What is the significance of the Cauchy horizon?

[yanniru]

I am intriqued by a recent series of three papers on black holes:

http://www.arxiv.org/abs/gr-qc/0411060
Title: The river model of black holes
Authors: Andrew J. S. Hamilton, Jason P. Lisle (JILA, U. Colorado)

http://www.arxiv.org/abs/gr-qc/0411061
Title: Inside charged black holes I. Baryons
Authors: Andrew J. S. Hamilton, Scott E. Pollack (JILA, U. Colorado)

http://www.arxiv.org/abs/gr-qc/0411062
Title: Inside charged black holes II. Baryons plus dark matter
Authors: Andrew J. S. Hamilton, Scott E. Pollack (JILA, U. Colorado)

In the first, space is flowing into the spherical black hole. At the event horizon, space inflow is at the speed of light so light cannot escape. If the black hole does not rotate, the River Flow model predicts, incorrectly as the first paper claims, that space drops through the Cauchy horizon and expands into a new universe. The reader is referred to the second paper for a correct prediction.

In paper two with charged baryonic matter inflow, along with space, the charged baryons either plunge directly into the central singularity if the black hole is uncharged, or if charged they are repelled, become outgoing and pass through the outgoing inner horizon, namely the Cauchy horizon. No mention of a new universe on the other side in this paper. Also no mass inflation so this solution is also presumably incorrect.

The correct solution is found in paper three where the outgoing, repelled baryons encounter an incoming flow of dark matter which leads to exponential mass inflation. The claim is that after mass inflation ceases, the outgoing baryons become ingoing and plunge to the spacelike singularity at zero radius, unless the baryons completely absorb the inflowing dark matter. In that case the baryons pass through the Cauchy horizon with no mention of a new universe.

Now what intriques me in all this is the prospect of predicting a new universe from GR. Is it correct to assume that whenever the Cauchy horizon is passed through, that a new universe is entered? And that this may happen whenever counterstreaming ceases?

Now what is even more intriquing is the relationship of these solutions to Bojowald's treatment of the spacelike singularity at zero radius using Loop Quantum Gravity (LQG). According to Coule, Bojowald's solution allows for the creation of baby universes in the singularity.

Counterstreaming certainly ceases in the singularity. Perhaps the physics captured by the LQG approximate model is essentially what Hamilton has also found outside the singularity, where GR is applicable.

Can someone tell me if any of these non-expert suppositions are correct?
Richard Ruquist

 

[pervect]

I haven't read the papers, but a Cauchy horizon can be more or less considered to be the surface of a domain of dependence.

If you have some set of points S, everything that a change in S can affect in the future defiines the domain of dependence of S.

The "surface" of this domain of dependence is the Cauchy horizon.

Example: the event horizon of a black hole is an example of a Cauchy horizon, because things that happen inside the black hole cannot effect or propagate outwards past the event horizon.

 

[Stingray]

An event horizon is not a Cauchy surface in the sense that I know it. It is the boundary to a region beyond which Cauchy surfaces are no longer sufficient to uniquely define the future (just a slight jumbling of your definition, but it makes a big difference).

More intuitively, it is a region where causality breaks down. You would usually think of specifying the initial conditions of the universe on a maximal spacelike hypersurface (Cauchy surface). This is sufficient to provide a unique future unless a Cauchy horizon is reached. There's no singularity at a Cauchy horizon, but for whatever reasons, initial data is not sufficient to give unique predictions anymore. I don't really have any good intuition for what these things mean, but they come up in the interiors of certain black hole solutions. I suppose that you might be able to say that you are entering another universe, but that depends on your definitions. I think a lot of people believe that Cauchy horizons are unstable (they shouldn't exist in any realistic spacetime), but this is just a conjecture (maybe an outdated one?).

 

[yanniru]

Thanks Stingray for your answer. It seems to be consistent with Hamilton's 'assumption' that when matter crosses a Cauchy horizon, it goes into a new space not in this universe. Hamilton did not say it was an assumption. But from the characterization in your post, it would appear that it is.

So I guess my next question is, does such a horizon exists in the singularity of the black hole? Such singularities can be treated by LQG but so far the available solutions do not have sufficient detail. A hint is given by Hamilton who says that his solution indicates that when the center of mass [of the inflowing dark matter and the outflowing baryonic matter in the black hole] turns inward, the Cauchy horizon shrinks back into the singularity. Perhaps it continues to exist there and contributes to the creation of new universes.

