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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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marked as duplicate by Jon Custer, Kyle Kanos, stafusa, Prof. Legolasov, Sebastian Riese Jul 7 '18 at 19:15

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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.

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  • $\begingroup$ This is misleading. The constancy of the speed of light is a theoretical feature of special relativity. Because this theory is validated by observation, it dictates that the speed of light is exactly constant, but not just constant within a measurement error as your answer implies. $\endgroup$ – safesphere Jul 5 '18 at 20:38
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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.

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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.

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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.

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  • $\begingroup$ This is incorrect. The vacuum permittivity and permeability are not measured constants of nature, but predefined numbers that serve as the definition of the standard for the Ampere unit of measurement. Also, a postulate is not a proof. $\endgroup$ – safesphere Jul 5 '18 at 20:23
  • $\begingroup$ @safesphere. Could you clarify on what they (permittivity and permeability) and physical constants are?. Are there physical constants at all?. If so, It doesn´t matter what type of constants you mean, or how you define a physical constant, measured or not. Equation (1) is a mathematical identity it just need constants on the right side for c to be constant. You are correct about the fact that postulates are not “proof”, however as every axiom they are thought as self proven. $\endgroup$ – J. Manuel Jul 11 '18 at 16:24
  • $\begingroup$ Numbers are different from physical constants. In the gravitational radius; $r_s=\dfrac{2GM}{c^2}$, G is a measured constant of nature, but 2 is a number. You don't measure a number, it is given. It is not 2.0001 or 1.9999, but exactly 2. The speed of light used to be a measured constant, but then we decided to use it in a standard of distance, so it became a number c = 1 ls/s where 1 is just 1, not 1.0001 or 0.9999. Then we define that 1 meter is exactly 1/299,792,458-th part of 1 light second. The denominator there is a whole number, not a measured constant. Same for your vacuum values. $\endgroup$ – safesphere Jul 11 '18 at 16:31
  • $\begingroup$ @safesphere I have MY own system of units where c was not “decided to be used in a standard of distance”. It is the value of the speed of light as measured by experiments like Fizeau’s experiment. This means that I’m wrong? So if a group of “guys” meet on a café and decide on a particular “constant” to be set as a number, it means that the physical constant disappear and it’s no longer a physical constant? How do you distinguish between the “measured constant” $\mathrm{G}$ and the “non-measured constant” $\mathrm{k_E}$? I truly don’t understand your definition of a physical constant. $\endgroup$ – J. Manuel Jul 11 '18 at 17:03
  • $\begingroup$ To measure the speed of light, you need a ruler and a clock. These are called "standards". Earlier the standard of meter was a platinum ruler kept somewhere in Europe. It was made to be equal to 1/40,000 part of the length of the equator. Later it was decided to use a different standard of length, such as the light-second. The platinum ruler is no longer in use. Now when you measure the speed of light in light seconds, it is always 1, so the very idea of measuring it is pointless. You sure can have your own system of units, but you cannot call your unit of length "meter", as its already taken. $\endgroup$ – safesphere Jul 11 '18 at 19:43

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