Holographic pressure and volume for black holes

This paper proposes a holographic definition of thermodynamic pressure and volume for black holes based on quasi-local gravitational thermodynamics, demonstrating that this framework allows for a consistent definition of extensivity and a large-system limit where both Schwarzschild and Anti-de-Sitter black holes transition from non-extensive to extensive behavior.

The Gist

The Big Idea: Black Holes Need a "Room" to Be Measured

Imagine you are trying to measure the temperature and pressure of a gas inside a balloon. In normal physics, you can easily say, "This gas has a certain volume (the size of the balloon) and a certain pressure (how hard it pushes against the walls)."

But for a long time, physicists had a hard time doing this with black holes. A black hole is a region of space where gravity is so strong that nothing escapes. When physicists tried to write down the "laws of thermodynamics" (the rules of heat and energy) for black holes, something was missing. The standard equation for a black hole looked like this:

Change in Energy = Temperature × Change in Entropy

Compare that to a normal gas, which looks like this:

Change in Energy = Temperature × Change in Entropy − Pressure × Change in Volume

The black hole equation was missing the Pressure and Volume part. It was like trying to describe a car engine without mentioning fuel or pistons. Furthermore, scientists were confused about whether black holes were "extensive." In simple terms, "extensive" means if you double the size of a system, you double its energy. If you double the size of a room full of air, you get twice as much air and twice the energy. But for black holes, this rule seemed to break.

The Solution: The "Holographic" Room

The authors of this paper propose a new way to look at black holes. They suggest we stop thinking about the black hole as an isolated object in empty space and start thinking about it as a system enclosed in a finite room (a boundary).

The Analogy:
Imagine a black hole is a hot soup.

• Old View: We only looked at the soup itself, ignoring the bowl it's in. We couldn't measure the pressure because we didn't have a container.
• New View (This Paper): We put the soup in a rigid, spherical bowl. Now, the soup pushes against the walls of the bowl. The "Volume" of the system isn't the soup itself; it's the surface area of the bowl. The "Pressure" is how hard the soup pushes against that bowl.

The authors use a concept called Holography. This is the idea that all the physics happening inside a 3D space (the soup) can be described by physics happening on the 2D surface of that space (the bowl).

• The Bowl's Surface Area = The Volume of the system.
• The Push on the Bowl = The Pressure of the system.

By using this "bowl" (which physicists call a "York boundary"), they can finally write the black hole equation with a Pressure and Volume term, just like a normal gas:

Change in Energy = Temperature × Change in Entropy − Pressure × Change in Volume

The Mystery of "Extensivity": Small vs. Large Black Holes

Once they had a proper definition of volume, they asked: "Are black holes extensive?" (i.e., if we make the bowl bigger, does the energy scale up nicely?)

They found the answer depends on two things: the type of black hole and how big the bowl is.

1. Flat Space Black Holes (The "Floating" Black Hole)

Imagine a black hole in empty space (no cosmological constant).

• The Small Black Hole: If you have a tiny black hole in a small bowl, it behaves strangely. It is not extensive. If you double the size of the bowl, the energy doesn't double in a simple way. It's like a small, wobbly balloon that doesn't follow the rules of normal gases.
• The Large Black Hole: If you have a huge black hole in a huge bowl, it starts to behave like a normal gas. It becomes extensive. The energy scales up linearly with the size of the bowl.
• The Catch: This only works if you look at the system from the "canonical" perspective (fixing the temperature). If you look at it from the "energy" perspective, even the large black hole acts weird. It's like a chameleon that changes its behavior depending on how you measure it.

2. Anti-de Sitter (AdS) Black Holes (The "Boxed" Black Hole)

Now imagine a black hole in a universe that naturally curves inward (like a box with elastic walls).

• The Result: Here, the rules are much more friendly. Both the small and large black holes eventually settle into a state where they are extensive when the bowl gets very large.
• The "Casimir" Effect: The authors found that at finite sizes, there is a "correction" term. Think of this like a small fee you have to pay to enter a large concert hall. When the hall is tiny, the fee is huge compared to the ticket price. But as the hall gets massive, the fee becomes negligible, and the ticket price (the energy) scales perfectly with the size of the hall. This "fee" is a sub-extensive correction that disappears in the limit of a very large system.

The "Smarr Formula" and the Missing Piece

The paper also re-examines an old equation called the Smarr formula, which relates the mass, temperature, and size of a black hole.

