When Gravity Balances the Lorentz Force
Figure 1 illustrates two resting, hollow spheres of charge viewed from their "rest" inertial frame, K. Each charge is held together by some agent, who also holds them at a constant distance from one another. The total force on each charge is zero. Viewed from any other inertial frame, the charges move at a common, constant velocity, which among other things means that (a) the distance between them is unchanging in all inertial frames and (b) the total force on each charge is zero in every inertial frame. Of course if frame K' moves to the left (so that the charges move to the right) then F'E, the combined electric and magnetic forces in K', is somewhat less than FE, the electrostatic force in K. This fact is fully consistent with the way forces in general transform relativistically.
Two Spherical Shells of Charge Held in Equilibrium
Let us now fill the spheres with neutral matter, at a density such that the gravitational attractive forces precisely cancel the electrostatic repulsive forces. Under these circumstances the agent need not hold the charges at a constant distance from one another (although each charge must still somehow be held to a constant radius). We know that the electromagnetic force, experienced by each charge, varies from frame to frame. But we also know that the total force on each charge must be zero in every frame. In effect, then, the force of gravity must transform from frame to frame precisely as other forces do.
The only theoretical alternative at present would appear to be provided by General Relativity, which proposes that there is no gravitational force, but rather that what is classically construed to be the force of gravity is really a matter of how space-time is curved. The problem is considerably more difficult to consider in General Relativity. But evidently, whatever the details may be, the curvature of space-time must transform in just such a way as to produce constant velocities in all inertial frames of reference.