An object's inertial mass is defined by the equation:
F = mi a
(Boldface indicates a vector quantity.)
While the gravitational mass is defined by the equation:
Fg = mg g
The equivalence principle states that mi =mg.
In order to incorporate the electromagnetic field (EM) into GR, first consider, in classical terms, a system composed of particles, each of which has a constant charge/mass ratio K > 0, so that the charge of the jth particle qj = Kmj. In such a system, the gravitational and the electric forces take on exactly the same form, except that the electric force is repulsive while the gravitational force is attractive. From a classical analysis of such a system, it is immediately apparent that the gravitational force Fg and the electric force Fe become indistinguishable as separate forces, and can be combined into a single (apparent) force, Fc, for which the equivalence principle holds.
Fg = ( G m1 m2 r12 ) / r2
Fe = - ( k q1 q2 r12 ) / r2
Fc = Fg + Fe = m1 m2 ( G - K2 k )r12 / r2
Fc = ( W m1 m2 r12 ) / r2
where W = G - K2 k
We can immediately see that the equivalence principle holds for the combined force Fc in the same manner that it holds for the gravitational force Fg. The only difference between Fg and Fc is the value of its coefficient, G or W. Thus, following the same logic which lead to the development of standard GR, we are lead to conclude that, in a system composed of particles which each have the same charge/mass ratio, the electric force "curves" spacetime in precisely the same manner that the gravitational force does. Remarkable, eh? [Surely someone has thought of this line of reasoning before. GR tells us that mass warps spacetime, and that m = E (by E=mc2); and since the electric field generates energy, we already know that the electric field warps spacetime. So perhaps the ideas presented above are just an alternate way of arriving at one of the conclusions that we already have by GR, namely, that the electric field warps spacetime.]
This would be a system composed of particles in which the gravitational force (which is attractive) and the electric force (which is repulsive for objects of like charges) exactly cancel each other out. A very uninteresting system, indeed.
This would be a system where the gravitational force Fg could essentially be "replaced" by the combined force Fc. GR would then take on exactly the same form as standard GR, except with the scalar constant G everywhere replaced by the new constant W.
In this case, the gravitational constant G is once again replaced by the new constant W, but since W < 0, some funky things are sure to happen which I haven't thought about yet. What would this do to GR???
(What would happen here? Don't know!)
Our next step is to attempt to generalize our argument to include systems in which particles do not have a constant charge/mass ratio. (That, of course, is the hard part, which I haven't done! But that's why I call this just the "kernal" of an idea, OK?????)
I think my next step might be to introduce into my system a "test" particle whose charge/mass ratio takes on an arbitrary value K* which is not necessarily equal to K. (All the other particles still have ratio = K.) I'll then need to see what this does to the framework of GR. (How does this affect the transformations to/from the reference frame of the test particle from any other frame of reference?)
Where might this be leading? I am trying to reduce the two values, charge and mass, into one value, which is the charge/mass ratio, and in so doing, achieve a unification of the gravitational and electric forces. But by such an approach, the "inertial mass" (mi) and the "combined gravitational + electric mass" ("mc") would take on different values, which would have to be specified by 2 numbers: either mi and K, or mi and mc. Thus I don't believe I'll be able to combine any variables here.