pith. sign in

arxiv: 2512.01916 · v2 · pith:NCYU5PLHnew · submitted 2025-12-01 · 🌀 gr-qc · hep-th

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

Yong Xiao , Yu-Xiao Liu , Yu Tian , Hongbao Zhang This is my paper

Pith reviewed 2026-05-21 17:46 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords extended black hole thermodynamicsthermodynamic volumecosmological constantcovariant formulaaction variationfirst lawblack hole thermodynamics
0
0 comments X

The pith

Thermodynamic volume in extended black hole thermodynamics receives an explicit covariant formula from action variation.

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

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.

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.

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.

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.

Where Pith is reading between the lines

These are 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.

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.

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.

read the original 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.

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

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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 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.

axioms (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.

pith-pipeline@v0.9.0 · 5733 in / 1461 out tokens · 38647 ms · 2026-05-21T17:46:04.292246+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Black Hole Entropy Beyond the Wald Term in Nonminimally Coupled Gravity: A Covariant Phase Space Decomposition

    gr-qc 2026-05 unverdicted novelty 6.0

    Black hole entropy in diffeomorphism-invariant nonminimal gravity decomposes as S_H = S_W + S_1 + ΔS, with the extra terms required for bumblebee and Weyl-vector Gauss-Bonnet solutions but not for regular Kalb-Ramond ...

  2. Dyonic Black Holes in Lorentz-Violating Gravity with a Background Kalb--Ramond Field

    gr-qc 2026-05 unverdicted novelty 5.0

    An exact dyonic black hole metric is derived in Lorentz-violating gravity with background Kalb-Ramond field and nonminimal EM coupling; geodesics and extended thermodynamics are analyzed showing parameter-dependent sh...

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · cited by 2 Pith papers · 24 internal anchors

  1. [1]

    Enthalpy and the Mechanics of AdS Black Holes

    D. Kastor, S. Ray and J. Traschen, “Enthalpy and the Mechanics of AdS Black Holes,” Class. Quant. Grav.26, 195011 (2009) [arXiv:0904.2765 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  2. [2]

    The cosmological constant and the black hole equation of state

    B. P. Dolan, “The cosmological constant and the black hole equation of state,” Class. Quant. Grav.28, 125020 (2011) [arXiv:1008.5023 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  3. [3]

    P-V criticality of charged AdS black holes

    D. Kubiznak and R. B. Mann, “P-V criticality of charged AdS black holes,” JHEP07, 033 (2012) [arXiv:1205.0559 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  4. [4]

    String theory corrections to holographic black hole chemistry,

    S. Dutta and G. S. Punia, “String theory corrections to holographic black hole chemistry,” Phys. Rev. D106, no.2, 026003 (2022) [arXiv:2205.15593 [hep-th]]

    work page arXiv 2022

  5. [5]

    Smarr Formula and an Extended First Law for Lovelock Gravity

    D. Kastor, S. Ray and J. Traschen, “Smarr Formula and an Extended First Law for Lovelock Gravity,” Class. Quant. Grav.27, 235014 (2010) [arXiv:1005.5053 [hep- th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  6. [6]

    Higher Order Corrections to Holographic Black Hole Chemistry

    M. Sinamuli and R. B. Mann, “Higher Order Corrections to Holographic Black Hole Chemistry,” Phys. Rev. D96, no.8, 086008 (2017) [arXiv:1706.04259 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  7. [7]

    Extended Black Hole Thermodynamics from Extended Iyer-Wald Formalism,

    Y. Xiao, Y. Tian and Y. X. Liu, “Extended Black Hole Thermodynamics from Extended Iyer-Wald Formalism,” Phys. Rev. Lett.132, no.2, 2 (2024) [arXiv:2308.12630 [gr-qc]]

    work page arXiv 2024

  8. [8]

    Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume

    M. Cvetic, G. W. Gibbons, D. Kubiznak and C. N. Pope, “Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume,” Phys. Rev. D84, 024037 (2011) [arXiv:1012.2888 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  9. [9]

    Black hole chem- istry, the cosmological constant and the embedding ten- sor,

    P. Meessen, D. Mitsios and T. Ort´ ın, “Black hole chem- istry, the cosmological constant and the embedding ten- sor,” JHEP12, 155 (2022) [arXiv:2203.13588 [hep-th]]

    work page arXiv 2022

  10. [10]

