Why is rapidity additive?

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manpreet Tuteehub forum best answer Best Answer 2 years ago

With the rapidity ϕϕ defined so that vc=tanhϕvc=tanh⁡ϕ, say you have 3 parallel moving reference frames SSSS′ and S′′S″ with a constant but different velocity/rapidity.

If the rapidity of SS′ in relation to SS is ϕ1ϕ1,
and the rapidity of S′′S″ in relation to SS′ is ϕ2ϕ2,
then the rapidity of S′′S″ in relation to SS is ϕ1+ϕ2ϕ1+ϕ2

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manpreet 2 years ago

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=1tanh⁡ϕc=1 is the same in all frames of reference.

Consider this case:

You observe a frame of reference SS′ moving with tanhϕ1=vtanh⁡ϕ1=v with respect to SS followed by a light beam with tanhϕc=1tanh⁡ϕc=1. we know the rapidity of light in the SS′ frame is the same as in SS. Given this fact we'd like to find out what is the relation f(ϕ1,ϕc)f(ϕ1,ϕc) between the old and new rapidities in the new frame SS′ such that:

tanh(f(ϕ1,ϕc))=1tanh⁡(f(ϕ1,ϕc))=1

 

2) The second constraint is that the rapidity of an object at rest is zero. So if there's an object moving with velocity vv in 

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