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As I understand from popular talks that gauge invariance is a problem in finding out exact solutions of Einstein's field equations (apart from the complication that it is non-linear), and gauge fixing is required. I'm a novice in the field of general relativity so I didn't quite understand the real problem. However, I know of the gauge invariance in classical electrodynamics, and we work e.g. in the Coulomb gauge or the Lorenz gauge, and the solution is independent of the choice of gauge. Moreover, in classical electrodynamics this gauge invariance doesn't trouble us unless we go to quantization of the radiation field. Is it possible to understand, why the problem of gauge invariance is so difficult to handle for Einstein's field equations?

Apologies for the question if it is too naive or ill-posed for the experts in the field.

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    $\begingroup$ Making this a comment because I can be neither comprehensive nor authoritative on this. There are three issues I can think of. First, gravity is non-abelian, so the gauge transform isn't as simple as E&M. Second, gravity has general coordinate invariance, which is a deal more difficult than just gauge invariance. Third, in gravity the metric is dynamical, other gauge theories have fixed metrics. $\endgroup$ – Sean E. Lake Sep 16 '16 at 12:45

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