In a hypothetical case take a hydrogen gas. If we start cooling it down to the point where its core temperature is 0 K, does the electrons become still and fall into the nucleus due to the fact there is no energy to maintain it in the electron cloud? Does this become a black hole? If yes, would this mean that the temperature of the core (singularity) of a black hole is 0 K? If you are frozen in subatomic stage, then you would experience slow time movement as if you were near a black hole (also due to gravitation pull). As we know, heat transfers from higher temperature to lower temperature, and the black hole is like a sink for all hot matter. That's why when matter falls into the black hole it becomes red hot, but since the black hole is more massive it is not quite enough to heat it up. It's like making a campfire on a iceberg. Can it be possible?
5 Answers
The OP contains a long chain of reasoning, with a lot of questions. I will try to cover everything
does the electrons become still and fall into the nucleus due to the fact there is no energy to maintain it in the electron cloud?
No, if you progressively lower the temperature of a hydrogen gas nothing so extreme would happen. The exact dynamic would depend on the pressure you keep the system at, but the following may be a rough description of the process at atmospheric pressure. As you cool the gas from a high temperature, first the electrons fall into the ground state, the lowest energy level where the electron has about $13.6$ eV of energy less than if it was free. At this point you can not extract any more energy from the electrons, if you want to cool the gas further, you can only take away the kinetic energy of the atoms. If you do that, at some point the hydrogen will become liquid and then solidify. Nothing more spectacular will happen as you cool it further.
Does this become a black hole?
No, it is very difficult to make a black hole. You have to concentrate a lot of mass inside its Schwarzschild radius $r_s = \frac{2GM}{c^2}$, which is typically extremely small. We don't have the technology to make a black hole (and probably never will).
If yes, would this mean that the temperature of the core (singularity) of a black hole is 0 K?
As others have already pointed out, there are ways to define the temperature of a black hole, but since it is not true that if you cool a hydrogen gas you obtain a black hole, the chain of reasoning of the post doesn't work. From this point on, every other conclusion the post comes to is flawed, since the premise is wrong.
If you are frozen in subatomic stage, then you would experience slow time movement as if you were near a black hole.
No, cooling a gas doesn't slow down time, it just slows down the molecules of the gas.
That's why when matter falls into the black hole it becomes red hot, but since the black hole is more massive it is not quite enough to heat it up.
Unfortunately black hole thermodynamics can be sometimes counter-intuitive. If you throw hot matter into a black hole, it actually cools down: the temperature of a black hole may be defined as $T = {\kappa \over 2\pi}$, where $\kappa$ is the surface gravity. For a Schwarzschild black hole $\kappa = {c^4 \over 4GM}$. Increasing the mass of the black hole, the temperature decreases.
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1$\begingroup$ Just to add: matter heats up before falling into a black hole due to its vast acceleration, subsequent massive friction, and tidal forces within the hole's accretion disk. $\endgroup$– Zac67Aug 5, 2021 at 20:12
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We don't have the technology to make a black hole (and probably never will).iirc, the radius of a black hole with the mass of Earth would be around 2-3mm. So yes, it seems rather unlikely that we will make even a tiny black hole. $\endgroup$– AnoEAug 6, 2021 at 13:25
Even at the lowest temperatures, a hydrogen atom would still be a hydrogen atom with a proton and an electron. The decay rate of electron capture can decrease with temperature, but this depends on whether we are considering an insulating or conducting material, and does not seem to be a general feature of the nuclear interaction. Further, as the temperature goes to zero (in practice, temperatures of ultracold gas clouds can be ~ 10$^{-12}$ Kelvin), electron capture is an active area of research. One example is the so-called Interatomic Coulomb Electron Capture in ultracold atomic systems, but this has not been achieved yet experimentally.
When a proton does absorb an electron, the result is a neutron and not a black hole. Gravity has negligible influence on nuclear reactions, because it is so weak compared to the other fields that determine the behavior of the particles that constitute atoms. Black holes are gravitational phenomena, and it one of the biggest open questions in theoretical physics to understand the quantum nature of gravity on the smallest scales.
The thermodynamic temperature of a black hole is an interesting question. Depending on how you interpret temperature, it could have a very low temperature-- only non-zero because of Hawking radiation-- or a negative temperature when you interpret temperature as a measure of the change in entropy when adding energy.
