Black Holes in Curved Spacetime

[Vanadium 50]

TL;DR Summary

Do Black Holes form in Curved Spacetime? What is and is not known about this?

We've been talking in another thread about supermassive black holes. That has me thinking about really, really big BH's - so large that the spacetime curvature and evolution of the universe matters.

Let's start by defining the density of a black hole as its mass divided by the volume enclosed by its event horizon. Thus is not technically correct, but it's good enough to support the next idea: as black holes get large, their required density drops. This follows from R ~ M.

Galactic SMBH's are not formed by a star's collapse. They are formed when enough (too many) stars get very close (too close) together, such that M > R.

One can ask how big do structures have to be before this happens. Galaxies aren't big enough. Clusters aren't big enough. Superclusters aren't big enough. However, aggregates of galaxies that are a substantial fraction of the size of the visible universe are big enough.

However - there's a big "but" here - the GR solutions that lead to BH's assume that the surrounding spacetime is static and asymptotically flat. If you have a region that is a good fraction of the visible universe, spacetime around it is anything but flat. It's not static either - the universe is expanding and its density is dropping.

This is not the same thing as "maybe the universe is a giant black hole". This is about the formation of very lartge black holes in the universe.

 

[Orodruin]

TL;DR Summary: Do Black Holes form in Curved Spacetime? What is and is not known about this?

However - there's a big "but" here - the GR solutions that lead to BH's assume that the surrounding spacetime is static and asymptotically flat.

… and typically also that there is vacuum. This is not the case for your typical cosmological scales and hence very distinct from the case you consider. What would be the closest to your setup that I am aware of would be the hypothetical formation of primordial black holes in the early Universe, which essentially require a sufficient overdensity to form.

 

[PeterDonis]

the GR solutions that lead to BH's assume that the surrounding spacetime is static and asymptotically flat.

Over the formation time of a typical stellar mass black hole, the surrounding spacetime is static and asymptotically flat to a good enough approximation. That is why idealized solutions like the original Oppenheimer-Snyder model, and more modern numerical simulations of more realistic collapses that lack the extreme symmetry of that model, work reasonably well.

For structure formation in the early universe (which is not limited to BHs, btw, the same problem exists for the formation of ordinary stars and galaxies), where, as you note, we can no longer take the surrounding spacetime to be static or asymptotically flat, AFAIK there are no exact or even approximate closed form solutions and numerical simulation is required. The basic idea, as I understand it, is that once an overdensity region is produced by density fluctuations, it will tend to clump further. But I am not very familiar with the details.

 

[Vanadium 50]

overdensity t

Yes. But at all times and places, somewhere in the universe is the place with the greatest overdensity. ("Someone's got to win the lottery")

But I don't see the value in considering the early universe unless I have to. The density then was greater than the density today, so if anything, the argument is even better.

numerical simulation

I am fairly sure these models don't use GR from first principles. The solve the equationns of motion on pads of paper and simulate that. I would not expect them to show emergent properties.

For technical reasons, these models don't usually expand. They set up an NxNxN grid and as the simulation progresses, the speed of light slows and spacetime flattens. So rather thah having a simulated "hypercluster" becomes BH, the conditions for BH formation evolves.

 

[PeterDonis]

I am fairly sure these models don't use GR from first principles.

A common method of simulation in GR is to use the ADM formalism, which is based on the Einstein Field Equation. That's about as "GR from first principles" as it gets. This method has been around for quite a while; see, for example, MTW Section 21.7.

The solve the equationns of motion

You can't do that unless you already know the spacetime geometry. But as I understand it, the simulations are trying to solve for the spacetime geometry. You have to use the EFE for that.

For technical reasons, these models don't usually expand.

I can see that grid size could be an issue in simulations of a non-stationary spacetime, yes.

 

[Vanadium 50]

You can't do that unless you already know the spacetime geometry

I think they simply perturb around FRW. In Newton-land, you could solve for what one voxel does in terms of the masses in the other voxels. This, as far as I know, does the same thing. But it doesn't resolve at every point - it plugs parameters into pre-calculated equations of motion.

