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The Bekenstein entropy for a black hole is proportional to the surface area $A$ of the black hole $$ S_{BH} = \frac{k_B}{4 l_P^2} A $$ with the Planck length $l_P = \sqrt{\frac{\hbar G}{c^3}}$.

The area is the surface of a sphere with Schwarzschild radius $r_s = \frac{2 M G}{c^2}$, so $$ A = 4 \pi r_s^2 = 16\pi \left(\frac{G}{c^2}\right)^2 M^2 $$ and the black hole entropy is therefore proportional to the mass of the black hole $M$ squared: $$ S_{BH} = \frac{4 \pi k_B G}{\hbar c} M^2. $$ But this quite unusual for an entropy. In classical thermodynamics entropy is always supposed to be an extensive quantity, so $S\sim M$. But the black hole entropy $S_{BH} \sim M^2$ is obviously a non-extensive quantity. Isn't a non-extensive entropy inconsistent within the framework of thermodynamics? Why is it, that the entropy of a black hole must be a non-extensive quantity? Shouldn't we better define an entropy for a black hole from e.g. the ratio of Schwarzschild radius to Planck length, which would give us an extensive entropy like $$ S_{BH, ext} \sim k_B \frac{r_s}{l_P} \sim k_B\sqrt{\frac{4G}{\hbar c}} M $$

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    $\begingroup$ thermodynamics in the presence of gravity is no longer extensive, even classical gravity, due to the long range nature of the force. That is one of the reasons why some people developed non-extensive thermodynamics, such as T'sallis statistics en.wikipedia.org/wiki/Tsallis_statistics $\endgroup$ – Wolphram jonny Jul 25 '16 at 22:07
  • $\begingroup$ @Wolphramjonny FWIW I think this comment could be developed into a very nice answer, if you can explain a little about Tsallis statistics (which are almost not mentioned before on this site). $\endgroup$ – Rococo Jul 25 '16 at 22:43
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    $\begingroup$ To the main point, a non-extensive entropy is certainly very interesting, and is the jumping off point for speculation about black holes, quantum gravity, and the holographic principle (en.wikipedia.org/wiki/Holographic_principle)... but it is certainly not inconsistent with thermodynamics. Specifying to entanglement entropy, one also sees this in condensed matter systems (usually near the ground state)- keyword is "area law." $\endgroup$ – Rococo Jul 25 '16 at 22:59
  • $\begingroup$ @Rococo please feel free to answer using my comment, perhaps I commented first, but I suspect you know more that I do about the subject $\endgroup$ – Wolphram jonny Jul 25 '16 at 23:05
  • $\begingroup$ @Wolphramjonny Thank you, but actually I really don't- I was hoping to learn something too :) $\endgroup$ – Rococo Jul 25 '16 at 23:07
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This is an answer adapted from Rococo and Wolphram jonny's comments plus a little googling.

Thermodynamics in the presence of gravity is no longer extensive (even classical gravity) due to the long range nature of gravity. This is one of the reasons why people developed non-extensive thermodynamics, like Tsallis statistics.

Tsallis Statistics

Tsallis statistics was originated by Constantino Tsallis, a Brazilian physicist working in Rio de Janeiro (though he was born in Greece in 1943 and grew up in Argentina). He introduced what is now known as Tsallis entropy and Tsallis statistics in his 1988 paper Possible generalization of Boltzmann–Gibbs statistics.

Tsallis statistics is considered to be a good (maybe even the best) candidate for a non-extensive theory of thermodynamics. It is intended to supplement Boltzmann-Gibbs statistics, not replace it. Tsallis statistics is a collection of mathematical functions and associated probability distributions that can be used to derive Tsallis distributions from the optimization of the Tsallis entropic form. They are also useful for characterizing complex, anomalous diffusion.

