# Puzzle: what's the entropy of BH horizon viewed from inside?

1. May 11, 2006

### marcus

In black hole thermo the famous entropy is that of the event horizon viewed from the outside.

does the standpoint of the observer make any difference to the entropy? and in particular does it make any difference to the entropy of the horizon if the observer falls in and is now viewing the horizon from inside?

comments including expressions of incredulous outrage welcome, have fun

2. May 11, 2006

### marcus

this question is related to a bothersome objection to QG posed by Roger Penrose----he objected to the loop quantum cosmology (LQC) replacement of bang singularity by bounce---and did so on grounds of the second law.

LQC unavoidably replaces the bang singularity with a bounce and a prior gravitational collapse, so if one is going to follow the LQC picture then at some point one confronts Penrose second law objection.

but Penrose objection possibly does not make any sense because it does not address the question of who is measuring the entropy. (are they before or after the planck regime "bridge")

casual background to this in Ashtekar's recent paper posted yesterday

http://arxiv.org/abs/physics/0605078
The Issue of the Beginning

Last edited: May 11, 2006
3. May 11, 2006

### NateTG

I'm sorry, but isn't that a bit like asking what the universe looks like if the observer is traveling at the speed of light?

4. May 11, 2006

### marcus

I expect you are right, that it is "a bit like" asking that. I am not sure in what way the questions are similar and would be happy if you want to explain in more detail how one is like the other.

there is also an obvious way that the questions are different. You couldnt take an ordinary observer and accelerate him to the speed of light---so what you are talking about is an impossibility.

However I imagine that a person COULD fall into a black hole. So it is not such an obvious impossibility as what you mention--and the question is distinctly more palpable

5. May 11, 2006

### f-h

Marcus, if you are in the black hole you can not view the horizon, it's in your causal past, right?. So the only question that makes sense is what does the observer see as it passes the horizon, and the common answer is: Nothing out of the ordinary.

6. May 11, 2006

### marcus

Right. So what becomes of Penrose thermo objection to Loop gravity bounce?

I've heard him make this objection twice now, once live at Berkeley and once on a Princeton video lecture.

Loop gravity inescapably predicts a deterministic evolution of the wave function through the ex-singularity from a gravitational collapse to an expansion. If that cannot happen (say for high 2nd law reasons as Penrose archly suggests) then Loop is in trouble.

he says that the gravitational collapse is into a state with a lot of entropy and WHOOPS! all of a sudden you have the beginning of an expanding universe with little or no entropy----he says that transition appears to violate the 2nd law. a kind of discontinuity where entropy jumps down to zero.

What I suspect is that Penrose objection is meaningless because the BH horizon entropy ceases to be defined as the observer passes through and the collapse becomes the past

7. May 11, 2006

### f-h

As far as I know nobody has given a good formulation of statistical dynamics for a general relativistic context, so you do not have a 2nd law except in the asymptotic region.

At the bounce region itself we expect the concept of time to make no sense for example.

Looking at a Blackhole the scenario as I've seen it described is that you pass through a region of a state that has no interpretation in terms of classical spacetime before you emerge to a smooth space time again.

And while LQG predicts a deterministic wavefunction evolution through the bounce not all information needs to be accessible at all timeslices in LQG either. (I should say it's an open question whether it is or not, Hartle says no, and he has the best method to make sense of these questions that I've seen so far).

8. May 11, 2006

### marcus

grazie tanto!

9. May 11, 2006

### hossi

marcus, are you psychic? That question comes damned close to the question I was about to bother the stringy guys with on Friday!

Anyway, here is my opinion:

I don't believe that there is a singularity inside the black hole, neither do I think that information gets lost in black holes. Instead, there should be some stable dense blob in the inside - the challenge for QG is to stabilize it, but lets just assume that works somehow.

Now, when matter collapses it can form a horizon (that can already happen at arbitrarily low densities). However, even if a trapped surface sourrounds the matter, it still has some entropy, particles moving around on trajectories, microstates etc. These remain there when the matter is by-whatever-mechanism stabilized. That is in perfect disagreement with Don's interpretation, which should make Friday fun.

To come to your question: the horizon is a surface where the vaccum is unstable and decays in particle pairs. These are entangled pairs, the total entropy is zero, but in the Hawking-radiation at infinity only one half of the pair is detected - thus the finite entropy. This means, from the inside the entropy of the horizon is the same as from the outside. (Leaving aside the question whether it can actually be 'seen', because that requires knowlegde about the stabilization).Best,

B.

10. May 11, 2006

### f-h

Hey B. That's a very cool reference, thanks for that!

11. May 11, 2006

### marcus

grazie tanto anche à Lei. Every bit of different perspective on something like this helps, I think.

No unfortunately

Rovelli understands the trialogue form re-invented by Gallileo. that trialogue with Rovelli Marolf and Jacobson was a good thing. I would like another please! F-H PLEASE SUGGEST TO ROVELLI THAT HE MAKE ANOTHER BLACK HOLE TRIALOGUE either with the same other two people or different people.

Marolf and Jacobson are both, so to say, extremely valuable as CREATIVE ADVERSARIES of Loop QG. Both of them have for 10 years or more often been thanked for conversations in the acknowledgments at the end of LQG papers.

Jacobson has been the phenomenological troublemaker and Marolf perhaps you could say has been the friendly-to-loopers shadow string theorist.

