Singularities/ringularities and their properties

Question

A black hole contains a singularity at its center. It is a zero-dimensional point, and it's where all its mass is located. So, my first question is:

If a singularity contains all a black hole's mass, when a black hole absorbs matter, the matter should collect at the singularity. Then why does the black hole increase in size? The matter it absorbs is all in the singularity, not in the space around it...

Second, we know that spinning black holes can't have a singularity at the center as a point, because points can't spin, and when they do, they aren't a point. So there is such thing as a ringularity where it is a one-dimensional line bent in a circle that spins, and contains all the black hole's matter.

However, is a ringularity really a bent one-dimensional line in a circle, or an infinite collection of points that form a circle-looking figure?

---

(My information on ringularities is from a paper published by Cornell University that compared the one-dimensional line to a bunch of zero-dimensional points, and videos by Kurgesagt.)

Answer 1

You need to be a bit cautious about statements like: A black hole contains a singularity at its center. It is a zero-dimensional point, and it's where all its mass is located. What you are referring to is a mathematical structure called the Schwarzschild metric. This is indeed a solution to the Einstein equations for a static black hole, but this particular solution takes an infinite time to form. So given that the universe is only a shade under 14 billion years old we can be confident that as I type this there are no Schwarzschild black holes in the universe.

Indeed, it is far from clear that a Schwarzschild black hole could ever form, even given infinite time, because black holes evaporate. This point is pursed in the question Why does Stephen Hawking say black holes don't exist?

Incidentally, the Schwarzschild black hole doesn't have all its mass at the centre. In fact it is a vacuum solution and contains no matter at all. The mass of the Schwarzschild black hole is actually a geometrical property called the ADM mass.

We run into the same problems with the spinning black hole. The structure with the ring singularity is the Kerr metric, and again while it's a perfectly valid solution to Einstein's equations, like the Schwarzschild metric it also takes an infinite time to form. Actually the Kerr metric is more troublesome still, because it is uncertain whether the structure is stable with respect to perturbations. It seems very likely that real collapsing stars would not form a structure resembling the Kerr metric. More precisely the region outside the event horizon would be well described by a Kerr metric, the region near the singularity would be different.

But if we put these doubts aside and return to what you asked, the answer is that in the Kerr metric the ring singularity really is a ring and not a collection of points.

Answer 2

A black hole contains a singularity at its center. It is a zero-dimensional point, and it's where all its mass is located.

This isn't an established fact. A lot of people say they don't think there is a point-singularity at the centre of a black hole. Google on black hole "no point singularity"

So, my first question is: If a singularity contains all a black hole's mass, when a black hole absorbs matter, the matter should collect at the singularity. Then why does the black hole increase in size? The matter it absorbs is all in the singularity, not in the space around it...

People typically talk about the extent of the event horizon when talking about the size of a black hole.

Second, we know that spinning black holes can't have a singularity at the center as a point, because points can't spin, and when they do, they aren't a point. So there is such thing as a ringularity where it is a one-dimensional line bent in a circle that spins, and contains all the black hole's matter.

That sounds like a ring singularity. See Wikipedia for something on that. Note however that a ring singularity is hypothetical, not proven fact.

However, is a ringularity really a bent one-dimensional line in a circle, or an infinite collection of points that form a circle-looking figure?

See the Wikipedia article. It says this: "the minimal shape of the singularity that can support these properties is instead a ring with zero thickness but non-zero radius, and this is referred to as a ringularity or Kerr singularity". I don't endorse that by the way. I think the original "frozen star" description of a black hole was right. Google on frozen star and Robert Oppenheimer and you can find things like the 1971 Physics Today article introducing the black hole by Remo Ruffini and John Wheeler. They said “in this sense the system is a frozen star”.