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arxiv: 2605.22429 · v1 · pith:F6TJG4XFnew · submitted 2026-05-21 · 🌀 gr-qc · hep-th

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

Jia-Zhou Liu , Shan-Ping Wu , Shao-Wen Wei , Yu-Xiao Liu This is my paper

Pith reviewed 2026-05-22 05:13 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords black hole entropyWald entropynonminimal couplingcovariant phase spaceKalb-Ramond fieldbumblebee gravityGauss-Bonnet gravitydiffeomorphism invariance
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The pith

In nonminimally coupled gravity, black hole entropy includes terms beyond the Wald entropy when matter fields cannot extend smoothly to the bifurcation surface.

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

The paper establishes that for static spherically symmetric black holes in diffeomorphism-invariant theories with nonminimal matter-curvature couplings, the entropy that enters the first law is not limited to the standard Wald entropy. Instead, when a matter field cannot be smoothly extended to the regular bifurcation surface, the horizon surface charge variation yields extra finite contributions that must be included. A sympathetic reader would care because this ensures thermodynamic consistency in a wider range of modified gravity models and identifies precisely when the Wald formula alone is insufficient. The decomposition is obtained by applying the covariant phase space formalism and subtracting ordinary work terms.

Core claim

For regular bifurcate Killing horizons the Iyer-Wald construction recovers the Wald entropy, but when matter fields fail to extend smoothly to the bifurcation surface the horizon surface charge variation contains additional finite pieces. After ordinary work terms are subtracted, the entropy entering the first law decomposes as S_H = S_W + S_1 + ΔS, where S_W is the Wald entropy, S_1 is the non-Wald part of the Noether charge, and ΔS is the remaining integrable part of the surface charge variation.

What carries the argument

The decomposition of the horizon surface charge variation in the covariant phase space formalism, after subtracting work terms, into the Wald entropy plus the non-Wald Noether contribution S_1 and the remaining integrable term ΔS.

If this is right

  • For the regular Kalb-Ramond branch the entropy reduces exactly to the Wald term.
  • Bumblebee branches produce either a nonzero ΔS with vanishing S_1 or a cancellation between S_1 and ΔS.
  • Weyl-vector extended Gauss-Bonnet examples require nonzero contributions from both S_1 and ΔS.
  • The criterion directly shows whether the Wald entropy density alone satisfies the first law or whether the full surface charge variation is needed.

Where Pith is reading between the lines

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

  • The same decomposition can be applied to other diffeomorphism-invariant theories with nonminimal couplings to test whether extra terms appear.
  • The results suggest that the smoothness of matter-field extension near the horizon controls whether thermodynamic relations receive corrections beyond the Wald formula.
  • Independent entropy computations for these specific solutions would confirm whether the decomposed expression or the pure Wald term matches other methods.

Load-bearing premise

The horizon is a regular bifurcate Killing horizon and the covariant phase space formalism applies without further restrictions on the matter-field extension or the choice of representative for the surface charge variation.

What would settle it

Perform an independent calculation of the entropy for a bumblebee black hole solution, for example via the Euclidean action or by direct integration of the first law, and check whether the result equals only the Wald term or requires the additional S_1 and ΔS contributions.

read the original abstract

We study the entropy of static, spherically symmetric black holes in diffeomorphism-invariant theories with nonminimal matter--curvature couplings, using the covariant phase space formalism. For regular bifurcate Killing horizons, the Iyer--Wald construction gives the standard Wald entropy. If a matter field cannot be smoothly extended to the regular bifurcation surface, however, the horizon surface charge variation can contain finite contributions that are not included in the Wald entropy density. In the representative obtained by directly varying the action, and after ordinary work terms are subtracted, we decompose the entropy entering the first law of black hole thermodynamics as \(S_{\mathrm H}=S_{\mathrm W}+S_1+\Delta S\). Here \(S_{\mathrm W}\) is the Wald entropy, \(S_1\) is the non-Wald part of the Noether charge, and \(\Delta S\) is the remaining integrable part of the horizon surface charge variation. Applying this criterion to Kalb--Ramond, bumblebee, and extended Gauss--Bonnet black holes, we find that the regular Kalb--Ramond branch has \(S_{\mathrm H}=S_{\mathrm W}\), the bumblebee branches yield either \(S_1=0\) with \(\Delta S\neq0\) or a cancellation between \(S_1\) and \(\Delta S\), and the Weyl-vector extended Gauss--Bonnet examples require both corrections. This gives a direct test of whether the Wald entropy density is sufficient, or whether the full horizon surface charge variation has to be used.

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 / 1 minor

Summary. The paper applies the covariant phase space formalism to static spherically symmetric black holes in diffeomorphism-invariant theories with nonminimal matter-curvature couplings. For regular bifurcate Killing horizons, it decomposes the entropy entering the first law as S_H = S_W + S_1 + ΔS when matter fields cannot be smoothly extended to the bifurcation surface, with S_W the Wald entropy, S_1 the non-Wald part of the Noether charge, and ΔS the remaining integrable contribution after subtracting ordinary work terms. The decomposition is evaluated on Kalb-Ramond, bumblebee, and extended Gauss-Bonnet examples, yielding S_H = S_W for the regular Kalb-Ramond branch, either S_1 = 0 with ΔS ≠ 0 or cancellation for bumblebee branches, and both corrections required for the Weyl-vector extended Gauss-Bonnet cases.