I am glad that there is a conversation among this three-some on the Arxiv and happy that you have read it and reminded us of it, B.

http://arxiv.org/hep-th/0501103
Black hole entropy: inside or out?
Ted Jacobson, Donald Marolf, Carlo Rovelli
42 pages, contribution to proceedings of Peyresq 9
Int.J.Theor.Phys. 44 (2005) 1807-1837

"A trialogue. Ted, Don, and Carlo consider the nature of black hole entropy. Ted and Carlo support the idea that this entropy measures in some sense 'the number of black hole microstates that can communicate with the outside world.' Don is critical of this approach, and discussion ensues, focusing on the question of whether the first law of black hole thermodynamics can be understood from a statistical mechanics point of view."

Dont get the idea that I understand your model of the pairs of particleantiparticle from near the horizon making the entropy look the same from inside as from outside. I am thankful to you for a very different perspective that is kind of orthogonal and every different view helps. Also I think it is extemely original. However am not saying that I understand it or could comment perceptively. Perhaps others can.

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12. May 11, 2006

### hellfire

Doesn't this assume a very specific interpretation of the Hawking radiation as particle pairs being split at the horizon? Anyway, you probably know the argument called the multiple species problem: if the black hole entropy is entanglement entropy, it should scale up with the number of fields on the gravitational background. As far as I know there exists an interesting result in flat spacetime (see e.g. this) showing that the reduced density matrix for a field in the exterior of a sphere (traced over the degrees of freedom of the field inside the sphere) gives an entanglement entropy proportional to the area of the sphere. But can this be extrapolated to spacetimes with horizons?

Last edited: May 11, 2006
13. May 11, 2006

### hossi

Yeah, probably. I don't like quantum mechanics and I don't like Hawking's papers, and that's the only interpretation I can somehow make sense of. After all, what is radiation when not particles detected?

No, didn't know that. Why is that a problem? Do you have any references?

Best,

B.

14. May 11, 2006

### hellfire

Yes you are right, Hawking radiation are particles. But I think the question is rather if there are splitted particle pairs at the horizon. I am not sure that this follows.

See section 2.3 of Bekenstein's Do We Understand Black Hole Entropy ?

15. May 11, 2006

### f-h

I have an old question of mine about Hawking radiation. Is it the same as the Unruh effect?

People who believe in the equivalence principle will tell you that an infalling observer does not see any Hawking radiation, and that the Hawking radiation a stationary observer sees is just the Unruh effect due to stationary=accelerated.

In this picture locally nothing exciting happens at the Horizon except that at the horizon an observer needs infinite acceleration to be stationary.

In Hawkings picture of splitting particle pairs the whole effect is local at the Horizon and absolutely distinct from stuff that happens away from the Horizon.

Can these two picture be mapped to each other?

16. May 12, 2006

### Careful

**I have an old question of mine about Hawking radiation. Is it the same as the Unruh effect? **

Good question. No, the calculations happen with respect to different vacuum states ( see Wald : QFT in curved spacetime and BH thermodynamics, pages 128-129 ).

**
In this picture locally nothing exciting happens at the Horizon except that at the horizon an observer needs infinite acceleration to be stationary. **

Concerning the Unruh effect : all it does is to calculate the restricition of the minkowski vacuum state to one section of Rindler spacetime. This boils down to taking a singular coordinate transformation´´ which by itself has not much of significance. However, one can use this transformation to conviently couple a local accelerated dector state (and field) to the KG field and see whether one measures (a) something or not (b) a thermal spectrum after a ridiculously high thermalization time (which of course depends upon detector at hand).

The Hawking effect on the other hand calculates the projection´´ of the evolved natural vacuum state on J^- (past null infinity) on the natural vacuum state at J^+ (there is only portions of regions I and III from the Rindler diagram here) for late times´´ (such that the Eikonal approximation can be made). The latter states are the natural Minkowski vacuum states (and not the Rindler one) - so here we have a situation where there are *no* radiation modes coming from J^- (in contrast with the Unruh effect), but thermal radiation is seen at J^+ coming from the black hole region.

Hope this helps.

Cheers,

Careful

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17. May 12, 2006

### f-h

J^+ is future null infinity right? Radiation emerging from the Horizon that escapes to future null infinity is infinitely redshifted though, so you don't actually get any radiation at infinity, or do you?

Edit:
Upon reflection, the modes don't emerge from the horizon, do they? Only close to the Horizon. So the Hawking temperature is the temperature as seen by an asymptotic observer?

Last edited: May 12, 2006
18. May 12, 2006

### Careful

** J^+ is future null infinity right? **

Right

**
Radiation emerging from the Horizon that escapes to future null infinity is infinitely redshifted though, so you don't actually get any radiation at infinity, or do you? **

The redshift for particles near the Horizon is very large though finite. In the calculation this fact is used in the other direction : there one starts from states on J^+ very close to the event horizon H^+ and notices that in propagating the signals backwards in time a huge blue shift occurs (with respect to an observer on the collapsing body) which is basically the motivation for the use of the eikonal approximation (mind also that the Hawking temperature is extremely small such that only very long wavelengths are to be detected).

You can find the details in :

Particle creation by black holes, S.W. Hawking, Communications in mathematical physics 43, 199-220 (1975)

On particle creation by black holes, Robert M Wald, communications in mathematical physics, 45, 9 -34 (1975)

There is also a more acessible course by Townsend : gr-qc/9707012 where this can be found around pages 125.

Cheers,

Careful

19. May 12, 2006

### Careful

**As far as I know nobody has given a good formulation of statistical dynamics for a general relativistic context, so you do not have a 2nd law except in the asymptotic region. **

Correct, it is indeed a formidable problem to identify the right dynamical degrees of freedom (although within causal sets and probably also spin-network/foam associated `spinoffs´´ it should be feasable). In some more specific line of thought, one may wonder about defining an *objective* entropy. Anyway, is Penrose's argument simply that in a bounce, you basically reverse the arrow of time or what ? Could you expand on this ?

Careful