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arxiv: 2510.08686 · v2 · submitted 2025-10-09 · ✦ hep-th · gr-qc

Non-closed scalar charge in four-dimensional Einstein-scalar-Gauss-Bonnet black hole thermodynamics

Romina Ballesteros , Marcela C\'ardenas , Eric Lescano This is my paper

Pith reviewed 2026-05-18 08:34 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords scalar chargeGauss-Bonnet gravityblack hole thermodynamicsspontaneous scalarizationshift symmetrydifferential formsSmarr formulahorizon generator
0
0 comments X p. Extension

The pith

Contracting the scalar equation with the horizon generator produces a non-closed 2-form scalar charge whose obstruction is a bulk 3-form in Einstein-scalar-Gauss-Bonnet gravity.

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

The paper constructs a covariant differential-form definition of scalar charges for stationary asymptotically flat black holes in four-dimensional Einstein-scalar-Gauss-Bonnet gravity with arbitrary scalar coupling. Contracting the scalar field equation of motion with the horizon generator yields a 2-form charge that fails to be closed; the failure is measured by an explicit 3-form bulk term. When the coupling possesses shift symmetry the 3-form vanishes, the charge becomes closed, and it obeys a Gauss law determined entirely by boundary data. The same construction supplies a covariant expression for the Smarr relation at more general couplings and interprets spontaneous scalarization as the point where the bulk term and scalar charge together signal the instability of the scalar-free solution.

Core claim

Contracting the scalar field equation of motion with the horizon generator k yields a non-closed 2-form scalar charge whose exterior derivative is a 3-form that encodes the obstruction to closedness. In the shift-symmetric limit the obstruction vanishes, the 2-form satisfies a Gauss law, and the charge is fixed by boundary data alone. This reproduces known topological results geometrically and supplies a charge-based account of the Smarr formula and of the onset of scalar hair.

What carries the argument

The 2-form scalar charge obtained by contracting the scalar equation of motion with the horizon generator k, whose exterior derivative equals a single 3-form bulk term that quantifies the obstruction to closedness.

If this is right

  • The Smarr formula acquires a covariant charge-based expression for arbitrary scalar couplings.
  • Spontaneous scalarization is captured by the simultaneous behavior of the scalar charge and the bulk 3-form.
  • In the shift-symmetric case the charge depends only on boundary data and reproduces known topological identities.
  • The framework unifies the treatment of scalar charges across different couplings in the same geometric language.

Where Pith is reading between the lines

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

  • The same contraction procedure could be applied to other scalar-tensor theories to isolate analogous bulk obstructions.
  • Numerical evolution of scalar perturbations on Schwarzschild or Kerr backgrounds might directly measure the growth of the bulk 3-form during scalarization.
  • The 3-form term may admit a topological interpretation that survives even when shift symmetry is weakly broken.

Load-bearing premise

The scalar field equation of motion can be contracted with the horizon generator to produce a well-defined 2-form whose exterior derivative is controlled by one 3-form bulk contribution.

What would settle it

For an explicit non-shift-symmetric solution, compute the exterior derivative of the constructed 2-form and verify whether it exactly equals the paper's proposed 3-form bulk term.

read the original abstract

We develop a covariant differential-form framework to define scalar charges for stationary, asymptotically flat black holes in $4$--dimensional Einstein-scalar-Gauss-Bonnet gravity with a general scalar coupling function. Contracting the scalar field equation of motion with the horizon generator $k$ yields a non-closed-form scalar charge, revealing a bulk contribution encoded in a $3$--form, which measures the obstruction to its closedness. In the presence of shift-symmetry, this obstruction vanishes and the $2$--form scalar charge satisfies a Gauss law, depending solely on boundary data. Geometrically, this reproduces known topological results in the shift-symmetric limit. This framework allows us to analyze the role of the non-closed scalar charges in black hole thermodynamics through the Smarr formula for more general couplings and provide a covariant, charge-based interpretation of the spontaneous scalarization mechanism, showing how the behavior of the scalar charge and the bulk term capture the instability of scalar-free black holes and the emergence of scalar hair. Our results offer a unified geometric understanding of the role of scalar charges and the mechanism of spontaneous scalarization in Einstein-scalar-Gauss-Bonnet gravity.

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 / 2 minor

Summary. The paper develops a covariant differential-form framework to define scalar charges for stationary, asymptotically flat black holes in 4-dimensional Einstein-scalar-Gauss-Bonnet gravity with a general scalar coupling function. Contracting the scalar field equation of motion with the horizon generator k yields a non-closed scalar charge 2-form, revealing a bulk contribution encoded in a 3-form which measures the obstruction to its closedness. In the presence of shift-symmetry, this obstruction vanishes and the 2-form scalar charge satisfies a Gauss law depending solely on boundary data. The framework is applied to the Smarr formula for more general couplings and provides a covariant, charge-based interpretation of the spontaneous scalarization mechanism.

