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Everything I read about black holes—discussions using Penrose diagrams and Kruskal coordinates, etc.—seems to be based on vacuum solutions to EFE. Sometimes it’s said that all trajectories entering a BH must end (or not end, as you prefer) at $r=0$, where curvature is infinite.

But my naive belief is that the mass-energy of a BH must still be there, or we wouldn’t be observing all the curvature in its vicinity. And if some region of $r>0$ contains this mass-energy, then a vacuum solution wouldn’t be valid there. What I want to know is, do some non-vacuum solutions (or approximations) determine geometries without the "everything goes to $r=0$" property, thus allowing for some kind of “pile-up” around the center of a BH? Is it possible that nature protects the singularity, not merely by covering its nakedness, but by preventing things from getting there?

Qmechanic
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gabe
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    What you're looking for is called a regular black hole; I'm pretty sure they don't exist in classical GR. – Javier Jul 25 '20 at 17:45
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    Related: https://physics.stackexchange.com/q/75619/2451 , https://physics.stackexchange.com/q/18981/2451 , https://physics.stackexchange.com/q/194391/2451 and links therein. – Qmechanic Jul 25 '20 at 20:59
  • After more reading I can reformulate my question more precisely as “Do the singularity theorems apply to all EFE solutions, only the vacuum ones, or just some of the non-vacuum ones?” I also think I’ve figured out that the answer is “almost all”, with exceptions being solutions with closed timelike curves or violations of some energy condition.

    I found this very helpful history of singularity theorems and their implications. I am still studying it. https://arxiv.org/abs/1410.5226

     – gabe Jul 27 '20 at 16:22
  • The original Schwarzschild solution has no interior, as the horizon is at $r=0$: https://arxiv.org/abs/physics/9905030 - If you redefine the radial coordinate to extend to the "inside" passed the point of $r=0$, you naturally get non-physical imaginary values for the radius. The mainstream interpretation today is that the radius "inside" becomes timelike. However, this interpretation is arbitrary, creates the "information paradox", and is just as non-falsifiable as life after death - no one can return from "the beyond" to tell us it really exists. – safesphere Jul 30 '20 at 21:31
  • Also please note that the volume inside a Schwarzschild black hole is zero: https://arxiv.org/abs/0801.1734 - This is consistent with the original interpretation of Karl Schwarzschild that no interior spacetime exists since the radius of the horizon is zero. It also is worth mentioning that the area of the horizon is not zero, even if both the radius and volume inside are zero. While this is counter intuitive, there is no contradiction, because the radial metric diverges at the horizon and keeps the surface area finite even at the zero radius. – safesphere Jul 30 '20 at 21:37

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