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A test body crosses the event horizon of a black hole. When can we say that GR breaks down? Only at the singularity? At any point CLOSE enough to the singularity? How much close? In summary, do we know when quantum gravity begins to be acting at distance X greater than the point when general relativity breaks down?

Details: I mean GR being not valid to calculate, e.g.:

  1. Firstly, curvature effects, geodesic motion and/or possible modifications of space-time at higher densities.
  2. The gravitational field and or derived quantities (is the metric well-defined in any point inside the black hole excepting the BH singularity?).

At what point the density of a black hole or the curvature is so strong that we can say GR needs a replacement? Of course, Hawking radiation is also there, ... And, the caveat is also if we can talk about “a point” inside a black hole or close to the singularity...

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    $\begingroup$ You'll have to clarify what you mean by GR failing. For example GR doesn't predict Hawking radiation. If you consider that failing then it fails at the horizon. $\endgroup$ – John Rennie May 15 at 9:23
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General relativity by itself does not provide a scale at which it becomes invalid. So, we do not know when it breaks down for the description of a black hole (or a black hole-like object). While we may suspect that the breackdown occurs inside a black hole when curvature approaches Planck values this is only a conjecture largely based on dimensional arguments. There are many alternative suggestions about how quantum gravity effects may modify compact gravitating objects in such a way that there are no black holes at all (in the sense that there are no event horizons) but instead some “black hole mimickers” or “exotic compact objects” with some quantum gravitational physics occurring at small (possibly microscopic) distances outside would be horizons. While such alternatives are not too popular among the proffessionals of the field they are largely not ruled out by either observations or by purely theoretical arguments.

For an overview of a “zoo” of black hole mimickers and possible tests that could constrain/falsify different theories have a look at this paper, a more recent review by the same authors as well as this answer.

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  • $\begingroup$ This seems reasonable. How do we quantify more carefully the notion that the "curvature approaches Planck values?" If we do this by testing the components of the Riemann tensor, then it's coordinate dependent. $\endgroup$ – Ben Crowell May 15 at 20:11
  • $\begingroup$ @BenCrowell something something Kretschmann scalar $\endgroup$ – AccidentalFourierTransform May 15 at 22:09
  • $\begingroup$ @BenCrowell: As AccidentalFourierTransform have said. We expect the effective action for QG contains terms quadratic in curvature, so when they become noticeable deviations from EFE would be significant. $\endgroup$ – A.V.S. May 16 at 4:56
  • $\begingroup$ Assuming the Schwarzschild metric, space inside a BH is a 3-cylinder with time pointing from its hypersurface to the axis of singularity. There are only 3 logical ways for timelike/lightlike geodesics to go. (1) End in the singularity regardless of whether or not GR holds near. Matter and spacetime end together as Fourier conjugates. (2) Turn through a wormhole to create a different universe from a white hole. (3) Turn again the way they were outside the horizon pointing along the axis of the 3-cylinder. This creates a causally disconnected eternal dense timelike object. Any logical flaw here? $\endgroup$ – safesphere Oct 25 at 16:39
  • $\begingroup$ Matter and spacetime end together as Fourier conjugates I do not know what that means. But an alternative that you have not included is that the “inside” of black hole is a background for completely different structures that could not be interpreted as Lorentzian geometry (or even as spacetime). $\endgroup$ – A.V.S. Oct 26 at 5:40
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It is difficult to say anything too definitive about quantum gravity but following the ideas of effective field theory, one often considers Einstein-Hilbert action as the first term in an infinite series $$ {\cal L} = \frac{1}{4\pi} \int d^4 x \sqrt{g} \Big( \frac{1}{G_N}R +R^2 +\ldots \Big) $$ where by $R^n$ I mean all curvature invariants which can be constructed which have $2n$-derivatives on the metric and one should include some unknown scalar constants.

So general relativity breaking down is then understood as the region where the approximation to the first term is no longer valid, which roughly speaking is when curvatures are of order the Planck scale (very large). Exactly where this happens inside a black hole is not yet particularly well understood but one can evaluate the curvatures at the horizon of an astrophysical size black hole and conclude that the curvatures are indeed much below the Planck scale, so one should say that general relativity is valid at the horizon. There are more sophisticated recent arguments (look up the "Firewall") which question this conclusion but require additional ingredients.

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Remember that to an outside observer, 'events' within the event horizon remain forever in the future as the external observer never sees the event horizon form.

From the outside, as matter falls towards the future forming horizon, time dilation will make it appear to slow down. At the same time, the same time dilation is responsible for extreme gravitational redshift, so any light coming from these falling objects is redshifted to invisibility. In other words, falling matter disappears from sight (becomes impossible to detect) but it is still there; it never reaches the horizon, which in turn never forms.

This is what an outside observer sees. Consequently, there is no breakdown in relativity theory, everything works just fine, and apart from Hawking radiation (which is so trivially small for an astrophysical black hole, it might as well not be there) general relativity describes the physics flawlessly.

So the only way to experience an event horizon is to fall through it. It is true that an falling observer reaches the horizon in a finite amount of time, as measured by his/her clock. (But it’s an infinite amount of time as measured by an outside observer’s clock.) Once this falling observer crosses the horizon, the singularity is no longer a location in space; rather, it is an unavoidable future moment in time. Any world-line that crosses the event horizon ends at the singularity.