Tsallis Entropy

Tsallis entropy is a generalization of the standard Boltzmann-Gibbs entropy. Also introduced by Constantino Tsallis in the same 1988 paper, it is identical in form to Havrda–Charvát structural α-entropy within information theory. From the year 2000 on, a wide variety of evidence has been accumulated that confirms Tsallis entropy's experimental predictions. A short list of the most notable confirmations is given below:

  1. The distribution characterizing the motion of cold atoms in dissipative optical lattices, predicted in 2003 and observed in 2006

  2. The fluctuations of the magnetic field in the solar wind enabled the calculation of the q-triplet (or Tsallis triplet)

  3. The velocity distributions in driven dissipative dusty plasma

  4. Spin glass relaxation

  5. Trapped ion interacting with a classical buffer gas

  6. High energy collisional experiments at LHC/CERN (CMS, ATLAS and ALICE detectors) and RHIC/Brookhaven (STAR and PHENIX detectors)$^1$

Implications

While not all of the implications of this theory can be completely known, it refines the Boltzmann-Gibbs definition of entropy, provides a further tool, Tsallis statistics, to explore non-extensive thermodynamics, it is a jumping off point for much speculation about black holes, quantum gravity, and the holographic principle, to name a few examples.

Bekenstein and Black Hole Thermodynamics

It is unusual that Bekenstein used a non-extensive quantity, namely the mass squared, and Tsallis statistics wouldn't have played a part in this. The reason for this, though, was really just a gut feeling on the part of Bekenstein.

It all started (so to speak) with Stephen Hawking's area theorem for black holes ($S = k A/4$). Right away (this is November of 1970), he noticed that his law bore an uncanny resemblance to the second law of thermodynamics. However, he thought it was nonsensical that this could be true - it didn't make sense that the two were related, and anyway, black holes were black.

Jacob Bekenstein was not convinced. Hawking saying that the two were not the same meant the violation of the second law of thermodynamics. Every scientist sided with Hawking in this argument except John Wheeler, Bekenstein's PhD advisor (because, according to him, "your idea is crazy enough that it just might be right"). In his paper (which can be read here) Bekenstein says,

All the analogies we have mentioned are suggestive of a connection between thermodynamics and black-hole physics in general, and between entropy and black-hole area in particular. But so far the analogies have been of a purely formal nature, primarily because entropy and area have different dimensions. We shall remedy this deficiency...by constructing out of black-hole area an expression for black-hole entropy with the correct dimensions.

It should also be noted that the area theorem proposed by Hawking (the event horizon area of a black hole cannot decrease; it increases in most transformations of the black hole) requires increasing behavior that is reminiscent of the thermodynamic entropy of closed systems, and as such it is reasonable that black holes should be a monotonic function of the area (and it is the simplest such function).

So matters remained until the next year, when Hawking showed that black holes do indeed emit radiation in the form of virtual particles, and the rest, as they say, is history. All of the other laws of black holes formulated were basically the laws of thermodynamics for black holes, resulting in black hole thermodynamics and the famous (sort of) equation $S_{BH} = \frac{k A}{4 l_p^2}$.

In the equation, $S_{BH}$ is the entropy of a black hole (or Bekenstein-Hawking, whichever you prefer), $k$ is Boltzmann's constant, $A$ is the area of the event horizon of the black hole, and $l_p$ is the Planck length, so $l^2_p$ is the Planck area. Interestingly, looking at your calculations, you use $l_p$ instead of $l^2_p$. I assume in your equation that you use $k_B$ as Boltzmann's constant, instead of $k$.

Next Steps

So, looking at the similarities between the laws of thermodynamics and the laws of black hole thermodynamics, I think it was a pretty reasonable (considering the results, which make sense) assumption, though of course we have the benefit of hindsight. The main consequences of these thoughts were in terms of information - one could ask how it is possible that all the information of the black hole is "coded" on its surface. This idea was formalized by the holographic principle. If this idea is true (and many theoretical calculations point to, at the very least, this making sense) the entropy of a black hole has to be proportional to the area of the black hole (@BobBee goes in-depth into this in his answer, and explains it very well).