    Higher-Dimensional Origin of Extended Black Hole Thermodynamics,

    A. M. Frassino, J. F. Pedraza, A. Svesko and M. R. Visser, “Higher-Dimensional Origin of Extended Black Hole Thermodynamics,” Phys. Rev. Lett.130, no.16, 161501 (2023) [arXiv:2212.14055 [hep-th]]

    work page arXiv 2023

  11. [11]

    Coupling Constants as Con- served Charges in Black Hole Thermodynamics,

    K. Hajian and B. Tekin, “Coupling Constants as Con- served Charges in Black Hole Thermodynamics,” Phys. Rev. Lett.132, no.19, 191401 (2024) [arXiv:2309.07634 [gr-qc]]

    work page arXiv 2024

  12. [12]

    Black hole chemistry: thermodynamics with Lambda

    D. Kubiznak, R. B. Mann and M. Teo, “Black hole chem- istry: thermodynamics with Lambda,” Class. Quant. Grav.34, no.6, 063001 (2017) [arXiv:1608.06147 [hep- th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  13. [13]

    Black hole chemistry: The first 15 years,

    R. B. Mann, “Black hole chemistry: The first 15 years,” Int. J. Mod. Phys. D34, no.09, 2542001 (2025) [arXiv:2508.01830 [gr-qc]]

    work page arXiv 2025

  14. [14]

    Insight into the Microscopic Structure of an AdS Black Hole from Thermodynamical Phase Transition

    S. W. Wei and Y. X. Liu, “Insight into the Microscopic Structure of an AdS Black Hole from a Thermodynamical Phase Transition,” Phys. Rev. Lett.115, no.11, 111302 (2015) [arXiv:1502.00386 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  15. [15]

    Wei, Y.-X

    S. W. Wei, Y. X. Liu and R. B. Mann, “Repulsive Interac- tions and Universal Properties of Charged Anti–de Sitter Black Hole Microstructures,” Phys. Rev. Lett.123, no.7, 071103 (2019) [arXiv:1906.10840 [gr-qc]]

    work page arXiv 2019

  16. [16]

    Thermody- namics of AdS Black Holes: Critical Behavior of the Cen- tral Charge,

    W. Cong, D. Kubiznak and R. B. Mann, “Thermody- namics of AdS Black Holes: Critical Behavior of the Cen- tral Charge,” Phys. Rev. Lett.127, no.9, 091301 (2021) [arXiv:2105.02223 [hep-th]]

    work page arXiv 2021

  17. [17]

    Holographic Dual of Extended Black Hole Thermodynamics,

    M. B. Ahmed, W. Cong, D. Kubizˇ n´ ak, R. B. Mann and M. R. Visser, “Holographic Dual of Extended Black Hole Thermodynamics,” Phys. Rev. Lett.130, no.18, 181401 (2023) [arXiv:2302.08163 [hep-th]]

    work page arXiv 2023

  18. [18]

    Zhang, W.-D

    M. Zhang, W. D. Tan, M. Lu, D. Bhattacharya, J. Yang and R. B. Mann, “Finite-cutoff Holographic Thermody- namics,” [arXiv:2507.01010 [hep-th]]

    work page arXiv

  19. [19]

    Holographic entanglement chemistry

    E. Caceres, P. H. Nguyen and J. F. Pedraza, “Holo- graphic entanglement chemistry,” Phys. Rev. D95, no.10, 106015 (2017) [arXiv:1605.00595 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  20. [20]

    Noether charge, black hole volume, and complexity

    J. Couch, W. Fischler and P. H. Nguyen, “Noether charge, black hole volume, and complexity,” JHEP03, 119 (2017) [arXiv:1610.02038 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  21. [21]

    Holographic Complexity and Thermody- namic Volume,

    A. Al Balushi, R. A. Hennigar, H. K. Kunduri and R. B. Mann, “Holographic Complexity and Thermody- namic Volume,” Phys. Rev. Lett.126, no.10, 101601 (2021) [arXiv:2008.09138 [hep-th]]

    work page arXiv 2021

  22. [22]

    Weak gravity conjecture,

    D. Harlow, B. Heidenreich, M. Reece and T. Rudelius, “Weak gravity conjecture,” Rev. Mod. Phys.95, 035003 (2023) [arXiv:2201.08380 [hep-th]]

    work page arXiv 2023

  23. [23]