Richard

 

[Stingray]

So I guess my next question is, does such a horizon exists in the singularity of the black hole? Such singularities can be treated by LQG but so far the available solutions do not have sufficient detail. A hint is given by Hamilton who says that his solution indicates that when the center of mass [of the inflowing dark matter and the outflowing baryonic matter in the black hole] turns inward, the Cauchy horizon shrinks back into the singularity. Perhaps it continues to exist there and contributes to the creation of new universes.

As I remember, the point of a Cauchy horizon is that you can pass through one (in theory). So a singularity would not be a Cauchy horizon. There are however, black hole solutions with Cauchy horizons in their interior (Schwarzschild is not one of these).

 

[yanniru]

Hamilton (http://www.arxiv.org/abs/gr-qc/0411062
Title: Inside charged black holes II. Baryons plus dark matter)
refers to the Cauchy horizon as a null singularity, whatever that means.
He then observes theoretically that the horizon falls back into the region of the central singularity when there is lessor density of baryonic outflow than dark matter inflow.
So I wonder if the horizon ceases to exist within the region where GR in not valid,
and I would look to more detailed LQG solutions to make such a determination.

 

[Stingray]

Hamilton (http://www.arxiv.org/abs/gr-qc/0411062
Title: Inside charged black holes II. Baryons plus dark matter)
refers to the Cauchy horizon as a null singularity, whatever that means.
He then observes theoretically that the horizon falls back into the region of the central singularity when there is lessor density of baryonic outflow than dark matter inflow.
So I wonder if the horizon ceases to exist within the region where GR in not valid,
and I would look to more detailed LQG solutions to make such a determination.

I don't really want to read through the paper (sorry), and I couldn't get much from the abstract. Where does he say that the horizon falls into the central singularity?

Also, he's not referring to a null singularity as a Cauchy horizon. The abstract specifically says that a null singularity does not form on the Cauchy horizon. So he's definitely distinguishing the two, which is correct.

 

[yanniru]

Page 14 of Hamilton's paper has a section that discusses the null singularity on the Cauchy horizon, claiming that several previous papers by other researchers had found one, but his assumption of self-similarity precludes such a case. My memory mistake.

 

[marcus]

This was an interesting thread started by yanniru (who signs himself Richard Ruquist) which raises the question if a new expanding region of space comes out the bottom of a black hole.

There has been some progress on that, made by several researchers, during the past couple of years. If yanniru were around PF now he might be interested to know, for instance, of the paper of Gambini Pullin of May 2008. Indeed they did find a horizon! But sort of the opposite from the usual inflow black hole horizon. The suggestion was a kind of exploding white hole horizon. And using the Loop formalism they found a new region would be created, at least in the simple case they studied.
So this goes back to what yanniru had been reading which was based on classical analysis (not quantum gravity) and it even goes back to a 1982 paper of Jim Hartle and Subrahmanyan Chandrasekhar (of the famous Chandrasekhar limit.)

It's kind of interesting how Gambini Pullin are in a way corroborating by quantum gravity something that people speculated about earlier (that in fact there might not be a singularity, a point where time-evolution just stops dead in its tracks---nature might go on---there might be something more---physics might just come to a dead halt down there in the hole).

Cool.
http://arXiv.org/abs/0805.1187
Black holes in loop quantum gravity: the complete space-time
Rodolfo Gambini, Jorge Pullin
4 pages, 2 figures
(Submitted on 8 May 2008)
"We consider the quantization of the complete extension of the Schwarzschild space-time using spherically symmetric loop quantum gravity. We find an exact solution corresponding to the semi-classical theory. The singularity is eliminated but the space-time still contains a horizon. Although the solution is known partially numerically and therefore a proper global analysis is not possible, a global structure akin to a singularity-free Reissner--Nordström space-time including a Cauchy horizon is suggested."Because Chandrasekhar was one of the great astrophysicists of all time and had incredible intuition I should show respect and get a link to Hartle and Chandra's 1982 paper. Here is the first page (a free read):
http://www.jstor.org/pss/2397225
"The geometry of the analytically completed Reissner-Nordström space time suggests that one may emancipate oneself from the past and escape into 'new worlds' by crossing the inner Cauchy horizon of the black hole. But...might be dangerous..."
http://rspa.royalsocietypublishing.org/content/384/1787/301.abstract

 

[cecilemangana]

Cauchy horizon is a light-like boundary of the domain of validity of a Cauchy problem (a particular boundary value problem of the theory of partial differential equations). One side of the horizon contains closed space-like geodesics and the other side contains closed time-like geodesics.Cauchy horizons are inherently unstable. However, cases of AWEC violation, such as the Casimir effect, do exist. If we can assume that the spacetime inside the Cauchy horizon violates AWEC, then the horizon becomes stable and frequency boosting effects would be canceled out by the tendency of the spacetime to act as a divergent lens.
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