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?


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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.

  • $\begingroup$ I knew that the charge to mass ratio of electron far exceeds the limit of extreme black hole. But can it be applied to more massive point charges, even more massive than muon or tauon? I mean, if the charge to mass ratio is not a problem, is the R-N black hole a viable model? $\endgroup$
    – Ballistics
    Commented Sep 22, 2015 at 14:12
  • $\begingroup$ Dear @Ballistics - elementary particles that may be modeled as a black hole with relatively small corrections have to be black holes, indeed. Their mass has to be comparable to the Planck mass or higher, and one needs the full theory of quantum gravity for those, anyway. For example, they are highly excited vibrating strings or something like that. For such heavy particles, one can't quite say that they are "elementary", because of ultraviolet-infrared mixing. My point is that it can never happen that both "elementary particle" and "black hole" description are valid simultaneously. $\endgroup$ Commented Sep 24, 2015 at 9:36
  • $\begingroup$ I don't see any evidence to model them this way. $\endgroup$
    – SDsolar
    Commented Apr 5, 2017 at 6:06
  • $\begingroup$ What's more important than whether you see it is whether the evidence exists and it surely exists and is very strong. $\endgroup$ Commented Apr 6, 2017 at 5:12

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