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From Schwarzschild metric, we know that every concentration of matter and energy such that all of its mass $M$ is concentrated in a radius $r_s\leq\dfrac{2GM}{c^2}$ turns into a black hole.

Nonetheless, the gravitational attraction pulls mass to its centre, and therefore its volume could be reduced over time. Does this happen? If so, can it go to an infinitesimal volume? Is there a maximum density that a black hole can have? (I assume its minimum is the one given by $r_s$) Can it come to a really high density, for example, Planck's one?

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  • $\begingroup$ Your opening statement is incorrect. It follows from the metric that the Schwarzschild radius is timelike while its spatial part is zero. So you must compress matter to a zero radius to create a black hole. The volume of he black hole is also zero: "There is zero volume inside the black hole in any Schwarzschild time slice of a Schwarzschild black hole spacetime. - arxiv.org/abs/0801.1734 - On the other hand, energy depends on the time dilation while the coordinate time stops at the horizon, so density is undefined. The geometry of curved spaces is very unusual and counterintuitive. $\endgroup$
    – safesphere
    Jun 29, 2023 at 18:13
  • $\begingroup$ As far as I know, we have $R_{\mu\nu}=0$, meaning no changes in volume (otherwise, the Ricci tensor would reflect it). It is only the shape that changes, reason why the Riemann tensor is not null. This means that the volume component $\sqrt{-g}$ is the same in Minkowski metric and in Schwarzschild one. The latter actually considered that his $g_{\mu\nu}\, \to \, \eta_{\mu\nu}$ when $r \, \to \, \infty$. So volume cannot be zero, for anyone in any point in space (maybe except in the singularity). And in Schwarzschild coordinates the horizon singularity is compelling (not in other coord syst) $\endgroup$
    – Antoniou
    Jun 29, 2023 at 22:28
  • $\begingroup$ The zero Ricci tensor refers to the constant infinitesimal spacetime hyper-volume, which reflects the relation between the time dilation and length contraction, but has absolutely nothing to do with the spatial volume. Good luck! $\endgroup$
    – safesphere
    Jun 30, 2023 at 5:29

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Modern physics does not know the answer to this. The structure of a black hole in General Relativity is based upon a pointlike singularly (or ringlike singularity for a rotating black hole). Inside the Schwartzchild radius, all future light cones point inwards towards the center, which seems to imply that any matter inside this region must collapse inward. It could not remain at a stationary radius any more than matter in normal space can remain stationary in time.

But, we do not have a quantum theory of gravity to tell us how matter will actually behave inside a region of such high spacetime curvature. Current quantum mechanical theories only deal with approximately flat spacetimes. It is possible that matter will reach a state of such compression that it will counteract the inward curvature and reach a stable configuration of finite size – a "collapsar" – but which is invisible to the outer universe behind a horizon.

With current physics this is pure conjecture.

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