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So Einstein's theory of special relativity as far as I know is essentially axiomatized, saying "Assuming the speed of light $c$ is constant in ALL frames of reference, what happens?" We know now of course that special relativity has a load of evidence behind it and quite clearly applies very well to many situations, but at the time period physicists didn't have that evidence that the speed of light was ALWAYS constant. It seemed it, but then again all of the "moving a fraction of the speed of light" analogies we use to explain special relativity had obviously never been seen experimentally. If physicists had no reason to believe the speed of light was constant to that level of consistency, why was special relativity accepted so widely and quickly? Or did they in fact have physical phenomenon which special relativity and only special relativity seemed to explain?

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    $\begingroup$ Einstein set out to show what the consequences would be if Maxwell's theory were true. That's a goal that he could have achieved regardless of whether or not Maxwell's theory actually was true and, regardless of whether or not he believed it to be true. $\endgroup$ – Solomon Slow Jul 10 '17 at 17:10
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Special Relativity was not widely accepted until long after it was first published as people believed it to be too much of a change and that's the one of the reasons why the 1921 Nobel Prize in Physics's statement to Einstein was made for his insight into photo-electric effect instead of relativity.

Now how Einstein took that the speed of light is observed constant for all observers as a postulate is for two main steps. One is that experiments done by Michelson and Morley in 1887 showed this in experiment. This seemed to violate the Galilean relativity etc. The second reason is that Maxwell had also shown in his equations that electromagnetic radiation must travel at a constant speed regardless of the observer measuring it, which is the speed of light. Thus Einstein found the notion of classical relativity to be flawed in some way and set out to formulate a new one which agreed with observation.

To add to that, Einstein's relativity actually started to gain attention when his Theory of General Relativity(first published 1915) on gravitation made a better model than Newton's model of gravitation as Arthur Eddington took pictures of stars during and after an eclipse to show light from the stars were bent by gravity(around the sun), providing massive evidence to support the predictions of General Relativity and also this theory answered the long standing mystery behind the exact orbit of Mercury. Thus then only was Einstein being just beginning to gain fame and acceptance, while today his theories are almost universally accepted.

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    $\begingroup$ The fact about michelson-morley experiment is historically incorrect since it seems that Einstein himself said that it played no role in the initial formulation of special relativity: physics.stackexchange.com/questions/89375/…. I agree that logically it is very pertinent and fundamental. $\endgroup$ – Claudio P Jul 10 '17 at 17:44
  • $\begingroup$ It's true, his major factor was due to Maxwell but Michelson's and Morleys experiment served as a stepping stone for the development of such a postulate. $\endgroup$ – Tausif Hossain Jul 11 '17 at 11:57
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The seeds of special relativity were sown when it was shown the in vacuo solutions of Maxwell's equations are not Galilean-invariant and predict their electromagnetic waves have speed $\dfrac{1}{\sqrt{\mu_0\varepsilon_0}}$. This speed matched experimental light-speed measurements, suggesting light is electromagnetic radiation. Most physicists at the time thought the solution was that Maxwell's equations are valid in a privileged reference frame with respect to which Earth has a non-relative velocity. However, this implied light-speed measurements could vary with the seasons, because of Earth's location in its orbit. Einstein realised the true was that physics needed to be Lorentz-invariant like Maxwell's equations, so there was no privileged reference frame. Einstein therefore could have phrased one of his axioms as, "Electromagnetic waves in vacuo are of speed $\dfrac{1}{\sqrt{\mu_0\varepsilon_0}}$ in all directions in all reference frames". For whatever reason, he happened to talk about light instead, since by then its electromagnetic status wasn't really disputed. However, it doesn't matter too much which thing is posited to behave so in a vacuum; as of general relativity we could even use gravitational waves instead.

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