The next step in black hole thermodynamics would be to calculate a theory of quantum gravity. Why? Well, black holes are at that intersection where both gravity and quantum mechanics are important. They have a singularity, and all of our laws of physics break down there. There are still problems to be solved in black hole thermodynamics, but I think that non-extensive entropies are consistent with the theory of thermodynamics.

It should be noted, when talking about consistency or inconsistency here, that entropy has "changed" a decent amount since it was first formulated. From Clausius' definition, through Boltzmann and Gibbs, Claude Shannon (in terms of information theory), Bekenstein and Hawking (in terms of black holes) and Tsallis, entropy has been found to have many connections to many fields. As WetSavannaAnimal aka RodVance says in his answer, we need to broaden what we mean by extensive.

Sources

Thanks to Wolfram jonny and Rococo for their great comments. I used the website linked below for the quote and for the section on Tsallis entropy. I used this website for the information on Constantino Tsallis. I used this website for my information on Tsallis statistics. For the very curious, here is a website where if you scroll down a tad you'll see a pdf of Dr. Tsallis' paper.

For the section on Bekenstein, I mainly used Black Holes and Time Warps by Kip Thorne; a copy of it on google books can be found here. The relevant pages are 422 through 427. I also used this website. Bekenstein's paper is cited within the text; that is where the quote is from. Another very informative website is this one.

Finally, both of the other answers here are very good. Thanks to BobBee for explaining how the holographic principle fits in, and recent developments (and, of course, about how we need to generalize our definition of extensive). Thanks to WetSavannaAnimal aka RodVance for expanding upon BobBee's answer, your explanation was also very insightful and helpful.

$^1$Quote from this website

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  • $\begingroup$ Bekenstein published his ideas about black hole entropy in 1973. I don't think Tsallis statistics from 1988 could have played a role in his reasoning, why the black hole entropy must be a non-extensive quantity. $\endgroup$ – asmaier Jul 26 '16 at 6:41
  • $\begingroup$ here is a nice example of how non-extensive statistics work better that the reagular one in self gravitating systems researchgate.net/publication/… $\endgroup$ – Wolphram jonny Jul 29 '16 at 23:35
  • $\begingroup$ @heather: I'm sorry, but I'm still missing the physical argument that made Bekenstein/Hawking think, that the entropy of the black hole cannot be proportional to it's mass (as it is normally the case for entropy), but must be proportional to mass squared. $\endgroup$ – asmaier Jul 30 '16 at 20:50
  • $\begingroup$ Funny thing though, Gibbs' entropy was already defined for general systems (extensive or non-extensive) in 1902. Was Tsallis entropy really necessary or is the entire motivation based on a misunderstanding of classic statistical mechanics? $\endgroup$ – Nanite Jul 30 '16 at 22:44
  • $\begingroup$ No. But I found a hint somewhere else. Somebody mentioned that mass is not conserved during a black hole merger, part of it is always converted to energy by gravitational radiation. So the mass of a merged black hole would be less than the mass of the two black holes before the merge. If entropy would be proportional to mass, this would mean that the entropy of the black hole after the merge would be lower than the entropy of the two black holes before. This would be a violation of the law that entropy should never decrease. $\endgroup$ – asmaier Jul 31 '16 at 21:23
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Actually it is an extensive quantity, but not in the way extensive is used in classical thermodynamics. It is proportional to area and not mass (or equivalently for a classical object, volume). Entropy is normally (but not always) proportional to mass, or energy, which is approximately proportional to the number of elementary particles and thus the number of possible states. For a black hole (BH), the number of possible states is proportional to the number of different Planck areas there are in the BH horizon. i.e., the states information is as though it was stored in the horizon, not the bulk

We don't know how to calculate the entropy of a singularity, the physics breaks down and it requires quantum gravity to calculate it. But Hawking has calculated the temperature of a BH, through his calculations, where he discovered that a BH radiates as a black body at a given temperature given by the inverse of its mass. From there it is easy to get the entropy.

