I am currently studying electrodynamics. And when looking at Maxwell's equations, I don't see any reason, why we cannot apply them to gravity.
We know that charges generate a force field that attracts opposite charges. From this and using special relativity we can deduce that there must be a magnetic field (see this nice video from minutephysics).
But masses also generate a force field that attracts other masses. Then by the same reasoning, moving masses must also generate some kind of field that interacts with other moving masses. And indeed, this happens. It's called Lense-Thirring effect.
So if these fields are so similar, why aren't there some kind of Maxwell's equations for gravity? What is the difference between electromagnetism and gravity?
EDIT: Thanks for the comments. So it basically boils down to the question why charge is invariant under Lorentz transformations while mass is not.