Could there be a Planck sized black hole with positive charge in the center of each atom, moving so fast on its way that it has nearly any mass?

Question

A few days ago, I asked if Planck-sized black holes at the centers of atoms might explain the positive charges in the Rutherford model of the atom.

Someone told me the Planck mass is much heavier than an atom, which makes the idea of a black hole incompatible as the source of the positive charges in atomic nuclei.

The Planck mass issue made me wonder if is a way that a source of positive charge could be heavier than the atom in which it resides, that is, that you could have this mass inside an atom yet not notice it. The best analogy I could come up with is magnetic levitation in bullet trains. When a bullet train is at rest, its entire mass resides on wheels on a conventional metal rail, just as with an ordinary train. However, electromagnetic levitation takes over when the train is in motion and levitates all that mass away from the metal rail. If you look only at the metal rails, the mass of the train is no longer detectable, but that does not mean the train has no mass. It's just that mass becomes imperceptible while the train is moving.

My question, then, is this: Could a Planck-sized black hole at the center of an atom move so fast that most of its mass becomes undetectable at the atomic scale, which would correspond to the metal rails of the levitating train?

Answer 1

A Planck sized black hole with the Planck mass ($\approx 2 \times 10^{-8}\,$ kg) would evaporate in a Planck time, releasing a Planck energy ($\approx 2\,$ GJ), which is a lot. That's that.

You can't say "well, imagine it's stable" because then you have to specify how you violate the laws of physics to make it not evaporate.

Moreover, motion doesn't cause an increase in mass. $E=\gamma mc^2$, so the mass can have a lot of energy, but so what? If $\gamma \gg 1$, then you're moving at a speed where Newtonian gravity is invalid, and you need to consider gravitoelectromagentism in any measurment.

Answer 2

Alexis, those are some fun and interesting speculations. Let's start with the positives:

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(1) Is it possible that charged particles are, in some fashion, tiny black holes with additional properties such as positive charge attached?

Sure. As long as the issue of electric charge is kept separate from that of mass, the idea of a point particle comes surprisingly close to saying that fundamental particles are just tiny black holes that cannot evaporate fully due to being "decorated" with absolutely conserved properties such as charge and spin.

Whether point-like particles are like black holes becomes almost a matter of definition since, pretty much by definition, any finite mass compressed into a point is a black hole. However, the experimental reality is that this definition is inherently paradoxical. That's because quantum uncertainty protects the electron from looking point-like without adding more energy — particle colliders are all about that. Thus, amusingly, if you want to see the black hole that is the point-like electron in a quantum wave function, you must first add so much energy that you end up creating a black hole! It's a cosmic Catch-22 that warns us that we may not be asking the right questions.

You might be interested to know you are by no means the first person to wonder whether point-like particles might be black holes. Here's a 2006 paper co-authored by the remarkable Czech physicist Luboš Motl, who has provided innumerable excellent answers here on the Physics Stack exchange:

The String Landscape, Black Holes and Gravity as the Weakest Force (2006)
Nima Arkani-Hamed, Lubos Motl, Alberto Nicolis, and Cumrun Vafa
https://arxiv.org/abs/hep-th/0601001

But it gets better! The earliest instance of which I am aware of a decently well-known physicist proposing that charged particles might be versions of black holes — wormholes between two parallel universes, to be precise — was this 1935 paper:

The Particle Problem in the General Theory of Relativity (1935)
Albert Einstein and Nathan Rosen
https://journals.aps.org/pr/abstract/10.1103/PhysRev.48.73  

Yes, that Einstein. It's a good thing I had not yet read this paper before watching the Marvel movie "Thor: Ragnarök" since if I had, I'm pretty sure I would have broken out laughing hysterically in the theater when Bruce Banner uttered the line "That looks like a collapsing neutron star inside of an Einstein-Rosen bridge!"

Ahem. The reality is that Einstein and Rosen proposed that at some incredibly tiny scale — quite a bit smaller than the visual on which Banner was commenting — all particles are black-hole-ish bridges between two parallel universes.

