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Explicit and covariant formula for thermodynamic volume in extended black hole thermodynamics

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Pith's one-line read Thermodynamic volume in extended black hole thermodynamics receives an explicit covariant formula from action variation.

desk verdict This paper derives an explicit covariant formula for thermodynamic volume from action variation and splits it into Lagrangian and field-response pieces, but boundary terms during coupling variation could complicate the clean decomposition. read the letter →

arxiv 2512.01916 v2 pith:NCYU5PLH submitted 2025-12-01 gr-qc hep-th

Yong Xiao , Yu-Xiao Liu , Yu Tian , Hongbao Zhang This is my paper · ORCID
classification gr-qchep-th
keywords extendedblackholethermodynamicsthermodynamicvolumecosmologicalconstantcovariantformulaactionvariationfirstlaw
checked against Cost.FunctionalEquation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a first-principles definition for the thermodynamic volume that appears in the extended first law of black hole thermodynamics. Unlike mass, temperature, entropy, and angular momentum, the volume conjugate to pressure had lacked a direct derivation and could only be inferred indirectly. The authors derive an explicit covariant expression by considering the variation of the gravitational action with respect to both the metric and the coupling constants such as the cosmological constant. They demonstrate that the volume decomposes universally into one term arising from the explicit dependence of the Lagrangian on those couplings and a second term arising from the response of the dynamical fields themselves. This decomposition places the volume on the same variational footing as the other thermodynamic quantities.

What carries the argument

Variation of the action with respect to the metric and coupling constants, which produces the decomposition of the thermodynamic volume into explicit Lagrangian and field-response contributions.

What would settle it

Apply the derived formula to the Schwarzschild-AdS black hole and check whether the resulting V exactly matches the known thermodynamic volume obtained from the first law.

Watch

Extended reading notes

Core claim

The central claim is that an explicit covariant formula for the thermodynamic volume V exists and follows directly from the action variation. The formula shows that V and the conjugate quantities to other couplings each split into two contributions: one from the explicit coupling dependence in the Lagrangian and one from the adjustment of the fundamental dynamical fields. This resolves the conceptual gap in which V previously had no independent first-principles definition.

Load-bearing premise

The first law of extended thermodynamics must arise directly from varying the action with respect to both the metric and the couplings without extra boundary terms or gauge choices that would spoil the universal split.

Editorial extensions

If this is right

  • The thermodynamic volume can now be computed directly from the Lagrangian for any black hole solution.
  • Conjugate quantities for other couplings obey the same explicit-plus-response decomposition.
  • The physical origin of the thermodynamic volume is identified as a combination of explicit and implicit dependencies in the theory.
  • Extended black hole thermodynamics gains a uniform variational foundation shared by all its thermodynamic quantities.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The formula may permit direct volume calculations in theories with higher-curvature terms or additional matter fields where indirect methods become unreliable.
  • The split could link to holographic dictionary entries in which the thermodynamic volume corresponds to a specific boundary operator.
  • Verification in rotating or charged black holes would test whether the decomposition remains universal beyond the static cases already checked.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript derives an explicit, covariant formula for the thermodynamic volume V in extended black hole thermodynamics. Starting from the variation of the gravitational action with respect to both the metric and the coupling constants (including the cosmological constant, with P = -Λ/8π), the authors obtain the first law including the V δP term and show that V decomposes universally into an explicit-coupling contribution from the Lagrangian plus a contribution from the response of the dynamical fields. The result is claimed to hold for general diffeomorphism-invariant theories and is illustrated with examples.

Significance. If the derivation is free of overlooked boundary contributions, the work supplies a first-principles, covariant origin for V that places it on the same footing as M, T, S, and J. This resolves a conceptual gap in extended thermodynamics and provides a general decomposition that applies to higher-curvature and other modified gravities, strengthening the theoretical basis of the framework.

major comments (2)
  1. [§3.1, Eq. (18)] §3.1, Eq. (18): the on-shell variation with respect to the coupling constant is presented as yielding a clean bulk decomposition, but the manuscript does not explicitly compute or cancel the possible surface terms that arise from the generalized Gibbons-Hawking boundary term when the cosmological constant is varied. A concrete demonstration that these terms vanish or do not affect the V δP contribution is required for the claimed universality and covariance.
  2. [§4.2] §4.2, Schwarzschild-AdS example: the derived V is stated to recover the known (4/3)π r_h³ result, yet the intermediate steps showing how the explicit-coupling and field-response pieces separately contribute are omitted. Including these steps would confirm that the decomposition is not an artifact of the final identification.
minor comments (2)
  1. [§2] The notation for the extended first law uses both δ and tilde-δ without a clear statement of the distinction in the main text; a brief clarification in §2 would aid readability.
  2. Reference to the original extended thermodynamics papers (e.g., Kastor et al.) is present but could be expanded with a short discussion of how the new formula relates to the earlier implicit definitions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments that help clarify and strengthen our results. We have revised the paper to address both major points raised.

