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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?

AoZora
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  • What force are you proposing that stabilizes matter inside the EH and stops that matter from collapsing to the centre? – PM 2Ring May 19 '19 at 09:48
  • @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? – AoZora May 19 '19 at 10:01
  • Remember, pressure also contributes to gravity, so once the pressure is high enough, you get a runaway forward-feedback process leading to collapse. The Tolman–Oppenheimer–Volkoff limit tells us the maximum size of a cold neutron star before that pressure feedback process becomes inevitable. – PM 2Ring May 19 '19 at 10:10
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    Possible duplicate of When does a singularity start to exist during a black hole formation? – StephenG - Help Ukraine May 19 '19 at 10:19
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    Penrose's singularity theorem guaranties that the spacetime inside the black hole will be geodesically incomplete. – MBN May 19 '19 at 12:01
  • pressure should oppose to gravity, so I don't get P2Ring's remark. On the other hand I think that @MBN point could lead to the correct explanation. But I am confused: does the theorem assume a trapped lightlike surface inside the horizon? Could this be avoided in the case of a BH with no empty space inside? Or maybe is just a matter of saying that at those energy scales gravity is way stroger than anything else we could reasonably come up with for matter UV interactions.. – AoZora May 19 '19 at 12:32
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    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. – PM 2Ring May 19 '19 at 12:35
  • @StephenG thanks for the lead but while the topic is reasonably similar (even though I'd like to discuss and understand my temptative argumentation), the answers are not taking into account possible problems with the creation of a singularity (one uses pressureless dust and the other is very qualitative) – AoZora May 19 '19 at 12:38
  • The EH is a trapped surface. That suffices for the theorem. The energy scales and the possible matter pressure or other properties are irrelevant here. All this can happen in vacuum. – MBN May 19 '19 at 13:53
  • The physical existence of singularities has not been verified, and there are various different ways of avoiding the naive singularity in ordinary GR. Firstly, you need to include higher curvature corrections to Einstein-Hilbert term in the action; these become important at very short distances. Secondly, you need to include quantum effects at these short length scales. We don't yet know what the correct, unambiguous, physical explanation for a singularity is. – Avantgarde May 19 '19 at 19:22

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