Ten Theses on Black Hole Entropy [Sorkin]

[tom.stoer]

There is a rather intersting paper from Sorkin:

http://arxiv.org/abs/hep-th/0504037
Ten Theses on Black Hole EntropyAuthors: Rafael D. Sorkin (Perimeter Institute and Syracuse University)
(Submitted on 5 Apr 2005 (v1), last revised 19 Oct 2011 (this version, v2))
Abstract: I present a viewpoint on black hole thermodynamics according to which the entropy: derives from horizon "degrees of freedom"; is finite because the deep structure of spacetime is discrete; is "objective" thanks to the distinguished coarse graining provided by the horizon; and obeys the second law of thermodynamics precisely because the effective dynamics of the exterior region is not unitary.

I just want to stress the following [Sorkin]

Thesis 1: The most natural explanation of the area law is that S resides on the horizon.

Thesis 4: The idea that the degrees of freedom are inside the black hole is wrong.

Thesis 9: To understand the generalized second law requires a spacetime approach, not a canonical one.

His reasoning regarding thesis 9 is striking and I would like to interpret as follows: 'locating' the degrees of freedom on a surface would require somehow to 'locate' this surface in spacetime; but for lightlike surfaces it is hopeless to 'locate' or 'embed' the surface in a spacelike slice or to reconstruct a non-spacelike horizon in a semiclassical approach on a spacelike slice which is used in canonical approach.

 

[czes]

According to holographic principle the space may be an illusion and it suggests that everything happens on a screen not in a space. The space is therefore a perfect mathematical matrix not a physical vacuum.
Therefore the space may be discrete and continuous as the information can.
The gamma ray is not disturbed by a fluctuation of the space because there isn't such a fluctuation of the mathematical space.

 

[PAllen]

I have increasing skepticism about over-emphasis on the horizon when looking for a quantum approach to black holes. My main sources of doubt:

1) Cosmic censorship is false in the strictly mathematical sense, in GR (Hawking already conceded there are mathematically valid naked singularity solutions). So far, these are not quite physically plausible, but the trend is finding ever closer to physically plausible naked singularity solutions. Efforts have been made to characterize distant observations that can distinguish a naked singularity from one with an event horizon.

2) The "No Hair" theorems only establish given same total energy, charge, angular momentum, that after a settling time, you cannot distinguish any features of the 'inside' from the 'outside'. However, that in no way establishes that the interior is identical.

On point (2), a key thing ignored in most such discussions is super-massive black holes. Here, at the time of horizon formation, you have perfectly ordinary stars, perfectly ordinary matter densities, throughout the interior region. One must presume perfectly normal, standard model physics continues until some high energy density is reached, where new physics applies.

This leads me to think of horizon theorems as analogous to classical thermodynamics versus statistical mechanics: globally true statements whose real explanation is microscopic, and in this case, the microscopic explanation has to deal with the interior.

 

[marcus]

Tom, I recall being fascinated by that Sorkin paper when it came out. It is deep and provocative. I want to offer a small excerpt of a paper by Bianchi that appeared about 11 months ago and seems related at least in a tangential way. I don't pretend to a firm grasp here: I just offer this IN CASE you can get something out of it. Just ignore it if you don't find it helpful:

==quote Bianchi page 2, section 2==
Black Hole entropy and the shape of the horizon

The idea that the microstates responsible for the entropy of a Black Hole correspond to fluctuations of the shape of its horizon goes back to Sorkin [12] and to York [13]. The idea is the following. Let us focus on the non-rotating case J = 0 for the moment. At equilibrium, the Black Hole is described by a Schwarzschild geometry. From a statistical mechanics point of view, this is to be understood as the macrostate. While the macrostate is spherically symmetric, the microstates don’t have to be. For instance, in presence of a planet orbiting around a Black Hole – because of tidal effects – a small bulge is produced on the horizon. Similarly, in the presence of a gas at a finite temperature, the horizon will be thermally fluctuating. Even in absence of a thermal bath, the horizon will be fluctuating because of quantum effects. The key idea is that these horizon degrees of freedom are accessible from the exterior of the Black Hole, contribute to heat exchanges of the Black Hole with its surroundings, and provide a microscopic explanation of its thermodynamic entropy (see [14] for a recent discussion of this idea).

The idea of horizon shapes resonates with another development started by Damour in the late seventies: the membrane paradigm [15]. This is a description of the interaction of a Black Hole with the outside world in terms of a horizon boundary-condition with an fascinating physical interpretation. The boundary dynamics turns out to be the one proper to a (fictitious) physical membrane with mechanical properties and with a finite surface viscosity.

The fact that the entropy of a Black Hole can be accounted for by considering the quantum fluctuations of the horizon was explored by Maggiore in the mid-nineties [16]. A perturbative Quantum-Field-Theory calculation of the fluctuations of the horizon/membrane leads in fact to an entropy proportional to the area of the horizon. However, the calculation requires an ultraviolet cut-off that shows up in the proportionality constant. When the cut-off is removed, the entropy diverges.