• Old View: Scientists thought the extra terms in this equation represented a new kind of "pressure" from the universe itself (the cosmological constant).
• New View: The authors argue that this extra term isn't a new pressure. Instead, it's a mathematical glitch caused by the system being finite. It's a "correction" that tells us the system isn't perfectly extensive yet. As the system gets infinitely large, this glitch vanishes, and the standard rules of thermodynamics take over.

Summary of Findings

1. We need a boundary: To define pressure and volume for black holes, we must imagine them inside a finite boundary (a "bowl"). The area of this bowl acts as the volume.
2. Flat space is tricky: Black holes in flat space are generally "non-extensive" (they don't scale simply), especially when they are small. They only act "normal" (extensive) when they are very large and we look at them in a specific way.
3. AdS space is nicer: Black holes in Anti-de Sitter space (with a cosmological constant) behave much more like normal matter. They become fully extensive as the system gets large.
4. The "Fee" disappears: The weird extra terms in the equations are just finite-size corrections. They vanish when the system is large enough, restoring the standard laws of thermodynamics.

In short, the paper argues that black holes can be understood as normal thermodynamic systems with pressure and volume, provided we stop looking at them as infinite objects in empty space and start treating them as systems enclosed in a finite boundary. When we do this, the strange non-extensive behavior of small black holes makes sense, and the large black holes reveal themselves to be perfectly normal, extensive systems.

Technical Summary

Technical Summary: Holographic Pressure and Volume for Black Holes

Problem Statement
Black hole thermodynamics lacks a well-defined notion of thermodynamic pressure and volume, leading to conceptual ambiguities regarding the extensivity of black hole variables. Standard formulations of the first law for Schwarzschild black holes (e.g., $dM = T_H dS$) lack a $PdV$ work term. Existing proposals to define a thermodynamic volume suffer from significant limitations:

1. Padmanabhan's proposal: Defines volume as the naive Euclidean volume enclosed by the horizon. However, for static, spherically symmetric black holes, both entropy and volume depend solely on the horizon radius, rendering them non-independent variables. This results in a degenerate thermodynamic state space where the fundamental equation is ill-defined.
2. Extended Black Hole Thermodynamics (Black Hole Chemistry): Promotes the cosmological constant $\Lambda$ to a thermodynamic pressure ($P_\Lambda = -\Lambda/8\pi G$). While this yields a consistent first law for AdS black holes, the conjugate volume coincides with the naive Euclidean volume, again fixing the volume in terms of entropy. Furthermore, treating $\Lambda$ as a state variable is conceptually problematic as it alters the theory itself rather than the equilibrium state. In the holographic context, varying $\Lambda$ corresponds to varying the central charge of the dual Conformal Field Theory (CFT), not a standard thermodynamic process.

Consequently, the question of whether black hole entropy is extensive remains unresolved, as standard definitions of extensivity (homogeneity of degree one) cannot be meaningfully applied without an independent thermodynamic volume.

Methodology
The authors advocate for a holographic definition of pressure and volume based on quasi-local gravitational thermodynamics (York's formalism). The core methodology involves:

• Quasi-Local Framework: Considering a black hole enclosed by a finite timelike boundary (the "York boundary"). In this setting, thermodynamic variables are defined quasi-locally rather than at infinity.
• Holographic Identification: Assuming a holographic duality between the bulk gravitational theory and a non-gravitational quantum theory living on the boundary. The authors identify the boundary area $A$ as the thermodynamic volume $V$ and the conjugate surface pressure $s$ (derived from the Brown-York stress tensor) as the thermodynamic pressure $P$.
  $$P \equiv s, \quad V \equiv A$$
• Covariant Phase Space Formalism: Deriving the quasi-local first law and a new quasi-local Smarr relation for AdS-Schwarzschild black holes using the covariant phase space (CPS) formalism. This allows for a rigorous treatment of boundary terms and background subtractions.
• Analysis of Extensivity: Investigating extensivity in two distinct thermodynamic representations:
  1. Internal Energy Representation: $E(S, V)$, with independent variables entropy $S$ and volume $V$.
  2. Canonical Representation: $F(T, V)$, with independent variables temperature $T$ and volume $V$.
     Extensivity is defined as the homogeneity of degree one of the thermodynamic potential with respect to the extensive variables in the large-system limit ($V \to \infty$).

