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My understanding of point particles is that they have mass and are dimensionless. If this is so then wouldn't the mass cause the particle to have a Schwarzschild radius which would then make the particle be within a black hole?

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It doesn't work for a number of reasons. Elementary particles like the electron, or quark, have spin and charge. When you calculate the contributions of those two quantities to the horizon radius (there's two in charged rotating black holes) one obtains a minimum mass that allows the horizons to exist. For the electron the mass is too small. So a horizon does not form and they then are naked singularities.

These are called super-extremal black holes, and calculations are that they can't form from a collapse - their angular momentum and also charge disallow them to form by throwing more charge or more angular momentum in when they start near extremal. Mass can be added in, but that gets them further away from being extremal. It's not been proven that they can't exist, but it is problematical.

The following Wikipedia article explain it calculating the Schwarzschild radius and then those due to angular momentum (rotation) and charge. The mass is just too small by many orders of magnitude. See Wikipedia at https://en.m.wikipedia.org/wiki/Black_hole_electron

The same is true for quarks and if you would ask also protons. Even neutrons, w/o charge, though neutrons and protons are known to be made up of quarks and gluons so the black hole idea is not even possible.

Naked singularities are a problem. They have closed timelike curves, meaning they are not causal spacetimes. Anything can come out of anywhere since there is no cause effect relationships.

There's also other reasons. The electron's Schwarzschild radius is so small, O($10^{-57}$) meters, smaller than the Planck length, that neither general relativity nor quantum theory apply. The proton, and approximately the quarks, are about 1000-100,000 times more massive, still more than approx. 15 orders of magnitude the mass needed for them not to be super-extremal if they were black holes.

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