# Logarithmic corrections to black hole entropy, especially for the standard model

1. May 8, 2012

### mitchell porter

Ashoke Sen has written a paper

http://arxiv.org/abs/1205.0971

which uses path integrals in Euclidean gravity to compute "Logarithmic Corrections to Schwarzschild and Other Non-extremal Black Hole Entropy in Different Dimensions".

Sen starts by listing several varieties of extremal black hole for which a string theory calculation of such corrections agrees with a Euclidean-gravity calculation of the corrections. He notes that there is no such stringy calculation for non-extremal black holes (such as the ones of astrophysical interest), but loop quantum gravity does make claims in this area, some of which have been discussed on this site (e.g. Bianchi 2010, Engle Noui Perez 2009). Sen performs the Euclidean gravity calculation and finds that it disagrees with the LQG values.

Lubos Motl writes that this is the end for LQG. Well, we'll see.

While I would like to see a discussion about how LQG could reproduce this result (or why it can be ignored), I also find it interesting that Euclidean gravity does agree with string theory here, several times over. Why exactly is that? Can you "see" the stringiness already in the Euclidean gravity, if you know what to look for?

A minor twist is that Sen's calculations are for Schwarzschild black holes in pure d=4 gravity, a theory which I believe is in string theory's "swampland" - the set of theories which can't be obtained as a low-energy limit of string theory (the name is meant to suggest the opposite of the "landscape" of field theories which can be so obtained). So it shouldn't actually be possible to reproduce Sen's exact calculation within string theory either. But this is a minor detail because the method of calculation should easily be adjusted to include the effects of the other standard-model fields, and if string theory does contain the standard model, then this adjusted result is definitely one it should be able to produce.

The primordial paper in which the prototype of Sen's calculation first appears is apparently

http://arxiv.org/abs/hep-th/9407001

You may see, for example, the origin of the denominator "45" in Sen's formula, in equation 87 of this paper from 1994. So if you're trying to use these Euclidean results as a guide to LQG theory construction, this is the part that you want to reverse-engineer.

Perhaps this paper could be fruitfully contrasted with "Bianchi 2010", linked above. And I'll also throw in 't Hooft's "The holographic mapping of the Standard Model onto the black hole horizon, Part I: Abelian vector field, scalar field and BEH Mechanism", which is not directly about the entropy, but which is aiming at the description of a black hole within the standard model.

edit #1: A bridge to 't Hooft's paper might be found in http://arxiv.org/abs/1104.3712 by Sergey Solodukhin, author of the 1994 calculation, when he discusses 't Hooft's 1985 "brick wall model", which is an important part of history both for the holographic principle and for black hole entropy calculations.

edit #2: Perhaps I should spell out that, although this is naturally another LQG vs string topic, the core calculation here does not originate in either framework, it comes from Euclidean gravity. Sen has calculated these corrections to the Bekenstein-Hawking entropy for a Schwarzschild black hole in d=4 pure gravity, and it should be easy to adjust his procedure to compute the corrections if there are other fields as well. The question for both LQG and string is then 1) what attitude to take towards these calculations, agree or disagree, and if agree, 2) explain how to produce them within the LQG or string framework. There is also the tangential (but very interesting) question of how these calculations relate to 't Hooft 1985 (the brick wall model) and 't Hooft 2005 (holography for a standard model black hole).

Last edited: May 8, 2012
2. May 8, 2012

### marcus

Nice to have the connections drawn and links laid out. Thanks! I'll add a possibly useful reference. Here is a review paper:
http://arXiv.org/abs/1101.3660
Detailed black hole state counting in loop quantum gravity
Ivan Agullo, J. Fernando Barbero G., Enrique F. Borja, Jacobo Diaz-Polo, Eduardo J. S. Villaseñor
(Submitted on 19 Jan 2011)
We give a complete and detailed description of the computation of black hole entropy in loop quantum gravity by employing the most recently introduced number-theoretic and combinatorial methods. The use of these techniques allows us to perform a detailed analysis of the precise structure of the entropy spectrum for small black holes, showing some relevant features that were not discernible in previous computations. The ability to manipulate and understand the spectrum up to the level of detail that we describe in the paper is a crucial step towards obtaining the behavior of entropy in the asymptotic (large horizon area) regime.
=============

These authors have a different log term (see table on page 30) from what Ashoke Sen refers to as characterizing the Loop BH entropy.
They say -(1/2)log a and he says (on page 28) -2log a.
Superficially different at least--perhaps reconcilable but I don't see how.
I'm not sure any of that will hold over the long term--still too much technical disagreement.

