When GR is not valid passing the event horizon?

Question

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...

Answer 1

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.

Answer 2

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.

Answer 3

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?

Answer 4

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.