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This is what my textbook says about the Michelson-Morley experiment:

"This invariance of the speed of light between inertial reference frames means that there must be some relativity principle that applies to electromagnetism as well as to mechanics. That principle cannot be Newtonian relativity, which implies the dependence of the speed of light on the relative motion of the source and observer."

Question: Why must there be a relativity principle that applies to electromagnetism and to mechanics? Why can't they be separate?

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Maxwell's equations imply a wave equation for light that has a speed in it $$ c = \frac{1}{\sqrt{\epsilon_0\mu_0}} \approx 3.0 \times 10^8 \,\mathrm{m/s} \,.$$

That speed must be measured relative something. The initial thought (consistent with the treatment of waves on a string, waves on the surface of a liquid and sound) it that it is measured relative a medium (dubbed the luminiferous ether).

The interferometry experiment, seems to imply the non-existence of such a medium (which has been confirmed to much high certainty since then) or some other moderately exotic options such as having Lorentz contraction with the ether.

By itself the Michelson-Morley result doesn't prove much.

Einstein sought and found a theory that encompasses the usual Galilean relativity at low relative velocities and the relativity implicit in Maxwell's equations (which had been found by Lorentz) in one neat bundle. That kind of neatness is appealing, and when the ability to test it came around it passed with flying colors.

In principle people could have tried to figure out how to get along with two relativities, but then you have endless corner cases. Frankly that is one reason neatness is desired when possible.

The reason the desire survived is, of course, because the Special Relativity agrees with the world we live in (up to the need for General Relativity to explain, for instance, the gravitational red-shift).

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Simply put, electromagnetism interacts with matter via energy and momentum exchanges. Since the nature of energy and momentum do not differ between electromagnetism and mechanics, any relativistic framework must be consistent with both aspects. The Michelson-Morley experiment showed that Newtonian relativity (which was strictly a mechanics relativity) did not apply to electromagnetics, and thus there had to be a more generalized theory which included the unusual behavior of electromagnetic waves. This is why Einstein developed his theory of relativity using the speed of light in vacuum as a constant.

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