Finding an in-between frame of reference | Special relativity | Physics | Khan Academy

Let's say I'm person A here in my ship, traveling through the universe at a constant velocity. So that is person A right over there. Let me write it a little bit bigger: person A.

And let's say that I have a friend, person B, and they are in another ship. In my frame of reference, so this is person B in my frame of reference, they are traveling. Their velocity vector looks like this: they are traveling at 8/10 of the speed of light, 0.8 C.

And so once again, this is all given in A's frame of reference, in my frame of reference. Now the question I have for you is, surely there must be some third party, let's call them C for convenience. Surely there must be some third party or some third party frame of reference.

We could imagine it might be someone in some ship where their velocity in my frame of reference is in between being stationary and traveling 0.8 times the velocity of light. Even more, from their frame of reference, A and B should be leaving or going outward from them at the same velocity.

So what am I talking about? This is where we're going to, in this column I'll write the frame of reference. So this first row is A's frame of reference that we just described, and C is going to be going with some velocity V away from A. Some velocity V, we haven't figured it out yet.

Now let's think about C's frame of reference. So frame of reference C, so C. In C's frame of reference, A will look like it’s moving to the left with a velocity of negative V, or its magnitude is the same but just in the other direction, and B will also be moving away with velocity V.

So B right over here is going to be moving away with velocity V. This is a really interesting question. Can we figure out what V is going to be? If we were dealing with the Galilean world, you might say, "Well, V is just going to be halfway in between these two things."

If we were just on the highway in a Galilean or Newtonian world, and B is going 80 mph and A is stationary, then if C goes halfway, if C is going 40 mph, then from C's point of view it looks like A is going backward at 40 mph, and it would look like B is going forward at 40 mph.

But we know by now that we aren’t living in a Newtonian or Galilean Universe; we’re living in one defined by special relativity. So I encourage you to pause the video and figure out what this in-between frame of reference, what its velocity needs to be relative to A.

And I'll give you a hint: it's going to involve the Einstein velocity addition formula. So let's work through this together. I'm just going to write down the Einstein velocity addition formula.

So it tells us that the change in X, ΔX', all change in X' with respect to T' is equal to U - V over 1 - UV over C². Now let's think about how we might apply it. The trick here is to really think about it from C's frame of reference.

If you think about it from C's frame of reference, you could say that V, right over there—let me do this in a different color—you could say that V is the velocity that A is moving away from C at. So velocity A is moving from C, and then you could say that U is the velocity that B is moving from C.

Remember, we’re dealing in C's frame of reference, so velocity that B is moving from C. And actually, let me make everything uppercase. This should be uppercase A and this should be uppercase C.

In that case, what is ΔX' over ΔT'? Well, that would be the velocity that B is moving away from A in A's frame of reference. I know this can get a little confusing, but I really want you to pause it, watch it in slow-mo, and really think about what we’re doing here.

I know I started this video in A's frame of reference, and this is really the trick of the problem. I'm now shifting over to C's frame of reference. I'm like, "Okay, V is the velocity A is moving away from C, B is a velocity, or U is the velocity B is moving from C."

We can view ΔX' over ΔT' as the velocity that B is moving away from A in A's frame of reference. So B moving from A in A's frame of reference, both of these are in C's reference.

Let me write that down: in C's reference, in C's reference. I really want you to think about this; this is a little confusing, but hopefully this helps you appreciate how this Einstein addition of velocity can be valuable.

Now we can substitute what we know. We know the velocity B is moving away from A; it is 0.8 C, so we can write 0.8 C is going to be equal to, is going to be equal to U, the velocity B is moving from C in C's frame of reference.

Well, we say that’s just going to be V; that’s going to be a positive V. So that right over there is going to be a positive V. Let me do it in that same color; it's going to be a positive V.

Then from that, we are going to subtract the velocity A is moving from C in C's frame of reference, so A is moving from C with the velocity negative V. I know it's kind of confusing to replace a V with a negative V, but this is the generalized formula while this is the actual value that we’re using in this case.

So minus negative V, all of that over 1 minus the velocity U times the velocity V, well that’s just going to be V times negative V, or negative V². So I’ll just write that as V² over C².

Just like that, we have set it up so we can solve for V. The key realization is that we said, "Okay, there must be some spaceship C that defines a frame of reference where A and B are moving away from it with velocities of equal magnitude."

We use that information to go into C's frame of reference and use the Einstein velocity addition formula. Instead of knowing what these are and then solving for this, we know what this is, and we’re assuming that these two have the same magnitude, and we’re able to solve for V.

So let’s do that right now. In fact, I will do that in the next video so that we have enough time. I encourage you to solve for V on your own before you watch the next video.