Significance. If the proposed construction is shown to yield the physically relevant conserved charge, the work supplies a geometric tool for analyzing non-closed scalar charges and bulk contributions in black hole thermodynamics within ESGB gravity. The reproduction of known topological results in the shift-symmetric limit serves as a useful consistency check. This could aid in understanding spontaneous scalarization beyond shift-symmetric cases.

major comments (1)
  1. [Section on scalar charge definition] The section introducing the scalar charge (around the definition via contraction of the scalar EOM with k): the 2-form is defined directly by this contraction without an explicit derivation or proof of equivalence to the Noether charge obtained from varying the action under the diffeomorphism generated by k. This equivalence is load-bearing for the central claim that the 3-form correctly measures the obstruction in the conserved quantity relevant for the Smarr formula and thermodynamic interpretation.
minor comments (2)
  1. [Notation and definitions] The notation for the 3-form obstruction and its explicit dependence on the coupling function could be expanded with a component expression to aid readability.
  2. [Introduction] Additional comparison to standard references on covariant charges (e.g., Wald's Noether charge formalism) would strengthen the positioning of the new framework.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment, which has prompted us to strengthen the presentation of the scalar charge construction. We address the point below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [Section on scalar charge definition] The section introducing the scalar charge (around the definition via contraction of the scalar EOM with k): the 2-form is defined directly by this contraction without an explicit derivation or proof of equivalence to the Noether charge obtained from varying the action under the diffeomorphism generated by k. This equivalence is load-bearing for the central claim that the 3-form correctly measures the obstruction in the conserved quantity relevant for the Smarr formula and thermodynamic interpretation.

    Authors: We agree that an explicit link between the contracted equation-of-motion 2-form and the Noether charge associated with the diffeomorphism generated by the horizon Killing vector k would make the construction more transparent. In the revised manuscript we have added a short derivation in the scalar-charge section. Starting from the diffeomorphism variation of the action, the associated Noether current is obtained in the standard way; on-shell, the scalar field equation reduces this current precisely to the 2-form defined by the contraction with k. Consequently the exterior derivative of the 2-form yields the 3-form that measures the obstruction to closedness, which enters the integral identity underlying the Smarr relation. This addition confirms that the thermodynamic interpretation rests on the relevant conserved quantity without altering any of the original results. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained from field equations with no circular reductions

full rationale

The paper defines the scalar charge via direct contraction of the scalar EOM with the horizon generator k, yielding a 2-form whose exterior derivative is controlled by an explicit 3-form bulk term. This construction follows immediately from the field equations of the Einstein-scalar-Gauss-Bonnet action and is used to obtain the Smarr formula and the geometric interpretation of spontaneous scalarization. In the shift-symmetric limit the 3-form vanishes by the symmetry assumption on the coupling, producing a boundary-only Gauss law. No step reduces a claimed prediction or uniqueness result to a fitted input, self-citation chain, or prior ansatz by the same authors; the framework remains independent of external benchmarks and does not rename known results under new coordinates. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on the standard Einstein-scalar-Gauss-Bonnet action, the assumption that the spacetime is stationary and asymptotically flat, and the use of differential-form identities that hold in four dimensions. No free parameters are introduced in the abstract; the scalar coupling function is kept general.

axioms (2)
  • domain assumption The spacetime is stationary and asymptotically flat with a Killing horizon generator k.
    Invoked to define the horizon generator and to apply the differential-form construction to black-hole solutions.
  • domain assumption The scalar field equation of motion can be contracted with k to yield a 2-form whose exterior derivative is a single 3-form bulk term.
    This contraction is the central definitional step presented in the abstract.
invented entities (1)
  • Bulk 3-form measuring obstruction to closedness of the scalar charge 2-form no independent evidence
    purpose: Quantifies the failure of the scalar charge to satisfy a pure Gauss law when shift symmetry is absent.
    Introduced as the mathematical object that encodes the non-closed character of the charge; no independent falsifiable prediction is given in the abstract.

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Forward citations

Cited by 1 Pith paper

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  1. Derivation of the Smarr formula from the Komar charge in Einstein-nonlinear electrodynamics theories and applications to regular black holes

    gr-qc 2026-05 unverdicted novelty 6.0

    A generalized Komar charge constructed via Lagrange multiplier promotion of the coupling constant yields a Smarr formula including that constant's contribution for asymptotically flat black hole and soliton solutions ...

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