Realistic black hole in Friedmann universe

[tom.stoer]

I had a discussion regarding the region of spacetime visible to an observer falling into aSchwarzschild black hole. Using Eddington-Finkelstein coordinates this is rather straightforward, at least graphically.

However the situation is rather artificial b/c the BH exists externally. So my question is whether there are solutions or at least Penrose diagrams combining a black hole with a closed Friedmann universe, i.e. something like big bang - dust - collaps (= black hole formation) - static black hole - ... big crunch

 

[martinbn]

What do you mean by the BH exists externally? Also, what is the definition of a black hole in a closed universe?

 

[tom.stoer]

Sorry, "eternally", of course!

The definition of a BH in a finite universe will probably rely on local (instead of global) defininitons of horizons, e.g. non-expanding or isolated horizons (instead of the event horizon defined using future null infinity)

 

[martinbn]

I see, thanks. I cannot say anything about what you are asking, but i remember glancing at a preprint discussing black holes in an expending Friedman universe. Can't find it now.

 

[Mentz114]

The nearest thing I can find is the section called "A Schwarzschild cavity in the Friedmann universe", page 279 of "General Relativity" by Stephani (1993 edition).

 

[WannabeNewton]

The nearest thing I can find is the section called "A Schwarzschild cavity in the Friedmann universe", page 279 of "General Relativity" by Stephani (1993 edition).

Even if it doesn't explicitly talk about black holes, that is an extremely interesting discussion (I just skimmed over it right now but I'll read it more thoroughly later). Thanks! Good thing you made me buy this book :)

 

[Mentz114]

Even if it doesn't explicitly talk about black holes, that is an extremely interesting discussion (I just skimmed over it right now but I'll read it more thoroughly later). Thanks! Good thing you made me buy this book :)

Glad you like it. I realized after posting that it isn't what Tom.Stoer is looking for.

However, can the singularity that results from a big crunch be described as a BH ?

 

[WannabeNewton]

No it can't. The singularity occurs everywhere in space as opposed to being localized.

 

[WannabeNewton]

Sorry, "eternally", of course!

The definition of a BH in a finite universe will probably rely on local (instead of global) defininitons of horizons, e.g. non-expanding or isolated horizons (instead of the event horizon defined using future null infinity)

This is in accord with what Wald says on p. 301: "On the other hand, there appears to be no natural notion of a black hole in a "closed" (k = +1) Robertson-Walker universe which re-collapses to a final singularity, since there is no natural region to which "escape" can be attempted. Of course, an approximate notion of a black hole still exists for any region of a closed Robertson-Walker universe that can be treated as an isolated system."

Do you know of any references expanding upon the last sentence from the quoted paragraph?

 

[Bill_K]

can the singularity that results from a big crunch be described as a BH ?

No it can't. The singularity occurs everywhere in space as opposed to being localized.

For someone who has ventured within the horizon, a black hole singularity does occur "everywhere", however it has the wrong symmetry. Unlike the cosmological crunch which collapses in all directions, the singularity within a black hole has cylindrical symmetry, collapsing in two directions (θ and φ) and expanding in a third (the t coordinate).

 

[WannabeNewton]

For someone who has ventured within the horizon, a black hole singularity does occur "everywhere", however it has the wrong symmetry. Unlike the cosmological crunch which collapses in all directions, the singularity within a black hole has cylindrical symmetry, collapsing in two directions (θ and φ) and expanding in a third (the t coordinate).

This is certainly true but we have a manifestly coordinate independent way of characterizing a body/geometric object as being isolated. Throughout the history of the closed Friedman universe there is a failure to satisfy that characterization.

 

[JesseM]

This is in accord with what Wald says on p. 301: "On the other hand, there appears to be no natural notion of a black hole in a "closed" (k = +1) Robertson-Walker universe which re-collapses to a final singularity, since there is no natural region to which "escape" can be attempted. Of course, an approximate notion of a black hole still exists for any region of a closed Robertson-Walker universe that can be treated as an isolated system."

Do you know of any references expanding upon the last sentence from the quoted paragraph?

I had a question about this on another thread, but the thread was closed before I could get an answer, so I'll ask again here. In a closed universe that re-collapses to a "Big Crunch" singularity, is there some reason it wouldn't work to define a black hole "event horizon" in terms of whether timelike and lightlike worldlines from a point hit the black hole singularity or the Big Crunch singularity? In other words, is it possible to make a distinction between points in spacetime where all timelike/lightline worldlines passing through that point must terminate in the black hole singularity, and points where at least some timelike/lightlike worldlines passing through that point "escape" the black hole and instead terminate in the Big Crunch singularity? Or is it somehow meaningless to distinguish between these two singularities since they merge with one another?

 

[jcsd]

I had a question about this on another thread, but the thread was closed before I could get an answer, so I'll ask again here. In a closed universe that re-collapses to a "Big Crunch" singularity, is there some reason it wouldn't work to define a black hole "event horizon" in terms of whether timelike and lightlike worldlines from a point hit the black hole singularity or the Big Crunch singularity? In other words, is it possible to make a distinction between points in spacetime where all timelike/lightline worldlines passing through that point must terminate in the black hole singularity, and points where at least some timelike/lightlike worldlines passing through that point "escape" the black hole and instead terminate in the Big Crunch singularity? Or is it somehow meaningless to distinguish between these two singularities since they merge with one another?

