I know that two consecutive Lorentz Boosts in different directions produce a rotation and therefore Lorentz Boosts don't form a group. But, my intuition tells me that, Lorentz Boosts in the same direction should form a group. Two boosts along the x axis should produce another boost along the x axis. Is that correct?
Asked
Active
Viewed 2,076 times
2 Answers
9
Yes, it is a one-dimensional subgroup generated by exponentiating an infinitesimal boost. Every one dimensional exponentiation of a generator forms an abelian group, because $e^{aG} e^{bG} = e^{(a+b)G}$, there is nothing to not commute. This result is the addition of velocities, you can explicity check that this is associative (it is always manifestly commutative).
1
Firstly, an excellent question. Never considered this before.
As has already been said, combining boosts along difference directions clearly doesn't form a group as the closure axiom is not met.
However, Lorentz boosts are nothing more than (hyperbolic) rotations in Minkowski space. So I think the set of boosts along the same axis should form a rotation group. I hope someone else can explain the reasoning without resorting to generators, as they're not my strong point.
See this YouTube video for some basic derivation.