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$\begingroup$ Electrons have to cross a high energy barrier No, this is wrong. $\endgroup$– user15381Aug 4, 2021 at 18:09
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1$\begingroup$ @BenCrowell good catch, I've tried to edit it. $\endgroup$ Aug 4, 2021 at 18:49
If you solve the Schrödinger equation for a hydrogen atom you will find that there is no reference to temperature at all. In a sense the solution actually corresponds to a 0 Kelvin hydrogen atom, and if you want to model a "warm" atom you need to use a more complicated model. Also, the probability of the electron fusing with the proton is essentially zero because of energy conservation: the resulting neutron would have more energy than the original atom. So zero Kelvin hydrogen is just a solid sitting there.
Black hole temperatures are (if Hawking et al. are right) nonzero and potentially quite hot for small holes. But this is the temperature of the event horizon as seen by outside observers. The singularity itself doesn't have any observable temperature since there is no way of getting a light signal from it to any observer, not even one falling towards it.
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2$\begingroup$ Black hole temperatures are (if Hawking et al. are right) nonzero and potentially quite hot. This is a little misleading as written. The temperature of a solar-mass black hole, as measured by a distant observer, is in the nanokelvin range. If "potentially" refers to microscopic black holes, then maybe you could say that more explicitly. Or if you mean the temperature as seen by a static observer near the horizon, you could explain that. $\endgroup$– user15381Aug 4, 2021 at 18:13
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$\begingroup$ Yes, that can be confusing, I updated it. I left out the near horizon observers since that raises a few more cans of worms of potential confusion, $\endgroup$ Aug 6, 2021 at 14:10
See here: https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator
The math is rather complex, but the general conclusion is not.
In short, one cannot completely stop a quantum oscillator from oscillating.
In the simple case of an e.g. hydrogen molecule, the vibration always retains the last half quantum of energy. (Tn contrast, the quantum rotator is pretty much stoppable)
The hydrogen molecule is a quantum oscillator. The hydrogen crystal lattice is a bunch of quantum oscillators, too. The hydrogen atom can be modeled in a similar fashion, just to find that one cannot take energy from the atom in its basic state.
Remember, an electron does not follow an orbit. It occupies an orbital. In a lot of possible orbitals it does even not carry an orbital angular momentum.
So no, the movement of the particles never really stops.
No.
As you cool a hydrogen atom, various quantum effects come into play that prevent the electron falling into the nucleus (proton). The whole idea of an "orbiting" electron that behaves as you'd expect when it loses energy, is incorrect. This was actually one of the issues that prompted the entire start of quantum theory, so its worth going back in history to the original problem.
In the macro (large scale/everyday) world, objects orbit objects. They gradually lose energy via friction (unless talking about huge masses or huge timescales when gravitational energy loss is another option), and spiral inwards, and collide.
An early concept of the atom had a central nucleus and orbiting electrons. But this raised some huge problems, including this one:
An orbiting particle is constantly accelerating, because change of velocity including change of direction is acceleration. That acceleration causes a gradual loss of energy, and inspiralling, as well. So how come all atoms havent already collapsed in the last few billion years? What sustains them?
The answer so far as we know, turns out to be that on a small (micro) scale of individual particles and atoms, it doesnt work as youd expect. At that scale, particles just dont "orbit" like large objects do. Instead they have a probability of being found at various positions, that over time average out to the position an orbiting particle might be at. And there they stay, eternally, unless/until some interaction, or chance quantum occurrence, changes the situation. They remain in an undefined position where you cant say exactly where it is, only where it is on average, or where the chances are that it is at some specific time.
This uncertainty isnt because we cant measure properly. It seems to be fundamentally part of how the universe works. You could measure perfectly, and you'd still get no exact answer. The more exactly you measure its position, the less sure you would be about its momentum - the direction or velocity it has, in simple terms. It seems this simply cannot be escaped.
So thats the first difference at the quantum level.
The second difference is that particles dont need energy to move, as such. If they did, then when they ran out of energy, theyd...... do what? And we would be back asking why all the atoms havent collapsed long ago.
So the answer to your first question is that an electron doesnt need energy to stay outside the nucleus or to stay in its fuzzy uncertain location around and near it, and cooling to absolute zero wont stop it moving, and it wont make the atom collapse inwards through being still, even if it could magically stop it.
As to your second question, a black hole requires a certain amount of mass, in a comparatively very small space. A black hole made out of the sun would be about 2 miles (3km) across. One made out of the entire earth would be about an inch across. But you have to ask, what could cause such a drastic collapse? For example, we can collide an electron and a proton easily. But why would we expect a black hole to form? What would cause its collapse? Gravity cant do it on that scale (too tiny). The particles are never still, so a hypothetical absolute zero wouldnt do it. Nothing we know of, could.
(We actually dont believe a black hole can form below a certain mass - and while the value of that mass is debated, its nowhere near the mass of any atom, its far far huger.)
