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?