Why does General Relativity predict that singularities are infinitely dense?

Question

From my understanding of general relativity, energy curves spacetime to produce an effect that we call gravity. The greater the concentration of energy, the greater the curvature and hence the greater the gravitational force we feel. However, I have repeatedly seen online that general relativity predicts that black holes have a 1D singularity of infinite density, which is an issue since the concept of infinitely small doesn't really exist physically.

What part of general relativity says that the singularity in a black hole must be infinitely dense? Why can't the explanation just be that the object has a sufficiently large mass to deform spacetime enough to produce an event horizon?

Answer 1

I think at least part of your question could be rephrased as

Why can't a black hole consist of a dense object with a nonzero radius less than its Schwarzschild radius?

If that's the case, the answer is that it's not possible to have a stable distribution of matter inside the Schwarzschild radius. One way to understand this is to note that inside a Swarzschild black hole, the "radial" coordinate (which is the surface area of the corresponding $2$-sphere divided by $4\pi$) is timelike. There is no force which could prevent matter from falling toward $r=0$ for precisely the same reason that outside the black hole, there is no force which could prevent matter from progressing towards next Wednesday.

The worldlines of massive particles are always timelike. Loosely speaking, outside of a black hole's event horizon this means that the worldlines can accelerate one way or another but they are always oriented in the direction of increasing $t$; inside of a black hole's event horizon, they can accelerate one way or another but are always oriented in the direction of decreasing $r$.

Answer 2

None of what you're saying is true. A black hole singularity isn't a one-dimensional point. Its dimensionality is undefined. It doesn't have infinite density according to GR, because it isn't part of the spacetime manifold, so GR doesn't define whether there is any matter there.

There is a notion of a strong curvature singularity. When you have this type of singularity, infalling matter's density approaches infinity as it approaches the singularity. This is the closest thing that GR has to what you've been thinking of as a singularity of infinite density.

However, the singularity of a Schwarzschild black hole is not a strong curvature singularity. Infalling matter gets spaghettified, but not infinitely compressed. Actually its density stays constant.

Answer 3

In my understanding, the black hole singularity is not a point in space but the description of physical quantities behavior for compactness parameter $\alpha~ (\equiv r_S/R)~$ approaching some critical value $\alpha_{c}~$. In the case of Schwarzschild interior solution $\alpha_{c}=8/9~$. The energy density at the singularity has finite value $3\alpha_{c}~$, while the pressure diverges. This behavior can be found in all static spherically perfect fluid solutions. The respective values of the critical parameter $\alpha_{c}~(\le 8/9)~$and the finite energy density at the singularity depend on the solution. Opposite to it, the pressure behavior is in all solutions the same. For $r \rightarrow 0~~$it diverges like $p \sim r^{-2}~$, generating the universal force equal $2~c^{4}/G$, as conjectured by Gibbons (arXiv: hep-th/0210109v1 [qr-qc]).