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arxiv: 2509.08390 · v2 · submitted 2025-09-10 · ✦ hep-th · gr-qc

Gravitational constant as a conserved charge in black hole thermodynamics

Wontae Kim , Mungon Nam This is my paper

Pith reviewed 2026-05-18 18:17 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords gravitational constantconserved chargesblack hole thermodynamicsEinstein-Hilbert theoryAbbott-Deser-Tekin formalismextended first lawSmarr formulascalar-gauge pairs
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The pith

Gravitational constant realized as conserved charge in modified Einstein gravity using scalar-gauge pairs.

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

The paper shows that the gravitational constant, which normally sets the overall scale of the Einstein-Hilbert term, can still be promoted to a conserved charge. This is done by supplementing a four-dimensional Einstein-Hilbert theory with two scalar-gauge pairs and then applying the quasi-local off-shell Abbott-Deser-Tekin formalism. The resulting charges for mass, cosmological constant, and the gravitational constant produce an extended first law of black hole thermodynamics together with a consistent Smarr relation. A sympathetic reader cares because the construction supplies a uniform symmetry origin for treating every coupling in the action, including the universal gravitational one, as a thermodynamic variable.

Core claim

In a modified four-dimensional Einstein-Hilbert theory supplemented with two scalar-gauge pairs, the quasi-local off-shell Abbott-Deser-Tekin formalism yields conserved charges for the mass, the cosmological constant, and the gravitational constant. These charges lead to the extended thermodynamic first law and the Smarr formula in a fully consistent manner, extending the conserved-charge interpretation to the universal gravitational coupling itself.

What carries the argument

Two scalar-gauge pairs combined with the quasi-local off-shell Abbott-Deser-Tekin formalism in a modified four-dimensional Einstein-Hilbert theory.

If this is right

  • Mass, cosmological constant, and gravitational constant each correspond to a distinct conserved charge.
  • The extended first law includes a term arising from variation of the gravitational constant.
  • The Smarr formula remains consistent once the new charge is included.
  • The construction introduces no additional physical degrees of freedom.

Where Pith is reading between the lines

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

  • Varying G can be incorporated into thermodynamic descriptions of black holes while maintaining consistency with the first law.
  • The same scalar-gauge technique may apply to overall normalization factors in other gravity theories.
  • Direct evaluation on known exact solutions offers a concrete check of the extended thermodynamic relations.

Load-bearing premise

The modified Lagrangian with the two scalar-gauge pairs preserves diffeomorphism invariance and permits the off-shell ADT charge construction without introducing new physical degrees of freedom or altering the asymptotic structure.

What would settle it

Compute the ADT charges explicitly for the Schwarzschild-de Sitter solution in the modified theory and verify that the first law holds when the gravitational constant is varied independently while holding other quantities fixed.

read the original abstract

Recent work has shown that couplings multiplying individual terms in a Lagrangian can be promoted to conserved charges by introducing scalar-gauge pairs. The gravitational constant, however, plays a qualitatively different role: $G^{-1}$ appears as an overall normalization of the Einstein-Hilbert sector rather than as the coefficient of a single term. In this paper, we show that the gravitational constant can nevertheless be realized as a conserved charge in a modified four-dimensional Einstein-Hilbert theory. Using two scalar-gauge pairs and the quasi-local off-shell Abbott-Deser-Tekin formalism, we construct the conserved charges associated with the mass, the cosmological constant, and the gravitational constant. The resulting charge assignment yields the extended thermodynamic first law and the Smarr formula in a fully consistent manner. Our result therefore extends the conserved-charge interpretation of couplings to the universal gravitational coupling itself in a concrete four-dimensional Einstein-gravity setting.

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 manuscript claims that the gravitational constant G, which appears as an overall normalization factor in the Einstein-Hilbert term rather than the coefficient of an isolated term, can be realized as a conserved charge. This is achieved by modifying the four-dimensional Einstein-Hilbert action with two scalar-gauge pairs and applying the quasi-local off-shell Abbott-Deser-Tekin (ADT) formalism to construct conserved charges for the mass, cosmological constant, and G. These charges are then shown to yield the extended thermodynamic first law and the Smarr formula in a consistent manner.

