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If we assume a universe where Galilean transformations are the correct transformations between inertial reference frames, would GR be any different ? Gravitational and inertial mass would still be equal, and thus the principle of equivalence between an accelerated frame of reference and a homogeneous gravitational field would still hold, at least for massive bodies. Would this inevitably lead us to conclude that mass induces curvature in space? It seems like Einsteins equations would still hold ? In other words, what role does a local Minkowski structure play in GR, and what would happen to GR if the world was locally Euclidian instead of Minkowski?

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    $\begingroup$ Don't know if this is equivalent but you might want to look up the Newton-Cartan formalism for classical mechanics : en.wikipedia.org/wiki/Newton%E2%80%93Cartan_theory $\endgroup$ – Slereah Oct 4 '18 at 12:15
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    $\begingroup$ Maxwell's equations are Lorentz Invariant not Galilean Invariant. So you break EM theory if you impose Galilean Invariance. This is why Maxwell's laws are so important beyond describing EM fields. $\endgroup$ – StephenG Oct 4 '18 at 15:02
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    $\begingroup$ if galilean transforms were the correct transforms then EM would definitely break, but my question is more about GR. do you think GR be significantly different @StephenG ? in particular, given that inertial and grav mass are still equal, do you think we would still conclude that gravitation is a manifestation of the spacetime metric ? $\endgroup$ – marjimbel Oct 4 '18 at 23:33
  • $\begingroup$ thanks @slerah, newton-cartan formalism is indeed equivalent to my question, and is quite interesting to follow. indeed, if we assume newtonian mechanics hold locally, we get a generalized version of gravitation taht is geometrically very similar to GR. it's essentially GR in the limit of "c-->infinity" :) $\endgroup$ – marjimbel Oct 6 '18 at 13:16
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the question is equivalent to generalizing gravity under the assumption that newtonian dynamics hold. a theory very similar to GR follows from the weak equivalence principle and it has been fully developed and is called Newton-Cartan formalism (see link from @slerah) or this great paper [link]"https://projecteuclid.org/download/pdf_1/euclid.cmp/1103899454". in that paper, it is shown that particles of this theory travel along geodesics in a curved space (time) as described by the connexion of that theory. there isn't a proper space-time as these two are kept as two separate tensors. the field equation is very similar to the GR one, where lambda is the cosmological constant :

$$R_{\alpha\beta} = (4\pi G \rho - \Lambda)t_\alpha t_\beta$$

i suspect this theory is equivalent to GR in the limit of "$c \to \infty$"

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