and the denominator by delta T. So we could multiply the numerator by 1 over delta T and the denominator by 1 over delta T, which is equivalent to dividing the numerator and the denominator each by delta T.

And why am I doing that? Well, if I do that, this first term right over here is going to be delta X over delta T, which we know. And then we have minus V. Minus V, because delta T over delta T is just going to be one.

And then our denominator, delta T over delta T is just 1, and then we're going to have minus V over C squared and then times delta X over delta T. Delta X divided by delta T.

So this is cool! We've just been able to figure out what our velocity in the primed coordinate system in the S Prime frame of reference is in terms of the relative velocity between the S Prime frame of reference and my frame of reference. Delta X over delta T—the velocity in my frame of reference—and, well, we know all of this other stuff too.

This is just delta X over delta T, and this is V. Or we could even write it like this: we said delta X over delta T is going to be equal to U. So it's equal to U minus V over 1 minus delta X over delta T is just U, so this is just U here.

So it's U times V, or V times U. All of that over c squared. So this is a really, really, really useful derivation, which we're going to apply some numbers to in the next video, so we can appreciate, uh, kind of how fun this is on some level.