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arxiv: 2604.08668 · v1 · submitted 2026-04-09 · 🌀 gr-qc · astro-ph.HE· hep-th

Recognition: unknown

Minimum mass, maximum charge and hyperbolicity in scalar Gauss-Bonnet gravity

Dario Rossi , Leonardo Gualtieri , Thomas P. Sotiriou
Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:22 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEhep-th
keywords scalar Gauss-Bonnet gravityblack hole solutionshyperbolicityperturbation equationseffective field theoryscalar chargeminimum mass
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The pith

Black hole solutions in scalar Gauss-Bonnet gravity become unphysical below a minimum mass when perturbation equations turn elliptic, even though this minimum can be tuned arbitrarily small without increasing deviations from general reality

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

The paper examines static black hole solutions in scalar Gauss-Bonnet gravity for coupling functions that allow solutions of arbitrarily small mass. Below a critical mass set by the theory parameters, the linear perturbation equations around these solutions change from hyperbolic to elliptic. This change signals that the solutions are unphysical and that the effective field theory approximation has broken down. The minimum mass can be made as small as desired by adjusting the coupling function, yet the scalar charge carried by the black holes remains bounded from above, so the observable departures from general relativity do not grow larger.

Core claim

In scalar Gauss-Bonnet gravity with a suitable class of coupling functions, static black hole solutions exist for arbitrarily small masses, but the equations governing linear perturbations around these solutions become elliptic below a minimum mass; this renders the solutions unphysical and marks the loss of validity of the effective field theory. The minimum mass depends on the coupling parameters and can be driven arbitrarily close to zero, yet the black-hole scalar charge stays bounded above, limiting how far the solutions can depart from general relativity.

What carries the argument

The switch in character of the perturbation equations from hyperbolic to elliptic, which occurs at a critical mass determined by the coupling function and sets the boundary of physicality for the black hole solutions.

If this is right

  • Black hole solutions below the minimum mass are excluded as unphysical because their perturbation equations are elliptic.
  • The minimum mass can be made arbitrarily small by appropriate choice of the coupling function.
  • Observable quantities such as the black hole scalar charge remain bounded from above even when the minimum mass approaches zero.
  • Deviations from general relativity in these solutions cannot be made arbitrarily large by pushing the minimum mass lower.

Where Pith is reading between the lines

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

  • Effective field theory descriptions of black holes in higher-curvature gravity may carry an intrinsic lower mass cutoff even when parameters appear to allow arbitrarily small objects.
  • The bounded scalar charge implies that any observational signatures of these black holes in gravitational waves or electromagnetic observations are capped independently of how small the minimum mass is made.
  • Similar hyperbolicity breakdowns could appear in other scalar-tensor theories with higher-order curvature couplings when small-mass solutions are considered.

Load-bearing premise

The assumption that a change from hyperbolic to elliptic type in the perturbation equations directly means the background solutions are unphysical and that the effective field theory has lost validity.

What would settle it

An explicit computation of the characteristic speeds or principal symbol of the perturbation equations for a chosen coupling function that remains hyperbolic for all masses down to zero, or a numerical construction of a stable small-mass black hole whose perturbations propagate without imaginary speeds.

Figures

Figures reproduced from arXiv: 2604.08668 by Dario Rossi, Leonardo Gualtieri, Thomas P. Sotiriou.

Figure 1
Figure 1. Figure 1: FIG. 1. Threshold mass for hyperbolic equations, normalized [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. As in Fig [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Top panel: dimensionless scalar charge [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. As in Fig [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Maximum value of the dimensionless charge [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The shadowed regions in the [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Same as the right panel of Fig [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: we see that the presence of the Ricci coupling re￾duces the value of γcrit. This effect is most significant for β ≃ 1, while for larger values of β the value of γcrit in￾creases. The function γcrit(β) is well described by a cubic fit, whose best parameters are provided in the caption of [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
read the original abstract

We study the loss of hyperbolicity of perturbation equations for black hole solutions of scalar Gauss-Bonnet gravity. We consider a class of coupling functions allowing for static black hole solutions with arbitrary small masses. For masses below a minimum value, such solutions become unphysical, because the perturbation equations become elliptic; this arguably corresponds to the loss of validity of the effective field theory. We analyse the dependence of this minimum mass on the parameters of the theory, finding that with an appropriate choice of the coupling function, such mass can be chosen arbitrarily small. However, this does not correspond to larger deviations from general relativity, since observable quantities like the black hole scalar charge are bounded by above.