It is almost certainly more complicated than the Newtonian case. For example, there will be a "force" pulling objects into a co-movibg frame, almost like a viscosity.

But the important thing is that this code will not come up with "hey! This supercluster is now a black hole!"

 

[PeterDonis]

I think they simply perturb around FRW.

If that is the case, then I would agree that such a model would never come up with a black hole, since that is way more than a small perturbation.

Using the ADM formalism, however, it shouldn't be necessary to assume anything about the global spacetime geometry. All you need to assume is the geometry and distribution of stress-energy on an initial slice. Then you can evolve the slice forward and compute both the spacetime geometry and the stress-energy distribution to the future of the slice. A model like that could conceivably compute collapse to a black hole from a small overdensity on the initial slice.

 

[Orodruin]

Yes. But at all times and places, somewhere in the universe is the place with the greatest overdensity. ("Someone's got to win the lottery")

Yes, so? It doesn’t affect my point at all. My point was that exactly that type of argumentation goes into the study of the formation of primordial black holes.

But I don't see the value in considering the early universe unless I have to. The density then was greater than the density today, so if anything, the argument is even better.

Again, I was talking about the techniques used. They should be equally applicable.

 

[Vanadium 50]

I undersstand. But that would argue that PBHs are not so "primordial" but are still being formed today : as time goes on, they just need to get bigger and bigger to form.

So where are they?

 

[Orodruin]

I undersstand. But that would argue that PBHs are not so "primordial" but are still being formed today : as time goes on, they just need to get bigger and bigger to form.

So where are they?

I am not an expert in the field, but my guess would be that the overdensities on the required scale are simply not large enough for them to form.

 

[Vanadium 50]

But that gets back to where we started.

The amount of mass in a large volume of spacetime - a few billion light years on a side - is sufficient for this mass to form a black hole in flat spacetime. Today. In the distant past, this would have been even easier as everything was denser.

Is this somehow blocked in curved spacetime?
What does it even mean to be a black hole in curved spacetime?

 

[Orodruin]

But that gets back to where we started.

The amount of mass in a large volume of spacetime - a few billion light years on a side - is sufficient for this mass to form a black hole in flat spacetime. Today. In the distant past, this would have been even easier as everything was denser.

Is this somehow blocked in curved spacetime?
What does it even mean to be a black hole in curved spacetime?

The point is that you need an overdensity of sufficient size, not just a particular amount of energy inside a given volume. The typical formation essentially tells you there is vacuum outside of the mass, which is not the case for the universe, which is largely homogeneous. We know the solution to the homogeneous and isotropic universe - the FLRW spacetimes, which do not contain black holes regardless of the density. Therefore you need overdensities to produce them and it is not sufficient with just enough density.

Even PBH models don’t predict an extreme amount of PBHs, which should be an indicator that you need quite a bit of overdensity in the early universe to produce them.

 

[Vanadium 50]

Thanks.

In a "more Newtonian" universe, the external mass doesn't matter of course. I can believe that dust in an overdense or underdense region experiences a radial acceleration, but struggle to see that the effect is large. Can you estimate the magnitude?

Argument for it to be small: in the Newtonian world, the 1/r potential has a conserved Runge-Lenz vector and a theorem telling you that a uniform external mass distribution does not cause a gravitational force, even along the radial direction. In GR, we know this isn't precisely true, but we know that the Runge-Lenz vector is almost conserved. For Mercury, the effect is about 75 ppb. (Tenth of an arcsecond per orbit) So the potential is approximately 1/r. and the external matter effect is approximately zero.

So where is the flaw in that reasoning?

 

[PeterDonis]

The amount of mass in a large volume of spacetime - a few billion light years on a side - is sufficient for this mass to form a black hole in flat spacetime.

I'm not sure what you mean by this. By what criterion is this mass sufficient?

