Monday, July 16, 2012

Bekenstein-Hawking entropy, strong and weak form

At the recent Marcel Grossmann meeting, I had been invited to give a talk about my 2009 paper with Lee on the black hole information loss paradox. (For a brief summary of the paper, see here.)

It occurred to me in some conversations after my talk that I lost part of the younger audience in the step where I was classifying solution attempts by the strong and weak form of the Bekenstein-Hawking entropy. Rarely have I felt so old as when I realized that the idea that the entropy of the black hole is proportional to its area, and the holographic principle which is based on it, has been beaten into young heads so efficiently that the holographic principle has already been elevated to property of Nature - despite the fact that it has the status of a conjecture, a conjecture based on a particular interpretation of the black hole entropy.

The holographic principle says, in brief, that the information about what happens inside a volume of space is encoded on its surface. It's like the universe is a really bad novel - just by reading the author's the name and the blurb on the cover you can already tell the plot.

The holographic principle plays a prominent role in string theory, gravity is known to have some "holographic" properties, and the idea just fits so perfectly with the Bekenstein-Hawking entropy. So there is this theoretical evidence. But whether or not quantum gravity is actually holographic is an open question, given that we don't yet know which theory for quantum gravity is correct. If you read the wikipedia entry on the holographic principle however you might get a very different impression than it being a conjecture.

The most popular interpretation of the Bekenstein-Hawking entropy is that it counts the number of microstates of the black hole. This interpretation seems to have become so popular many people don't even know there are other interpretations. But there are: Scholarpedia has a useful list that I don't need to repeat here. They come in two different categories, one in which the Bekenstein-Hawking entropy is a property of the black hole and its interior (the strong form), and one in which it is a property of the horizon (the weak form). If it is a property of the horizon there is, most important, no reason why the entropy of the black hole interior, or the information it can store, should be tied to the black hole's mass by the Bekenstein-Hawking formula. If the weak interpretation is true, a black hole of a certain mass can store an arbitrary amount of information.

If Hawking radiation does indeed not contain any information, as Hawking's calculation seems to imply and is the origin of the black hole information loss paradox to begin with, then you're forced to believe in the weak form. That is because if the black hole loses mass then, according to the strong form of the Bekenstein-Hawking entropy, its capacity to store information decreases and that information has to go somewhere if it's not destroyed. So it has to come out, and then one has explaining to do just how it comes out.

There is a neat and simple argument making this point in a paper by Don Marolf, the "Hawking radiation cycle"
"[O]ne starts with a black hole of given mass M, considers some large number of ways to turn this into a much larger black hole (say of mass M′), and then lets that large black hole Hawking radiate back down to the original mass M. Unless information about the method of formation is somehow erased from the black hole interior by the process of Hawking evaporation, the resulting black hole will have a number of possible internal states which clearly diverges as M′ → ∞. One can also arrive at an arbitrarily large number of internal states simply by repeating this thought experiment many times, each time taking the black hole up to the same fixed mass M′ larger than M and letting it radiate back down to M. We might therefore call this the ‘Hawking radiation cycle’ example. Again we seem to find that the Bekenstein-Hawking entropy does not count the number of internal states."

Let me also add that there exist known solutions to Einstein's field equations that violate the holographic bound, though it is unclear if they are physically meaningful, see this earlier post.

While I admit that the strong form of the Bekenstein-Hawking entropy seems more appealing due to its universality and elegance, I think it's a little premature to discard other interpretations. So next time you sit in a talk on the black hole information loss problem, keep in mind that the Bekenstein-Hawking entropy might not necessarily be a measure for the information that a black hole can store.

For a good discussion of these both interpretations and their difficulties, see "Black hole entropy: inside or out?" by Ted Jacobson, Donald Marolf and Carlo Rovelli.

22 comments:

Ulla said...

How do you COMPRESS bosons?

Vince said...

>> Scholarpedia has a useful list

when I click on this link I see a high level overview of BH entropy but no list or different categories of entropy.

Mitchell said...

"It's like the universe is a really bad novel - just by reading the author's the name and the blurb on the cover you can already tell the plot."

That's hilarious.

Vince said...

I guess you mean this listhttp://www.scholarpedia.org/article/Bekenstein-Hawking_entropy#What_is_behind_the_black_hole_entropy...

Bee said...

Vince: Sorry, I was assuming you're able to scroll down yourself. Best,

B.

Uncle Al said...

"taking the black hole up to the same fixed mass M' larger than M and letting it radiate back down to M." Real world black holes have event horizon temperatures below that of the cosmic microwave background. They only net accrete.

(T_H) = [(h-bar)c^3]/[8(pi)GM(k_b)]
P = [(h-bar)c^6]/[15360(pi)G^2M^2]
(r_s) = 2GM/c^2

CMB temp is 2.725 K or 2.348x10(-4) eV. A black hole must have ~0.8% or less of Earth's mass to net radiate. Schwarzchild radius is 1.48×10^(−27) m/kg, giving it a maximum radius of 71 microns. Surface-to-volume ratio effects will be invisible.

Plato Hagel said...
This comment has been removed by the author.
Plato Hagel said...

Blackhole wars(The Cosmic Landscape, by Leonard Susskind, Page 339, Para 2) no longer exist when we've located a peaceful realm in dialogue.

Best,

Plato Hagel said...

“Whereas Plato envisioned common perceptions as revealing a mere shadow of reality, the holographic principle concurs, but turns the metaphor on it’s head. The Shadows-then things that are flattened out and hence live on a lower-dimensional surface-are real, while what seemed to be more richly structured, higher dimensional entities (us; the world around us) are evanescent projections of the shadows.” Brian Greene-The Fabric of the Cosmos, pg. 482

You can't get away from the question of the way in which one looks at the surface. You know Lee called it the Thing.:)

Oh I see you will have to respond to Musser's article by linking your experiences in relation to his?

