Interesting is that this formulation, called Gravitoelectromagnetism, was formulated by Heaviside before the advent of GR, in 1893.
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One question that I've had, but not solved is that, in case Quantum gravity could be defined, what role would've had the fact the graviton and photon have different spins. Does this information decouple and that's why the analogy works?
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IIRC there's a factor of two in the equations hidden somewhere that takes that into account
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