Why black holes are "infinitely small"?

[ran cohen]

Is that only because the actual physical content of any atom and distance between them is so great?

If so,  given enough stars collapsing to the black hole, can it reach for example couple of meters? Or are there others laws preventing that?

Best Regards

Ran

[Markus Hanke]

I am not sure what you are asking here - what do you mean by black holes being ‘infinitely small’? Usually, the size of a (non-rotating, electrically neutral) black hole is given by the radius of its event horizon, which is finite and well defined. For example, if our sun was to undergo gravitational collapse, the resulting black hole would have a Schwarzschild radius on the order of ~3km.

[ran cohen]

Hi Marjus,

Thank you for the answer. But i am not referring to the size of the event horizon but to the actual size of the physical material inside.

And what does it include exactly?, is it just the elementary particles compressed?  Does the physical size increase when more stars collapse to it?

I guess no one know for sure but i would like to know what is expected.

ps: will the event horizon increase as more stars collapsing? I would expect so.

Best Regards

Ran

[Markus Hanke]

Ok, I get you now. This is not a trivial question, and the short truth is that we do not really know the answer, based on currently known physics.

The trouble is this - the established model currently available to describe gravity (General Relativity, or GR for short) is purely classical, meaning it does not and cannot account for quantum effects. When we describe the process of gravitational collapse, then in the beginning stages of that process quantum effects can be neglected, so up to a certain point GR does a really good job in describing things. We can even cheat a bit, and extend the range for which our description is valid by considering already known quantum effects simply as classical pressures that counteract gravity. For suitable initial conditions, this may yield an equilibrium state such as a dwarf star, or a neutron star, or something more exotic like quark stars. However, once the total mass of the collapsing object exceeds a certain limit, there is no known mechanism by which the collapse could be stopped - in these cases the object keeps collapsing under its own gravity, and eventually becomes so dense that quantum gravitational effects can no longer be ignored. At that point General Relativity quite simply stops being a valid model. And this is where we get stuck, because we do not yet have a model of quantum gravity, so we simply do not know what happens in the final stages of such a collapse, and what happens to the mass of the original object. There are a few speculations, hypotheses and candidate models, but none of them is sufficiently well understood, or tested in any way.

If we naively consider GR on its own, the end result of this collapse is a singularity - all the mass of the collapsing object becomes concentrated in a single point of infinite density, and infinite spacetime curvature. The ‘size’ of that infinity is always zero, regardless of how much mass you start out with, and regardless of how much mass falls into it later on. However, this is not to be understood as a physical prediction - in physics, when a model becomes singular and infinite, then that simply means that we have wrongly extended that model beyond its domain of applicability. In this particular case, we have attempted to apply a purely classical model to a physical situation that is decidedly not classical, so obviously the answer we get is not physically meaningful.

Note that the singularity itself, for mathematical reasons, wouldn’t be part of the spacetime manifold, so counterintuitively the entire spacetime in and around a black hole of this kind would be completely empty. The mass of such a black hole is actually a global property of the entire spacetime, and cannot be localised anywhere.

1 hour ago, ran cohen said:

the actual size of the physical material inside

The volume of a singularity - in so far as that concept makes sense (it doesn’t, really) is zero. This is true for both point singularities, and ring singularities.

1 hour ago, ran cohen said:

And what does it include exactly?, is it just the elementary particles compressed?

Yes, in the purely classical picture of GR it would be matter compressed to infinite density. But we know (see above) that this is not a physical meaningful concept, since it cannot happen in the real universe. Even the already known laws of quantum physics prohibit such a state (ref e.g. the Pauli exclusion principle).

1 hour ago, ran cohen said:

Does the physical size increase when more stars collapse to it?

Again, in the classical picture of GR the answer is no - the singularity remains point-like or ring-like. What does change though is the radius of the event horizon.

1 hour ago, ran cohen said:

I guess no one know for sure but i would like to know what is expected.

When we look at current attempts to write out a model of quantum gravity (a very difficult problem!), then three main themes emerge, depending on which model is used:

1. Below a certain length scale, a new symmetry emerges that turns the collapse into a rebound - so the collapsing matter will never become singular, but instead begins to ‘bounce’ back out while the event horizon shrinks. However, due to the extreme time dilation in that region, this process would take a very long time (~100s of billions of years) as seen by an outside observer, which is why it has never been observed.
2. You end up with some sort of exotic degeneracy state below the horizon, such as a fuzzball.
3. Spacetime itself becomes quantised below a certain length scale, so the question as to what happens to the matter or where it goes becomes meaningless

There is no telling at present if any of these possibilities describes what actually happens in the real world.

1 hour ago, ran cohen said:

will the event horizon increase as more stars collapsing?

Yes, but at the same time it will also continue to evaporate via Hawking radiation.

Edited September 6, 2020 by Markus Hanke

[Eise]

@Markus Hanke: Is a quantum theory of gravity the only kind of solution for the singularity? E.g. just like when a neutron star is formed by combination of protons and neutrons, wouldn't it be possible that in a black hole all matter is compressed into a very compact form we do not know? In other works, instead of quantum gravity, might we not need an extended form of quantum mechanics?

[Markus Hanke]

44 minutes ago, Eise said:

Is a quantum theory of gravity the only kind of solution for the singularity?

No, there are other ways to avoid the singularity - namely by making adjustments to the laws of (classical) gravity. The most straightforward amendment would be to allow what is called torsion in your spacetime. GR is explicitly constructed to use only curvature to describe gravity, so torsion always vanishes. If one allows torsion (simply by choosing a different connection), then we end up with a model called Einstein-Cartan gravity (ECG). In this model, the singularity never happens, and the collapse actually turns into a bounce, so after a very long time everything that fell into the black hole will bounce back out (but in scrambled form). This would also be true for the Big Bang singularity, so the BB model is naturally replaced by a ‘Big Bounce’ scenario.

The issue with this is that when you allow torsion on your spacetime, then this has consequences not just for gravity, but for some other laws of physics as well - specifically, it makes the Dirac equation for spin-½ particles non-linear, and also has other impacts on the Standard Model. The resulting effects would be too small to be detectable at currently available energies (at least as far as I know), so ECG remains a potentially viable model that can’t be ruled out (but also not confirmed) at present.

57 minutes ago, Eise said:

wouldn't it be possible that in a black hole all matter is compressed into a very compact form we do not know?

Yes, this is what would happen if String Theory is physically viable - the interior of black holes would then simply be the highest form of degeneracy, being what is called a fuzzball (see link in previous post). The singularity is again avoided.

1 hour ago, Eise said:

In other works, instead of quantum gravity, might we not need an extended form of quantum mechanics?

Well, instead of having a model of gravity that accounts for quantum effects, you can of course look for an extension of the Standard Model that accounts for gravitational effects. The end result is the same  

[joigus]

Before Planckian scales are reached, the theory must be changed. Whatever the theory is, point-like object will have no place in it.

That much I can say.