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arxiv: 2606.17171 · v1 · pith:7Q62RUXPnew · submitted 2026-06-15 · ✦ hep-th · gr-qc

Black hole equations of state and response functions

Silvester G.A. Borsboom , Manus R. Visser This is my paper

Pith reviewed 2026-06-27 02:35 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords black hole thermodynamicsquasi-local thermodynamicsthermodynamic representationsequations of stateresponse functionsSchwarzschild black holethermodynamic stabilityholographic duality
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The pith

Black hole stability depends on which thermodynamic variables are held fixed under perturbations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a framework for equilibrium thermodynamics that treats each representation as a choice of thermodynamic potential together with one variable from every conjugate pair. First derivatives of the potential supply equations of state while second derivatives supply response functions. When the framework is applied to a Schwarzschild black hole inside a spherical cavity, with cavity area interpreted as thermodynamic volume, the stability properties of the black hole change according to the representation. The large black hole branch is thermally stable at fixed volume yet mechanically unstable under isothermal compression, mechanically stable under adiabatic compression, and thermally unstable at fixed pressure, while the thermal expansion coefficient remains negative everywhere.

Core claim

For the quasi-local Schwarzschild black hole, stability depends on the thermodynamic representation, or equivalently on which variables are held fixed under equilibrium perturbations. The large black hole branch is thermally stable at fixed volume but mechanically unstable under isothermal compression, while the system is mechanically stable under adiabatic compression everywhere. At fixed pressure, the black hole is thermally unstable throughout the physical state space. The thermal expansion coefficient is negative everywhere and isenthalpic expansion always cools the black hole.

What carries the argument

Thermodynamic representations, each consisting of a potential and one independent variable chosen from each conjugate pair, from which equations of state and response functions are obtained as first and second derivatives.

If this is right

  • The large black hole branch remains thermally stable when volume is held fixed.
  • The same branch is mechanically unstable under isothermal compression.
  • Mechanical stability holds under adiabatic compression for every black hole state.
  • Thermal instability occurs at fixed pressure in the entire physical state space.
  • Isenthalpic expansion cools the black hole because the expansion coefficient is negative.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same representation-based analysis could be applied to other quasi-local black hole solutions to test whether their stability also varies with the choice of fixed variables.
  • The reported negative thermal expansion coefficient may translate into a corresponding property of the dual holographic fluid.
  • Specifying the thermodynamic representation removes apparent contradictions between different statements about black hole stability.

Load-bearing premise

The cavity area and its conjugate surface pressure can be interpreted as the thermodynamic volume and pressure of a holographically dual system.

What would settle it

A direct computation of the thermal expansion coefficient for the cavity-enclosed Schwarzschild black hole that yields a positive value would contradict the reported negative coefficient.

Figures

Figures reproduced from arXiv: 2606.17171 by Manus R. Visser, Silvester G.A. Borsboom.

Figure 1
Figure 1. Figure 1: Energy representation. From left to right: the fundamental relation E(S, V ) and the equations of state T(S, V ) and P(S, V ). The physical domain is 4GS < V [PITH_FULL_IMAGE:figures/full_fig_p031_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Helmholtz representation. From left to right: the fundamental relation F(T, V ) and equations of state S(T, V ) and P(T, V ). The physical domain is V > 0, T ≥ Tmin(V ) = √ 27 4 √ πV . The black curve marks the merger of the small and large black holes, corresponding to x = 2/3 and T = Tmin(V ). The free energy F remains finite there, but the surface develops a cusp [PITH_FULL_IMAGE:figures/full_fig_p031_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Gibbs representation. From left to right: the fundamental relation G(T, P) and the equations of state S(T, P) and V (T, P). The physical domain is 0 < P < T /4G. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Enthalpy representation. From left to right: the fundamental relation H(S, P) and equations of state T(S, P) and V (S, P). The physical domain is S > 0, P > 0. The zero￾pressure edge corresponds to the limit V → ∞ at fixed entropy; along this edge H approaches the asymptotically flat Schwarzschild mass and therefore remains non-vanishing, as seen in the left plot [PITH_FULL_IMAGE:figures/full_fig_p032_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Named thermodynamic response functions of the Schwarzschild black hole. From [PITH_FULL_IMAGE:figures/full_fig_p032_5.png] view at source ↗
read the original abstract