Significance. If the decomposition is robust, the work supplies a practical criterion for deciding when the Wald entropy density suffices and when the full horizon surface charge variation must be retained. The concrete results for three model classes furnish falsifiable predictions and illustrate how non-smooth matter extensions generate additional integrable contributions. This strengthens the covariant phase space approach for nonminimally coupled theories.

major comments (2)
  1. [Abstract] Abstract (paragraph beginning 'In the representative obtained by directly varying the action'): the decomposition is performed in one specific representative of the surface charge. The covariant phase space formalism permits addition of exact forms to the Noether current without changing the equations of motion. No demonstration is given that the split S_H = S_W + S_1 + ΔS or the total S_H remains unchanged under such additions. Because the central claim concerns the necessity of terms beyond S_W, invariance under representative choice is load-bearing and should be shown explicitly.
  2. [Applications to models] Applications section (Kalb-Ramond, bumblebee, and extended Gauss-Bonnet examples): the reported outcomes (S_H = S_W for regular Kalb-Ramond; S_1 = 0 with ΔS ≠ 0 or cancellation for bumblebee; both corrections for Weyl-vector Gauss-Bonnet) rest on the chosen representative and on the assumed regularity of the bifurcate horizon. Explicit expressions for the surface charge variations and the subtracted work terms would allow verification that the reported patterns are not artifacts of the representative choice.
minor comments (1)
  1. [Notation] The abstract introduces S_1 and ΔS without an equation number; a numbered display of the decomposition S_H = S_W + S_1 + ΔS in the main text would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments correctly identify that the manuscript presents the entropy decomposition in a specific representative of the Noether current and that the applications would benefit from more explicit intermediate expressions. We address both points below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph beginning 'In the representative obtained by directly varying the action'): the decomposition is performed in one specific representative of the surface charge. The covariant phase space formalism permits addition of exact forms to the Noether current without changing the equations of motion. No demonstration is given that the split S_H = S_W + S_1 + ΔS or the total S_H remains unchanged under such additions. Because the central claim concerns the necessity of terms beyond S_W, invariance under representative choice is load-bearing and should be shown explicitly.

    Authors: We agree that an explicit demonstration of invariance is needed. In the revised manuscript we will add a short subsection (in the general formalism section) showing that the addition of an exact form dα to the Noether current does not change the decomposition or the total integrable entropy S_H. On a closed bifurcation surface the integral of the exact term vanishes identically, and any residual boundary contributions cancel against the subtracted work terms when the first law is assembled. This establishes that the split S_H = S_W + S_1 + ΔS is representative-independent for the class of theories and horizons considered. revision: yes

  2. Referee: [Applications to models] Applications section (Kalb-Ramond, bumblebee, and extended Gauss-Bonnet examples): the reported outcomes (S_H = S_W for regular Kalb-Ramond; S_1 = 0 with ΔS ≠ 0 or cancellation for bumblebee; both corrections for Weyl-vector Gauss-Bonnet) rest on the chosen representative and on the assumed regularity of the bifurcate horizon. Explicit expressions for the surface charge variations and the subtracted work terms would allow verification that the reported patterns are not artifacts of the representative choice.

    Authors: We accept that the applications section would be more transparent with the intermediate expressions. In the revision we will add an appendix containing the explicit forms of the horizon surface charge variation δQ_H and the subtracted work terms for each of the three models. These expressions will be derived from the same representative used in the main text, allowing direct verification that the reported results (S_H = S_W for the regular Kalb-Ramond branch, S_1 = 0 with ΔS ≠ 0 or cancellation for bumblebee branches, and both corrections for the Weyl-vector Gauss-Bonnet cases) follow from the general decomposition and are not artifacts of the representative choice. The regularity assumption for the bifurcate horizon is the standard one employed in the Iyer-Wald construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard formalism to specific representative

full rationale

The paper applies the covariant phase space formalism and Iyer-Wald procedure to decompose the horizon surface charge variation for regular bifurcate Killing horizons in nonminimally coupled theories. The split S_H = S_W + S_1 + ΔS is obtained after subtracting work terms from the representative chosen by direct action variation; this partitioning does not reduce any claimed result to a fitted parameter, self-defined quantity, or load-bearing self-citation by construction. The central claim is tested on concrete examples (Kalb-Ramond, bumblebee, extended Gauss-Bonnet) where the presence or absence of extra terms is exhibited explicitly. No equation equates the final entropy expression to its inputs tautologically, and the construction remains self-contained against external benchmarks in the literature.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes the standard covariant phase space formalism and the existence of regular bifurcate Killing horizons; no free parameters, ad-hoc axioms, or new postulated entities are introduced or fitted.

axioms (2)
  • domain assumption The horizon is a regular bifurcate Killing horizon to which the Iyer-Wald construction applies.
    Stated in the opening sentence of the abstract as the setting in which the standard Wald entropy is recovered.
  • domain assumption The covariant phase space formalism yields a well-defined horizon surface charge variation after ordinary work terms are subtracted.
    Implicit in the decomposition S_H = S_W + S_1 + ΔS presented in the abstract.

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