Inside a black hole, nothing can hold together. At the "event horizon" then something traveling the speed of light can stay there, anything else falls in further. But within the black hole, even light collapses to the center and can't escape.
So there is no surface, you just keep falling and falling. Anything whether stationary or moving, even a light beam moving at the speed of light away from the center, is still on an inevitable trajectory towards the center of the black hole.
However for really big black holes, then crossing the event horizon can be quite a gentle process. You are doomed at that point, there, is no way you can get back out again, but you don't notice it for a while later
Example here BLACK HOLES by Ted Bunn where he says that if you started, say, from a distance of ten times the black hole's radius, for a million solar mass black hole, then it would take 8 minutes to reach the horizon, and then, only another seven seconds to hit the singularity at the center. But you'd survive for a short while over the horizon.
For a truly gigantic super cluster of galaxies mass black hole, then you could fall for some time before you hit the center.
As to what is in the center - well it's a mathematical point. So it is theoretically of no size at all. Everything just ends up being a single point.
What that means "physically" goodness knows. How can a mathematical point have a mass of a million suns? And how does it differ from a mathematical point with a mass of a single sun?
It's quite a crazy kind of an idea when you try to make sense of it intuitively and try to connect it to intuitive ideas of matter.
Though mathematically it seems to make sense apparently.
We don't need to develop a detailed physics for what happens inside the black hole, because in our universe all we ever know about it is what we can see from outside. And as seen from outside, a black hole has only three properties - its mass, its charge, and its spin rate and direction of spin axis.
Once you know those three things (plus of course its position and velocity in space) you know everything you can possibly ever discover about the black hole. All you can do after that is to add a few more decimal points to your measurements of those properties.
So they are the simplest objects to describe in the entire universe. See Black Holes and Neutron Stars
Everything else is hidden behind the event horizon - because nothing can escape from it.
Spinning black holes however have some interesting properties - these are the so called Kerr Black Holes.
If a black hole spins fast enough, then its event horizon gets smaller - and in very special circumstances, you may get a so called Naked singularity which would then make it possible to actually observe an infinitely dense mathematical point of mass in our universe.
In 1994 Mathew Choptuik showed that you can, theoretically, get a naked singularity forming during rather special conditions - the perfectly spherical collapse of a rotating black hole.
His paper is here: "Examples of Naked Singularity Formation in the Gravitational Collapse of a Scalar Field,"
See What's so scandalous about a naked singularity?
Though whether those special conditions can occur in reality is another question. Roger Penrose's Cosmic Censorship Conjecture hypothesizes that they never can be seen in our universe - that naked singularities can't exist but are always hidden behind an event horizon.
Some people wonder if black holes really exist in our universe, and there are Black hole Alternatives to explain some of the observations - or it might be that the rules of physics can be changed so that they are not quite what they seem to us to be.
See also Chris Craddock's comment to this answer - he makes some interesting points :).
You can get this and many more of my answers now as a kindle book:
Simple Questions - Surprising Answers - In Astronomy
COULD WE EVER PROVE WE HAVE FOUND A BLACK HOLE?
if we had a black hole that we could study close up, we could drop matter into it, and see it disappear below the event horizon. If it was small we could spot Hawking radiation. We could drop a torch into it and see the light red shift as it falls towards the center. We could map the geometry of the space around the black hole.
None of that would actually prove that what we have inside is a single mathematical point with properties such as stellar mass, charge and a spin rate (how can a dimensionless point spin?) But it would prove that it looks like a black hole as far as the event horizon.
Also black holes as we understand them are a prediction of general relativity. But we know that GR is not the final answer because it is inconsistent with quantum mechanics, indeed the theory has no explanation of how matter is possible at all, just assumes its existence.
So in that sense, what we observe can't really be black holes in the sense of General Relativity - so what are they really? Perhaps the paradoxical seeming idea of a black hole as a mathematical point with mass, spin and charge just results because of this incomplete nature of GR, and because they aren’t really black holes in that sense. General Relativity is probably an approximation just as Newton Gravity was an approximation to GR. But a very very accurate approximation.
The alternatives for a stellar mass black hole involve some form of exotic matter able to resist collapse. Alternatives for a supermassive black hole would involve large masses orbiting each other perhaps.
It could also be that the theory of gravity has to be modified in some way. For instance quantum gravity leads to the elegant idea of a Planck star - ordinary stellar sized black holes would "rebound" before they can get small enough to be a black hole, but the rebound from seen from outside would outlast the age of the universe so far so that it would seem like a black hole to us.
We can only observe very distant black hole candidates. I think the most interesting evidence perhaps is the faintness of the accretion disk around an X ray binary during quiescence - as it seems to prove that there is no solid surface for the matter to land on. But even so - that's hardly very direct evidence. And if it was, say, a Planck star, would that not maybe have a similar signature?
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