# Matter inside a black hole

The question I'm asking is the following: at which state is the matter inside a black hole? I know that supermassive black holes have a density very low, while stellar black holes have a very high density. However, the concept of density cannot be defined within a black hole because the metric tensor diverges, and therefore the concept of distance cannot be defined. But I think it is not even a question of density, but of defining whether matter is composed, for example, of atoms or elementary particles.

• – Ben Crowell Nov 13 '19 at 20:32
• "supermassive black holes have a density very low, while stellar black holes have a very high density" - This is a common error. The volume of a black hole observed from outside is zero, observed from the inside is infinite. So the average density does not depend on the size of the black hole. – safesphere Nov 14 '19 at 17:25
• "The volume of a black hole observed from outside is zero" means that the event horizon that an external observer see, is a point? – blackhole Nov 14 '19 at 21:41
• @blackhole You should address your comments with the @ sign for users to be notified. In differential geometry of curved spacetimes, radius, area, and volume of a sphere relate to each other differently from ordinary flat space. The area of the event horizon is the same $4\pi r^2$, but the volume is 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 (page 6) - So no, it's not a point, but a sphere with a finite radius and area, but a zero inside volume, as observed from outside. – safesphere Nov 16 '19 at 5:28

I know that supermassive black holes have a density very low, while stellar black holes have a very high density.

This is not true. The interior of a black hole can have matter with any density you like, including zero. The standard black hole models that people normally study are vacuum solutions, so the density is zero everywhere.

However, the concept of density cannot be defined within a black hole because the metric tensor diverges,

I would put this in a different way. Everything misbehaves at the singularity, but we can still define the notion of a singularity that is or is not a strong curvature singularity (SCS). As matter approaches an SCS, its density goes to infinity. As matter approaches a non-SCS singularity, it can be subjected to infinite tidal forces (spaghettification), but its density stays finite. Astrophysical black holes probably have non-SCS singularities, as do all the simple mathematical models of black holes that we normally study. Therefore the density remains finite at all times -- in fact, it stays constant as the matter infalls. The matter that reaches the singularity simply doesn't exist in our spacetime manifold anymore.

But I think it is not even a question of density, but of defining whether matter is composed, for example, of atoms or elementary particles.

The infinite tidal forces would certainly destroy atoms and nuclei. We don't know what happens at the Planck scale.

• "The matter that reaches the singularity simply doesn't exist in our spacetime manifold anymore." - Very true. This shows that thinking of a singularity as having mass is naive. +1 – safesphere Nov 14 '19 at 17:37