    Thermodynamics with Pressure and Volume under Charged Particle Absorption

    B. Gwak, “Thermodynamics with Pressure and Vol- ume under Charged Particle Absorption,” JHEP11, 129 (2017) [arXiv:1709.08665 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  24. [24]

    Cardoso, M

    V. Cardoso, M. Kimura, A. Maselli and L. Senatore, “Black Holes in an Effective Field Theory Extension of General Relativity,” Phys. Rev. Lett.121, no.25, 251105 (2018) [arXiv:1808.08962 [gr-qc]]

    work page arXiv 2018

  25. [25]

    First order corrections to the black hole ther- modynamics in higher curvature theories of gravity,

    Y. Xiao, “First order corrections to the black hole ther- modynamics in higher curvature theories of gravity,” Phys. Rev. D106, no.6, 064041 (2022) [arXiv:2207.00967 [gr-qc]]

    work page arXiv 2022

  26. [26]

    Gauss-Bonnet Black Holes in AdS Spaces

    R. G. Cai, “Gauss-Bonnet black holes in AdS spaces,” Phys. Rev. D65, 084014 (2002) [arXiv:hep-th/0109133 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  27. [27]

    Background subtraction method is not only much sim- pler, but also as applicable as the covariant countert- 9 erm method,

    W. Guo, X. Guo, X. Lan, H. Zhang and W. Zhang, “Background subtraction method is not only much sim- pler, but also as applicable as the covariant countert- erm method,” Phys. Rev. D111, no.8, 084088 (2025) [arXiv:2501.08214 [hep-th]]

    work page arXiv 2025

  28. [28]

    Higher derivative corrections to Kerr-AdS black hole thermodynamics,

    W. Guo, X. Guo, X. Lan, H. Zhang and W. Zhang, “Higher derivative corrections to Kerr-AdS black hole thermodynamics,” Phys. Rev. D112, no.6, 064038 (2025）[arXiv:2504.21724 [hep-th]]

    work page arXiv 2025

  29. [29]

    Quadratic curvature corrections to 5-dimensional Kerr- AdS black hole thermodynamics by background subtrac- tion method,

    G. Chen, X. Guo, X. Lan, H. Zhang and W. Zhang, “Quadratic curvature corrections to 5-dimensional Kerr- AdS black hole thermodynamics by background subtrac- tion method,” [arXiv:2508.18171 [hep-th]]

    work page arXiv

  30. [30]

    Validity of the Background Subtraction Method for Black Hole Thermodynamics in Matter-Coupled Gravity Theories

    Y. Xiao and A. Zhang, “On the Validity of the Background Subtraction Method for Black Hole Ther- modynamics in Matter-Coupled Gravity Theories,” [arXiv:2511.07209 [gr-qc]]. 8

    work page internal anchor Pith review Pith/arXiv arXiv

  31. [31]

    Action Integrals and Partition Functions in Quantum Gravity,

    G. W. Gibbons and S. W. Hawking, “Action Integrals and Partition Functions in Quantum Gravity,” Phys. Rev. D15, 2752-2756 (1977)

    work page 1977

  32. [32]

    Thermodynamics Of Black Holes In Anti-De Sitter Space,

    S. W. Hawking and D. N. Page,“Thermodynamics Of Black Holes In Anti-De Sitter Space,” Commun. Math. Phys. 87, 577 (1983)

    work page 1983

  33. [33]

    The First Law of Thermodynamics for Kerr-Anti-de Sitter Black Holes

    G. W. Gibbons, M. J. Perry and C. N. Pope, “The First law of thermodynamics for Kerr-anti-de Sitter black holes,” Class. Quant. Grav.22, 1503-1526 (2005) [arXiv:hep-th/0408217 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  34. [34]

    On Euclidean and Noetherian Entropies in AdS Space

    S. Dutta and R. Gopakumar, “On Euclidean and Noethe- rian entropies in AdS space,” Phys. Rev. D74, 044007 (2006) [arXiv:hep-th/0604070 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  35. [35]

    A Stress Tensor for Anti-de Sitter Gravity

    V. Balasubramanian and P. Kraus, “A Stress tensor for Anti-de Sitter gravity,” Commun. Math. Phys.208, 413- 428 (1999) [arXiv:hep-th/9902121 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  36. [36]

