Black Hole Electron
en.wikipedia.org
en.wikipedia.org
Until we do, all such (semi)classical calculations just demonstrate that we are in dire need of a new theory of QG.
No, an electron is not a point particle (QED with renormalization is just a preposterous hack).
No, an electron is not a black hole.
No, QM entanglement is not literally a GR wormhole (ER != EPR).
Something new is needed. We don't have it yet.
Why couldn’t a theory of quantum gravity remove the need for such renormalization without making electrons not point particles?
Points mean infinite precision and infinite information for continuous real-valued coordinates.
These infinities just cannot exist.
The need for renormalization is telling us we are wrong. We cannot remove renormalization and leave the bare infinities, we have to change the substrate.
QFT is all waves, no particles. Particle localization is just Fourier Analysis over those fields. Heisenberg's Uncertainty Principle is just basic Fourier Analysis on wave packets.
QFT says fields propagate as waves, but interact as particles. A particle is just an interaction event, not implying a continuous localized existence between interactions.
But QFT is formulated over a pre-existing background spacetime, which GR denies.
The naive combination of QFT and GR just does not make any sense at all. We need a new theory of QG.
My hunch (prejudice perhaps) is that QG will be discrete. No points. No infinities.
A theory of quantum gravity won't eliminate the fact that electrons act that way when actually observed (they have a "smeared out" effect in a region near but outside of the nucleus that has physical dimensions greater than zero).
Assume electrons are black hole electrons. Why are there so many more of them than positrons?
Assume all charged leptons are black holes. Why are the higher-mass tau and muon so much less stable than the electron, even though (anti-)muons are stable enough that muonium can participate in chemical reactions (there's useful real-world applications for that since the muon's magnetic properties distinguish from atomic hydrogen)? Why is muonium still perfectly fine linear theory (in bound-state QED theory (BSQED) they're just two pointlike leptons) while black hole electron + black hole antimuon almost certainly isn't?
(Similar questions can be asked about positronium if only the first generation of charged leptons can be tiny black holes).
Extra credit: why aren't neutral leptons also black holes?
It seems everything is very weird.
Honestly, black hole electron sounds extremely exciting :)
https://en.wikipedia.org/wiki/Hawking_radiation
"Hawking radiation reduces the mass and rotational energy of black holes and is therefore also theorized to cause black hole evaporation. Because of this, black holes that do not gain mass through other means are expected to shrink and ultimately vanish.
For all but the smallest black holes, this would happen extremely slowly. The radiation temperature is inversely proportional to the black hole's mass, so micro black holes are predicted to be larger emitters of radiation than larger black holes and should dissipate faster per their mass."
I agree with the first half but not the last. We know that the methods used to predict Hawking radiation are no longer valid in the regime of tiny black holes (curvature becomes extremely high near the event horizon but Hawking's computation assumes near-zero curvature), so the claim of black holes vanishing is a guess at best (and one that leads right to the black hole information paradox). Another option would be that we'll be left with a black hole remnant. Or maybe something else happens. Without a quantum theory of gravity we can't possibly know.
If you divide a small number by a much smaller one, you end up with something resembling a infinity, eventually.
There are two kinds of singularities related to black holes.
The first kind are coordinate singularities, for example at the event horizon in the Schwarzschild metric. Those are mathematical artifacts of the chosen coordinates and we can get rid of them by using different coordinates for calculations.
The second kind are the ones at the center, where, for example, the curvature blows up to infinity. For those we do not know how to get rid of them. Still very few physicist will consider them any more real than the first kind, they are most likely an artifact of relativity not accounting for quantum physics or something like that.
When it comes to naked singularities, we are concerned with the second kind of singularity, those at the center. They are usually hidden behind the event horizon because - in case of a non-rotating uncharged black hole - the Schwarzschild radius is positive and nothing from inside the Schwarzschild radius, where the event horizon is located, can reach the outside.
For rotating or charged black holes the event horizon can however disappear, I think the Schwarzschild radius or its equivalent becomes complex. No event horizon, nothing hiding the singularity, i.e. a naked singularity.
So the uncharged non-rotating electron has a mass and I would guess also a singularity in the gravitational field but its Schwarzschild radius is positive hiding this singularity behind the event horizon. That is, if you wanted to get close enough to the electron to have a look at the gravitational singularity, you would first have to pass through the event horizon of the electron and then you could unfortunately no longer tell us what you saw. Also the "electron" would now be 70 kg, I guess.