The simplistic version then is that since $dS = dQ/T$, with $Q$ the heat absorbed by the BH and $T$ its black body radiation-equivalent temperature, if you take Hawking's result as $T = k/M$ for some constant $k$, then $dS = kMdQ$. Since the heat absorbed into the BH is energy, it increases the BH mass (in natural units) as $dQ = dM$. So, substituting, $dS = kMdM$, and then integrating $S = kM^2$. You can see other derivations, but when Hawking found his BH radiation temperature dependent on mass, there's then no two ways around it that they could think of, nor anybody else so far.

This of course was proven by Hawking-Bekenstein based on finding that the Hawking radiation is black body with temperature proportional to the so called surface gravity at the horizon, which is inversely proportional to the BH mass, and with Bekenstein then led to entropy proportional to the horizon area, and in fact equal to the number of Planck areas multiplied by $k_b/4$ ($k_b$ is the Boltzmann constant), as per the Hawking-Bekenstein equation, also discussed in @Heather’s answer, and written (slightly off) in the question. This relates BH entropy and physics to thermodynamics – and indeed BH entropy can grow or stay the same, but never decrease, by the second law of thermodynamics. The two BH’s which merged on 9/14/15 had a final entropy greater than the sum of the two before, plus some entropy was radiated away in the gravitational radiation. The BH entropy laws also lead to equations for the maximum energy that may be extracted from merging BH’s. Bekenstein also showed that BH entropies are the maximum entropy possible for any volume of space with the same volume as the BH.

But the question still remains: why, and how? How is it that the possible states of the BH are encoded in the horizon, if indeed the statistical interpretation holds? For only if it holds would the statistical interpretation be on firm ground, regardless of the thermodynamic relationships. So there’s been attempts and some success (but no proof or certainty yet) in two separate results in physics.

One is the Holographic principle by 't Hooft and Susskind, and others, now a few years old, and still no proof, but some ocassional developments on it. They conjecture that the quantum gravity solution of a spacetime in d+1 dimensions is in a 1-1 correspondence to a Conformal Field Theory (CFT) without gravity, in the d dimensional boundary of the spacetime.

They based this on generalizing the proven results of the AdS/CFT correspondence found by Maldacena and others, who proved the results for a string quantum gravity in anti-de Sitter (AdS) spacetime. AdS see is a vacuum solution of the Einstein Field Equations with a negative cosmological constant (de Sitter is with a positive cosmological constant, the limit of our known universe as its age goes to infinity and the cosmological constant completely dominates). The AdS/CFT correspondence is also called gauge/gravity duality, gauge for the CFT. CFT’s are quantum field theories.

The Holographic conjecture and the AdS/CFT correspondence have led people to think that information on the quantum states of the bulk, in some cases or in general, is stored in its boundaries. Or surfaces, similar to the way holography works for a 3D object. But there are also some counter-examples, and so in general it is still an interesting approach, but not well understood or accepted. Still, if the general case is valid, the ideas is that the states of the BH would be encoded, or imprinted, on its horizon. Possibly that could provide a mechanism for the matter/energy that falls in, and the quantum information that was thought lost to the BH, to be in the horizon and not lost, and possibly later encoded in the Hawking blackbody (with something extra then) radiation.

There is another more recent finding that indicates that the information on the state of the BH may be stored at the horizon. It’s by Hawking, Perry and Strominger, from January of 2016. See the paper in arXiv, and the Phys. Rev. article in June 2016 (I’ve not read the Phys Rev version but the abstracts are the same).