Did something solid come out of the conference?

Best,

Uncle Al said...

"taking the black hole up to the same fixed mass M' larger than M and letting it radiate back down to M." Real world black holes have event horizon temperatures below that of the cosmic microwave background. They only accrete.

(T_H) = [(h-bar)c^3]/[8(pi)GM(k_b)]
P = [(h-bar)c^6]/[15360(pi)G^2M^2]
(r_s) = 2GM/c^2

CMB temp is 2.725 K or 2.348x10(-4) eV. A black hole must have ~0.8% or less of Earth's mass to net radiate. Schwarzchild radius is 1.48×10^(-27) m/kg, giving it a maximum radius of 71 microns. Surface-to-volume ratio effects will be invisible.

Eric said...

I don't know why people are so intent on using the BH entropy paradox as the clue to the quantum gravity question. There is obviously a vital piece of information that is missing that makes it a paradox. I don't have an answer or a clue about it and I don't think anyone else who has published about it has one either. I could be wrong.

The point being is that when a person tries to piece together a puzzle there often comes a time when you realize that you just don't have enough information to find a solution to one aspect of the puzzle. Instead you must move on to other unresolved questions (in quantum gravity)for which you have a better hope of finding a solution. The hope is always that the answer to these other easier unresolved questions will give more information to solve the current unanswerable question. Crossword puzzle anyone? Why are people being so obstinate about working on this question to the detriment of other more tractable problems?

Bee, this philosophy of tackling intractable problems for which more information is needed directly leads to people assuming answers which they don't have enough information to assume - holographic information on the surface of the BH. As the policemen at a horrible crash says, the leaders in physics should just say to the rest "just keep moving along, nothing to see here folks".

Zephir said...

The dense aether model prefers the steady state Universe model, so it requires to find some model for black hole evaporation - or we would have the Universe already full of black holes. Recently the axion condensation based theory gained some merit. But I do believe, that black holes can emanate neutrinos or even heavier particles directly and continuously through polar jets a gravitational waves (tachyon condensation).

Alex Schneider said...

Could it be that the holographic principle is the result of something analogous to mistaking an energy for a torque because they happen to have the same units?

Bee said...

Hi Uncle,

Your comment was stuck in the spam filter, sorry about that. Maybe check your numbers, I believe I recall it being giant black holes, masses of 10^27g or so, that have a temperature comparable to the CMB. Best,

B.

Bee said...

Hi Alex,

No. Best,

B.

Uncle Al said...

Apology for the double post tendered in turn. "8^>) Thermodynamics proposes, kinetics disposes. Theory employing processes that cannot empirically obtain is uncomfortable.

Earth's mass is 5.9723x10^27 g or 5.9723x10^24 kg. Let's calculate explicitly given a full Earth mass (limited by the uncertainty in G),

(T_H) = [(h-bar)c^3]/[8(pi)GM(k_b)]
[(1.054571726x10^(-34) J·s) (299,792,458 m/s)^3] / [(8)(3.141593)(6.6742x10^(-11) m^3/kg-s^2) (5.9723x10^24 kg) (1.3806503x10^(-23) m^2-kg/K-s^2) = 0.02054 K; way too low a temperature!

(0.02054 K)/(2.725 K) = 0.00754 of Earth's mass is the maximum mass for BH Hawking radiation net decay when immersed in the CMB. Corrected Schwarzchild radius is maximum 67 microns (twice that of a coarse human hair). Surface-to-volume-ratio effects will be very slow to evolve.
(We refrain from mentioning machine shop distances smaller than a smidgeon and a skosh.)

rickyjames said...

Questions I've wondered about - say a pair of entangled photons are produced and one crosses the event horizon of a black hole. If the "free" photon is then "polarized", does the "trapped" one follow suit? What if the "trapped" photon is polarized "first" by equipment that was previously sent across the event horizon - does the "free" photon follow suit? Just what happens when you run a John Bell / Alain Aspect experiments where there is an event horizon in the flight path of the photons?

Sam said...

"Let me also add that there exist known solutions to Einstein's field equations that violate the holographic bound, though it is unclear if they are physically meaningful, see this earlier post."

It has been known for some time that the naive bound claiming that any 2-sphere contains at most a finite entropy is violated. The (as far as I know) correct bound is the covariant entropy bound proposed by Bousso. Cf. the nice review
http://arxiv.org/abs/hep-th/0203101
From a quick look at their Penrose diagrams, I would suspect the solutions Hsu and Reeb only violate the naive bound. They don't seem to discuss the relation to the covariant entropy bound on their slides.

Bee said...

Sam,

I don't see how this would make a difference. From the outside these things look like normal black holes. Put your surface there and calculate the entropy inside. It can be arbitrarily large. How does using the covariant bound change anything about that? Best,

B.

Bee said...

Hi rickyjames,

Yes, that's normally assumed to happen. Read for example section II of this paper, which discusses exactly this example. Best,

B.

Bee said...

Hi Uncle,

Right. I should have looked up the Earth mass before writing nonsense, it seems we're saying the same anyway. Best,

B.

Zephir said...

I would react to this article: Complementarity or Firewalls?  Why not both? These two animations (1, 2) are illustrating it clearly. Just the evaporation of massive object inside of black hole may be perceived as a complementary dual behaviour: the massive remnant of object is compressed and spaghettised, but the other part of it evaporates and expands. Mathematically these approaches indeed aren't consistent because they're switching intrinsic and extrinsic observational perspectives.