As is well-known, Maxwell's equations were already consistent with Special Theory of Relativity while the Newtonian law of Gravitation wasn't. It can be shown by solving the equations of Maxwell for a dynamic system that the information of something done on a charged particle over here reaches somewhere else in time no shorter than it would take for the light to travel between the two places - without explicitly using any of the Relativity.
Noticing the remarkable resemblance between the Coulomb's law and the Newtonian law of Gravitation, it is tempting to ask whether one can formulate something like Maxwell's equations (which contains the Newtonian law as a special case just like Maxwell's equations contain the Coloumb's law as a special case) for describing the Physics of Gravitation. I am curious as to whether such an attempt was ever made or not. There are chances that before General Relativity, people tried making the laws of Gravitation consistent with Special Relativity exactly along these lines. I know that General Relativity reduces to a set of four equations in the weak field limit (in some particularly chosen frames) which resembles Maxwell's equations to a great extent.
But I am asking if such attempts were made before the formulation of General Relativity and whether they had got the right equations - the equations that we call today the gravitomagnetic equations (the equations that look like Maxwell's equations and are the weak field approximations of General Relativity)?