Is the self energy divergence problem of point charge resolved in the context of general relativity?

Question

The point charge model of electron became problematic in the context of electrodynamics/special relativity, because if we calculate the mass/energy of the electric field, it becomes divergent in the case of a point charge. Although the classical theory is insufficient for the physics of electron, can we think about the problem in another way? Can we treat the point charge as a charged black hole? Then we get a R-N solution which doesn't result in an infinite total mass. Does it mean the problem can be resolved in the context of classical theory alone?

Answer 1

The charged black hole that would have the mass and charge of the electron violates the extremality bound. So classically, it's forbidden. In the Planck units, the mass is less than $10^{-22}$ (times the Planck mass) but the charge is of order one.

So any description of the electron as a black hole is inadequate. The corrections are much larger than the "leading terms". By the way, the fact that some elementary particles have to be "superextremal, classically forbidden black holes" has been conjectured to be a general principle of physics, the Weak Gravity Conjecture.

More generally, one may consider the electron to be a charged black hole with huge quantum gravity corrections. The electron is one of the "lightest microstates of a charged black hole". However, we need to use the full exact theory of quantum gravity to calculate anything that includes these corrections, and without this exact calculation, we can't reliably make even qualitative conclusions with any certainty. The classical R-N solution is such a bad approximation for the electron that the huge quantum gravity corrections may change all the properties, including the very qualitative ones.

Quite generally, the infinite self-energy problem may be "cured" by various nonlinearities – and decades ago, physicists were proposing them, like the Dirac-Born-Infeld theory. But the progress in quantum field theory made it clear that it's not necessary. Renormalization allows all predictions to be consistent even though the self-energy is naively divergent. The divergent pieces get subtracted.

At the end, there exists a microscopic theory of quantum gravity – e.g. string theory (or "necessarily" string theory) – where the electron is "regulated" at some finite length scale so that the infinite contribution doesn't really ever arise. Like you said that the electron is a black hole, in perturbative string theory, it is an extended vibrating string. But the details of the cancellation of the infinite piece are different in string theory.