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We know how to define black holes as trapped surfaces and solutions from gravitational theories. However, if we define black holes as vacuum solutions/charged solutions to Einstein Field Equations, can we have a well defined black hole interior? Classically, it seems not be problematic unless you realize time-like coordinate flips after passing the black hole horizon.

Do we know if black hole interiors are vacuum too? I ask this because, e.g., there were some trials to define rotating black holes as fluids of some type but it failed as far as I know. So, is black hole well-defined after all? Is it "matter"?

Qmechanic
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riemannium
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A time-line coordinate flips, only if one assumes that Schwarzschild's vacuum solution is valid in the interior of black hole, too. In my view, it is not correct. A solution of Einstein field equations spans the whole spacetime. In case of compact mass it can be cast into two parts: the interior and the exterior one. It is admissible to work with only one of them, and glue it to other, but it is not admissible to extend them over their validity domain. Possible, there is no interior spacetime at all. See for example Aharon Davidson's paper entitled 'Hollowgraphy Driven Holography: Black Hole with Vanishing Volume Interior' https://arxiv.org/abs/1007.1170, or t'Hooft lecture https://webspace.science.uu.nl/~hooft101/lectures/GtHBlackHole_latest.pdf, where he introduced his idea of vacuole on page 42. I believe to have proved that within the framework of classical general relativity.

JanG
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