Theoretical proof of the constant of speed of light $c$ in vacuum in all frames of references Is there any theoretical proof for the constant speed of light in all frames of references? I know it is experimentally proven but just curious.
 A: As WillO says, one has to state one's theory precisely through a definition of one's axioms (and allowable rules of inference).
But something you may find intellectually fulfilling is the following. Beginning from very basic symmetry principles - homogeneity and isotropy of spacetime as well as continuity of transformations between frames and continuous dependence on relative velocity and, finally, causality, see the Pal and Levy-Leblond papers I cite in my special relativity resource recommendation answer here. One can prove from these assumptions that there must exist a frame-invariant speed $c$ (it could be infinite i.e. Galilean relativity is included in the possible outcomes) and also one derives the form of the Lorentz transformations.
One then experimentally finds that $c$ is finite because the speed of light is experimentally found to behave in this frame invariant way.
A: There is a theoretical "prove" that $c$ is constant, though it could be considered "weak". Let me explain.
It is well known from mathematical physics that any quantity $f=f(x,t)$ obeying the equation $\frac{∂^2f}{∂t^2}-v^2\nabla^2 f=0$ represents a wave moving with velocity $v$. Now, if one uses Maxwells equations for vacuum, it can be shown that the eletric and magnetic fields obey exactly an equation of that type, that is (lets use just the electric field for simplicity)
$$\frac{∂^2E}{∂t^2}-\left(\frac{1}{\mu_0 \epsilon_0}\right) \nabla^2 E=0$$
This imply an electric field wave moving with speed $c=\frac{1}{\sqrt{\mu_0\epsilon_0}}$. That is the electric component of light.
Now, as you see, both $\mu_0$ and $\epsilon_0$ are constants coming from completely unrelated experiments like Coulomb's and Biot-Savart's experiments. They were added as "force transforming" constants. $c$ here is constant and there is no way to remedy it. It doesn't admitted any other value and it came from the theory of electromagnetics (electrodynamics).
It would be over simplifying to say electrodynamics summarizes just to the wave equation. However, this wave is at the core of the entire theory. The slightest change here affects all the theory. Nonetheless, any one accepting Galileo's transformation could in principle say that Maxwell's electrodynamics is wrong and needs change. That why I say it is "weak".
At the end of 19th century it became clear that the following very well established theories: a)Newton's mechanics; b)The (Galileo's) relativity principle; and c)Maxwell's electrodynamics; could not be simultaneously true. One could choose 2 of then but the remaining would request for reforms.
One could make $c$ constancy as a postulate, saving electrodynamics and strengthening it. If you admit the relativity principle also, then voila! you'd be a genius (Einstein to be precise).
From this 2 postulates one can get Lorentz transformation. You can see from this post that it keeps the speed of the light wave constant and equal to $c$, as expected.
In summary: electrodynamics is the "prove" theory for $c$ to be constant.
PS: Einstein actually made this twice in his relativity theory. He also transformed the weak correspondence principle, considering the evidence from Eötvös experiments that gravitational mass must be the same as inertial mass into the correspondence principle (making it strong) by postulating it.
A: The proof might have two steps totally based on geometry.
Given an arbitrary metric for curved spacetime in general relativity you can always find a local inertial frame. It is an approximation to first degree that helps you switch from general relativity to special relativity around a point for which the first partial derivatives of the metric all vanish. The metric is now equal to the Minkowski metric at this point and around.
This was the rigorous wording of the basic insight by Einstein that "the physics of curved spacetime must reduce over small regions to the physics of simple inertial mechanics."
Now consider the Minkowski metric for the inertial frame that remains invariant under Lorentz transformations.
This metric demands that there should be a constant multiplier of timelike dimension with the opposite sign of spacelike dimensions, i.e. signature, in the geometry of the four dimensions, if you want to have an invariant definition for the interval between points or events in spacetime after rotations, boosts, and translations.
This constant of the geometry of spacetime, with the dimension of space over time, in the inertial frame of reference is identified with the speed of light.
This constant should be there theoretically if you want to have an invariant notion of distance or interval in spacetime undergoing the Lorentz transformations.
A: When Einstein proposed a thought experiments and then by assuming that speed of light is constant irrespective of all inertial frame of reference. That theory had lots of consequences, like length contraction, time dilation and mass-energy equivalence relation.. You of course cannot proof a theory rather it's consequences can be tasted experimentally. And as the consequences of STR are proven experimentally, hence the assumption was correct.
