I have read this question:
The fundamental confusion many have about black holes is thinking that they are discrete "things" surrounded by horizons and other phenomena. But they are actually extended spacetime curvature structures (that imply the various phenomena). The singularity is not doing anything and is not responsible for the gravitational field, it is a consequence of the field.
What are tidal forces inside a black hole?
As far as I understand, as per general relativity, spacetime curvature is caused by stress-energy (not mass). This answer is using a vacuum solution to describe black holes, and you can read in the comments to that question that there is no need for any matter (or mass) to be present inside the black hole, it is just a vacuum, but spacetime itself is curved, and the gravitational field itself has the energy needed for the curvature itself. This includes the singularity itself, which in this answer is described as being "off the metric", that is not part of our spacetime, hence, it cannot cause the curvature.
Now, if the interior of the black hole, is a vacuum (the model is a vacuum solution), meaning the collapsed star's gaseous matter is not there (as far as I understand it is in the singularity), and the singularity is not part of our spacetime, then neither can cause curvature.
Again, GR describes curvature as being caused by stress energy. If there is no matter, no mass, nothing with stress-energy inside the black hole, except the singularity, but the singularity is not part of our spacetime, then what causes the curvature?
There are suggestions in the comments, that the collapsing star's gaseous matter transforms into the energy of the gravitational field itself. But I do not understand how electrons and quarks can transform into gravitons.
Still, how can the gravitational field itself cause the curvature, or how can it sustain itself? Gravity sustains itself, curvature means stress-energy in the gravitational field, and this energy causes curvature?
Question:
$1$. If black holes are just an empty vacuum of space inside, then what causes the curvature?
Best Answer
This statement is slightly wrong and is the cause of your confusion here.
Technically, in GR the stress energy tensor is the source of curvature. That is not quite the same as being the cause.
An easy analogy is with Maxwell’s equations. In Maxwell’s equations charge and current density are the sources of the electromagnetic field. However, although charges are the source of the field there exist non trivial solutions to Maxwell’s equations that involve no sources. These are called vacuum solutions, and include plane waves. In other words Maxwell’s equations permit solutions where a wave simply exists and propagates forever without ever having any charges as a source.
Similarly with the Einstein field equations (EFE). The stress energy tensor is the source of curvature, but just as in Maxwell’s equations there exist non trivial vacuum solutions, including the Schwarzschild metric. In that solution there is no cause of the curvature any more than there is a cause of the plane wave in Maxwell’s equations. The curvature in the Schwarzschild metric is simply a way that vacuum is allowed to curve even without any sources.
Now, both in Maxwell’s equations and in the EFE the vacuum solutions are not particularly realistic. Charges exist as does stress energy. So the universe is not actually described by a vacuum solution in either case. So typically only a small portion of a vacuum solution is used to describe only a small portion of the universe starting at some matching boundary. A plane wave can match the vacuum region next to a sheet of current, and the Schwarzschild solution can match the vacuum region outside a collapsing star.
So realistically, the cause of the curvature would be stress-energy that is outside of the vacuum solution, in the part of the universe not described by the Schwarzschild metric. This would be in the causal past of the vacuum region including the vacuum inside the horizon. Since it is in the causal past it can be described both as the cause and the source of the curvature, with the understanding that it is strictly outside of the Schwarzschild metric which is a pure vacuum solution in which the curvature has no source.