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I am trying to reformulate the axioms of Special Relativity. It seems intuitively true that all inertial frams should be equivalent (*), but there seems to be no philosophical justification that light should travel at constant velocity to all inertial frames (+).

Could someone show me, without sacrificing too much detail, the proofs for the following:

1. the principle of least time for light, which states that light always travels in paths of least time. Does the proof need the premise of condition (+)?

2. given the principle of least action/time, derive that light always travel in geodesics of the space to which it is confined.

Thanks, all.

--Shiro