Can the electric potential be expressed by the space-time geometry?

Answer 1

I dont know exactly what you mean, but in microwave circuitry, operating at say 600 MHz and above, there is definitely a time space thing going on , big time. The voltage in a 2 GHz (2E9 hertz) circuit is vastly different one inch away on a circuit board , at least at one snapshot in time. A fraction of a nanosecond later, both points will have changed appreciably. The voltage travels in waves at these frequencies.

Answer 2

I see what you're asking. The simple answer is yes. We could use the charge as source for the curvature instead of mass, and electrostatic attraction could be modelled in this way, as it follows the same 1/r^2 dependence as gravity. But why? We use electromagnetic theories IN general relativity . . . and it can be argued electrodynamics is a more succesful theory than relativity (it predicts light travels at c to begin with!). Also, the point in GR is that photons, which are massless, bend tragectories in the presence of mass. This question got me thinking so I refreshed my memory a bit . . . the electric fields of a charge would be able to produce a gravitational force--since the space-time curvature depends on the Energy-Momentum tensor. So to better answer your question, we need GR to account for the gravitational effects of the electric potential. So the electric potential already has an effect on the spacetime geometry. Nice question.

Answer 3

An electric field can vary in space and time. But a scientist wouldn't refer to it with the words 'space-time geometry' because that would convey an entirely different meaning. That's used in relativity, string theory, M-brane theory, etc.