Comparing the simples form of the forces of both phenomena: the law of Newton for gravitation $V\propto \frac{1}{r}$, and the Coulomb law for electrostatics $V\propto \frac{1}{r}$, one might think that if one can be extended relativistically to a curvature in space-time, that the other one would lend itself to a similar description. Is this so?
Thanks for any useful thoughts and/or suggestions!
Answer
I don't really follow the text underneath the title, but the answer to this question is most certainly, yes. (For example, we witness electromagnetic waves traveling through curved space.) Here is how electromagnetism is described mathematically (I fear this answer is slightly beyond the level of the questioner -- sorry -- but perhaps not of other readers):
Maxwell's equations are best expressed in terms of the field strength tensor. That tensor is the curvature of a connection on a circle bundle (over spacetime). The connection is the four-vector electromagnetic potential, in physics terms. In this set up, two of Maxwell's equations are automatic (for example, saying that the magnetic field is locally the curl of a 3-vector is saying that it is divergence-free). The other two are equations for the "divergence" of this tensor. This way of phrasing the problem makes sense in any metric, i.e. on curved spacetimes.
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