### Does General Relativity really predict Black Holes?

I'm doing some research on black holes for a science video contest. I want to explain the physics of how they work, but also want to have a little background on how they're formed. As far as I've searched, black holes are predicted by general relativity (GR). But I saw this site that said 'general relativity is inaccurate at very small sizes' and that it kind deviates from GR. So I want to confirm whether black holes really are a prediction of GR or an inconsistency with GR. Can someone please help? Thanks.

The linked site that says "general relativity is inaccurate at very small sizes" is relatively inaccurate. That would imply having observations that are inconsistent with GR. A confirmed observation that showed GR to be inaccurate at any scale would be worthy of a Nobel Prize. All observations of the universe to date are consistent with general relativity. The key problem with GR is that it is a classical (i.e., non-quantum) theory. The key problem with quantum mechanics is that it can't yet describe gravity. That's a double-ended impasse, at least for now.

@DavidHammen I agree that in isolation, his statement is a little misleading, but the linked page does make it clear that we need some kind of quantum correction to GR to properly model what happens at the core of a BH. The author appears to be well-educated in QM topics, but I suspect he's not a GR expert. See https://wtamu.edu/~cbaird/sq/mobile/faqs/

Presumably you've read A Brief History of Time? (I've read it several times, and each time I understand it a little less than the previous time :-( )

Might note that "black holes" are theoretically allowed in Newtonian gravity, and were predicted by Laplace and others in the 18th century: https://phys.org/news/2017-04-black-holes-theorized-18th-century.html

@Strawberry: That's clearly because you understand more real physics.

@DavidHammen But the reason why no experiment has shown any inconsistency with general relativity is precisely because we can’t do experiments with gravity on very small scales. Measurements of the gravitational field of a single electron in a double-slit experiment would be enormously informative if they were in any way feasible.

Viktor Toth Correct answer

one year agoWell, yes, but we must be careful with the meaning of "predict".

The Schwarzschild solution, developed by Karl Schwarzschild in 1916 [1], is the first closed-form, explicit solution of Einstein's field equations for gravitation. It describes a spherically symmetric, static, vacuum spacetime. The solution goes singular at a specific radius (the Schwarzschild radius). In the weak field limit, it correctly replicates the Newtonian gravitational field of a compact object.

Though Birkhoff's theorem (general relativity's version of the shell theorem) was not yet known in 1916, the Schwarzschild solution was nonetheless recognized as the general relativistic description of the gravitational field outside a compact gravitating body. The fact that it went singular at the Schwarzschild radius was either ignored or taken to imply that objects that are as small as, or smaller than, this radius cannot exist.

In any case, just because a solution exists in general relativity does not mean that objects described by that solution exist in Nature. (E.g., the field equations admit solutions that blatantly violate causality; yet I don't see any time machines out there.) For all they knew, the Schwarzschild solution was nothing more than a mathematical curiosity, an idealized case that does not describe reality.

For this reason, I'd suggest that a much more groundbreaking paper is that of Oppenheimer and Snyder from 1939 [2]. This paper demonstrates that a spherical cloud of dust initially at rest will undergo "continued gravitational contraction". An observer that is falling with the collapsing matter would see total collapse in a finite amount of time, but to a distant observer, the collapse will continue forever, the object asymptotically approaching, but never quite reaching, its Schwarzschild radius. And it was in 1957 I believe that Regge and Wheeler first demonstrated that a Schwarzschild singularity is stable under small perturbations [3], i.e., perfect symmetry is not required. (Wheeler, of course, was also the first to popularize the name, "black hole".)

Lastly, as I was reminded in a comment, we should not forget Penrose's 1965 singularity theorem [4], which introduces the concept of a trapped surface and shows that gravitational collapse is indeed a very generic feature of general relativity. After all, this is the result that earned Penrose the 2020 Nobel prize in physics.

In light of that, I think we can confidently state that general relativity predicts black holes, but only because we know not only that black hole solutions exist, but also that physically realizable configurations of matter can collapse into black holes (or, at the very least, to objects observationally indistinguishable from black holes, which of course may evaporate in stupendously long but finite timeframes due to Hawking radiation [5]) and that the solutions are stable under small perturbations, i.e., perfect symmetry is not a prerequisite.

[1] https://arxiv.org/abs/physics/9905030

[2] https://journals.aps.org/pr/abstract/10.1103/PhysRev.56.455

[3] https://journals.aps.org/pr/abstract/10.1103/PhysRev.108.1063

[4] https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.14.57

[5] https://www.nature.com/articles/248030a0

Thanks for your answer, good to learn a bit more about the history of BH research. Could you place last year's Nobel, which was partly awarded to Penrose for showing that collapsars can form blach holes, into the context of your answer? I think that would help many people who are not super-familiar with the topic..

Thanks for reminding me. I added a paragraph about Penrose.

Nice answer, Viktor! I didn't know that you're a member here. (FWIW, I've used & linked to https://www.vttoth.com/CMS/physics-notes/311-hawking-radiation-calculator *numerous* times, both here & on the Physics stack). Perhaps you could add something to this answer about the $r=0$ singularity of the Schwarzschild black hole and the relevance of quantum mechanics and quantum gravity.

Thanks. I mostly just lurk here. Regarding the $r=0$ business, the Schwarzschild solution inside the horizon is a disconnected, separate coordinate patch with $r$ (or rather, $-r$) playing the role of timelike coordinate. As to what its relevance is to quantum physics... I don't know! (Do horizons even form or does Hawking evaporation "win" first?) I wish I had a nice working quantum theory of gravity to help craft a sensible answer, but I don't.

"Schwarzschild radius" is almost always used (in modern discussions) for the location of the event horizon, which turns out to be just a coordinate singularity (an artifact of Schwarzschild's coordinates which can be removed by using different coordinates), not a true physical singularity like the one at $r = 0$.

Hi, thanks for the clear, detailed input! Helpful to know about such studies.

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PM 2Ring one year ago

You may find my answer here helpful. Also see this question, especially Florin's answer. The Schwarzschild black hole was actually the very first solution found to the Einstein Field Equations of GR (which aren't easy to solve), but it's not quite the same as a black hole formed by star collapse.