Is there a fundamental upper limit to electric charge density?
If not, is it possible to concentrate enough charge in a small region of space such that an electron wouldn't be able to escape unless it were moving at the speed of light — effectively creating an electric-field-based analog of a black hole, not due to gravity but due to the strength of the electric field?
Additionally, since any form of energy density contributes to spacetime curvature in general relativity, could a high enough charge density potentially lead to formation of an actual black hole as well?
In particular, could a system like a capacitor — whose mass density is below the critical threshold for black hole formation — nevertheless form a black hole due to the stored electromagnetic energy from its charge?
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2 Answers
is it possible to concentrate enough charge in a small region of space such that an electron wouldn't be able to escape unless it were moving at the speed of light — effectively creating an electric-field-based analog of a black hole, not due to gravity but due to the strength of the electric field?
No. Not in the same sense that gravitational fields create black holes.
An "electric-field based black hole" you'd think would require massive electrical forces, but when electric fields approach or exceed the Schwinger limit $$V=\frac{m_e^2c^3}{q_e\hbar}\approx 1.32\times 10^{18}\ Vm^{-1}$$ then QFT effects become significant.
At such intense electric fields, there will be spontaneous electron-positron pair production, which will quench the electric field making it short-lived.
It is not a question of whether we can have an electric field strong enough to prevent electrons from escaping (like a black hole does for light), but more that quantum electrodynamic effects fundamentally set an upper bound on how strong a continued electric field can be.
Edit: good point made in comment by Yukterez.
It is generally accepted that electromagnetic energy with extremely high intensities will exert a repulsive gravitational force. This is because the electromagnetic field itself would contribute to the stress-energy tensor whose spatial components correspond to tension (negative pressure), which opposes the inward force of gravity, thus preventing collapse.
Kyathallous asked: "Can a high enough charge density alone lead to the formation of a black hole?"
If, in natural units of $\rm G=c=k_e=1$, the charge $\rm Q$ is larger than the total energy equivalent ADM mass $\rm M=E/c^2=\sqrt{(16m^4+8m^2Q^2+Q^4)/(16m^2-4a^2)}$, where the irreducible mass is $\rm m =\sqrt{2M^2 - Q^2 + 2\sqrt{M^2 (M^2 - Q^2 - a^2)}}/2$, you can't have a black hole.
You can not have charge $\rm Q$ without mass $\rm M$ by the way, otherwise a finite force $\rm F$ would lead to infinite proper acceleration $\rm a=F/M$, so the charge (and if there is also the spin) is in units of $\rm M$. Therefore massless particles can not be charged, and you can not have a charge without a mass.
Charged black holes are described by the Reissner Nordström metric and including spin the Kerr Newman metric, the best you could get with that is a naked singularity when $\rm a$ or $\rm Q>M$.
The charge itself has its own gravitational field, but it's gravitationally repulsive, so that doesn't really help for creating a black hole out of it since for that the attraction must dominate, not the other way around.
Kyathallous asked: "Is it possible to concentrate enough charge in a small region of space such that an electron wouldn't be able to escape unless it were moving at the speed of light?"
An electron (from the relativistic perspective) has also spin $\rm a$, and for a black hole you need $\rm a^2+Q^2 \leq M^2$. For an electron $\rm a=3\cdot 10^{44} \ M, \ Q=2\cdot 10^{21} \ M$, so that can't be a black hole since the horizon of a spinning and charged black hole is at $\rm r=M+\sqrt{M^2-a^2-Q^2}$, which has no real solution if $\rm a^2+Q^2 > M^2$. In order to turn that into a black hole you would need to add at least as much $\rm E$ or $\rm m$ so that the total $\rm M$ exceeds the $\rm \sqrt{a^2+Q^2}$.
The mutual electromagnetic attraction of oppositely charged particles requires only finite force to overcome it, in contrast to gravitational attraction where it can also take infinite force to overcome the attraction, for example at a black hole's horizon.
You can push the electron into the proton and thereby turn both into a neutron, but that probably goes beyond the scope of your question since the tags are general relativity and black holes, not particle physics.
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