As I guess you are well aware, the question of black hole entropy is not settled in LQG.
Even in the pre-2012 work, where the authors think that they must specify a value of the Immirzi parameter in order to recover Bek.Hawk semiclassical, they use different enough methods so that some get γ=0.237 and others get γ=0.274.
Again see the table on page 30 of the Agullo et al paper. http://arXiv.org/abs/1101.3660 Crisp summary of differences.
And then Bianchi posted a paper last month (April 2012) which finds the entropy to be quite different from either group. Basically proportional to area with coefficient 1/4 without fixing the value of Immirzi at all!

If I had to bet, I'd guess that Bianchi is closer to being right---that the BH entropy relation does not require fixing a particular value of Immirzi (a radical innovation in context of earlier work). And Bianchi has not yet worked out the quantum corrections, or any way not posted. His paper does not specifically mention a log term at all. So we'll just have to wait and see if there is a log term and if so what it is.

Last edited: May 8, 2012
3. May 8, 2012

### fzero

Sen (and earlier Solodukhin) argues that the logarithmic correction to the entropy depends only on infrared physics and not the details of the UV completion of gravity. One of the ways used to argue this is the fact that it is generated at one-loop only in Euclidean gravity. So any theory that purports to provide a correct description of quantum gravity must reproduce the Euclidean result.

In the string computations, the framework is typically a supersymmetric QM for the lightest degrees of freedom. These are "stringy," because they contain wrapped strings and branes, but they are point particles and do not include higher oscillator states. In the super QM. the number of states is computed using a topological index, which is protected from quantum corrections.

The string description must be thought of as dual to the gravity description, in the sense that microscopic states in a gauge theory are dual to extended objects in a gravitational theory. On one side we have the D-brane picture, on the other we have solitonic p-branes. The AdS/CFT duality is a precise version of this duality, but it was known already before Maldacena studied the near-horizon limit.

In the case of the extremal black holes, the microscopic description is particularly nice (super QM with an index theorem). For non-extremal BHs, quantum corrections to the energy of states is no longer under complete control, which is a primary reason for the lack of concrete results in those cases.

I've gotten off track to add some details of the stringy description, but the least puzzling way to look at this is not that the Euclidean gravity sees "stringy" degrees of freedom. It is that the dual string description in terms of gauge QM properly computes gravitational physics in the appropriate regime.

For Sen, $a_\mathrm{Sen}$ is the length scale of the black hole. In 4d, the BH area goes like $\sim a_\mathrm{Sen}^2$. In the other paper, $a$ is the area. Sen accounts for this properly, see e.g., the discussion around eq (4.1). I think there is also a complication from comparing the grand canonical ensemble with the microcanonical ensemble that he also explains.

I've already explained carefully that Bianchi's result is independent of the Immirzi because he does not sum over the internal structure of the BH state. So it is hidden in the area constraint which is used to rewrite a microscopic expression (that depends on the Immirzi) in terms of a macroscopic constant (that does not). Every computation of the entropy in LQG that uses an ensemble relies on the Immirzi taking a fixed value.

It is unlikely that Bianchi can even obtain the log correction using his latest methods. In the standard analysis, the log comes from the character formula for the Chern-Simons representation for the states. In Bianchi's polymer formalism, he gives a plausible explanation of how to reproduce this. But it will only appear if you compute the distribution of states, which is only going to be apparent from studying an ensemble.

4. May 8, 2012

### tom.stoer

LQG is dead. LQG remains dead. And Lubos has killed it. How shall he comfort himself, the murderer of all murderers?

5. May 8, 2012

### marcus

:rofl:
Indeed how shall he comfort himself? You think then he imagines himself as a kind of murderer-in-chief?

btw has anyone pointed out the difference in sign?

Sen's log a term is about +1.7 log a. And if we take A~a2 then that would be roughly +0.85 log A

But in Loop context the term tends to be negative, about -0.5 log A.

The +1.7 is what he compares to the Loop results he cites, more precisely he gives it on page 29, equation 4.5 as
212/45 - 3
That comes to 1.7111...
"This is different from (4.3), showing that the loop quantum gravity result for logarithmic correction to the entropy does not agree with the prediction of the Euclidean gravity analysis."

Last edited: May 8, 2012
6. May 8, 2012

### atyy

I'm not sure one can kill the undead.

7. May 8, 2012

### tom.stoer

God is dead. God remains dead. And we have killed him. How shall we comfort ourselves, the murderers of all murderers?
Friedrich Nietzsche, The Gay Science

Gott ist tot! Gott bleibt tot! Und wir haben ihn getötet! Wie trösten wir uns, die Mörder aller Mörder?
Friedrich Nietzsche, Die fröhliche Wissenschaft

8. May 8, 2012

### marcus

So that's where it's from! Nietszche! I thought it had a ringing classic sound to it.