 

[PeterDonis]

a uniform external mass distribution does not cause a gravitational force, even along the radial direction. In GR, we know this isn't precisely true

Depends on what you mean. GR has a shell theorem just like Newtonian mechanics: if you are inside a spherically symmetric shell surrounded by a spherically symmetric external stress-energy distribution, that external distribution has zero effect on the spacetime curvature inside the shell.

In our actual universe the external distribution is not precisely spherically symmetric, but it's pretty close.

we know that the Runge-Lenz vector is almost conserved

Yes, but this has nothing to do with the external distribution; it has to do with the field of the central mass.

 

[Orodruin]

a uniform external mass distribution does not cause a gravitational force

If you consider the Newtonian case to be the solution of the Poisson equation for a constant density case, then it is incompatible with a uniform scenario. If you have a homogeneous and isotropic mass density, then any compatible boundary consitions will necessarily break homogeneity.

 

[Vanadium 50]

By what criterion is this mass sufficient?

The mass is enough to form a BH in flat spacetime.

If you want to say "No, no, no, you can't use that. You're not in flat spacetime: I agree. I said so in Message #1. So what criteria should I use for a black hole in curved spacetime?

 

[PeterDonis]

The mass is enough to form a BH in flat spacetime.

If you want to say "No, no, no, you can't use that. You're not in flat spacetime: I agree. I said so in Message #1.

Yes, you did, but I'm not sure you've fully thought through the implications. See below.

So what criteria should I use for a black hole in curved spacetime?

I think you're asking the question backwards. "Black hole" is a name for a particular curved spacetime geometry (or category of such geometries). So the criterion is simple: is the actual spacetime geometry in the category that "black hole" is the name for?

You can't answer that question by looking at rules of thumb. You have to actually look at the spacetime geometry.

For example, if I consider a region in our universe billions of light-years wide, and the math tells me that the total mass enclosed in the volume of that region (which, if we assume our universe is spatially flat, we can indeed calculate using the Euclidean formula for the volume of a sphere) is greater than the mass of a black hole whose horizon area is equal to the area of the enclosing sphere, that in itself says nothing about the spacetime geometry of the region. If we just go by the FRW solution we use for the universe as a whole, it's obvious that the geometry of any region of that solution is the FRW geometry, which is not in the "black hole" category of geometries. For a region that large, even if the geometry is not exactly FRW, since the matter distribution is not actually uniform, the average geometry is basically FRW, and the perturbations are small, so we're still nowhere near a geometry that's in the "black hole" category.

We can approach this from another direction as well: what would we expect a black hole billions of light years wide to look like from the outside? Well, for starters, we would expect to not see any light coming from it, since light can't escape from a black hole. So if we do see light coming from such a region in our actual universe, obviously that region is not a black hole. That means that, whatever spacetime geometry that region has, it can't be a black hole geometry.

 

[PeterDonis]

the average geometry is basically FRW, and the perturbations are small, so we're still nowhere near a geometry that's in the "black hole" category

To restate this in more technical language: on a scale of billions of light-years, the individual galaxies or galaxy clusters are moving, to a good approximation, on comoving worldlines. Comoving worldlines are expanding--more precisely, the expansion scalar of the congruence of comoving worldlines is positive. The perturbations of the individual motions are small compared to that overall expansion scalar. If the region in question were a black hole, the expansion of that congruence of worldlines would have to be negative (because inside a black hole everything is falling inwards). That's way beyond a small perturbation.

 

[Ibix]

Butting in here, but I thought that the formal definition of an event horizon was the boundary of a region which cannot send signals to future null infinity. But in spacetimes that aren't asymptotically flat there is no future null infinity, so there can be no event horizons by that definition. So formally, there are no black holes in our universe, at least if you define a black hole to be an object enclosed within its event horizon.

There do appear to be compact objects that we call black holes in the universe. So is there a question here of "what is the proper mathematical description of these objects, then", alongside how they form?

