Schwarzschild metric black hole

Question

Schwarzschild metric solution presents two singularities. An apparent one at $r=2GM$ and a real one at $r=0$. It is known that everything freezes at the event horizon from an outside observer point of view. What are the physical interpretations or theories that describes the inner observations when $0<r<2GM$?

Answer 1

The "theory" describing the black hole interior (in the classical approximation) is the same theory that implies the existence of the black holes, namely the general theory of relativity.

As the OP correctly said, the singularity at the event horizon is a coordinate singularity – one that is an artifact of a bad choice of coordinates.

When a coordinate transformation is performed (into new coordinates that are related to the Schwarzschild coordinates in a singular way near the horizon), the event horizon becomes a smooth light-like surface in the spacetime – much like the $ct+z=0$ hyperplane in the normal Minkowski space.

The Penrose (causal) diagram above is a sketch preserving all causal relationship in the spacetime. Each point of the diagram corresponds to a sphere $S^2$ of the spherically symmetric spacetime - the angular coordinates are suppressed (and the radius of the sphere isn't pictured). The remaining two coordinates, say $t,r$, are reparameterized in such a way that the $rt$-part of the metric is, in the diagram $rt$ coordinates, $$ ds^2 = (-c^2 dt^2 +dr^2) K(r,t) $$ Up to a place-dependent scaling, it's just the Minkowski geometry. $K$ implies that some places of the diagram contain more volume of concentrated spacetime than others. But the correct relative coefficient between $r,t$ says that all lines on the diagram that are tilted by 45 degrees are null, all "more vertical" lines are timelike, and all "more horizontal ones" are spacelike. Massive objects may only move along timelike paths.

The spacetime with a black hole then literally looks like the "Penrose diagram" on the picture. There is nothing special about the interior. It's just a curved region of the spacetime. The curvature is larger there than outside and the curvature invariants go to infinity near the singularity – which is a future event for anyone who ever crosses the event horizon (dark green 45 degree line).

The small tooth at the very top of the picture is only there if one discusses Hawking evaporating black holes.

So all the classical observations by any observer may be perfectly calculated using the universal laws of GR. In particular, no information can ever escape from the purple triangle interior to the exterior region because one would need superluminal, spacelike trajectories for that (given by lines that are closer to horizontal ones than to vertical ones in the Penrose diagram).

Note that the normal black hole is normally thought of as a static object. But as soon as one chooses any coordinates that cure the fake singularity at the event horizon, the static character becomes invisible. The picture above is in no way translational invariant under vertical translations. Also, in the black hole interior, the Schwarzschild "translations" are spacelike, so the perceptions of any observer inside are unavoidably time-dependent – for example, after some time, every observer unavoidably dies.

At the quantum level, the identification of the fields, information, and relationship of the interior with the rest of the spacetime is a difficult question that is the topic of active research of quantum gravity experts. There are some genuine insights about the rough sketch. But even very well-known physicists disagree and some physicists have gone so far to argue that the calculation of GR that implies that the interior exists and has a specific geometry is invalidated in quantum gravity – so that no interior exists at all. They say that the natural resolution of some paradoxes they believe to exist is that any infalling observer is burned by a "firewall" already at the event horizon. Most experts disagree with that.