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I am looking for what motivated Albert Einstein in the direction of his Special Theory. I have read that it is unclear if he was set on that path by the Michelson-Morley experiments. Was AE aware that the Maxwell equations did not transform covariantly under the Galilean transformation? Does anyone know?

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    $\begingroup$ Welcome to Physics! Questions about the history of science are probably better-suited for History of Science and Mathematics. I've flagged this question for possible migration. $\endgroup$ Jan 19 at 19:31
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    $\begingroup$ As of Wikipedia page, he was well aware. $\endgroup$
    – User123
    Jan 19 at 19:32
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    $\begingroup$ Oh, he definitely knew. I don't know the exact dates or who did what, but it went like this: Maxwell's equations turn out not to be Galilean invariant (e.g. they imply speed-$(\mu_0\varepsilon_0)^{-1/2}$ wave equations); they turn out to be Lorentz invariance; it's proposed the "right" reference frame is that of the aether; the Michelson-Morley experiments show that won't work; Einstein decided to make Lorentz transformations the new standard for physics. $\endgroup$
    – J.G.
    Jan 19 at 20:46
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    $\begingroup$ I would not be surprised if the concept of "Galilean transformation" had been born only after the "Lorenz transformation" was already invented. Of course it was known since Newton, but not this was its name. Instead, it was considered as the only possible formula (or algorithm) to switch inertial systems. Much more later result, far after Einstein, that actually 3 possible group structures can satisfy a homogenous and isotropic 4D spacetime, beside Lorenz and Galilei group, there is also a third (it might be a timeless 4D space but not sure). $\endgroup$
    – peterh
    Jan 19 at 22:12
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    $\begingroup$ @J.G. Afaik there was yet another line of thinking at the time to explain the Michelson-Morley failure, it was some like an "aether dragging" theory. In the QM world, there was another line to explain the wave-particle duality, the "waverider theory": essentially, it assumed that both the wave and the particle are separate, interacting entities. I am curious, maybe combining the waverider theory with the aether dragging, maybe a mathematically equivalent theory to the QFT of the today could have been created. $\endgroup$
    – peterh
    Jan 19 at 22:23

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There is an interesting historical paper dealing with this issue:

John Norton: Einstein’s Investigations of Galilean Covariant Electrodynamics prior to 1905

Einstein was aware before he wrote his famous 1905 special relativity paper, that the negative result of the Michelson-Morley experiment can be explained by an emission theory of light, in which there is no aether, but in which the speed of light depends on the velocity of the source by the ordinary Galilean velocity addition.

Einstein dismissed the emission theory of light for the following reason as described by Norton:

The decisive consideration, Einstein tells us, that spoke to him against an emission theory prior to his 1905 paper was his conviction that light should be characterized by frequency and intensity (and polarization) alone. He was then rather uninterested in the fussy details of how a variety of distinct emission theories might be devised to accommodate to various sorts of processes of deflection or reflection. They seem to have come to the fore in the literature emerging around 1912 that sought to test an emission theory experimentally, for just those details decide how the experiments are to be done.

Of course, many modern experiments show that the speed of light does not depend on the velocity of the source, see Refutations of emission theory.

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