    Holographic Renormalization

    M. Bianchi, D. Z. Freedman and K. Skenderis, “Holo- graphic renormalization,” Nucl. Phys. B631, 159-194 (2002) [arXiv:hep-th/0112119 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  37. [37]

    Thermodynamics of Asymptotically Locally AdS Spacetimes

    I. Papadimitriou and K. Skenderis, “Thermodynamics of asymptotically locally AdS spacetimes,” JHEP08, 004 (2005) [arXiv:hep-th/0505190 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  38. [38]

    Black hole entropy is the Noether charge,

    R. M. Wald, “Black hole entropy is the Noether charge,” Phys. Rev. D48(1993) no.8, 3427-3431 [arXiv:gr- qc/9307038 [gr-qc]]

    work page arXiv 1993

  39. [39]

    Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy

    V. Iyer and R. M. Wald, “Some properties of Noether charge and a proposal for dynamical black hole entropy,” Phys. Rev. D50, 846-864 (1994) [arXiv:gr-qc/9403028 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  40. [40]

    A comparison of Noether charge and Euclidean methods for Computing the Entropy of Stationary Black Holes

    V. Iyer and R. M. Wald, “A Comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes,” Phys. Rev. D52, 4430-4439 (1995) [arXiv:gr-qc/9503052 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  41. [41]

    A short note on the boundary term for the Hilbert action,

    T. Padmanabhan, “A short note on the boundary term for the Hilbert action,” Mod. Phys. Lett. A29, no.08, 1450037 (2014)

    work page 2014

  42. [42]

    A Boundary Term for the Gravitational Action with Null Boundaries

    K. Parattu, S. Chakraborty, B. R. Majhi and T. Pad- manabhan, “A Boundary Term for the Gravitational Ac- tion with Null Boundaries,” Gen. Rel. Grav.48, no.7, 94 (2016) [arXiv:1501.01053 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  43. [43]

    A Novel Derivation of the Boundary Term for the Action in Lanczos-Lovelock Gravity

    S. Chakraborty, K. Parattu and T. Padmanabhan, “A Novel Derivation of the Boundary Term for the Action in Lanczos-Lovelock Gravity,” Gen. Rel. Grav.49, no.9, 121 (2017) [arXiv:1703.00624 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  44. [44]

    Surface term, corner term, and action growth in F(Riemann) gravity theory

    J. Jiang and H. Zhang, “Surface term, corner term, and action growth inF(R abcd) gravity theory,” Phys. Rev. D 99, no.8, 086005 (2019) [arXiv:1806.10312 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  45. [45]

    Covariant phase space with boundaries,

    D. Harlow and J. Q. Wu, “Covariant phase space with boundaries,” JHEP10, 146 (2020) [arXiv:1906.08616 [hep-th]]

    work page arXiv 2020

  46. [46]

    Equiv- alence of Noether charge and Hilbert action boundary term formulas for the black hole entropy in F(Rabcd) gravity theory,

    W. Guo, X. Guo, M. Li, Z. Mou and H. Zhang, “Equiv- alence of Noether charge and Hilbert action boundary term formulas for the black hole entropy in F(Rabcd) gravity theory,” Phys. Rev. D110, no.6, 064071 (2024) [arXiv:2406.15138 [hep-th]]

    work page arXiv 2024

  47. [47]

    In this case, the sym- plectic currentw[δϕ,L ξϕ]≡δC[L ξϕ]− LξC[δϕ] must be added to Eq.(10)[27, 45, 46]

    For readers wishing to retain thedCterm, a well- established technique is available. In this case, the sym- plectic currentw[δϕ,L ξϕ]≡δC[L ξϕ]− LξC[δϕ] must be added to Eq.(10)[27, 45, 46]. This cleverly transforms thedCterm into aδCterm, which can be absorbed into δM(the key step to rewrite Eq.(15)into the Brown- York mass form). Anyway, all theseCterms ...

    work page

  48. [48]

    Massive Gravity in Three Dimensions

    E. A. Bergshoeff, O. Hohm and P. K. Townsend, “Mas- sive Gravity in Three Dimensions,” Phys. Rev. Lett.102, 201301 (2009) [arXiv:0901.1766 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2009