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Michelson and Morley's experiment, together with other experiments, was determinant to establish one of the postulates of Einstein's special relativity, namely that the speed of light is the same in all inertial frames (I know that this experiment's aim was to detect the presence of aether, but the historical consequences went much beyond that).

But... Michelson and Morley's experiment was performed on Earth! So why could Einstein deduce from it conclusions regarding the speed of light in special relativity, if this theory by definition can be applied only in ABSENCE of gravity?

Did he simply assume that the effect of gravity on the experiment (for example, the bending of light rays) could be neglected, or something like that?

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    $\begingroup$ Light barely bends around the Sun. It has no measureable bending in the MM experiment, traveling only a short distance through a weak and almost-uniform field. $\endgroup$ – G. Smith Sep 18 at 19:11
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    $\begingroup$ And, both arms were in the same gravitational potential. $\endgroup$ – Jon Custer Sep 18 at 19:12
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In the quotation below Banesh Hoffmann, the collaborator of Einstein, clearly explains that, "without recourse to contracting lengths, local time, or Lorentz transformations" (as was the situation in 1887), the Michelson-Morley experiment is compatible with Newton's variable speed of light, c'=c±v, and incompatible with the constant (independent of the speed of the emitter) speed of light, c'=c, posited by the ether theory and "borrowed" by Einstein:

Banesh Hoffmann, Relativity and Its Roots, p.92: "Moreover, if light consists of particles, as Einstein had suggested in his paper submitted just thirteen weeks before this one, the second principle seems absurd: A stone thrown from a speeding train can do far more damage than one thrown from a train at rest; the speed of the particle is not independent of the motion of the object emitting it. And if we take light to consist of particles and assume that these particles obey Newton's laws, they will conform to Newtonian relativity and thus automatically account for the null result of the Michelson-Morley experiment without recourse to contracting lengths, local time, or Lorentz transformations. Yet, as we have seen, Einstein resisted the temptation to account for the null result in terms of particles of light and simple, familiar Newtonian ideas, and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether. If it was so obvious, though, why did he need to state it as a principle? Because, having taken from the idea of light waves in the ether the one aspect that he needed, he declared early in his paper, to quote his own words, that "the introduction of a 'luminiferous ether' will prove to be superfluous." https://www.amazon.com/Relativity-Its-Roots-Banesh-Hoffmann/dp/0486406768

Wikipedia: Newton's variable speed of light, c'=c ± v, explains the result of the Michelson-Morley experiment:

"Emission theory, also called emitter theory or ballistic theory of light, was a competing theory for the special theory of relativity, explaining the results of the Michelson–Morley experiment of 1887. [...] The name most often associated with emission theory is Isaac Newton. In his corpuscular theory Newton visualized light "corpuscles" being thrown off from hot bodies at a nominal speed of c with respect to the emitting object, and obeying the usual laws of Newtonian mechanics, and we then expect light to be moving towards us with a speed that is offset by the speed of the distant emitter (c ± v)." https://en.wikipedia.org/wiki/Emission_theory

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