What they claim is new, and based on new re-discovered asymptotic symmetries at conformal infinity. They claim that based on those symmetries BH's have conserved quantities, they call soft hair. That is, that BH’s have some hair that is over and beyond the mass, angular momentum and charge proved by Hawking years ago. Where it is not the same he 'proved' before (yes, there's reason this new symmetries break on of his assumptions back then, that the vacuum was non-degenerate, was unique) is that BHs actually have what the authors call soft hair, very low energy hair that in the limit is zero energy, but still is there. The soft hair is due to soft photons or soft gravitons, that reside in the horizon. They claim that, from part of their abstract:

This Letter gives an explicit description of soft hair in terms of soft gravitons or photons on the black hole horizon, and shows that complete information about their quantum state is stored on a holographic plate at the future boundary of the horizon. Charge conservation is used to give an infinite number of exact relations between the evaporation products of black holes which have different soft hair but are otherwise identical. It is further argued that soft hair which is spatially localized to much less than a Planck length cannot be excited in a physically realizable process, giving an effective number of soft degrees of freedom proportional to the horizon area in Planck units.

Thus they claim they have argued or shown that the entropy is due to the degrees of freedom, or possible states of, the horizon, in accord with BH thermodynamics. Still, they do state, in the body of the arXiv paper, that they have not proven that indeed there’s enough degrees of freedom to really store all the information, and that work remains. Their degrees of freedom, or the soft hair, comes from new symmetries that they re-discovered on the conformal infinity of asymptotically flat spacetime (think of a BH in asymptotically flat spacetime), and which lead to new conserved quantities, the soft hair. Since the BH horizon is one boundary of the asymptotically flat spacetime (i.e., outside the horizon), they show using Penrose diagrams and can calculate the conserved soft hair over the BH horizon.

The soft ‘charges’ (the conserved entities of the symmetries) they re-discovered had been identified by Weinberg in 1965, based on conformal symmetries at infinity called the BMS symmetries (Bondi, actually also van der Burg, Metzner and Sachs), found and published by those first 3 in a paper, and Sachs in another, in 1962. See the Living Reviews article. Those 4 demonstrated in 1962 that there are additional symmetries at conformal infinity in asymptotically flat spacetime, besides the Poincare group. The BMS group was also used by them to define the BMS mass, in asymptotically flat spacetime, at conformal infinity. They found a family of symmetries called supertranslations, and another called superrotations, generalizations of the Poincare group which also inlcudes it, basically coming out of the conformal invariant structure. They also found that those symmetries were non-trivial, that were physical, and cannot be transformed away. Those symmetries turns out were an infinite set of diffeomrphisms, and lead to the soft hair. They occur for gravitational fields and for electromagnetic fields, so for soft photons and gravitons. Hawking et al. did their calculations for electromagnetic fields in a BH, but argued for the same effect for the gravitational field. They admit there’s still a lot to calculate.

So, the most promising (with all the unknowns and unproven theories yet) explanations for the entropy of a BH to be proportional to the horizon area is that the information on the state of the BH, at a quantum level, is imprinted in its horizon. If so it has to be proportional to area and not mass. [BTW, a personal aside, I’m proud to have had Sachs as my advisor in General Relativity, but it was about 5-6 years later, and he was not doing the gravitational wave work he did earlier anymore, not of course did I]