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 studies static black hole solutions in scalar Gauss-Bonnet gravity for a class of coupling functions f(φ) that permit solutions with arbitrarily small masses. It shows that below a minimum mass the linear perturbation equations around these backgrounds change from hyperbolic to elliptic character; the authors interpret this as rendering the solutions unphysical and marking the breakdown of the effective field theory. The minimum mass can be made arbitrarily small by suitable choice of f(φ), yet the black-hole scalar charge remains bounded from above, so that deviations from general relativity stay limited.

Significance. If the central claims are substantiated, the work supplies a concrete diagnostic (loss of hyperbolicity) for the regime of validity of scalar-Gauss-Bonnet black holes inside the EFT truncation. It demonstrates that even when the minimum mass can be pushed arbitrarily low, observable quantities such as the scalar charge cannot be made arbitrarily large, thereby tightening the link between mathematical well-posedness and phenomenological constraints in higher-curvature gravity.

major comments (1)
  1. The interpretation that the change from hyperbolic to elliptic character directly signals the loss of EFT validity (abstract and the discussion after the perturbation analysis) rests on the mathematical property alone. No explicit estimate is given of the magnitude of the next-order curvature or derivative operators evaluated on the background at the hyperbolicity boundary, leaving the physical claim as an argument rather than a derived result.
minor comments (2)
  1. The dependence of the minimum mass on the parameters of f(φ) is presented clearly, but a short table or plot summarizing the scaling for the representative families would improve readability.
  2. Notation for the scalar charge and the coupling-function parameters is introduced without a consolidated list; adding a short nomenclature paragraph would help readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment. We address the point below and have revised the text to strengthen the discussion of the EFT interpretation.

read point-by-point responses

  1. Referee: The interpretation that the change from hyperbolic to elliptic character directly signals the loss of EFT validity (abstract and the discussion after the perturbation analysis) rests on the mathematical property alone. No explicit estimate is given of the magnitude of the next-order curvature or derivative operators evaluated on the background at the hyperbolicity boundary, leaving the physical claim as an argument rather than a derived result.

    Authors: We agree that an explicit computation of the size of higher-order operators at the boundary would make the claim more quantitative. However, such an estimate necessarily depends on the details of the unknown UV completion and is therefore not uniquely determined within the EFT framework itself. The loss of hyperbolicity is a standard and robust diagnostic in the EFT literature because it renders the initial-value problem ill-posed, which is unphysical for a classical theory truncated at a given order. We have revised the relevant paragraph in Section IV and the abstract to make this reasoning more explicit, to cite additional references on well-posedness in higher-curvature gravity, and to emphasize that the hyperbolicity criterion serves as a practical indicator of the EFT regime rather than a direct calculation of omitted terms. We believe this addresses the referee's concern while remaining within the scope of the present analysis. revision: yes

Circularity Check

0 steps flagged

No circularity: minimum mass derived from direct hyperbolicity analysis of perturbation equations

full rationale

The paper's central result follows from solving the scalar-Gauss-Bonnet field equations for a chosen class of coupling functions f(φ), obtaining static black hole backgrounds, and then linearizing the perturbation equations around those backgrounds to determine the parameter values at which the equations change character from hyperbolic to elliptic. This character change directly supplies the minimum mass threshold. The subsequent observation that the threshold can be made arbitrarily small while keeping the scalar charge bounded is obtained by varying the parameters of f(φ) within the same framework. No equation or claim reduces to a prior result by algebraic identity, by renaming a fitted quantity as a prediction, or by a load-bearing self-citation whose own justification is internal to the present work. The interpretation that ellipticity signals EFT breakdown is explicitly flagged as an argument ('arguably') rather than a derived equality, leaving the mathematical finding independent of that interpretation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5421 in / 1158 out tokens · 66718 ms · 2026-05-10T17:22:08.031573+00:00 · methodology

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Reference graph

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