Electromagnetic waves, in the classical sense, are due to oscillations in two different fields. They propagate at the speed of light, and they can be described by a set of differential equations.
Could there be a field coupled to gravity just like the magnetic field is coupled to the electric field? It would make sense that we haven't detected it yet (recall that the magnetic field of a wave is $c$ times smaller than the electric field); it would also make sense for this secondary field to have something to do with moving masses (just like the magnetic field) since relativity is all about masses moving relative to each other - take this expression with a side of salt.
Does any physical theory prove there cannot be such a field in the case of gravity? Does quantum field theory say something about this?
Don't be afraid to delve into the technical details, since I'm interested in knowing a lot more.
