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Is there an argument, apart from experiments, that we know this is true?

And if we only know it by experiment, how do we know the experiments are precise enough to conclude this?

Stated differently, what argument is there that prevents us from replacing c in the Lorenz transformations by $c' = c + \epsilon$

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    $\begingroup$ The premise of your question's title would be better expressed as "How do we know the limiting speed of conventional matter is the same as the speed of light?", since according to SR the speed of anything with non-zero rest mass must be less than $c$. $\endgroup$ – PM 2Ring Jun 12 '18 at 10:11
  • $\begingroup$ Actually, it's better to think of $c$ as the space / time scale factor, that is, 299792458 metres has the same magnitude as 1 second; that's how $c$ is used in the Lorentz transformations, and in the energy equation $E^2=(pc)^2+(mc^2)^2$. The fact that light happens to travel at $c$ is a consequence of those principles. $\endgroup$ – PM 2Ring Jun 12 '18 at 10:31
  • $\begingroup$ Which technical possibilities do you have to accelerate matter to c? You will end with electromagnetic radiation. Basically because any push or pull on the atomic level are interactions between electrons. Try to find any other interaction as EM radiation. $\endgroup$ – HolgerFiedler Jun 12 '18 at 10:38
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We know from Maxwell's equations that the speed of light is constant: $$c=\frac{1}{\sqrt{\epsilon_{0}\mu_{0}}}$$ We know from Galilean Relativity that the laws of motion are the same in all inertial frames. So for each observer to measure the same value of the speed of light, it must be the same no matter how fast you're moving relative to it. If you work out the math, you will get some equations that predict that particles with mass are constrained to move slower than the speed of light. Photons are allowed to travel at the speed of light because they have no mass - meaning that they cannot ever be at rest. Having mass means that you can always find a frame relative to which you are at rest.

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