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Consider a star, for simplicity a non rotating one. The corresponding spacetime will be similar to a Schwartzschild one (if the star is static and spherically symmetric).

Outside the star we will have exactly the Schwartzschild fall-off, while inside the star we will not have a horizon or a singularity, since the matter content of a sphere contained in the star is proportional to the sphere volume itself.

If this star collapses into a black hole, for the same reason I would not expect the black hole to necessarily contain a naked singularity.

I would say that the distribution of matter related to the black hole is all inside the Schwartzschild radius, but couldn't it be that once you enter it the matter spatial distribution prevent the singularity at the center?

The argument seems very similar to the one used in the stellar case.

I think that any motivation on the line of: "inside the horizon of a Schwartzschild metric all the geodesics will crush into the singularity" should be reviewed because you could have a non point-like matter distribution inside and a non Schwartzschild metric, like in the stellar case.

I think that maybe, if you have even the thinnest shell of empty space right inside the horizon photons there will start falling and therefore anything else at smaller radii will be falling, hence bringing to a pointlike singularity.

But even if that was correct, I still think we could have cases where there is no empty space inside the horizon.

So, I would very much appreciate any insight on how are we sure that star collapse in General Relativity brings to a singularity. If it is true, how can it be fully proved? And in particular, where the reasoning in the case of a BH with no empty space would fail?

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marked as duplicate by StephenG, Qmechanic May 19 at 12:38

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  • $\begingroup$ What force are you proposing that stabilizes matter inside the EH and stops that matter from collapsing to the centre? $\endgroup$ – PM 2Ring May 19 at 9:48
  • $\begingroup$ @PM2Ring yeah I am assuming that stability can be achieved. Best guess would be that Strong and EM forces become strong enough or that some other underlying UV interactions will do the job. Is there any reason to think that once you have matter inside a Schwartzschild radius only gravity matters? $\endgroup$ – AoZora May 19 at 10:01
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    $\begingroup$ Possible duplicate of When does a singularity start to exist during a black hole formation? $\endgroup$ – StephenG May 19 at 10:19
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    $\begingroup$ Penrose's singularity theorem guaranties that the spacetime inside the black hole will be geodesically incomplete. $\endgroup$ – MBN May 19 at 12:01
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    $\begingroup$ Sure, for normal stars, pressure opposes gravity, and we can mostly ignore the other components, apart from energy, of the stress-energy tensor when calculating the spacetime curvature. But eventually, the gravity caused by the pressure must be taken into account. $\endgroup$ – PM 2Ring May 19 at 12:35