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  • $\begingroup$ I just have to ask: how does this answer the question? I don't think it really does. $\endgroup$ – heather Aug 2 '16 at 14:38
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    $\begingroup$ The question is what are the physical states that provide for the possible states of the system. Once you have that you count and if equiprobabke you take p x log(p) and add them. It is usually extensive because the sum is over all possible states. So the question is where is that state information stored? In the BH horizon, if those hypothesis are right. No on the bulk, but in the surface area. That was the point of it all. I tried to give details as to why some physicists claim that. $\endgroup$ – Bob Bee Aug 2 '16 at 19:00
  • $\begingroup$ The question is about the logic behind Bekenstein's choice to use a non-extensive quantity instead of an extensive quantity in his equation for black hole entropy. $\endgroup$ – heather Aug 2 '16 at 19:01
  • $\begingroup$ No. You are missing a lot. He figured that would do it. Hawking proved that indeed it reproduces the black body radiation spectrum and temperature where the entropy is proportional to area. Mass didn't do it. The others, from AdS to the soft hair and other things back it up. $\endgroup$ – Bob Bee Aug 2 '16 at 19:08
  • $\begingroup$ I know that Hawking proved that, and everything. I'm trying to explain that to the OP! That is, however, the question. $\endgroup$ – heather Aug 2 '16 at 19:08
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I'd like to add to BobBee's insightful answer, which one can summarize as: we need to broaden our notion of extensive for systems Gibbs, Boltzmann and all the others could not conceive of.

Another point, which is implicit in the notions that BobBee's answer discusses is that entropy is always extensive in the following broadened sense:

Shannon entropy is additive for the composite of statistically independent systems

simply by construction (the definition). For the pedantic, let's say we multiply the Shannon entropy by the Boltzmann constant, to make it reduce to classical thermodynamic entropy when this latter notion is used (those who say that thermodynamic entropy and Shannon entropy are not the same, please read my answer here about how they are postulated to be the same modulo the Boltzmann constant).

In causal set theory (which I only have the fleetingest knowledge of) I understand, the assumed "atoms" of spacetime causally influence one another and you of course have pairs of these atoms that are entangled but which lie on either side of the Schwarzschild horizon: one of the pair is inside the black hole and therefore cannot be probed from the outside, whilst the other pair member is in our universe. The outside-horizon pair member observable in our universe therefore has "hidden" state variables, i.e. encoded in the state of the pair member inside the horizon that add to its von Neumann entropy, as we would perceive it outside the horizon. So the theory foretells an entropy proportional to the horizon area (the famous Hawking equation $S = k\,A/4$) because it is the area that is proportional to the number of such pairs that straddle the horizon.

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  • $\begingroup$ @ArtBrown That's a good point, and you are correct. I guess I tend to think of the modern conception of the notion as Shannon's because he was the first to clearly think of the notion of information content. Certainly one needs, for example, the noiseless coding theorem to show that the Boltzmann entropy is proportional to the minimum number of bits needed to encode, with arbitrarily small coding error probability, the full state of a system conditioned on the knowledge of the macrostate when that system comprises statistically independent constituents with identical probability distribution. $\endgroup$ – WetSavannaAnimal Aug 7 '16 at 6:47
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Although the existing answers are extensive (sorry for the pun) I want to add the following thought, which I found in Susskinds book https://en.wikipedia.org/wiki/The_Black_Hole_War :

The reason why entropy $S_{BH}$ of a black hole is proportional to $M^2$ is not that the entropy for a black hole is counted differently, but is in the definition of mass for the black hole.

When we talk about entropy of a black hole, the mass $M$ means the so called gravitational mass. But one can also define a so called baryonic or free mass, by taking all the (baryonic) particles (Susskind is actually talking about strings) an object consist of and weight them separately, then adding up all the mass. Counterintuitively the gravitational mass is lower than the free mass (because of the negative gravitational binding energy) and for very dense objects like a neutron star the gravitational mass $M$ is already quite a lot less (20%) than its so called baryonic or free mass $M_{b}$, see also

Susskind explains that for a black hole this effect is even more pronounced, namely as I understand it $$ M = \sqrt{M_b} $$ And so we come to the conclusion that entropy $S$ of a black hole only looks like a non-extensive quantity because in the Bekenstein-Hawking formula we relate it to the gravitational mass. In fact if we relate it to the free mass (gravitational mass + gravitational binding energy) the entropy of a black hole is still an extensive quantity $$ S \sim M^2 \sim M_b $$

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protected by Qmechanic Aug 3 '16 at 15:46

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