Alright here's a very crude heuristic(definitely not a f="https://forum.tuteehub.com/tag/proof">proof) on why(without using velocity composition) rapidity is additive.
Impose these two constraints:
1) the constancy of speed of light in all frames. This means that the rapidity of light tanhϕc=1 is the same in all frames of reference.
Consider this case:
You observe a frame of reference S′ moving with tanhϕ1=v with respect to S followed by a light beam with tanhϕc=1. we know the rapidity of light in the S′ frame is the same as in S. Given this fact we'd like to find out what is the relation f(ϕ1,ϕc) between the old and new rapidities in the new frame S′ such that:
2) The second constraint is that the rapidity of an object at rest is zero. So if there's an object moving with velocity v in
manpreet![Tuteehub forum best answer]()
Best Answer
2 years ago
With the rapidity ϕϕ defined so that vc=tanhϕvc=tanhϕ, say you have 3 parallel moving reference frames SS, S′S′ and S′′S″ with a constant but different velocity/rapidity.
If the rapidity of S′S′ in relation to SS is ϕ1ϕ1,
and the rapidity of S′′S″ in relation to S′S′ is ϕ2ϕ2,
then the rapidity of S′′S″ in relation to SS is ϕ1+ϕ2ϕ1+ϕ2