The idea explored in this paper is that LQG provides a very specific physical cut-off for the horizon-shape degrees of freedom. The physical cut-off is due to quantum geometry effects [17] proper of LQG, and makes the entropy finite.
==endquote==

In case anyone wants to look at more of the article, here is the link:
http://arxiv.org/abs/1011.5628
Black Hole Entropy, Loop Gravity, and Polymer Physics
Eugenio Bianchi
(Submitted on 25 Nov 2010)
Loop Gravity provides a microscopic derivation of Black Hole entropy. In this paper, I show that the microstates counted admit a semiclassical description in terms of shapes of a tessellated horizon. The counting of microstates and the computation of the entropy can be done via a mapping to an equivalent statistical mechanical problem: the counting of conformations of a closed polymer chain. This correspondence suggests a number of intriguing relations between the thermodynamics of Black Holes and the physics of polymers.
13 pages, 2 figures

 

[marcus]

I have increasing skepticism about over-emphasis on the horizon when looking for a quantum approach to black holes...

Just a thought. Could it be that your objections arise not from over-emphasis on the horizon but from overemphasis on classical GR and on a particular definition of the horizon?

In quantum gravity work it has turned out to be convenient to replace the earlier notion of BH horizon by the "isolated horizon". Your comment about supermassive BH is interesting and I wonder it it applies if we are using the isolated horizon (IH) definition.

PF-member Stingray had a post about IH in the Relativity forum:
https://www.physicsforums.com/showthread.php?p=1130939#post1130939

==quote Stingray==
Isolated horizons have their own notion of mass which only requires knowing quantities on the horizon itself. Also, all of the classical laws of black hole mechanics have been shown to hold true with isolated horizons (or dynamical ones where appropriate).

One reason for doing all of this is that event horizons are very difficult to use in general. Their definition requires knowing the entire history of the universe. There are also examples of spacetimes where portions inside an event horizon are actually completely flat. Isolated (and more generally dynamical) horizons are a completely local definition. They can only exist in a strong-field region, and identifying them doesn't require knowing the future.

The original laws of black hole "thermodynamics" often had somewhat difficult interpretations owing to the nonlocal definitions of event horizons. This situation goes away when looking at the formulation of these laws in the isolated/dynamical horizon framework.

There are actually precise flux laws that one can write down describing the evolution of the hole's mass and angular momentum based on the flux of (what are essentially) gravitational waves and matter across its horizon. The fact that all of this was possible in exact GR was a complete surprise to almost everyone. And that includes the people who developed this formalism.
==endquote==

 

[marcus]

Here is an early paper (2000) on IH by Ashtekar Baez Krasnov:
http://arXiv.org/abs/gr-qc/0005126
==quote introduction==
Isolated horizons are a generalization of the event horizons of stationary black holes to physically more realistic situations [1,2]. The generalization is in two directions. First, while one needs the entire spacetime history to locate an event horizon, isolated horizons are defined using local spacetime structures. Second, spacetimes with isolated horizons need not admit any Killing field. Thus, although the horizon itself is stationary, the outside spacetime can contain non-stationary fields and admit radiation. This feature mirrors the physical expectation that, as in statistical mechanics of ordinary systems, a discussion of the equilibrium properties of black holes should only require the black hole to be in equilibrium and not the whole universe. These generalizations are also mathematically significant. For example, while the space of stationary solutions to the Einstein-Maxwell equations admitting event horizons is three dimensional, the space of solutions admitting isolated horizons is infinite dimensional. Yet, the structure available on isolated horizons is sufficiently rich to allow a natural extension of the standard laws of black hole mechanics [2,3]. Finally, cosmological horizons to which thermodynamic considerations also apply [4] are special cases of isolated horizons.
==endquote==

PAllen, what I'm thinking is that some of your misgivings about focusing on the BH horizon might be allayed by the shift to the IH definition. (This occurred in quantum gravity BH research some years ago--it is taken for granted and not always stressed that the IH definition is being used.)

Also, if you don't base analysis on the horizon, what can you base it on? As far as we know the "singularity" is a fictional idea which depends on classical GR. But the whole point of BH research is to develop a quantum gravity treatment. When this is attempted one typically loses the singularity. People have proposed different geometric objects to put in its place, but this has not stabilized yet. To date, AFAIK, it continues to be more mathematically tractable and interesting to deal with the (IH) horizon than to try to study what one presumes to be in the pit of the hole.

 

[tom.stoer]

The interesting thing about Sorkin's theses is that they contradict both LQG (in the canonical formulation) and string theory (fuzzball proposal - d.o.f. NOT on a horizon)

 

[atyy]

I just want to stress the following [Sorkin]

Thesis 1: The most natural explanation of the area law is that S resides on the horizon.

How about more prosaic examples like http://arxiv.org/abs/0704.3906?

2) The "No Hair" theorems only establish given same total energy, charge, angular momentum, that after a settling time, you cannot distinguish any features of the 'inside' from the 'outside'. However, that in no way establishes that the interior is identical.

