Electrostatic Acceleration as the Result of Spacetime Curvature
The description of electrostatic acceleration as a result of spacetime curvature seems reasonable since the formulas for gravitational and electrostatic accelerations have exactly the same form, and this suggests that the latter are also produced by some kind of spacetime curvature. (Here we will avoid the term "force" since the impression of the existence of a force seems to be an artefact. Instead of saying that we have a force that produces an acceleration, it seems to be closer to reality to say that we have an acceleration that produces a force, or the impression of one.)
The main difference between the two forms of acceleration, gravitational and electrostatic (besides the units and constants, and the difference in magnitude between the two accelerations), is the fact that in electrostatic acceleration we may have either attraction or repulsion.
An extra dimension
One way of producing such a phenomenon would be to assume that electrostatic acceleration is due to a curvature of the spaceline (the space dimension) not in relation to time, but in relation to an extra fifth dimension. I will try to describe things verbally, but there will be a simulation that will present this idea. (The idea is in the form of an initial working hypothesis that may need to be modified.)
Instead of having the spaceline curve in relation to the time dimension (which means that different points of the spaceline have different time coordinates; see Forms of Spacetime Curvature and Spacetime Curvature and Gravitation), we assume the existence of a third dimension (for a simple spacetime of one dimension of space and one of time) that is perpendicular to the other two. Then we can have the spaceline curve in relation to this dimension also, which means that different points will have different coordinates in relation to this dimension.
Positive and Negative Charge as Different Directions of Curvature
We can assume that the curvature in this dimension towards the one direction represents positive charge and towards the other the negative charge. We shall use the working term "electrostatic curvature" for this. When two particles with the same charge approach one another, the "depth" of the curvature must increase, and this seems to produce an acceleration away from one another, that is perceived as "repulsion". When two particles of opposite charge approach, the curvature towards the two directions cancels, and this seems to produce acceleration towards each other, which is perceived as "attraction".
Nature of the Extra Dimension
Of course, in such a formulation we are faced with a question of what this dimension is exactly. It could be an additional dimension of space or an additional (second) dimension of time. The second case seems more probable, but we will have to see. If this proves correct, maybe light moves along the "second" time dimension only (see The Proper Time of Photons and the Nature of Light).
A second question is, "If there is an extra dimension, why do we not see it?" This question seems reasonable, but maybe it is not. Why should it be considered certain that if there was an extra dimension we would be able to see it? There is also a scenario where the extra dimension exists, it is non-perceivable, and the fact that the perceivable dimensions curve within it produces a series of phenomena.