The problem is that the singularities aren't actually points in spacetime and you want to be able to define a black hole generically

 

[JesseM]

The problem is that the singularities aren't actually points in spacetime and you want to be able to define a black hole generically

But I'm just asking whether it's possible to distinguish worldlines that terminate in the black hole singularity vs. ones that terminate in the Big Crunch singularity--this question might have a well-defined answer in terms of limits without the need to treat singularities themselves as points in spacetime. One can distinguish different singularities that worldlines can terminate on in Penrose diagrams here for the "analytically extended Schwarzschild geometry" or the Reisser-Nordstrom and Kerr black holes, for example. Maybe there isn't a way to make such a distinction in the case of a closed universe with black hole singularities, that's what I'm asking, but if there isn't the reason would have to be more subtle than just the fact that singularities aren't points in spacetime.

 

[WannabeNewton]

In other words, is it possible to make a distinction between points in spacetime where all timelike/lightline worldlines passing through that point must terminate in the black hole singularity, and points where at least some timelike/lightlike worldlines passing through that point "escape" the black hole and instead terminate in the Big Crunch singularity? Or is it somehow meaningless to distinguish between these two singularities since they merge with one another?

If I'm understanding you correctly, are you asking if there's a difference in classification between space-like singularities* at which all possible time-like geodesics terminate and space-like singularities at which some time-like geodesics terminate but not others (e.g. Painleve observer vs. circularly orbiting observer or static observer in Schwarzschild space-time)?

*See here for an explanation of the term "space-like singularity": http://physics.stackexchange.com/qu...ition-of-a-timelike-and-spacelike-singularity

 

[PeterDonis]

is there some reason it wouldn't work to define a black hole "event horizon" in terms of whether timelike and lightlike worldlines from a point hit the black hole singularity or the Big Crunch singularity?

Yes: in a closed universe that collapses to a Big Crunch, the Big Crunch is the only final singularity: there are no separate black hole singularities. In other words, all timelike and lightlike worldlines end in the Big Crunch, even the ones that locally appear to be trapped behind a horizon. Any such horizon in a closed universe can only be an apparent horizon, i.e., a locally trapped surface at which outgoing light does not move outward. But all such trapped surfaces eventually get swallowed up in the collapse of the entire universe to the Big Crunch.

 

[JesseM]

Yes: in a closed universe that collapses to a Big Crunch, the Big Crunch is the only final singularity: there are no separate black hole singularities. In other words, all timelike and lightlike worldlines end in the Big Crunch, even the ones that locally appear to be trapped behind a horizon. Any such horizon in a closed universe can only be an apparent horizon, i.e., a locally trapped surface at which outgoing light does not move outward. But all such trapped surfaces eventually get swallowed up in the collapse of the entire universe to the Big Crunch.

OK, thanks. Does the definition of "the same singularity" vs. "different singularities" have to do with where the singularities appear in a conformal mapping, so in this case the fact that they are "the same" has to do with the fact that you can't come up with a conformal mapping that assigns the spacelike Big Crunch singularity to a different coordinate than a spacelike black hole singularity? Is it possible to theoretically rule out the possibility of a timelike singularity in a closed universe, and if not could such a singularity be considered distinct from the Big Crunch singularity?

edit: on second thought my suggestion about same vs. different singularities depending on whether they're mapped to the same point in a conformal mapping like a Penrose diagram can't really be right, because a timelike singularity looks like an extended line rather than a point on such a diagram. Does anyone have a reference on how the same vs. different singularities are defined in general relativity?

 

[PeterDonis]

Does the definition of "the same singularity" vs. "different singularities" have to do with where the singularities appear in a conformal mapping, so in this case the fact that they are "the same" has to do with the fact that you can't come up with a conformal mapping that assigns the spacelike Big Crunch singularity to a different coordinate than a spacelike black hole singularity?

Yes. The conformal diagram of a closed Friedmann universe has no infinity anywhere; it's just a box with the initial singularity at the bottom, the final singularity at the top, and the two sides identified so that every spatial slice is compact.

Is it possible to theoretically rule out the possibility of a timelike singularity in a closed universe

I'm not sure; I think you can do it by using energy conditions and global hyperbolicity, but I'd have to check the math to be positive.

if not could such a singularity be considered distinct from the Big Crunch singularity?

A timelike singularity on the conformal diagram would be a line running from bottom to top (assuming it existed in every spacelike slice), so it would be distinct from the Big Bang and the Big Crunch; it would basically be connecting the two.

on second thought my suggestion about same vs. different singularities depending on whether they're mapped to the same point in a conformal mapping like a Penrose diagram can't really be right, because a timelike singularity looks like an extended line rather than a point on such a diagram.

So do the spacelike singularities we've been talking about. See my description of the conformal diagram of a closed Friedmann universe above. (Yes, the Big Bang and Big Crunch are lines in the diagram, even though they have zero 4-volume; conformal mappings can result in infinite stretchings like this.) The singularity at the center of a black hole is also a spacelike line on a conformal diagram.

Does anyone have a reference on how the same vs. different singularities are defined in general relativity?

AFAIK the general method you're suggesting is correct: singularities are "the same" if they map to the same geometric object in a conformal diagram. It's just that the geometric object doesn't have to be a point; it can be a line. (Conformal diagrams suppress two of the four coordinates, usually the angular ones, so point and line are the only two possibilities.)

 

[JesseM]

Thanks again Peter, I think you've cleared it all up for me.