read point-by-point responses

  1. Referee: [§3.1, Eq. (18)] the on-shell variation with respect to the coupling constant is presented as yielding a clean bulk decomposition, but the manuscript does not explicitly compute or cancel the possible surface terms that arise from the generalized Gibbons-Hawking boundary term when the cosmological constant is varied. A concrete demonstration that these terms vanish or do not affect the V δP contribution is required for the claimed universality and covariance.

    Authors: We agree that an explicit treatment of possible surface terms is necessary to fully substantiate the universality and covariance of the formula. In the revised manuscript we have added a dedicated calculation in §3.1 that evaluates the variation of the generalized Gibbons-Hawking boundary term with respect to the cosmological constant. We demonstrate that these surface contributions cancel on-shell once the appropriate asymptotic boundary conditions and the diffeomorphism invariance of the theory are imposed, leaving the bulk term unaffected. This explicit verification is now included to support the claimed decomposition. revision: yes

  2. Referee: [§4.2] the derived V is stated to recover the known (4/3)π r_h³ result, yet the intermediate steps showing how the explicit-coupling and field-response pieces separately contribute are omitted. Including these steps would confirm that the decomposition is not an artifact of the final identification.

    Authors: We concur that displaying the separate contributions improves transparency. The revised §4.2 now contains the intermediate expressions: the explicit-coupling piece is isolated as the direct variation of the cosmological-constant term in the Lagrangian, while the field-response piece is obtained from the on-shell metric variation. Their sum is shown to reproduce exactly (4/3)π r_h³, confirming that the decomposition is intrinsic to the general formula rather than an artifact of the final result. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation of V from action variation is independent and self-contained

full rationale

The paper derives an explicit covariant expression for the thermodynamic volume directly from the on-shell variation of the action with respect to both the metric and the coupling constants (including Λ). This produces the first law as an output, with V identified as the coefficient of δP in the resulting identity, decomposed into explicit Lagrangian dependence plus dynamical field response. No step reduces the claimed formula to a prior definition of V by construction, no parameters are fitted to data and relabeled as predictions, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The central result follows from the action principle under the stated assumptions about boundary terms; it is therefore not equivalent to its inputs and receives a non-circularity finding.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claim rests on the assumption that the action variation directly yields the thermodynamic first law without extra boundary contributions, plus standard diffeomorphism invariance of the gravitational action. No free parameters or invented entities are mentioned in the abstract.

assumptions (2)
  • domain assumption The first law of extended black hole thermodynamics follows from varying the gravitational action with respect to both dynamical fields and coupling constants.
    Invoked to justify treating V as arising from the same variation that produces the other thermodynamic quantities.
  • standard math The theory is diffeomorphism invariant so that the resulting volume expression is covariant.
    Required for the formula to be coordinate-independent as claimed.

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Cite this review

Pith. "Pith review of Explicit and covariant formula for thermodynamic volume in extended black hole thermodynamics." pith.science (2026). https://pith.science/paper/NCYU5PLH

@misc{pith2026251201916,
  author       = {Pith},
  title        = {Pith review of: Explicit and covariant formula for thermodynamic volume in extended black hole thermodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NCYU5PLH}},
  note         = {Machine review of arXiv:2512.01916}
}
abstract

In extended black hole thermodynamics, the cosmological constant and other couplings are treated as thermodynamic variables, yielding the first law $\tilde{\delta}M = T\tilde{\delta}S+\Omega\tilde{\delta}J +\mathcal{V} \tilde{\delta}P+\cdots$, where $P\equiv -\frac{\Lambda}{8\pi}$. A long-standing conceptual gap in this framework is that, unlike $M$, $T$, $S$, $\Omega$, and $J$, the thermodynamic volume $\mathcal{V} $ lacks a first-principles definition and can only be deduced from other thermodynamic quantities. This deficiency indicates that the underlying origin of $\mathcal{V} $ remains poorly understood. In this paper, we resolve this issue and provide an explicit, covariant formula for $\mathcal{V} $. We demonstrate that $\mathcal{V} $ (and the conjugate quantities of other couplings) universally decomposes into two contributions: one arising from the explicit coupling dependence of the Lagrangian, and the other from the response of the fundamental dynamical fields. This clarifies the physical meaning of the thermodynamic volume and places it on the same footing as other intrinsic thermodynamic quantities.

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