We systematically develop a framework for equilibrium thermodynamics in terms of thermodynamic representations, which are choices of thermodynamic potential and independent variables, one from each conjugate pair. In each representation, equations of state arise from first derivatives of the potential and response functions from its second derivatives. We apply this framework to ideal gases and to Schwarzschild black holes in a spherical cavity in various representations, interpreting the cavity area and its conjugate surface pressure as the thermodynamic volume and pressure of a holographically dual system. For this quasi-local Schwarzschild black hole, stability depends on the thermodynamic representation, or equivalently on which variables are held fixed under equilibrium perturbations. The large black hole branch is thermally stable at fixed volume but mechanically unstable under isothermal compression, while the system is mechanically stable under adiabatic compression everywhere. At fixed pressure, the black hole is thermally unstable throughout the physical state space. We also find that the thermal expansion coefficient is negative everywhere and show that isenthalpic expansion always cools the black hole. More broadly, the framework provides a systematic route to deriving equations of state and response functions for a wide class of black hole systems using quasi-local gravitational thermodynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript develops a general framework for thermodynamic representations, where each representation consists of a thermodynamic potential and a choice of independent variables from each conjugate pair. Equations of state are obtained from first derivatives of the potential, and response functions from second derivatives. This framework is applied to ideal gases as a consistency check and to quasi-local Schwarzschild black holes in a spherical cavity, where the cavity area and its conjugate surface pressure are interpreted as the thermodynamic volume and pressure of a holographically dual system. The central results concern representation-dependent stability and response functions for the black hole, including thermal stability of the large branch at fixed volume but mechanical instability under isothermal compression, mechanical stability under adiabatic compression, thermal instability at fixed pressure, negative thermal expansion coefficient, and cooling under isenthalpic expansion.

Significance. If the identification of cavity area with thermodynamic volume is justified, the paper offers a systematic approach to computing equations of state and response functions for black holes using quasi-local thermodynamics. The finding that stability depends on the thermodynamic representation provides new insight into black hole thermodynamics and could have implications for holographic duals. The framework itself is general and could be applied to other systems.

major comments (1)
  1. [Application to Schwarzschild black holes] The substitution of the cavity area for the thermodynamic volume V (and conjugate pressure for P) in the holographic dual interpretation is presented without an explicit derivation of the first law in the form dE = T dS - P dV or verification that Maxwell relations hold under this identification. Since the stability conclusions (e.g., thermal stability at fixed volume but not at fixed pressure) and response functions rely on this substitution being thermodynamically consistent, this justification is load-bearing and should be provided.
minor comments (1)
  1. The abstract clearly summarizes the results but could benefit from specifying the exact representations considered for the black hole.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for explicit thermodynamic consistency in the application to Schwarzschild black holes. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The substitution of the cavity area for the thermodynamic volume V (and conjugate pressure for P) in the holographic dual interpretation is presented without an explicit derivation of the first law in the form dE = T dS - P dV or verification that Maxwell relations hold under this identification. Since the stability conclusions (e.g., thermal stability at fixed volume but not at fixed pressure) and response functions rely on this substitution being thermodynamically consistent, this justification is load-bearing and should be provided.

    Authors: We agree that an explicit derivation of the first law and verification of the Maxwell relations under the cavity-area identification would strengthen the presentation and address the load-bearing nature of this step. In the revised manuscript we will add a dedicated subsection deriving dE = T dS − P dV from the quasi-local energy expression for the Schwarzschild solution in a spherical cavity, following the standard boundary-term construction. We will then compute the relevant Maxwell relations explicitly for the chosen representations and confirm that they are satisfied. This material will be inserted in Section 3 prior to the stability analysis, ensuring the response functions and stability conclusions rest on a fully documented thermodynamic foundation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework derivations are self-contained

full rationale

The paper first defines thermodynamic representations as choices of potential and one variable from each conjugate pair, then derives equations of state from first derivatives and response functions from second derivatives using standard thermodynamic identities. These steps are applied to ideal gases (recovering known results) and to the quasi-local Schwarzschild black hole by substituting the cavity area for volume under the stated holographic interpretation. No load-bearing claim reduces by construction to a fitted parameter renamed as prediction, a self-citation chain, or an ansatz smuggled via prior work; the stability conclusions follow directly from the representation choice and Maxwell relations without circular reduction. The interpretation of area as volume is an explicit modeling assumption rather than a definitional equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that equilibrium thermodynamics in multiple representations applies to quasi-local black holes and that the cavity quantities map to a holographic dual volume and pressure. No free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Equilibrium thermodynamics applies to Schwarzschild black holes in a spherical cavity using quasi-local quantities.
    The framework is applied directly to the black hole system without additional justification in the abstract.
  • domain assumption Cavity area and surface pressure correspond to thermodynamic volume and pressure of a holographically dual system.
    This mapping is invoked to interpret the black hole results in holographic terms.

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Reference graph

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