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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.

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    $\begingroup$ 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. $\endgroup$
    – PM 2Ring
    Apr 7 at 20:27
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    $\begingroup$ 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. $\endgroup$ Apr 8 at 3:47
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    $\begingroup$ @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 wtamu.edu/~cbaird/sq/mobile/faqs $\endgroup$
    – PM 2Ring
    Apr 8 at 4:11
  • $\begingroup$ 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 :-( ) $\endgroup$
    – Strawberry
    Apr 8 at 11:04
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    $\begingroup$ Might note that "black holes" are theoretically allowed in Newtonian gravity, and were predicted by Laplace and others in the 18th century: phys.org/news/2017-04-black-holes-theorized-18th-century.html $\endgroup$
    – jamesqf
    Apr 8 at 15:44
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Well, 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

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    $\begingroup$ 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.. $\endgroup$ Apr 8 at 0:05
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    $\begingroup$ Thanks for reminding me. I added a paragraph about Penrose. $\endgroup$ Apr 8 at 0:23
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    $\begingroup$ Nice answer, Viktor! I didn't know that you're a member here. (FWIW, I've used & linked to 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. $\endgroup$
    – PM 2Ring
    Apr 8 at 3:58
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    $\begingroup$ 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. $\endgroup$ Apr 8 at 4:46
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    $\begingroup$ "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 $r = 0$. $\endgroup$ Apr 9 at 10:38
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Its worth being careful about black holes, because its common that some points get mixed up or overlooked.

So I'll tackle your question in a few parts - an outline to help you get the basics right, some common things that are "features" often described wrongly or misunderstood, then what exactly about this is predicted by GR and what that means, and last, how they form.

Basic outline of black holes

General relativity predicts that matter (technically matter/energy: because of E=mc2 they're the same) distorts spacetime. If there is enough matter in a small enough region of space, it distorts spacetime so much that all futures in that region point inwards. What that means is anything in that region of space - whether its gross matter, or light (photons), or anything else, will be unable to avoid its future being closer to the "centre" of that region. Hence why light and matter cant escape, hence why its "black".

So far, thats a prediction of General Relativity (GR).

There are three main physical features of a black hole worth noting at this point.

  • The boundary of space within which all futures point inward and nothing can escape, is known as the event horizon. Because like the horizon, we can't see events beyond it.
    To be clear, nothing physical exists at the event horizon, it's a bit like the border of a country in a way, you could cross it without realising.
  • Somewhere at the centre of that region, is the point (or in some theories, line etc) of space which everything is drawn towards: the gravitational centre. GR predicts that this is a singularity - a point of zero size. But we don't actually know if that's correct, perhaps quantum effects prevent that from happening and the centre of a black hole doesn't go all the way to literally zero size, perhaps it has some tiny size on a quantum/planck scale. We just don't know.
  • Because nothing can escape, black holes have very little to distinguish between them. Theory says that the only properties of a black hole that allow us (from outside) to tell them apart, are their mass, rotation, and charge. Again, we currently believe this but we don't 100% know.

When we refer to a black hole, we usually mean the region of space for which the event horizon is a boundary.

Other key behaviours and features we expected to observe

Its worth mentioning one common misunderstanding, that the event horizon is where gravity becomes infinite. It isn't. The event horizon is the boundary of a region in space, where gravity becomes intense enough that nothing can travel from inside it to outside, and anything inside can only travel "more inside" in its future. If the black hole is enormous, then in theory, the event horizon could be so far from the singularity (or whatever's at the centre), that gravity at the event horizon is intense but not enough to rip you apart, so you wouldn't notice when you cross it. Or at least a smaller particle or molecule wouldn't - you might as you're larger!

From outside, as we watch an object approaching the event horizon, it looks completely different. Light finds it harder and harder to escape, and in effect is slowed down slower and slower. As it's slowed, it also redshifts - which doesnt mean its literally red, it means whatever the wavelength is, it becomes longer and longer wavelength, so it could be redshifted to microwaves, radio waves etc. Closer to the event horizon this effect becomes stronger and stronger, light is slowed more and more, becomes longer and longer wavelength.... So to us it looks like time has frozen. We see the object getting closer and close to the event horizon, dimmer and dimmer, but we never see it passing the event horizon. It looks like its just frozen eternally moving sloooower. That's an illusion. The object passes the event horizon without necessarily noticing anything unusual at that point in space, and continues moving/accelerating inward within the black hole, just like any freely falling object in an intense gravitational field. But we from outside never see that part though.

