General Relativity downfall point/radius in BH physics [closed]

Question

Studying GR, sometimes one asks himself/herself where GR fails to describe physics. For simplicity, take a Schwarzschild solution (nonrotating chargeless black holes), what of the following conditions are expected to be true and/or equivalent in order to solve the downfall of GR that GR itself predicts? Consider BH is an effective theory very close to "the true theory" but not the whole story. Then, are the following cases equivalent?

a) Does if fail at the event horizon? Case 1: BH horizons are only apparent or quantum atmospheres make EH fuzzy.

b) Does if fail whenever you cross the event horizon, since metric spacetime signature flips at that boundary? Case 2: after crossing BH EH the metric flips space-like and time-like coordinates, thus signature is relative and we should seek a theory of relativity for metric signatures.

c) Does it fail only at the center, when we find out a spacetime singularity? After all, a change of coordinates shows us that only r=0 is a problem for curvature and density. Can we really think a BH has an infinitely dense object when it is yet extended over the event horizon scale? Case 3: the only real problem is at the center, in the black hole singularity, and quantum gravity will erase this problem.

d) Maybe GR does not fail but only gets wrong answers since we are not yet armored with quantum gravity and the proper TOE to understand the fate of BH singularities, beyond the scope of a GR approximation. Case 4: GR is only an effective field theory, thus, this question is irrelevant.

e) Does GR fail in the firewall before the EH reaches? Case 5: firewalls are inevitable.

f) Does GR fail whenever we reach the fuzzball scale from the string theory proposal? Case 6: fuzzballs will solve the divergences and the information problem

g) Does GR fail whenever we reach the spin network scale, close to the planck length(radius)? Case 7: Planck scale and/or Loop Quantum Gravity will enter into the game to clarify the GR downfall.

Should we disregard BH singularities as they would imply and infinite resource or infinite information recording machine?

BONUS: Can the point where GR fails be related to the information paradox or is that an independent problem?

Answer 1

When we say "Theory X fails," the context we typically have in mind is that there is a more complete Theory Y, to which Theory X is an approximation. For concreteness, suppose Theory X is a good approximation to Theory Y when some parameter $\epsilon \ll 1$. Then "Theory X fails" when $\epsilon \gtrsim 1$, or in other words when $\epsilon$ takes a value where $\epsilon \ll 1$ is no longer a good approximation.

In the case of GR, we typically have in mind that a "more complete theory" would be a theory of quantum gravity. There are a few complications:

(a) There is no experimentally tested theory of gravity. So we don't know precisely what the correct theory that replaces GR is.

(b) Even for specific theories of gravity, it is often not known how to treat a black hole. The effective theory of gravity would tell us that GR "should" work as a good approximation until the curvature becomes of order the Planck scale. However, in string theory, there are many proposals (such as fuzzballs or complementarity or the firewall), where GR will begin to breakdown at the horizon. And in loop quantum gravity, as far as I understand it is not known how to take a classical limit to recover GR, which would mean it is not even known if GR is an approximation to LQG in some regime.

All of this is to say that there is no "first principles" way for us to say when GR breaks down, in the sense of saying that GR as a description of physics breaks down. There is not a unique theory for what replaces GR, and exactly what happens in specific theories is not always known.

However, there are places where we can say that the predictions of GR don't make physical sense. In the case of the Schwarzschild black hole, the singularity is the point where the solution breaks down mathematically. Physicists generally expect a more complete theory to explain what happens at the singularity in a more satisfactory way, but whether the true theory of Nature will agree with GR until "near the singularity" or will begin to disagree at a larger length scale (such as "near the horizon") is not known.