Most people know that the speed of light in vacuum is independent of relative motion but I want to know the proof of it.
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By "light is independent of relative motion" I'm going to assume you mean "the speed of light is the same in every frame". This is "proven" by observation and experimentation. As a postulate of the theory of relativity, it is not proven anywhere, it is only observed. There are a myriad of experiments (see, e.g. the Michelson-Morley experiment) that are consistent with this fact, and so it is a good postulate.
enumaris
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There is no proof, unless you considere experiments as a proof. In this case, Michelson and Morley experiment proves that no eather exists and therefore light moves at the same velocity, c, in any reference systems.
raskolnikov
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We know this is true because we have determined experimentally that it is true. It is an assumption in the theory of relativity, stated as one of the fundamental postulates that is used to prove everything else. In this case, we can test both the assumptions themselves (with tests like the Michelson-Morley experiment) and the rest of the theory that follows (with tests like measuring the time dilation on atomic clocks, gravitational lensing, the observation of muons at Earth's surface, and many others). As of yet, there has been no experiment that has been able to show that the fundamental assumption that the speed of light is direction-independent to be wrong.
If we were to eventually find that this assumption wasn't true, then we would have to revise our assumptions about reality. Depending on what the revised set of assumptions were, the theory that explains spacetime might at the outset look completely different. However, whatever new theory we come up with must reduce to relativity under the right conditions (just as relativity reduces to Newtonian mechanics under the right conditions), or else it would stand in contradiction to the many experiments we've done showing that, in the situations we've tested, relativity is an extremely good description of reality. And even then, that theory will have some assumptions in it that must be tested, and so the process continues.
probably_someone
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From electrodynamics the speed of light $c$ can be proven to be
$$c=\sqrt{\frac{k_B}{k_E}}=\frac{1}{\sqrt{\mu_0 \epsilon_0}} \tag{1}$$
Where $k_E$ is Coulomb's law constant and $k_B$ are Biot-Savart's law constant.
The right hand side of this equation is formed by fundamental constants only, and unless you want to mess with one of the 4 fundamental forces of the universe, and unbalance it :-) you should admit that the speed of light is constant.
PS: This is a weak proof. The actual proof is Einstein's postulate itself.
J. Manuel
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