Are there any physically relevant examples where the topology is not space X time? That is an assumption made in the canonical framework.

 

[skeptic2]

For instance, in presence of a planet orbiting around a Black Hole – because of tidal effects – a small bulge is produced on the horizon.

Is this correct? It would seem that for an object between the horizon and the orbiting planet, the planet's gravity would partially cancel some of the BH's gravity resulting in a depression in the EH instead of a bulge.

 

[marcus]

Is this correct? It would seem that for an object between the horizon and the orbiting planet, the planet's gravity would partially cancel some of the BH's gravity resulting in a depression in the EH instead of a bulge.

I can't resolve this. You have me stumped. I hope someone else will be able to respond. Eugenio's paper was published in May 2011 in Classical and Quantum Gravity (CQG journal). Here is the published version:
http://iopscience.iop.org/0264-9381/28/11/114006/

It happens to be free online access. I will check to see if there was any change following peer-review.
I see no change. It says "bulge". Here is the PDF:
http://iopscience.iop.org/0264-9381/28/11/114006/pdf/0264-9381_28_11_114006.pdf

 

[PAllen]

Is this correct? It would seem that for an object between the horizon and the orbiting planet, the planet's gravity would partially cancel some of the BH's gravity resulting in a depression in the EH instead of a bulge.

I can't resolve this. You have me stumped. I hope someone else will be able to respond. Eugenio's paper was published in May 2011 in Classical and Quantum Gravity (CQG journal). Here is the published version:
http://iopscience.iop.org/0264-9381/28/11/114006/

It happens to be free online access. I will check to see if there was any change due to peer-review.

I'm not familiar with the orbital case. However, for a body approaching a horizon, it is definitely true that at some point the horizon expands outwards to encompass the infalling body, and you have an irregular shaped horizon for a while. This is true for both the apparent horizon and the true horizon.

Separately, I don't know what the difference is between apparent horizon (local definition, almost always inside the true horizon), and isolated horizon. My guess is some detailed technical difference, or a separate term would not be used.

I don't think it is a trivial matter to jump from the infall behavior to orbital behavior. The infall behavior comes about future influence on horizon that Marcus mentioned - that fact that the infall will occcur affects the boundary of what light emitted now will ultimately be trapped. Interestingly, while apparent horizons are not sensitive to 'all of the future', the analyses I've seen show apparent horizons having similar shape evolution to true horizons, just a little inside the true horizon.

 

[PAllen]

PAllen, what I'm thinking is that some of your misgivings about focusing on the BH horizon might be allayed by the shift to the IH definition. (This occurred in quantum gravity BH research some years ago--it is taken for granted and not always stressed that the IH definition is being used.)

Also, if you don't base analysis on the horizon, what can you base it on? As far as we know the "singularity" is a fictional idea which depends on classical GR. But the whole point of BH research is to develop a quantum gravity treatment. When this is attempted one typically loses the singularity. People have proposed different geometric objects to put in its place, but this has not stabilized yet. To date, AFAIK, it continues to be more mathematically tractable and interesting to deal with the (IH) horizon than to try to study what one presumes to be in the pit of the hole.

I don't think the IH will affect my misgivings if it is at all similar to apparent horizon. There are two parts to my misgivings:

1) I am less and less convinced that Cosmic Censorship is robustly true. Physics will then have to deal with the actual process of catastrophic collapse and what prevents the ultimate singularity, without help of horizons.

2) For super-massive black holes, again, I don't think the IH changes anything. You have the apparent disappearance of billions of stars at perfectly ordinary energy density, with the stars still discrete, ordinary stars at the time you last could see them. For me this really nails it that horizons are fundamentally a gravitational optical effect in same category as lensing. The fact that, when present, you can characterize bulk properties by the horizon alone is interesting, and a useful theoretical device, but not a solution to the singularity problem

I am led to think that the core achievement required of a quantum gravity theorem is modeling the collapse itself. This success would cover the naked singularity possible problem, as well as being presumably related (time reversed) to the big bang itself.

 

[atyy]

I am led to think that the core achievement required of a quantum gravity theorem is modeling the collapse itself. This success would cover the naked singularity possible problem, as well as being presumably related (time reversed) to the big bang itself.

I don't think the singularity is a problem, since quantum GR breaks down near it, and so singularities are presumably not predicted.

OTOH, the event horizon, when it exists, creates Hawking's information loss problem.

 

[marcus]

..., since quantum GR breaks down near it, and so singularities are presumably not predicted.
...

I don't understand . The quantum geometry I'm familiar with does not break down but different versions evolve differently, where the classical BH singularity used to occur.

I think "break down" is too strong, and gives a misleading impression. It suggests "fail to compute", some kind of system crash, glitch, breakdown of the model. That's not what happens.

But different papers make different assumptions and arrive at different conclusions. BH including some modeling of BH collapse has been studied quite a bit in LQG but no one "official" version has prevailed as yet.