The gravitational field of a point mass and the electric field of a point charge are structurally similar. In each case, the force has the same vector sense as the field. The electric field may point radially toward or away from the source charge, depending on the sign of the source charge. The gravitational field points toward the source mass in all known cases.
Energy density is a measure of the energy stored in the field per unit volume of space.
3 represents the energy density for the electric field, and a similar expression represents the energy density for the magnetic field, no such energy density term has ever been defined for the gravitational field. But one suspects that it could be, and possibly even should be. Let us use the similarity between the gravitational and electric fields to construct a gravitational energy density term. Historically, attempts to modify Newton's law of gravitation to account for the observed motion of Mercury have proved unsatisfactory. It is reasonable, then, that the present argument will not be an exact solution, but a rough approximation.