I What exactly happens when the Schwarzschild radial r is....

[Istegosaurus]

I know black holes are not that well understood but if someone can explain this I'd be grateful.

Please correct me if I have anything wrong, I don't know much about this.

The Schwarzschild solution of

comes from the Einstein field equation

I think I have that right. Now I don't understand any of that stuff and I'm not really interested in how it's solved but I did notice one thing that seems to suggest that actually dividing by 0 is in fact possible, in the right conditions.

What exactly happens when the Schwarzschild radial

is exactly 0? Dividing by 0 isn't allowed... Or is it?

I'm still trying to wrap my head around the singularity, is it actually truly dimensionless and the radial is literally 0, or is it just incredibly close to 0?

Or the more likely explanation is I'm an idiot and don't understand what I'm talking about

 

[mfb]

It is expected that general relativity cannot accurately describe what is going on "at the singularity" - and that this is not actually a singularity. We'll probably need a combination of GR with quantum field theory to solve this puzzle. We cannot observe it...

 

[Maxwell's Demon]

I know black holes are not that well understood but if someone can explain this I'd be grateful.

Please correct me if I have anything wrong, I don't know much about this.

The Schwarzschild solution of

comes from the Einstein field equation

I think I have that right. Now I don't understand any of that stuff and I'm not really interested in how it's solved but I did notice one thing that seems to suggest that actually dividing by 0 is in fact possible, in the right conditions.

What exactly happens when the Schwarzschild radial

is exactly 0? Dividing by 0 isn't allowed... Or is it?

I'm still trying to wrap my head around the singularity, is it actually truly dimensionless and the radial is literally 0, or is it just incredibly close to 0?

Or the more likely explanation is I'm an idiot and don't understand what I'm talking about

This question is a little confusing. The Schwarzschild radius is actually very simple - it's just:

And that just defines the event horizon.

The equation you cited is called the Kretschmann scalar:

K =

Just to be clear, all non-rotating black hole singularities are considered to have zero radius, but most if not all black holes are rotating so they describe the singularity as a circular ring with zero volume. And like mfb said, this description is considered to be a placeholder: a black hole is a quantum object and until we have a quantum theory of gravity there's no way to know the precise nature of the singularity. I'd imagine that at the very least there's some "fuzziness" around the singularity because the position of anything at the quantum scale has some intrinsic level of uncertainly.

The good news is that last year the Laser Interferometer Gravitational-wave Observatories (aka LIGO) picked up two signals of gravitational waves, generated by pairs of black holes coalescing into one, and the properties of that kind of signal can give us new insight into what's going on inside the event horizon of a black hole, because the "ringdown" of the collision will tell us something about the process of the two singularities merging. Perhaps sufficient improvements in our gravitational wave detectors will even help us arrive at a unified theory of gravitation and quantum mechanics.

Even with the fairly low-resolution gravitational wave signals we have so far, there seem to be questions emerging about the GR model of black holes that not only pertain to the singularity, but perhaps to the event horizon as well:
http://www.nature.com/news/ligo-black-hole-echoes-hint-at-general-relativity-breakdown-1.21135