# Definitions of Thermodynamics and Holography

There are many differences between the laws of thermodynamics and the laws of black hole thermodynamics (BHT):

Zeroth Law: In thermodynamics, the Zeroth Law establishes the notion of thermal equilibrium and defines temperature. It states that if objects A and B are in thermal equilibrium, and B and C are in thermal equilibrium, then A and C are also in thermal equilibrium. This allows for the definition of an empirical temperature scale. In BHT, there is no notion of different black holes being in "equilibrium" with each other. The statement that surface gravity is constant over the event horizon is more analogous to a consequence of the Zeroth Law, not the law itself.

First Law: The thermodynamic First Law defines internal energy and relates it to heat and work. In BHT, the mass of the black hole plays the role of internal energy, but there is no clear notion of heat or work. The First Law relates changes in mass to changes in area and angular momentum, but the physical meaning is unclear.

Second Law: In thermodynamics, the Second Law states that entropy increases in any spontaneous process. In BHT, the area theorem states that the area of the event horizon never decreases, which is analogous but conceptually different than entropy increase. There is also no underlying statistical mechanical explanation for why black hole area should increase.

Third Law: The thermodynamic Third Law says entropy approaches zero as temperature approaches absolute zero. There is no parallel statement in BHT, and surface gravity can approach zero independently of area.

E.T. Jaynes said the word "entropy" has been very abused and documented six different types:

• $$S_E$$ - Experimental entropy of Clausius, Gibbs, and Lewis, defined based on macroscopic laboratory measurements.
• $$S_I$$ - Information entropy, defined as $$-∑p_i log(p_i)$$.
• $$S(A)$$ - Maximum of $$S_I$$ subject to constraints $${A_1...A_n}$$.
• $$S_B$$ - Boltzmann entropy, defined with single-particle distribution function.
• $$S_G$$ - Gibbs entropy, defined in terms of N-particle distribution function.
• $$S_{BEP}$$ - Entropy of Boltzmann, Einstein and Planck, defined as $$klogW$$.

Is Bekenstein-Hawking entropy another?

Question: What might this imply about holography and the AdS/CFT correspondence?

(Aside: Frankly, I never really understood what was so profound about holography, given that the fundamental field equations are second order. Many ordinary boundary value problems involve data from a dimension lower. The holographic duality between quantum gravity theories on anti-de Sitter space and conformal field theories on the boundary is mathematically elegant but whether our universe exhibits anti-de Sitter geometry is another matter.)

• There are some things you are missing fundamentally, since BH entropy is a kind of coarse-grained entropy, whereas your remark about holography is somewhat naive. Also, our universe cannot be AdS, since our universe exhibits a positive $\Lambda$. Commented Sep 29, 2023 at 17:10
• I still don't get what you mean -- what are we missing? And AdS/CFT is great because it was the string theoretic description of holography, which has many reasons to appreciate it. De Sitter on the other hand is a more philosophical discussion, and like I said -- dS/CFT exists. Commented Sep 29, 2023 at 17:45
• The system in this case is the entire horizon + exterior matter fields system. This is why there exists a coarse-grained description of entropy increase from the generalized second law. What problems are there with GSL? As far as I have worked, I see that GSL has some very strict proofs that work quite well in all sorts of dynamics. There do exist some questions, but not ones we cannot answer nonetheless. Commented Sep 29, 2023 at 18:09
• I do not know the answer to your question. However, I am pretty sure Beckenstein's entropy is a different beast with respect to the whole list by Jaynes. Certainly, it has an additional profound difference from thermodynamic entropy: it is a convex function of the surface area, while thermodynamic entropy must be a concave function of its extensive variables. Commented Sep 29, 2023 at 20:37
• @MatthewCory Ah, I see this is what you are referring to. Well, it is true that Bekenstein-Hawking entropy seems to be unnatural, but this is not a property of holography like AdS/CFT, but the general superset of this, called the holographic principle, advocated by the covariant entropy bound. See Bousso's papers on this to see what the BH entropy looks like in general. Let me edit my answer to include this as well. Commented Sep 30, 2023 at 4:23

