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Based on the Bekenstein-Hawking Equation for Entropy, hasn't the relationship between quantum mechanics and gravity already been established.

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I thought I had written a comment answering this on 18 June 2013 21st century 3 milenium? ? Where did it go? Let me re - write it. (1) It only is about a specific scenario, for black holes. (2) It only calculates entropy nothing else. (3) Gravity is treated as a classical field, but matter as quantum,. 4 / . –  Dimensio1n0 Jul 17 '13 at 18:00
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@Dimension10 even good and important comments disappear, if they displease some people for the one or the other reason from this site these days ... –  Dilaton Jul 17 '13 at 18:32
    
Besides, this simple area law does not taking into account charge, angular momentum, etc. So even in the restricted domain of semi-classical gravity, it is still a incomplete description –  lurscher Jul 17 '13 at 18:58

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up vote 11 down vote accepted

The macroscopic Beckenstain-Hawking entropy formula

$$ S_{BH} = \frac{k A}{4 l_p^2} $$

with the Planck length given by

$$ l_p = \sqrt{\frac{G\hbar}{c^3}} $$

gives a hint that quantum gravity is needed to determine the entropy because it contains both, the gravity constant $G$ and Plancks constant $\hbar$.

However, this formula does NOT say what the correct quantum gravity is, that is needed to correctly describe the microstates of the black hole. Assuming a certain quantum gravity and calculating the entropy from a statistical mechanics point of view by counting the microstates

$$ S = -k \sum\limits_i P_i \ln P_i $$

where $P_i$ is the probability that the system is in the microstate $i$, the Beckenstein-Hawking formula must be reproducable.

If it does not, the quantum gravity applied is wrong.

In summary, the Beckenstein-Hawking formula is not a quantum gravity theory, but it can be used as a test of all wannabe quantum gravities.

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How can the correspondence principle apply to the B-H formula? Doesn't Hawking Radiation have to be detected for the theory to be validated. To my knowledge Hawking Radiation have never been detected. –  user4884 Mar 24 '13 at 3:11
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@user4884 I do not know why the correspondance principle (between QM and classical mechanics you mean?) should not hold in the case of a black hole. Quantum mechanics is fundamental and holds for everything, however in certain cases (large systems for example) it is valid to assume the classical limit and use classical calculations. I do not exactly understand why you mention Hawking radiation in this context; for which theory to be validated do you think it need to be detected? The Hawking radiation as such does not discriminate between different quantum gravities either as far as I know. –  Dilaton Mar 25 '13 at 12:46
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Yes: arxiv.org/abs/0903.5321v1.pdf. Another similar one: arxiv.org/abs/1103.6044.pdf. This one was supposedly cited as a serious paper: arxiv.org/abs/hep-th/0503249v2.pdf. –  Dimensio1n0 Jul 17 '13 at 19:48
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@Dimension10 Lolz. I'm going to use "Supersplit Supersymmetry" from now on. :) –  Michael Brown Jul 26 '13 at 0:25
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@Dilaton: Setting dimensionless constants to 1 may be dangerous (e.g. coupling constants). –  Dimensio1n0 Jul 26 '13 at 12:53

To add to Dilaton's correct answer: The black hole area law is a result in classical gravitational physics. It tells us something about the macroscopic behavior of gravity, but it doesn't tell us anything directly about quantum gravity. It isn't even formulated in quantum mechanical terms. (This is what makes quantum gravity such a puzzle. The best constraint we have only constrains the correspondence limit.)

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Oh yep, nice important addition ... –  Dilaton Mar 23 '13 at 20:12
    
First, the term law in science is reserved for experimentally tested theories. The limit of these laws are based on instrumentation (i.e better instrumentation better measurements). Second, quantum gravity is either a difficult puzzle or a wrong approach. Einstein never tried to quantize gravity for a good reason. –  user4884 Mar 25 '13 at 15:27
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@user4884: "area law" is commonly used terminology. If you have a problem with this, you should register a complaint with the International Board for The Control of Scientific Terminology. –  user1504 Mar 25 '13 at 15:36

The Bekenstein-Hawking formula is obtained in the so-called "black hole thermodynamics", which is based in pseudo formal analogies with real thermodynamics. Some mistakes are reported by thermodynamic physicist Lavenda in his recent What's Wrong With Black Hole Thermodynamics?, but there are more...

Even if we accept the formula as if was correct, it does not establish "the relationship between quantum mechanics and gravity" because it precisely ignores quantum gravity effects and treats the black hole in a classical or 'semi-classical' fashion. When quantum gravity corrections are included, the event horizon (a purely classical concept) disappears. An introduction to the kind of quantum gravity corrections expected is given in Small, dark, and heavy: But is it a black hole?

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Please, link to arXiv abstract pages, not actual pdfs. This the norm at our place. –  Slaviks Jul 17 '13 at 19:42
    
@Slaviks Please cite the norm. Thanks! –  juanrga Jul 25 '13 at 18:55
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@juanrga When you go to arxiv.org/abs/1110.5322 and look at "Cite as:" the link it to the Abstract and not to the paper. –  ungerade Jul 25 '13 at 19:55
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While this norm is not written down anywhere in particular, a search like site:physics.stackexchange.com abstract pdf will reveal a plethora of comments indicating that we prefer abstracts rather than pdfs in links. This is in large part to help users of this site whose browsers/data connections (think mobile devices) are not so good with pdfs. @Slaviks for minor enough things that don't hurt the post, you can always suggest (or soon enough make) edits. –  Chris White Jul 25 '13 at 23:29

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