Entropy of Non-Extremal Black Holes from Loop Gravity

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Referee this paper: [arXiv:1204.5122]

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Summary thanks to Lubos Motl (PhysicsOverflow posts) who also contributed the review

The paper goes approximately like this.

We want to calculate the right entropy of black holes from loop quantum gravity, with the ultimate goal to show that loop quantum gravity is a competitor of string theory as a theory of quantum gravity. How do we do that?

Take a few tons of spin foam and construct a little model of Stephen Hawking, Jacob Bekenstein, and Bill Unruh.

Now, make the spin foam model of Bill Unruh accelerate with a constant acceleration. Because loop quantum gravity has "quantum gravity" in its name, his observations will match those in the real world which contains quantum mechanics as well as gravity. In particular, this spin foam model of Bill Unruh must be able to calculate the temperature he experiences. In the same way, the spin foam models of Stephen Hawking and Jacob Bekenstein will do the same calculations as their real-world counterpart and they will obtain the result

$S=\frac{A}{4G}$

for the black hole entropy. It's great because we have just proved that loop quantum gravity obtains the right result for the black hole entropy, regardless of the value of the Barbero-Immirzi parameter that was promoted to fix factors in previous calculations of the entropy – calculations whose results were proportional to this parameter.

summarized
paper authored Apr 23, 2012
recategorized Aug 27, 2014

1 Review

+ 3 like - 0 dislike

Thanks to Lubos Motl (posts on PhysicsOverflow) for allowing to us to use his blog post as a review.

Just a week ago, I discussed the incompatibility between the black hole thermodynamics and loop quantum gravity. In a new paper, Ashoke Sen updated the list of black holes whose logarithmic corrections to their entropy are calculable.

For the Schwarzschild black hole in $d=4$, the right formula contains terms such as $(212/45-3)\ln(a)$while the LQG folks have confidently claimed that their theory predicts the coefficient equal to $-2$ or $-3$. Sorry, guys, that didn't work well. ;-)

But a "slightly less intelligent" paper was pointed out by Backreaction and Physics Forums:

It's great because we have just proved that loop quantum gravity obtains the right result for the black hole entropy, regardless of the value of the Barbero-Immirzi parameter that was promoted to fix factors in previous calculations of the entropy – calculations whose results were proportional to this parameter.

Or have we? ;-)

Obviously, the paper has nothing to do with a microscopic calculation of the black hole entropy which requires that you count microstates. Instead, it is just a rewriting of the usual macroscopic or thermodynamic derivation – equivalent to the semiclassical derivations that emerged from the work by Bekenstein and Hawking – into a language that tries to sound as much loopy as possible. But the actual microscopic theory, the loop quantum gravity, isn't used anywhere. So the paper obviously doesn't do anything beyond the work done in the 1970s.

As Sabine has equally noticed, the situation is actually much worse than that. By rephrasing the semiclassical arguments – which are not quite correct but they're close enough to be called a spin foam caricature of the right calculation – in the loopy language, one is forced to say many things about the spin foams and their number of states that clearly and brutally contradict statements about the same things that exist in the loop quantum gravity literature.

In particular, the entropy of the black hole should depend on the Barbero-Immirzi parameter but Bianchi derives that it doesn't. So one gets two different values predicted "by the same theory" for the same object. That's clearly a contradiction. Obviously, the contradiction doesn't imply that mathematics is inconsistent. Instead, what it implies is that some of the assumptions had to be incorrect. The assumption that was incorrect was that the structures that are constructed from the spin foam toys ever resemble physics in a nearly flat space. They never do. That's why you can never construct an accelerated loopy Bill Unruh in a Rindler space and that's why you can't use this impossible Bill Unruh to prove a contradiction in mathematics.

Even though the author himself works hard to keep his head in the sand and not to see the flagrantly obvious fact, Bianchi's paper is a proof that loop quantum gravity is inconsistent with physics in a nearly flat space. We don't even have to discuss any curvature – the Rindler space doesn't have and doesn't require any. Even in the flat space, we may derive a contradiction. That's because the flat space, as an approximation, isn't among the environments that are allowed by loop quantum gravity.

The discussion at the Physics Forums is hilarious, too. There may be hundreds of rather detailed comments including formulae in which the participants try to make sense out of the loop quantum gravity literature on black hole thermodynamics. Obviously, certain technical yet elementary questions arise such as:

1. Does the leading LQG result of the entropy of a large black hole depend on the Barbero-Immirzi parameter?
2. May the degenerated faces with the spin foamy spin j>1/2 be neglected?
3. Can the correlations between groups of faces be neglected when one computes the entropy at the leading order?
4. Do the leading and subleading terms depend on physics beyond the semiclassical approximation?
5. Is the paper XY in the LQG literature – where XY is pretty much any element of a large set – correct?
6. Does the Barbero-Immirzi parameter run?
7. If it runs, is this fact consistent with the discreteness of the level of the Chern-Simons theory?
8. Should the Barbero-Immirzi parameter which was inserted as a fudge factor to mask a wrong result be correct by another fudge factor?

and many others. You may see that the answer to absolutely every elementary question of this type becomes totally open by the end of the Physics Forums discussion. The situation is totally fuzzy when it comes to absolutely everything and absolutely anything in LQG. It's clear what the right answers to these questions (except for those that depend on the nonsensical LQG magic toolkit) are in the correct theory – most of them follow from approximations to quantum gravity and every consistent full theory of quantum gravity must respect these answers.

It's clear that the analogous qualitative questions in a functional theory of quantum gravity would be instantly answered – otherwise a sensible physicist wouldn't dare to suggest that he has any theory of these phenomena at all. The term $S=A/4G$$S=A/4G$ is universal and derivable from semiclassical gravity, the logarithmic corrections also depend on the low-energy spectrum only, some higher-order corrections may need you to know more, and so on. String theory has calculated the entropy for lots of diverse multi-parameter classes of black holes – with different dimensions, different topologies, different charges, different dual descriptions, different levels of deviations from the extremality, and so on – and whatever had to agree with the macroscopic calculations has always exactly agreed although the agreement has always looked miraculous. Despite these initial feelings of a miracle, no one is surprised anymore. Those things have to hold in a consistent theory of quantum gravity and string/M-theory is one (and quite certainly, the only one).

But none of them get answered over there, in loop quantum gravity. There aren't any answers because there isn't any fixed theory here.

Instead, what they have are just some vague ideas about mathematical tools that could be used in the desired theory – and there is a lot of wishful thinking that such a theory could be found. But every, arbitrarily basic question about the "right way" to combine the proposed "building blocks" and how to choose the right assumptions remains unresolved. Whatever combination of building blocks and assumptions is chosen always leads to some immediate contradictions. In fact, it is trivial to show that any and every combination of the assumptions and building blocks used by the LQG folks is inconsistent with physics of gravity.

reviewed May 25, 2014 by (10,278 points)
edited Jun 2, 2014

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