 

[PeterDonis]

I thought that the formal definition of an event horizon was the boundary of a region which cannot send signals to future null infinity. But in spacetimes that aren't asymptotically flat there is no future null infinity, so there can be no event horizons by that definition.

Yes, this is true. However, see below.

is there a question here of "what is the proper mathematical description of these objects, then", alongside how they form?

Technically, yes, but there appears to be a pretty standard answer used by cosmologists: for "asymptotically flat", substitute "isolated from other concentrations of mass by a large enough distance that spacetime in between approaches flat to a good enough approximation".

Heuristically, this works because if we have a matter distribution that is isolated well enough in the above sense (whether it's a planet, star, solar system, galaxy, or galaxy cluster), we can draw a "world tube" around it and separate our analysis of what goes on inside the world tube (where we approximate spacetime as asymptotically flat) and what goes on outside (where we approximate the stuff inside the world tube as an isolated mass $M$ and ignore its internal structure). The shell theorem, which I mentioned before, helps here for the analysis inside the world tube because, if the matter distribution outside the world tube is spherically symmetric to a good enough approximation (as it should be in our universe), we can ignore its effect on the spacetime geometry inside the world tube--which equates to treating spacetime inside the world tube as asymptotically flat.

 

[PeterDonis]

If the region in question were a black hole, the expansion of that congruence of worldlines would have to be negative

We can use this sort of observation to come up with at least a first try at a criterion for black hole formation in an approximately FRW spacetime. If we assume that the matter is cold (i.e., zero pressure dust) and obeys the relevant energy conditions, then the various theorems of Hawking and Penrose from the 1960s and 1970s tell us that if we have a congruence of geodesics describing the matter whose expansion is negative, that matter will form a black hole. So if we have a large clump of matter in an FRW spacetime that detaches from the Hubble flow and reaches a point where it has negative expansion, it will form a black hole as long as the dust (zero pressure) approximation remains valid.

For a star-sized clump of matter, that won't be true--pressure will become important while the matter is still much larger than a black hole of its mass, and fusion will ignite. Even when fusion energy is exhausted, it might collapse to a white dwarf or neutron star instead of a black hole.

But for a clump of matter that is millions or billions of solar masses, the zero pressure approximation might remain valid--the matter might be a big gas cloud or it might have sub-clumped into stars, but the stars will be far enough apart that they don't coalesce so the matter as a whole can still be approximated as a zero pressure "dust" made of stars. So if such a clump in the early universe formed and reached a point of negative expansion, it could form an early universe SMBH.

 

[Vanadium 50]

was the boundary of a region which cannot send signals to future null infinity. But in spacetimes that aren't asymptotically flat there is no future null infinity

I think you are on to something, but I think this throws the baby out with the bathwater. This definition prohibits black holes in FRW spacetime, but we see black holes. The argument says they are merely very, very dark gray.

However, whay I think we can do is define a characteristic time of R/c (i.e. M in geometriced units) for a BH. For ordinary BHs and SMBH;s the external geometry is static on this timescale. For the cosmological BHs I am discussing, it is not. For example, one can orbit an ordinary BH: we are doing it right now. It may not be possible to orbit a cosmological black hole at all, if space expands so much you cannot return to your starting point.

 

[PeterDonis]

we see black holes

We can't know for sure that the compact, non-light-emitting regions we see are black holes by the definition @Ibix gave, because we can't know for sure that such regions have actual event horizons. They could be, for example, Bardeen "black holes", which have no actual event horizons. (We have had previous PF threads on these; basically the spacetime geometry in their deep interior becomes de Sitter or something like it, which prevents the formation of either a singularity or a true event horizon--the only horizons in these spacetimes are apparent horizons, i.e., trapped surfaces, which eventually evaporate away.)

In other words, to know for sure that a compact, non-light-emitting region is an actual black hole, rather than something that looks like one for a long time but isn't one, we would need to know the entire future of our actual spacetime. And of course we can't.