BTW a few posts back there was some speculation about the validity of Bianchi's new handling of Loop BH entropy, so a propos that, I should pass along the news that he has been invited to give the PI Colloquium three weeks from now!

Julian Barbour will give a Colloquium talk on Shape Dynamics tomorrow 9 May, and then Bianchi gives the next one: Wednesday 30 May, at 2PM.

I'm curious to know what topics he will cover. His invited talk at the April meeting of the APS, in Atlanta, was a general survey of Spin Foam QG. The main talk he gave out here at UC Berkeley physics department was on Loop QG as the dynamics of topological defects. He already gave a talk at Perimeter about quantum polyhedra and approach to black hole entropy. http://pirsa.org/10110052/ So maybe the colloquium will not be about BH. But of course it's always possible that he will touch on this new work.

http://www.perimeterinstitute.ca/Scientific/Seminars/Colloquium/ [Broken]

Last edited by a moderator: May 6, 2017
9. May 8, 2012

### tom.stoer

anyway - Lubos is definitly right: LQG was not able to settle this issue

10. May 8, 2012

### marcus

So say I as well: it's clearly not settled yet, what the log term should be in the Loop BH context. The 2011 survey paper I linked has a small table showing the different groups' conclusions.

Also I have no great confidence in Ashoke Sen's number 212/45 - 3 = 1.7111...

The log term is speculative, up in the air, for the time being AFAICS.

I found the pirsa link for Bianchi's upcoming colloquium talk. Must remember to check it out around 30 May to see if there's some reference to the BH research.

Last edited: May 8, 2012
11. May 8, 2012

### fzero

I'm not sure quite what you're getting at here. $C_\mathrm{local}$ is the trace anomaly in the Euclidean gravity theory. Even if some minus sign relative to it and the zero mode contribution were wrong, it would not agree with the LQG result.

Marcus, science is not a spectator sport. When someone writes a paper on a subject, that makes it possible for others to see what they've done, understand and reproduce (or not) their results.

When I said that it was unlikely for Bianchi to obtain the log correction, I meant that it was unlikely for him to compute with any certainty any log correction using the Clausius relation and the semiclassical argument with the pure state. This has nothing to do with Sen's paper. Where the log correction comes from can be easily seen from looking at the LQG state counting papers.

In Bianchi's approach, he adds a single facet to his pure state, adding an entropy $\delta S =2\pi\gamma j_f$. By summing this over the facets of the pure state he finds $S = 2\pi\gamma \sum_f j_f = A/(4G\hbar)$. The only possible way to obtain a different result is to add the physics of the ensemble, removing the semiclassical aspect of the method.

Besides, unless there is some demonstrable error in all of the LQG state counting papers, how could Bianchi obtain something different? Perhaps the polymer model would give some different result, but the whole point of that program was to provide an alternative picture of the same physics as appears in the ordinary treatment.

It is not necessary to wait for some future paper to appear. There is enough information out there to actually look at the details of how these computations were done.

Sen's result is in agreement with earlier work by Solodukhin and Fursaev. Do have some objective reason to doubt it? His analysis is presented in impressive detail and he explains why it should be believed.

I don't think that an expert like Vidotto would agree, since she considers the papers that found the result to be authoritative, e.g., https://www.physicsforums.com/showthread.php?p=3888082#post3888082 I obviously disagree with her interpretation of swapping the Immirzi parameter for the CS level, but I have gone through the various calculations in sufficient detail to not doubt the "meat" of the calculations. That is, assuming the model of microstates to be correct, I don't think that there are trivial or nontrivial math errors in the derivation of the entropy. But you don't even have to take my word for it, since there are at least a dozen LQG experts that have produced these results in many papers. The only disagreements between the coefficients of the log term are explained by whether they sum over U(1) or SU(2) invariant states, which was appreciated by Sen when he presented the comparison.

Last edited by a moderator: May 6, 2017
12. May 8, 2012

### tom.stoer

As far as I understand the string theory perspective the problem is neither that LQG is not ready to predict the log-correction correctly, nor that there are different normalizations applied to the Immirzi parameter. The problem is that in the LQG framework the proportionality to the area is TRIVIAL b/c of the definition of the horizon (it is introduced and fixed classically). Therefore S ~ A is no prediction at all. But S = const * A with a definite value for const would be.

So the fact that LQG predicts S ~ A means that LQG only reproduces the input. What has to be calculated once and for all is the const!

Last edited: May 9, 2012
13. May 8, 2012

### atyy

Is it possible the model of microstates is wrong? I remember thinking the old calculations were a bit strange, where essentially only the horizon microstates were counted. I think the new papers use something different?

