I’ve heard black holes described as “more diameter than the circumference suggests” — to paraphrase Kip Thorne.
Is there a point where you’re at “quantum scale” around the singularity but not across the singularity? — like a quantum Einstein ring?
Is there a point where you’re at “quantum scale” around the singularity but not across the singularity? — like a quantum Einstein ring?
I'm not sure where Thorne said whatever you are paraphrasing, but this description does not apply to black holes. It applies to ordinary gravitating bodies like stars and planets, but unlike them, black holes do not have a well-defined "diameter". The singularity inside the hole at r = 0 is not a place in space, it's a moment of time. So there is no well-defined "spatial distance from the singularity"; that would be like having a well-defined "spatial distance from tomorrow", which makes no sense.
In the case of the Earth, the coordinate singularity in longitude at the poles is an artifact of the coordinates: the Earth's poles are perfectly good parts of the Earth's surface and nothing physically goes wrong there. You just need to pick different coordinates if you want to cover the poles.
The case of the horizon of a black hole in Schwarzschild coordinates is similar: there is a coordinate singularity, but it's an artifact of the coordinates. You can find other coordinate charts that have no coordinate singularity at the horizon.
The black hole singularity at r = 0, however, is not an artifact of coordinates. It is a genuine physical singularity, where the spacetime curvature goes to infinity.
To be fair we don't know the physical curvature goes to infinity--the math says it does.
More precisely, the math of the theory that works everywhere we can test it, when extrapolated to r = 0 inside a black hole, where we cannot directly test it, says the curvature increases without bound as r = 0 is approached. The mathematical process of extrapolating a well tested theory into a domain where we cannot directly test it is used all the time in science; there's nothing inherently questionable or unreliable about it.
The issue in this particular case is that the extrapolation gives us an answer that has a problem: it doesn't seem physically reasonable to have curvature increasing without bound in a finite time, along a finite arc length on the worldline of some observer, at which point spacetime just ends and the observer is destroyed. So most physicists believe that GR, the theory, will actually break down somewhere before this point is reached, and we will need a new theory, which most physicists believe will be a theory of quantum gravity, to tell us what actually happens in this regime.
But you're correct that we won't know for sure unless and until we have some way of testing what happens in this regime.
Where are you getting this from? It makes no sense.
In the 2D example, is there some area where quantum effects happen in an annulus rather than in a disc because that central stretched region is “too big”.
There is no such thing. The singularity is a moment of time, not a place in space.
Saying that an event horizon or photon sphere isn’t around the singularity of a black hole doesn’t make sense, either.
What facts specifically are you trying to articulate about spacetime geometry around a black hole?
I already have: the singularity is a moment of time, not a place in space.
> Saying that an event horizon or photon sphere isn’t around the singularity of a black hole doesn’t make sense, either.
Yes, it does, for the reason I've already given: because there is no such thing as a sphere around a moment of time.
> What facts specifically are you trying to articulate about spacetime geometry around a black hole?
The fact that the singularity is not a place in space, it's a moment of time.
What fact about spacetime geometries are you trying to convey?
You should try explaining the relevance of what you’re repeating in terms of spacetime geometries rather than just repeating the same statement over and over — which isn’t a constructive reply.
No, it isn't. At least not in GR (perhaps your definition is valid in some other context).
In GR, a singularity is present in a spacetime if the spacetime has non-extendable geodesics. Strictly speaking, the singularity itself is not even part of the spacetime manifold; but if we attach it to the spacetime manifold as a sort of boundary, it does not have to be a point. In the case of the singularity at r = 0 in Schwarzschild spacetime, the singularity is a spacelike line which is to the future of all events inside the hole. No geodesics can be extended past that line because spacetime curvature increases without bound as it is approached (and curvature is "infinite" on the line itself, which is why the line itself is not and cannot be part of the spacetime manifold, since curvature invariants must be finite everywhere on the manifold).
> You should try explaining the relevance of what you’re repeating in terms of spacetime geometries
What I am saying is a well known property of a well known solution in GR. I am describing it in as non-technical language as I can. Perhaps what I said above will help.
No, it isn't. It's a spacelike line that is to the future of every event inside the hole. It's a moment of time, not a place in space.