Magnitude of relative velocity between two frames in special relativity

Physics Asked by Darkenin on December 3, 2020

It seems to be taken trivially, and it is indeed intuitive, that the relative velocity $v$ of some frame $O$ with respect to $O’$ is the same in magnitude to the relative velocity of $O’$ with respect to $O$. Is there any way to prove this? I’ve seen some tricks with rotating the relative motion axis by 180 degrees, but i’m not sure I get that.

One Answer

It's a symmetry argument.

There is no absolute difference between $O$ and $O'$.

But if the speed of $O$ as seen from $O'$ is not the same as the speed of $O'$ seen from $O$, $|v'| ne |v|$, then it must be larger or smaller: $|v'|>|v|$ or $|v|>|v'|$. That would give an absolute difference between the two frames, so it can't happen and the speeds, $|v|$ and $|v'|$, must be the same.

When you extend this to consider velocities the minus sign is allowable as that's just about the choice of direction of the $x$ axis

Correct answer by RogerJBarlow on December 3, 2020

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