How deep is the analogy between gravity and electrodynamics? When I was first taught about electrostatics I was taught about it by analogy to gravity. Specifically, the force due to gravity between two objects is, $$F_g=G\frac{m_1m_2}{r^2}$$and similarly the force between two charged objects is, $$F_q=k\frac{q_1q_2}{r^2}.$$When you keep going, however, you learn that gravity isn't really a force in the traditional sense but is more of a measure of the curvature of some field we call spacetime. This then tells us that what we mean by "mass" is how much a particular object deforms spacetime. (Is this true?)
Does this relativistic interpretation of gravity have an analogous interpretation using some other field theory? I suspect this is probably answered, if it is true, in quantum field theory.
I fear that question might be very broad. So, to narrow the question, is there some field, for which a curvature represents how charged an object is? Additionally, can this curvature be used to explain the long-range nature of coulombic forces?
I hope there's a good question buried somewhere in there.
 A: I can wax poetic for hours about the similarities and dissimilarities between gravity and electrodynamics. 
Nevertheless, the resemblance you noticed between 
$$
F_g=G\frac{m_1m_2}{r^2}
$$
and 
$$
F_q=k\frac{q_1q_2}{r^2}.
$$
has a simple explanation: both gravity and electrodynamics are governed by second order differential equations which reside in 4 dimensional space-time. For static solutions, a second order differential equation reduces to 3D Poisson's equation
$$
\Delta \phi = \rho,
$$
which gives rise to (for point-like source)
$$
\phi \sim \frac{1}{r}
$$
potential that is translated to 
$$
F \sim \frac{1}{r^2}
$$
force. 
A: In gravity field all bodies with same initial position and velocity move almost the same. This makes it possible to describe gravity as curved spacetime where bodies move along shortest trajectory.
In EM field, there are bodies with same initial position and velocity that do not move the same way. Positively charged particle will move differently than negatively charged particle. This "signedness" makes EM field different from gravity field. It is difficult to think of a way the motion of charged particles could be due to curved space.
A: Well, GR is very successful in giving a quantitative analysis of gravity and associated effects and it has been experimentally tested and scrutinized to match that quantitative analysis. GR does not consider gravity a force.
Said that, I like your question - GR does not consider gravity a force, but in reality that may not be the case. Think about it this way. To counter gravity, you have to apply a force. A force would only counter a force. So, at core it has to be a force, or alternatively, a curved space would only counter a curved space. That means all forces have to be curving of space in some way, but I would not go that far.
Actually "not considering gravity a force" may be acting as a barrier to unification. How can one unify forces with something that itself is not a force.
But in advanced science, I think all are treated as fields. Whether a field is a force field? If they have some influence, then they are influence fields at the least, if not force fields.
