# Composition of 2 Boosts in Special Relativity

1. Aug 3, 2011

Most of what I've learned about Special Relativity is self taught using Google and Wikipedia because I'm still in high school, so forgive me if I'm saying something wrong.
Does anyone have the exact equation for the composition of two boosts, without the rotation that's induced by combining two boosts (according to what I've read, a boost followed by a boost is not a pure boost but a boost followed by or preceded by a rotation, which makes sense to me because of effects like relativistic aberration). The article mentioned gyrovector spaces being used to compose two boosts into a pure boost, but I can't find any information on gyrovector spaces. Would anyone mind explaining this to me?

2. Aug 4, 2011

### mathfeel

Two boosts in the same direction obvious equals to just a single boost in the same direction. Although, the relationship between the boost speed is not simply: $v \neq v_1 + v_2$.

Any general Lorentz transformation is a rotation + a boost: you first orient your axis along the direction of the moving object, then boost into its frame.

3. Aug 4, 2011

### JDoolin

I was pondering something like this in posts 12 and 13 of https://www.physicsforums.com/showthread.php?t=430956" From what I am able to gather, neither rotations nor boosts are commutative. But two rotations amount to a single rotation in another direction. Likewise, two boosts amount to a single boost in another direction.

If you have motors powering rotations on different axes, it appears to be rotating along several axes at once, but in reality, at any given time, the central object is only rotating along one axis. (that is, unless my conclusion in post 13 from the other thread is somehow flawed.)

Last edited by a moderator: Apr 26, 2017
4. Aug 4, 2011

### JDoolin

Equation 11.98 in Classical Electrodynamics by J.D. Jackson (third edition) has three boosts. But I'm not in the mood to type the whole thing in right now.

It looks commutative, but I think in the following sense: If the rapidities are figured in the original reference frame, then they can be applied in any order (right, forward, up). But if the rapidities are figured in the reference frame of the body which is accelerating, in order (right, forward, up) I don't think the operations are commutative.

For instance, if you do a rapidity change of (right 5, forward 5, up 5) That is effectively rotating to point a certain direction, and then accelerating that way in one boost. But if you consecutively execute rapidity changes (right 5) then (forward 5), then (up 5) in your space-ship, this will have a significantly diferent result

5. Aug 4, 2011

### Fredrik

Staff Emeritus
I strongly doubt that learning about gyrovectors will make it easier to learn special relativity, but if you're interested just out of curiosity, I would guess that this book is the place to start. I haven't read it myself.