If the singularity is actually a singularity, GR predicts it's never visible/detectable to anyone, as it's only ever in the future for light and visibility purposes wherever you are in the black hole (technically it would only ever be in the future light cone for all points, so no signals can reach any observer from it, however close). So you don't even know you get there until you do. In fact, technically its not even correct to call it a point - its more like an infinitesimal or zero size gap in the geometry of spacetime, if GR is correct. (This paragraph with thanks to PM 2Ring, see comments)

If you survive long enough to look around, after crossing the event horizon, you'd notice a few other odd things. You'd notice that the event horizon still seems to appear to be in front of you. That's because even within the event horizon, light still can't escape any further outwards (all of a photon's futures are directed inward, like yours). So the same effect that stopped you seeing objects beyond the event horizon before, stops you seeing objects in front of you now, even when you're inside the event horizon. Another effect is, you can't avoid the singularity, and spacetime is so distorted, that the singularity is in fact in any/every direction for you, wherever you are. Every direction you go, or try to move in, takes you closer to it. However you try to move, whatever direction, however fast, you'll end up there in a finite amount of time. But because everything's futures are inward directed, you just won't know you're there until you actually are (because signals can't reach you from there).

Also its worth noting that black holes can lose energy ("Hawkings Radiation"). This can be an extremely slowwwwww process, which for large black holes can take trillions of trillions of years (its very fast for microscopic size black holes). But the energy isn't leaked from inside the event horizon,whatever it may seem. In effect, and described simply, on a quantum scale, very close to the event horizon, gravity can cause negative energy to pass the event horizon from outside, leaving what to us looks like positive energy typically in the form of a photon travelling away from the event horizon. So it looks like the black hole has lost mass/energy, and emitted radiation. But nothing has travelled from inside to outside in reality.

Predictions of General Relativity

So far, those are the understandings we have, and they are mostly predictions made by GR. By that, we mean, when we solve the GR equations and try to interpret the results, this is what they seem to be saying we might expect.

The GR equations for a black hole are not easy to solve in most cases, but some simple cases have known solutions - sometimes exact solutions. (Because of their complexity we don't know exact solutions in all cases).

The fact that GR predicts things, doesn't mean those things actually exist. Equations can have solutions that correspond to things that really exist, and things that don't really exist (whatever real means!). For example if a square has area 4, what are its sides, the equation says there are 2 possibilities: the square has sides -2 or +2. But only one of those actually exists, one does not.

In short, a theory like GR can predict the potential for something to exist, and that if it exists it should behave in specific ways.

We check those match our observations, to see if our theory seems to hold up, and to help us develop theory further. If they don't match, it may mean our observations were wrong, we forgot to include something, or the theory is incomplete/wrong.

So although GR predicts these things, we need to test experimentally, do we find evidence these things really exist. There's 2 parts to that, when we ask if GR "predicts" a phenomenon - can the phenomena/behaviours that our mathematical equations describe or suggest, actually exist in the real world? (Are the solutions actually able to occur or are they "phantom" solutions that don't correspond to anything that can exist in reality). And, if they can exist, do they, can we observe them? Indeed, we do find very strong evidence that they can and do exist for real. For example:

  • We can observe gravitational distortion of light around a region of space where nothing can be visibly seen.
  • We can photograph a black hole and the photo shows huge energy effects around a central dark (lightless) space.
  • We can detect intense gravitational waves by their distortion of spacetime, and match those up with other observations and theoretical models of what a black hole collision would look like according to GR.
  • We can track the orbits of stars at the centre of our galaxy and find they orbit a tiny point where nothing can be seen, but which is too small and gravitationally intense for anything but a black hole.
  • When we measure these things, they seem to match what the GR equations say they should look like.
  • And so on.

So these are ways we test if the predictions of GR about black holes are real, and if black holes are real. So far as we can tell, yes.

Formation

Your last question is how they are formed in nature. That's much easier to summarise.

