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The Schwarzschild solution is the standard example used to describe a black hole, its important points being the event horizon and the central singularity. But this solution is derived by assuming an empty spacetime around a point with a "mass" parameter.

All geodesics inside the horizon point inexorably inward for this solution. But why does this preclude a clump of matter or "collapsar" at the center? For example, the Schwarzschild solution also describes the field around a planet, except that the planetary matter occupies the field's central region. At the surface of the planet, the field description transitions from the exterior to the interior Schwarzschild solution (assuming idealized geometry, obviously).

What is it that preludes the predicted field, even at $r<r_{Schwarzschild}$, from simply terminating at the surface of a collapsar of dense matter, and taking some other form inside it?

(Obviously the field could be solved numerically for such a structure, rather than fitting it to a theoretical metric like Schwarzschild, but I presume something happens in the equations that makes this unphysical.)

Qmechanic
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RC_23
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Well the famous singularity theorems show (very very roughly speaking) that in the theory of classical GR, collapse beyond horizon implies a singularity. Classical GR is not the true theory of physics, and quantum gravity effects might do something like what you say. But probably not for $r$ outside the region of Planck scale curvature, which is effectively almost a singularity anyway. Also, too dense "planets" require infinite pressure. Look up singularity theorems and Buchdahl limit. There are metrics like the one you want e.g. Bardeen, Hayward. Whether they exist in physics is another question.

Joe Schindler
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What is it that preludes the predicted field, even at $r<r_{Schwarzschild}$, from simply terminating at the surface of a collapsar of dense matter, and taking some other form inside it?

This is rather simple. The radial direction inside the horizon is timelike. Nothing can stay at a constant radius inside for exactly the same reason why we cannot stay in time at noon without moving to afternoon. No matter what structure you can imagine, a collapsar or whatever, inside the horizon it must move forward in time toward the region known as singularity where time ends.

The shape of this region is a moment of time along an infinitely long spacelike Euclidean line in the coordinate chart - this region is not a part of the spacetime manifold. In other words, the singularity is a coordinate location that does not exist in the real spacetime. See a visual diagram and geodesics here for details: https://math.stackexchange.com/questions/2929400

Please note that commonly the term collapsar refers to a collapsed star whose matter remains outside the horizon in the coordinates of a remote observer. Since the radial direction outside is spacelike, matter can appear to remain at approximately the same radius indefinitely.

safesphere
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According to Thorne and Susskind, the fact that a black hole's mass is distributed as a 2-dimensional surface (the event horizon), and not a volume, indicates that the black hole has no interior.

You can also look at it as a singularity smeared out across space (but not time).

You want to impose a euclidian geometry on space that really isn't euclidian because of its metric signature.

The fact that spacetime is non-euclidean only becomes noticeable at unusual extreme instances, but a BH is one of those instances. Another is the time dilation experienced by GPS satellites. And the precession of Mercury's perihelion.

A black hole is a topological hole in the manifold. Spacetime bends around it. There is no "there" there. You just have to accept reality the way it is, and stop trying to figure out what's "inside" the black hole.

If that seems somehow unsatisfying and even irritating, take it up with God. I just work here (as his humble PR explainer).

Miss Understands
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