If I've plugged in the right figures, the Schwarzschild radius would be something like 22 orders of magnitude smaller than the Planck length. (edit - ah the result of this is in the wiki article)
The problem is that the maths of general relativity assumes spacetime is differentiable everywhere, while also predicting non-differentiable singularities from benign starting conditions. This is still a problem no matter the Schwarzschild radius, because the unwanted infinity is still there.
It's also not a problem, because everyone already knows by this point that neither GR nor QM are complete, and stuff on this scale is exactly the sort of thing where a future replacement for both of them takes over.
"Problem" or "not" is just "do you work on finding the next theory, or do you build stuff at scales where we can pretend both (or either one plus Newton) are the complete truth?"
That the Schwarzschild radius is less than the Planck length might mean something or not, all the Planck quantities are just scales derived from fundamental constants. At those scales different effects from our current theories would become comparable if our theories would hold at those scales, but there is currently not really any known significance to them.
It is about the limitations of describing positions, not an implication that spacetime is discrete.
We don't know if spacetime is discrete, but we also have no evidence to suggest it is right now.
If you consider at what energy a photon (E = hc/λ) will be able to resolve, i.e. have a wavelength equal to, the Schwarzschild radius of that energy (r = 2GM/c² with E = mc²), then you get the Planck length up to a factor of root two [1], implying that measuring a Planck distance requires a photon so energetic that it turns that region into a black hole. If, very big if, the known laws still apply at those scales.
[1] To get the Planck length, the wavelength must be doubled, i.e. be equal to the diameter instead of the Schwarzschild radius.
r = 2GM/c² → m = rc²/2G → E = rc⁴/2G (with E = mc²)
hc/λ = rc⁴/2G → hc/2r = rc⁴/2G (with λ = 2r) → r² = hG/c³ → lₚ = r = √hG/c³I don't see how this could work if an electron would be a black hole.
It doesn't, because the "Planck length" and all other "Planck" units ARE NOT fundamental, universal limits. They are merely a differently defined unit that makes some of the math simpler. It's just a really small yard stick, they don't mean anything.
I think this could be very dangerous if this water is extremely poisonous.
On the other hand, imagine something like extremely tough stuff. We know this trope from science fiction stories. Let's say graphene with black holes wandering around its sheet. Perhaps this will increase its strength massively? Science fiction stuff and imagination is only the limit till we have solid data. Perhaps when we know more about these black hole electrons we could simulate stuff containing them? Of course maybe this needs processing power not yet available.
So, basically after you replaced the electron with a black hole, it's chemistry.
Moreover, the idea is not to substitute some QED electrons with black hole electrons, but that all electrons are charged spinning black holes. If that breaks chemistry, it also breaks the idea of black hole electrons.
The original (1938) idea behind black hole electrons was mainly to simply some of the calculations of charges moving in curved spacetime (computers not being available yet), before the development of quantum electrodynamics (1965 Nobel prize in physics), and before the development of an exact analytical solution to the Einstein Field Equations (of General Relativity) for spinning black holes (1963, and 1965 for spinning with electric charge). In particular, they wanted to understand how electromagnetic interactions could change the trajectories of charged particles in both an electromagnetic and gravitational field (and conversely, how the mass of an electron affects the development of an uncharged cloud of gas around it). Einstein and his 1938 coauthors certainly knew some things about electrons, and in particular that they have intrinsic angular momentum (Stern-Gerlach experiment in 1921). Knowing how many details had to be minimized or even ignored in their ease-of-calculation approach, they were surprised it worked as well as it seemed to. "Ease" here is compared to what they thought they would have to work with. Here's their very-much-not-easy paper: <https://sci-hub.se/https://www.jstor.org/stable/1968714?orig...>.
That surprising "why does this work at all?" (it doesn't work especially well) is sometimes thought-provoking, but attempts to formalize the idea that fundamental particles are geometrical (black holes are a feature of spacetime geometry, so a black hole electron must be mostly geometrical) rather than excitations in quantum fields have rarely made progress. In general, the attempts that have come closest to reproducing successes of quantum field theory have had the most difficult to calculate equations.