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Do other fields remain unaffected by gravity and only the spacetime field gets curved? Or does gravity bend other fields too? What is the way to prove that other fields don't get bent?

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We need to be careful with the terminology here as your question is somewhat vaguely phrased.

In GR the curvature is described by a tensor field called the metric. The gravitational field doesn't exist - what appears to be a gravitational force is just motion on the curved spacetime. So when you say in your question title gravity “bends” spacetime this is meaningless. It is mass, or more precisely the stress-energy tensor, that causes spacetime to be curved.

In principle the energy in electromagnetic fields also curves spacetime so for example we get charged variants of the stationary and rotating black holes called the Reissner-Nordström and Kerr-Newman black holes. However in practice the net charge of astronomical objects is too small to cause any significant curvature so we treat the EM field as if it is propagating on a static curved background. This done using Maxwell's equations in a curved spacetime.

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  • $\begingroup$ I believe the question is about other fields curving with spacetime, even if they don't have energy. I think the answer is yes if spacetime is the "stage" for all other fields, as many popular science books put it. $\endgroup$ – Jus12 Jun 13 '18 at 21:28
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In general relativity it is space time that generates the observed attraction between two massive bodies which we have named gravitational attraction. The curvature of space time is dependent on the energy momentum tensor and thus when studying a system , whatever has energy and momentum, interlocks and affects the curvature of spacetime.

Thus the answer is that yes , electric and magnetic fields are affected and may affect the curvature of space , electromagnetic radiation as well as gravitational waves will contribute to the curvature and changes in the curvature.

As the curvature becomes important for very large masses the effects of electromagnetic fields will need special solutions dependent on the boundary conditions of the problem, their energy content , for example, with respect to the masses in the problem. Note that the Newtonian gravitational field is replaced by the curvature of space.

The curvature of space generated by the existence of the massive earth affects the GPS measurements. The contribution of the electromagnetic waves which transfer the information is infinitessimally small and does not have to be computed.

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