Significance. If the central construction is valid, the result would meaningfully extend prior work on promoting Lagrangian couplings to conserved charges to the case of the universal gravitational coupling itself. The systematic use of the off-shell ADT formalism to define quasi-local charges for all three quantities is a clear strength, as it provides a uniform framework without relying on ad-hoc choices. This approach could inform broader questions in extended black-hole thermodynamics where fundamental constants enter the first law.

major comments (2)
  1. [§2] §2 (Modified Lagrangian): The introduction of two scalar-gauge pairs to handle the overall G normalization is load-bearing for the entire claim. The manuscript must explicitly show that the equations of motion for these auxiliary fields are pure constraints with no propagating degrees of freedom and that the asymptotic fall-off conditions required by the ADT formalism remain unchanged; without this verification the quasi-local charges for mass, Λ, and G cannot be guaranteed to be well-defined.
  2. [§4] §4 (Charge construction and thermodynamic relations): The paper states that the ADT procedure yields the extended first law and Smarr formula, but does not supply the explicit expression for the G charge nor perform the verification on a concrete metric (e.g., Schwarzschild-AdS). This absence leaves the central claim that the thermodynamic relations are recovered in a fully consistent manner without direct support.
minor comments (1)
  1. [§2] The notation distinguishing the two scalar-gauge pairs could be made more uniform across sections to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below and will revise the manuscript to incorporate the requested clarifications and verifications.

read point-by-point responses

  1. Referee: §2 (Modified Lagrangian): The introduction of two scalar-gauge pairs to handle the overall G normalization is load-bearing for the entire claim. The manuscript must explicitly show that the equations of motion for these auxiliary fields are pure constraints with no propagating degrees of freedom and that the asymptotic fall-off conditions required by the ADT formalism remain unchanged; without this verification the quasi-local charges for mass, Λ, and G cannot be guaranteed to be well-defined.

    Authors: We agree that an explicit verification of these properties is necessary to rigorously support the construction. In the revised manuscript we will expand Section 2 with a dedicated subsection deriving the equations of motion for the two scalar-gauge pairs and showing that they reduce to non-dynamical constraints with no additional propagating degrees of freedom. We will also confirm that the asymptotic fall-off conditions imposed by the off-shell ADT formalism remain unchanged for the metric and auxiliary fields, thereby ensuring the quasi-local charges are well-defined. revision: yes

  2. Referee: §4 (Charge construction and thermodynamic relations): The paper states that the ADT procedure yields the extended first law and Smarr formula, but does not supply the explicit expression for the G charge nor perform the verification on a concrete metric (e.g., Schwarzschild-AdS). This absence leaves the central claim that the thermodynamic relations are recovered in a fully consistent manner without direct support.

    Authors: We acknowledge that the absence of the explicit G-charge expression and a concrete verification weakens the presentation of the central claim. In the revised version we will supply the explicit form of the conserved charge associated with G as obtained from the off-shell ADT formalism. We will additionally perform and display the explicit calculation for the Schwarzschild-AdS metric, verifying that the extended first law and Smarr formula are recovered in a fully consistent manner. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via explicit model construction

full rationale

The paper introduces an explicit modification to the Einstein-Hilbert Lagrangian using two scalar-gauge pairs, then applies the standard quasi-local off-shell ADT formalism to assign conserved charges including one for the gravitational constant. The extended first law and Smarr formula follow directly from the general properties of ADT charges in this setup. This is a constructive procedure rather than any reduction by definition, fitted prediction, or self-citation chain. The modification is stated as the input, and no load-bearing step collapses to a prior result by the authors or renames a known pattern. The approach remains independent of the target thermodynamic relations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of a diffeomorphism-invariant modified Lagrangian whose ADT charges reproduce the extended first law. No free parameters are fitted to data; the construction is purely formal.

axioms (1)
  • domain assumption The modified Einstein-Hilbert action with two scalar-gauge pairs remains diffeomorphism invariant and admits a well-defined quasi-local off-shell ADT charge construction.
    Invoked when the authors state that the conserved charges for mass, Lambda, and G can be constructed.
invented entities (1)
  • Two scalar-gauge pairs no independent evidence
    purpose: To promote G^{-1} from an overall normalization to a coefficient that can be associated with a conserved charge.
    The pairs are introduced specifically for this purpose; no independent evidence outside the construction is provided.

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Reference graph

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