The idea didn't go far, to say the least, since it violates Einstein's theory of general relativity by invoking incredibly flexible spacetime at no particular energy cost. The paper was not one of his high points. But the same paper has gotten much more press lately after David Susskind began promoting it. That doesn't change the problem: Einstein and Rosen's ideas violate Einstein's General Relativity. Details, details!

I also brought up the idea of electrons as black holes, independently and far more naively than any of the above, in this 2013 question here on stack exchange:

Are electrons just incompletely evaporated black holes? (2013)
https://physics.stackexchange.com/questions/75911/are-electrons-just-incompletely-evaporated-black-holes

Finally, you don't need to invoke the Planck mass for this. That goes back to the hiding issue: If an electron is point-like, then when you pin it down close enough to see that pointiness in all its glory, by then, you've added so much mass-energy that there's no particular need to invoke the Planck mass. It's quantum uncertainty that allows this hypothetical black hole to exist in principle — as a limit — yet nonetheless exhibit only a very modest mass at ordinary energies. Quantum uncertainty can be pretty weird that way.

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(2) Can rapid motion hide mass?

Alas, most definitely not! It's the other way around: The faster an object moves, the more gravitational mass it exhibits. As someone already noted, what's going on in maglev trains is just a mass transfer from the metal rails to the electromagnetic coils around the track. They experience all the wear and tear of that mass moving over them, which is one reason (among many) why maintaining bullet trains gets costly.

The positive part of this one is that there are mechanisms in quantum mechanics and particle physics that can hide component mass.

I've already mentioned one of them, which is quantum uncertainty hiding of particles that seem to be, at some infinite or near-infinite limit, precisely point-like. But that one gets to be a bit of a shell game since you must add energy to see the "hidden" energy. I've my mental model aggressively to a "soft universe" view in which particles are bundles of energy, properties, and rules with asymptotic limits, some of which make them more point-like with a sufficient injection of energy. Soft universes have fewer mathematical paradoxes because they don't have to assume it to be infinitely smooth.

The other mechanism for making parts seem lighter than the whole is to convert some of their mass into energy and radiate it away. Nuclear energy is the most conspicuous example. A deuteron is the nucleus of the deuterium or "heavy water" isotope of hydrogen, consisting of one proton and one neutron. If you whack two deuterons together, they sometimes stick to give you a helium-4 nucleus. Even though this nucleus has the identical two protons and neutrons you had before fusion, the resulting new nucleus has substantially less total mass than the original two deuterons.

Finally, where there are powerful force fields, such as in the interior of a proton or neutron, you can get what is probably the weirdest form of heavy particles, which is virtual particle pairs. One member of such a pair can have more mass than the entire proton or neutron! However, this one is a bit bogus in much the same way as the quantum hiding of point-like particles that I mentioned earlier, since the total mass of these very transient, uncertainty-hidden pairs doesn't show up at the complete proton or neutron level.

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OK, on to some negatives:

(1) Positive charge is not gravity! I'm not sure if that was your intent, but your question at least suggests that idea. Gravity and electric attraction have very different properties.

(2) While one can speculate pretty plausibly that a positively-charged point-like positron — that is, an antimatter electron — is "like" a black hole in specific ways, you flatly cannot do that for protons, neutrons, or nuclei. Those have remarkably complex and well-studied structures and finite volumes and are not at all point-like.

I like your enthusiasm for physics. However, like any topic with lots of past work, it's a bit like learning how to drive a car: Until you understand the rules of the road — and, for that matter, where the roads are — it's hard to get traction and move very far. On the other hand, the more you know the "roads" of physics, the better your questions become.

Several online training services, including free ones, can help. I recommend the free Khan Academy service, which makes sure to cover absolutely everything, from basic math up to pretty detailed particle physics:

https://www.khanacademy.org/

Such services not only help you get a feel for what's already out there in physics, but they can also be great fun if you enjoy learning how nature works.