  • Very massive stars (but not the very very largest!) form black holes at the end of their life. The intense pull of gravity inward is balanced by the energy of fusion processes as they burn fuel. When they reach the end of their burning life, and run out of fuel, this balancing force vanishes, and the star catastrophically implodes. Literally, its core falls inward at a large fraction of the speed of light. Part of that core rebounds and forms a supernova. Part of it is so dense it reaches the limit of how much mass inside how small a space, and constitutes a black hole.
    (The exception for the very largest absolutely enormous stars is that they can be so hot and unstable when fusion ceases, that they blow themselves apart instead of imploding, so no black hole forms)
  • Stars slightly under the size needed to collapse directly into a black hole, can still do so, but they do it in 2 stages. In stage 1 the same thing happens as above, but the result isnt quiiite dense/massive enough to form a black hole itself. But it is massive enough for some of the debris/ejected matter from the supernova, to fall back after exploding outwards, and when enough ejecta falls back onto the remnant, then it will collapse in stage 2, into a black hole.
  • A very dense object such as the remnant of a star that wasn't big enough for a black hole to form, could over time pull in mass ("accretion") or merge/collide with other stars or massive stellar remnants, and the combined object could have enough mass in a small enough space for a black hole to form. The most common object here is a neutron star - a remnant which is incredibly massive and dense, but not quiiiiite enough to form a black hole.
  • High energy collisions could give rise to enough density of energy in a small enough space, to form a black hole.
  • According to some theories, at the beginning of the universe, primordial black holes could have been created, during the Big Bang, when mass/energy densities were much higher than today, and if so, some could still exist now.
  • Many/most galaxies have a central huge black hole, like ours does. That could form over time.
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  • $\begingroup$ +1 My quibble would be your statement that the formation of black holes is easier to explain. Then you haven't explained it right! Forming black holes is a much more complex process to understand than the properties of non-astrophysical, "eternal" black holes, which is what the first part of your answer is about. $\endgroup$
    – ProfRob
    Apr 9 at 11:41
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    $\begingroup$ I meant it in this sense. The properties and behaviours of BH are subtle complex and easy to misunderstand as a layperson. Simplification tends to distort. But the formation of BH can be simplified in a way that doesn't make big misunderstandings so likely. Also looking at the level of explanation, this is about a video contest, and checking basic understandings are on track. Are there any factual corrections you'd like to see? $\endgroup$
    – Stilez
    Apr 9 at 11:45
  • $\begingroup$ Hi, this explanation was very easy to understand, being a high schooler myself. Thanks so much. $\endgroup$
    – AdiBak
    Apr 9 at 17:01
  • $\begingroup$ Added a little on what it means to.say GR "predicts" these, just for clarity. $\endgroup$
    – Stilez
    Apr 9 at 21:54
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    $\begingroup$ Nice work. It's hard to properly understand what GR says about black holes because our geometric intuition didn't evolve to handle curved four dimensional spacetime. ;) Technically, the singularity at the core of a BH isn't even a point. From physics.stackexchange.com/a/144458/123208 "A singularity in GR is like a piece that has been cut out of the manifold. It's not a point or point-set at all". Also, the singularity is never in the past lightcone of any observer, even an observer inside the BH. So the singularity doesn't exist for you until you get there. $\endgroup$
    – PM 2Ring
    Apr 11 at 10:11
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A black hole is defined as something with an event horizon. General relativity does predict that black holes can be formed as the end-products of stellar evolution.

But I saw this site that said 'general relativity is inaccurate at very small sizes' and that it kind deviates from GR.

This is about a singularity. Whether there is also a singularity is a separate issue. The Penrose singularity theorem guarantees that, in GR, there is a singularity. However, the existence of a singularity is interpreted by many people as evidence that GR breaks down.

Observational astronomy of black holes is rather mature at this point:

https://en.wikipedia.org/wiki/Event_Horizon_Telescope

https://en.wikipedia.org/wiki/Gravitational_wave

There is no real doubt that black holes exist and that GR's description of them at scales of ~1 km is very good.

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  • $\begingroup$ Hi, thanks for the input! $\endgroup$
    – AdiBak
    Apr 9 at 17:08

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