So the place where physics breaks down is not the centre but rather, this future moment, accessible only to observers who fell through the horizon. Physics 'breaks down' because as this moment approaches, the intensity of the gravitational field increases without limit. Since we do not like infinities, there is also the issue that eventually, the gravitational field reaches the point when it will be comparable in strength to other forces, and when its putative quantum nature can no longer be ignored.

There are some interesting questions to ponder here, including the possibility that Hawking evaporation may mean that no horizon forms in the first place… but the nature of this breakdown of physics is similar to the breakdown of physics in the very early universe, near the initial singularity (aka. the Big Bang), and there is no horizon that would hide the Big Bang from us. So never mind black holes, in the extreme early universe, when energy levels were very high and the gravitational field was very strong, we have the same problem: the quantum nature of gravity can no longer be ignored.

As there is no proper quantum theory of gravity. There are many proposals, but none really satisfactory. In fact, some people wonder if gravity is a quantum field theory in the first place. And there are very few ways to test these ideas.

The infamous BICEP2 observation of polarisation of the cosmic microwave background, presumably representing an imprint of primordial gravitational waves, was seen as a sign that gravity is a quantum theory; unfortunately, the observation was invalidated when it became clear that they were not able to isolate signal from noise. So we remain kind of in the dark: Should we look for a quantum theory of gravity? Something more radical? Or keep gravity classical? Perhaps contend ourselves with “semi-classical gravity”, which, after all, works very well in all domains accessible to us by observation or experiment?

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Quantum gravity is not "acting", it is the attempt to harmonize general relativity with quantum mechanics. Most theories of quantum gravity are based on the attempt of quantization of spacetime, this implies that quantum mechanics is the universal concept for the description of the universe, and GR becomes a part of quantum mechanics. If we say that "GR breaks down", this means that we are reaching the limits of GR, but this must not imply that GR is wrong (there is no reason for this because GR has been widely proved experimentally), it simply means that we have a problem with the harmonization of GR and QM.

Observations on the basis of GR are limited by the principle of cosmic censorship. We cannot see beyond the event horizons of black holes. But this is not obligatorily a "failure" of GR, because our observations may go until the end of time. The main example is an infalling object. From the moment the object is reaching the event horizon GR does not describe this object any longer, but from the point of view of outside observers, the object will never reach the event horizon (only at "infinity"), and the object is perfectly described by GR until such infinity. In this sense, GR is a complete system for the description of the objects of the universe, and there is no failure because it is evident that GR will not describe phenomena which are happening after "the end of time" from the point of view of all outside observers.

There are attempts to describe the inside of black holes, such as the Kruskal metric, and we know that from the point of view of an infalling observer he will reach the event horizon within finite time. But none of these metrics is permitting the derivation of what is happening inside the black hole. For this purpose we have to take into account that everything beyond the event horizon is happening after the end of our time (!), and for this reason it is only normal that GR does not describe such events. We may speculate that beyond the event horizon there is a universe after the end of our universe, a spacetime after the end of our spacetime, and that ¬- if there are observers in this new universe - a new kind of GR does apply which has no common point with our universe. But this is only speculation.

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  • $\begingroup$ I downvoted because the text is repeatedly muddled. e.g. "beyond the event horizon is after the end of our time" (but Schwarzschild time is not the only possible measure); "none of these metrics is permitting the derivation of what is happening" but a metric is a statement of what is happening; "GR does not describe these events" but it does. And then finishing in wild speculation. $\endgroup$ – Andrew Steane Jun 27 at 8:36
  • $\begingroup$ @ Andrew Steane: Thank you very much for your feedback! Yes, you got the key point of my answer: I refer to the time of any clock of any outside observer. The introduction of other metrics may be useful for certain purposes of theoretical physics/ theories of quantum gravity, but why say GR is failing, referring to cases where all clocks of all outside observers have reached infinity? For me, such conclusions (of failure of GR) are drawn too easily. $\endgroup$ – Moonraker Jun 27 at 9:03
  • $\begingroup$ @AndrewSteane It is unfortunate, that you dhad to ownvote a good answer. Let's analyze your objections. "but Schwarzschild time is not the only possible measure" - The Schwarzschild coordinates cover the entire 4D physical manifold to the infinity of space and the eternity of time. This manifold is static and consists of spacetime events. (Here "static" is not in the sense of "static spacetime", but in the sense that worldlines don't change.) Please note that in this entire manifold, the event of crossing the horizon does not exist. Sure you can use different coordinates by mapping... [cont] $\endgroup$ – safesphere Oct 25 at 7:52
  • $\begingroup$ [...] them to the Schwarzschild coordinates in a mathematically valid way. However, no change of coordinates, as our description of reality, changes the actual reality. The Schwarzschild coordinates cover the entire spacetime. This spacetime does not include the event of crossing the horizon. Therefore no other valid description, no other valid choice of different coordinates would "create" an event that simply does not exist in reality. In no other "measure" of time the event of crossing the horizon can be observed. If you disagree, please kindly specify a valid frame observing this event. $\endgroup$ – safesphere Oct 25 at 8:01
  • $\begingroup$ @AndrewSteane Furthermore, your objection on the metrics does not hold either. Since the horizon cannot be crossed, there is no physical process that can lead to its creation. You could assume the existence of "eternal" black holes, except the universe is not eternal. This throws the validity of any inner metric out of the window. All these metrics describe what would happen, but not what actually happens. And GR indeed does not describe what is based on incorrect assumptions. Finally, the answer ends on the consensus idea that the inner spacetime is causally disconnected from our universe. $\endgroup$ – safesphere Oct 25 at 8:19

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