## 2 Answers

To go bottom top: our universe is not AdS. AdS/CFT is wonderful because it was the first description of holography that explicitly could be used to do bulk reconstruction and subregion duality. People like me find it amazing because of the algebraic structure (emergence of the bulk from type III$$_{1}$$ algebra on the boundary side), but the general idea is that AdS/CFT is the convenient description, with a timelike boundary made of distinct points that can be joined by spacelike geodesics and a unitary CFT. However, there is also dS/CFT, which is amazing because our universe is approximately de Sitter. I am, in fact, involved with de Sitter quantum gravity and dS/CFT, but it is too technical to include here. So your remark is somewhat naive. However, one does have the nice emergence aspect in dS/CFT as well, where the bulk dS emerges from entanglement between two CFT copies.

The Bekenstein-Hawking entropy is reproduced from the Ryu-Takayanagi prescription for holographic entanglement entropy. If you find the corresponding RT surface for the full boundary subregion, you would get precisely the Bekenstein-Hawking entropy. The Bekenstein-Hawking entropy is arguably the first hint towards the holographic principle -- that information about a surface is encoded on its boundary. See Bousso, Wall and other papers on this, particularly Bousso's covariant entropy bound, because the proper way to see the BH entropy in AdS/CFT closely follows the notion of Engelhardt and Wall's holographic screens description using the coarse-grained formulation using Hubeny-Rangamani-Takayanagi (HRT) entropy. Regarding your second law remark, one has the generalized second law, due to Bekenstein, Bousso and Wall's prominent works, which has a good holographic description, again, in terms of holographic screens, where the entanglement entropy calculated by the von Neumann entropy of the exterior contributions and the Bekenstein-Hawking area decrease is always increasing. There are other aspects with this, like pre- and post-Page time calculations, holographic islands, and so on. The BH entropy is a black hole entropy calculation -- see for instance, Hawking's calculations and Bekenstein's works.

• The emergence of the Bekenstein-Hawking entropy from the Ryu-Takayanagi prescription in the context of AdS/CFT doesn't resolve the disconnects between regular thermodynamics and BHT. While it offers a way to calculate black hole entropy from a holographic perspective, it doesn't fully explain the microscopic origin of black hole entropy or bridge the conceptual gaps between black hole thermodynamics and traditional thermodynamics.
– user143181
Commented Sep 29, 2023 at 17:24
• I'm sorry -- what "conceptual gaps"? BH entropy is a very well understood thing, I am not sure what you mean. Also -- it does give the origin of black hole entropy in the holographic perspective -- what do you mean by the difference between these two? See also HRT, since it offers the correct dynamics associated with calculations of BH entropy. RT is not fully the right description since it is not a coarse-grained description, so no GSL, etc. Commented Sep 29, 2023 at 17:27

The notion of "equilibrium" in black hole thermodynamics would be between the radiation background and the black hole, not between two merging black holes.

The ADM mass of the black hole is literally the energy of the black hole, and generally the first law is thought of as a statement about the equation of state of the black hole, rather than the version that is broken up into heat and work

For the second law, I think this is a direct desire to equate the entropy with some proportion of the black hole area. Some versions of quantum gravity have done this explicitly by counting microstates

idk why you are bothered by just saying "no physical process can bring the surface gravity to zero" being the statement of the third law, which is equivalent to the "you can't reach absolute zero' version of the third law in ordinary thermodynamics.

• So you are agreeing with me?
– user143181
Commented Sep 29, 2023 at 17:51
• @MatthewCory: I think the correspondence is pretty good? It's mostly just missing a fundamental picture in terms of microstates, but that pretty much requires a clean quantum gravity Commented Sep 29, 2023 at 20:01