Key Contributions and Results

1. Derivation of Quasi-Local Laws
The paper derives the quasi-local first law for static, spherically symmetric black holes:
$$dE = TdS - s dA$$
where $E$ is the Brown-York quasi-local energy, $T$ is the Tolman temperature, and $s$ is the surface pressure. Under the holographic identification ($P=s, V=A$), this recovers the standard thermodynamic form $dE = TdS - PdV$. Crucially, $S$ and $V$ are independent variables even for Schwarzschild black holes, resolving the degeneracy of previous proposals.

The authors also derive a quasi-local Smarr relation for AdS-Schwarzschild black holes:
$$(d-3)E = (d-2)(TS - PV) - \frac{\Lambda \tilde{V}_\xi}{4\pi G N_B}$$
where $\tilde{V}_\xi$ is a background-subtracted Killing volume. The authors interpret the final term not as a new conjugate pair (like $P_\Lambda V_\Lambda$), but as a subextensive correction signaling the failure of exact homogeneity at finite volume.

2. Extensivity of Asymptotically Flat Black Holes

• Energy Representation: The quasi-local energy $E(S, V)$ is non-extensive even in the large-volume limit. It satisfies a quasi-homogeneous scaling relation rather than a homogeneous one.
• Canonical Representation: The system exhibits two branches: small and large black holes.
  • The small black hole branch is non-extensive.
  • The large black hole branch becomes extensive in the large-volume limit for the Helmholtz free energy $F(T, V)$ (i.e., $F \propto V$). However, the internal energy $E(T, V)$ remains subextensive.
• Conclusion: Extensivity for asymptotically flat black holes is representation- and branch-dependent. The non-extensivity in the energy representation suggests an obstruction to realizing a holographic dual of flat-space gravity as a local quantum field theory.

3. Extensivity of AdS-Schwarzschild Black Holes

• Energy Representation: The quasi-local energy decomposes into a subextensive part and an extensive part $E_{ext}(S, V)$ that dominates in the large-system limit. The authors derive an explicit closed-form expression for this extensive energy:
  $$E_{ext}(S, V) = \frac{(d-2)V}{8\pi G L} \left( 1 - \sqrt{1 - \left(\frac{4GS}{V}\right)^{\frac{d-1}{d-2}}} \right)$$
  This function is homogeneous of degree one. In the large-system limit, the Euler relation $E_{ext} = TS - PV$ is recovered.
• Canonical Representation: Similar to the flat case, the small black hole branch is non-extensive. However, for the large black hole branch, both the Helmholtz free energy and the internal energy become extensive in the large-volume limit.
• Significance: Unlike the flat case, AdS gravity admits a thermodynamic regime (the large black hole branch) where extensivity holds consistently across different representations. This is consistent with the locality of the dual CFT.

4. Interpretation of the AdS Smarr Formula
The paper offers a thermodynamic interpretation of the AdS Smarr formula distinct from "extended black hole thermodynamics." The extra term involving the Killing volume is identified as a finite-size, subextensive correction (analogous to a thermal Casimir energy in the dual CFT) that vanishes in the large-system limit. This interpretation avoids treating the cosmological constant as a thermodynamic state variable.

Significance and Claims
The paper claims that quasi-local gravitational thermodynamics, combined with the holographic principle, provides a physically meaningful and non-degenerate definition of pressure and volume for black holes.

• It resolves the issue of variable independence, allowing for a standard thermodynamic analysis of extensivity.
• It demonstrates that the extensivity of black hole thermodynamics is not a universal property but depends on the asymptotic structure of spacetime (flat vs. AdS), the thermodynamic representation, and the specific equilibrium branch (small vs. large black hole).
• The non-extensivity of asymptotically flat black holes in the energy representation is presented as a concrete constraint on any holographic description of flat-space gravity formulated on a finite boundary, suggesting such a dual may not be a local quantum field theory.
• The recovery of extensivity for large AdS black holes supports the consistency of the AdS/CFT correspondence in the thermodynamic limit.

The authors do not claim to have solved the microscopic origin of black hole entropy but rather provide a macroscopic thermodynamic framework that clarifies the conditions under which black holes behave as extensive systems. They note that their analysis is strictly at the level of equilibrium thermodynamics and does not assume the existence or uniqueness of underlying microscopic ensembles.