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Apologies if this question was already asked a few times but i could only find proofs of the invariance of $ ds^2 $.

Is there any way of proving the 2nd postulate (that $c$ is invariant in all reference frames) by using the Lorentz transformations or the Lorentz group?

Or more explicitly, is there any way of deriving the second postulate just by assuming that 4-vectors transform under the Lorentz group $ SO^+(1,3) $ in 3+1 spacetime?

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  • $\begingroup$ The logical flow seems to be the opposite. We postulate the invariance of c, and then discover that the correct linear transformation group to keep the invariance, is the Lorentz group. Why are you seeking to go the other way around? $\endgroup$
    – naturallyInconsistent
    Commented Apr 3 at 9:09
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    $\begingroup$ The conservation of lightcones, i.e. the locus where $ds^2=0$ is equivalent to the conservation of $c$. Since $ds^2$ is conserved this automatically leads to the conservation of $c$. $\endgroup$
    – LPZ
    Commented Apr 3 at 9:12
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    $\begingroup$ The only way for $ds^2$ to be zero is if the velocity is $c$. Since we know $ds^2$ is invariant that means if the velocity is $c$ in one frame it must be $c$ in all frames. $\endgroup$
    – John Rennie
    Commented Apr 3 at 9:50

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For simplicity, consider (1+1)-Minkowski spacetime.

Find the eigenvectors of the boost transformations.

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