[JP]

What is the energy density within the eye of a hypersonic cyclone? 

Depends on the cyclone, but it's a finite number.

[Bill S]

the predecessor to a black hole is a finite object becoming a singularity

Within this definition, is a singularity infinite? If so, are you saying that something finite becomes infinite?

[JP]

Yor_on might have something else in mind, but I think the explanation is that general relativity treats it as infinitely dense, but in reality it probably isn't.  Once it gets small enough, general relativity probably isn't an accurate theory anymore, and the infinite density that it gives is probably a result of the theory breaking down.

That said, you can talk about the singularity, to mean the point at which the theory breaks down.

[syhprum]

I cannot see why what exists beyond the event horizon needs to of quasi infinite density, the matereal of a Neutron star has nearly a high enough density to produce an event horizon so why does the contents of a black hole have to be vastly greater

[JP]

I cannot see why what exists beyond the event horizon needs to of quasi infinite density, the matereal of a Neutron star has nearly a high enough density to produce an event horizon so why does the contents of a black hole have to be vastly greater

A neutron star doesn't have enough density to produce an event horizon.  Assuming it does is due to mistakenly applying a solution to GR which is valid in empty space to the region within the neutron star as well.

An analogy to this would be the Shell theorem in Newtonian gravity.  If you're outside the earth's surface, you can apply the shell theorem to predict the gravitational field anywhere by assuming all of the earth's mass is concentrated at its center.  This isn't physically true, but it's a useful mathematical trick.  Of course, if you start digging towards the center of the earth, this solution isn't valid anymore.  You have to recalculate the force of gravity taking into account only the mass contained within a shell whose radius is equal to your distance from the earth's center.

In the same way, you can solve Einstein's field equations to get a solution for a non-rotating, spherically symmetric object.  By making an analogous solution to the shell theorem, where all the object's mass is concentrated at it's center, you can come up with a solution to these equations.  This solution has an event horizon, or Schwarzschild radius.  Obviously if this radius lies within the object, it doesn't really exist, since this solution is only valid outside of the object.  I suspect that's the event horizon you're referring to.  Solve the equations within the object, you should see that no horizon actually exists.

That the Schwarzschild radius does tells us is that if you take all the mass in your object and squash it to within the Schwarzschild radius, you do suddenly get a physical event horizon appearing.  Because any force keeping the object from collapsing further would have to be communicated faster than light, it has to keep collapsing at this point, forming a black hole with a singularity at its center.

[yor_on]

Thanks JP

I think an infinite density is an equivalence to energy. And I don't think you can reach that density, as little as matter can reach the speed of light in a vacuum. Before that it have to become that scarlet pimpernel 'energy', but I don't know of course

[JP]

Mass is equivalent to energy and mass density is equivalent to energy density.  So you have infinite energy density, but finite energy.  How can this happen?  It's infinite density squashed into an infinitely small point, so the infinities "cancel" in a sense.  In reality, things becoming infinitely large and small at that point is probably a sign that GR doesn't work for tiny things.

[yor_on]

Huh, Are you trying to scare me?
"infinite energy density, but finite energy"

I just knew there was something

So a density can't be equivalent to a energy then?
==

Mass has then no defined density?
Ahem

(and how the he* will I be able to sleep after this?)

[JP]

Mass density is mass/volume.  Energy density is energy/volume.  Mass or energy density can become infinite in two ways.  Either mass or energy becomes infinite or volume decreases towards zero.  In the case of a black hole it's mass/energy is finite, but the volume of the singularity decreases to a point, which is zero volume.

This isn't exactly the same as dividing mass/0, which is mathematically illegal.  It's taking mass/x as x->0, which is mathematically legal.  But this whole idea of dividing by volume that's going to zero is a sign that the black hole singularity probably isn't a physical thing.

[yor_on]

Okay, that makes sense.
And yes, that's how I understand the center.
As not describable.

Very nice and concise explanation JP.
==

I think I can sleep now

[Bill S]

the volume of the singularity decreases to a point, which is zero volume.

Can anything be smaller than the Planck mass and still exist?
How do you distinguish between having "zero volume" and not existing?

[JP]

the volume of the singularity decreases to a point, which is zero volume.

Can anything be smaller than the Planck mass and still exist?

No one knows.  We don't have high enough energy experiments to test what happens at such small scales, and our best tested theory (the standard model) doesn't give answers for things that small.  It might be that smaller things do exist.  It might also be that the fabric of space takes on a totally different character at such lengths, so size as we know it doesn't mean the same thing (I read that somewhere on a writeup of quantum loop gravity).  But no one knows.

How do you distinguish between having "zero volume" and not existing?

Easy.  On exists and one doesn't! 

I get your point, though.  Zero volume, as far as we know, is a mathematical construct that's probably accurate as a physical model up to a certain point, and then it has problems.  We won't know what it really means in this case until we figure out a theory of quantum gravity.

If I model a black hole as a singularity, I get an answer that makes sense and appears to model black holes pretty well.  If I model a black hole as not existing... well, I have no black hole.

[Bill S]

On exists and one doesn't!

You say that "one exists", but then qualify that by saying that it is a "mathematical construct".  However, as discussed with respect to mathematical infinities, something that can be said to exist mathematically does not necessarily exist in reality. 
I suppose that causes no problems as long as we realise we are talking about models.

[yor_on]

I think that in a way it's like JP pointed out to me before. Some things we can describe mathematically, like the idea of gluons, and find evidence for indirectly. But never being able to prove first handedly by direct observations. A black holes center just got to be one of those things

It falls back to how rigorously you expect mathematics to mimic 'reality', and that seems to have a lot to do with your definitions, and also conceptions, of reality. I avoid the word preconceptions because I think most people do want their mathematics to be as close to what is observed as possible. But in hindsight we've always had preconceptions, it's hard to avoid I guess, one can only build on what one know, and for me that changes constantly I'm afraid