14. May 8, 2012

### fzero

I haven't bothered addressing this question, since my interest in the thread on the recent Bianchi paper was in understanding why his method gave a different answer from the other LQG calculations. It was clear that the problem there could be addressed by breaking things into

input -> machine -> output

and simply understanding what the "machine" was. Now, to really understand what is different between papers written in 96, 05 and today, one must get into the details of the input first.

The identification of what states should be counted is heavily influenced by Rovelli's picture in gr-qc/9603063. This is the paper where he argues that only the horizon degrees of freedom should be counted. His logic there is that the horizon is the only thing that an outside observer can interact with. So far, this is picture is consistent with the general ideas of holography: some microscopic theory on the horizon should capture the physics of the entire BH spacetime.

Now, Rovelli goes on to argue that the degrees of freedom are those of a spin network whose tetrahedra intersect the horizon, $\Sigma$. Here is where a problem comes up. We have taken a tesselation of the BH spacetime $\Delta$. There are a large number of microscopic degrees of freedom associated with the graph. Rovelli claims that only the degrees of freedom living in $\Delta \cap \Sigma$ contribute to the entropy of the black hole. He then "derives" that the entropy is proportional to the area of the horizon. But as Tom alludes to, when we do this procedure, there's no way that the leading behavior of the entropy could be anything but proportional to the area. So this much wasn't a test of the model.

Now, let's think more about the degrees of freedom. We threw away degrees of freedom living in $\Delta - \Sigma$ and kept those in $\Delta \cap \Sigma$. The dynamics of the horizon degrees of freedom are obtained from reducing the LQG machinery to the boundary. This is certainly not what holography would have suggested about the degrees of freedom. Holography would suggest that the horizon degrees of freedom are some dual formulation. Computing some number by summing over bulk degrees of freedom must agree with the appropriate computation done by summing over horizon degrees of freedom, but there is no straightforward way to imagine the horizon degrees of freedom being some subset of the bulk degrees of freedom. They are an equivalent, but different, description.

The Strominger-Vafa string theory computation is completely in line with the lessons of holography. There all gauge theoretic degrees of freedom that could contribute to a BH of the specified mass and charges were included and the area law came out anyway. The type of gauge-gravity duality exhibited was a precursor to the AdS/CFT correspondence .

As far as I can tell, all mainstream LQG approaches to BH entropy follow Rovelli's example. They use the same degrees of freedom, but there are some differences in the constraints placed upon them. The Agullo et al review that marcus cited above is probably a good place to look. To the extent that I understand that part of the methods, I would not do a great job of briefly explaining it.

Bianchi's 2010 paper uses the polyhedral language to describe the BH entropy calculation, but it is just another way to parametrize the same variables.

So to answer your question, yes, the model of microstates could be wrong. As I said, the "machine" part of the LQG computation seems to have settled into a well-understood steady state. So one is left with questioning the "input" if one wants to explain any disagreement with Euclidean gravity.

15. May 9, 2012

### tom.stoer

I remember two papers, one regarding coarse-graining and the semiclassical limit (Rovelli) and one (author ???) regarding an effective intertwiner reproducing the spin net network intersection with the horizon. So essentially a BH in LQG could simply be represented as one single huge intertwiner. Therefore the counting of horizon intersections should not depend on the way how the intersections are created within the horizon. In a sense this may relate coarse graining and averaging over bulk degrees of freedom.

But again it relies on the classical definition of a horizon.

16. May 9, 2012

### MTd2

And, here, I wonder where is science in all this.

I wonder if there isn't any super strong force that would bind particles, or anything relevant, at a scale close to Planck scale and whose excited states are just the particles we see. This would make everything non trivial since no calculation without computers would not accomplish even approximations. Like calculating excited states of nucleons or molecules by hand, but worse.

17. May 9, 2012

### tom.stoer

I am not so sure about that.

The BH entropy issue is mostly a geometrical / kinematical one. It seems that details of the dynamics are not required, only the state space itself, i.e. the microscopic d.o.f. and the geometry are required.

18. May 9, 2012

### MTd2

The point it is, for example, you mention quantum properties when even classical black holes are hardly studied. It's just like people consider that obvious. It shouldn't.

19. May 9, 2012

### tom.stoer

I expect the theory of quantum black holes to be simpler than the classical one. Look at the hydrogen atom.

20. May 9, 2012

### MTd2

I am not sure a log correction would be a good analogy with the hydrogen: to calculate p,d,f sub orbitals in relation to to the only s orbitals of Bohr atom, or if it would be a step beyond and calculate the hyper fine corrections. Or perhaps avoiding this sort of analogy with orbitals and considering a statistical analogy, if distinguishing between log corrections would be like choosing between Einstein or Debye models for specific heat.

What I had in mind, anyway. was that we are clueless about what is happening at levels of microstates. Maybe it is not complicated. Maybe it is extremely complicated. How to figure that out?