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arxiv: 2601.17115 · v2 · submitted 2026-01-23 · 🌀 gr-qc · hep-th

Effective geometrodynamics for renormalization-group improved black-hole spacetimes in spherical symmetry

Johanna Borissova , Ra\'ul Carballo-Rubio This is my paper

Pith reviewed 2026-05-16 11:26 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords renormalization group improvementblack hole spacetimesHorndeski theoryspherical symmetryeffective gravitational couplingMisner-Sharp massgeometrodynamics
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The pith

RG-improved black-hole spacetimes with scale-dependent coupling are exact vacuum solutions to two-dimensional Horndeski theories.

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

The paper develops a systematic operational approach to renormalization-group improvement of the spherically reduced Einstein-Hilbert action and its associated field equations, using a scale-dependent Newton coupling. It employs master field equations constructed from two-dimensional Horndeski theory to retain selected higher-curvature contributions from the effective action while preserving the second-order nature of the dynamics. This procedure produces static black-hole solutions in which the effective gravitational coupling varies with areal radius and Misner-Sharp mass. The same framework embeds earlier RG-improvement constructions and demonstrates that the resulting spacetimes differ according to whether the improvement is imposed at the level of the action, the field equations, or the Schwarzschild metric itself.

Core claim

Static RG-improved black-hole spacetimes with an effective gravitational coupling depending on the areal radius and the Misner-Sharp mass are derived as vacuum solutions to these master field equations, and are thereby identified as solutions to generally covariant two-dimensional Horndeski theories.

What carries the argument

Master field equations for spherically symmetric gravitational fields constructed from two-dimensional Horndeski theory, which retain partial higher-curvature contributions from the effective action while remaining second-order.

If this is right

  • Earlier RG-improved black-hole models can be recovered as particular cases inside the same covariant formalism.
  • Distinct physical metrics arise when the RG improvement is applied at the action, at the field equations, or directly to the Schwarzschild solution.
  • The resulting spacetimes satisfy generally covariant two-dimensional Horndeski theories exactly.
  • Selected higher-curvature terms are kept in the dynamics without introducing higher-order derivatives.

Where Pith is reading between the lines

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

  • The same master-equation technique could be applied to time-dependent or axisymmetric configurations to obtain consistent RG-improved metrics beyond static spherical symmetry.
  • The dependence of the effective coupling on both radius and Misner-Sharp mass suggests a possible route to incorporating mass-dependent quantum corrections into horizon thermodynamics.

Load-bearing premise

The master field equations constructed from two-dimensional Horndeski theory allow retention of partial contributions from higher-curvature truncations of the effective action while preserving the second-order nature of the resulting field equations.

What would settle it

A direct substitution of one of the derived RG-improved metrics into the master field equations that yields a nonzero residual would show the solution does not satisfy the Horndeski dynamics.

Figures

Figures reproduced from arXiv: 2601.17115 by Johanna Borissova, Ra\'ul Carballo-Rubio.

Figure 1
Figure 1. Figure 1: FIG. 1: Effective average action Γ [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Schematic representation of interrelations between RG-improvement at the level of solutions, field [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Gravitational collapse into a regular black-hole spacetime with time-time component of the metric [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
read the original abstract

We consider the spherically reduced Einstein-Hilbert action, Einstein field equations and Schwarzschild spacetime modified by a renormalization-group (RG) scale-dependent gravitational Newton coupling, and present a systematic and operational approach to such an RG-improvement. The master field equations for spherically symmetric gravitational fields, recently constructed from two-dimensional Horndeski theory, allow us to retain partial contributions from higher-curvature truncations of the effective action, while preserving the second-order nature of the resulting field equations. Static RG-improved black-hole spacetimes with an effective gravitational coupling depending on the areal radius and the Misner-Sharp mass are derived as vacuum solutions to these master field equations, and are thereby identified as solutions to generally covariant two-dimensional Horndeski theories. We discuss explicitly the embedding of previous key works on RG-improvement into the newly developed formalism to illustrate its broad range of applicability. This formalism moreover allows us to establish explicitly the discrepancies in the outcomes of RG-improvement when implemented at the level of the action, in the field equations, or in the Schwarzschild solution.

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

0 major / 2 minor

Summary. The paper develops a systematic formalism for renormalization-group improvement of spherically symmetric black-hole spacetimes. It employs master field equations derived from two-dimensional Horndeski theory to retain selected higher-curvature contributions while preserving second-order equations. Static RG-improved solutions are constructed with an effective Newton coupling G depending on areal radius and Misner-Sharp mass; these are shown to solve the master equations and are thereby identified as solutions of generally covariant 2D Horndeski theories. The formalism is used to embed prior RG-improvement studies and to compare outcomes when the improvement is performed at the level of the action, the field equations, or the Schwarzschild solution.

Significance. If the derivations hold, the work supplies a covariant effective-theory framework that consistently incorporates scale-dependent gravitational couplings into black-hole geometrodynamics. It clarifies ambiguities among different RG-improvement prescriptions and provides an explicit bridge between asymptotic-safety-inspired metrics and second-order Horndeski dynamics, which may facilitate further analytic and numerical studies of quantum-corrected black holes.

minor comments (2)
  1. Abstract: the central claim that the RG-improved metrics solve the master equations is stated without a one-sentence indication of the explicit substitution or verification performed in the body; adding this would improve readability.
  2. Section 3 (or equivalent): the dependence of G on both r and M is introduced; a short remark on whether this dependence is uniquely fixed by the RG flow or remains a choice within the formalism would help readers assess the predictive power.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were provided in the report, so we have no specific points requiring rebuttal or clarification at this stage. We will incorporate any minor suggestions during the revision process.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper constructs master field equations from two-dimensional Horndeski theory that preserve second-order structure while allowing selected higher-curvature terms. It then derives the RG-improved spacetimes (with effective G depending on areal radius and Misner-Sharp mass) as explicit vacuum solutions to those equations, thereby identifying them as Horndeski solutions. This is a direct solving step rather than a reduction of outputs to inputs by definition or fit. No load-bearing self-citation chain, ansatz smuggling, or renaming of known results is evident; the formalism is used to embed prior RG-improvement works without forcing the central claim tautologically. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Ledger extracted from abstract; full paper likely specifies the RG flow function and truncation details.

free parameters (1)
  • scale-dependent Newton coupling G(r, M)
    Effective gravitational coupling taken as function of areal radius and Misner-Sharp mass; specific functional form chosen from RG considerations.
axioms (2)
  • domain assumption Spherically reduced Einstein-Hilbert action with RG-improved Newton coupling
    Starting point for the modification in spherical symmetry.
  • standard math Master field equations from two-dimensional Horndeski theory retain partial higher-curvature contributions while remaining second-order
    Invoked to construct the effective geometrodynamics.

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discussion (0)

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Forward citations

Cited by 5 Pith papers

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

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    hep-th 2026-03 conditional novelty 7.0

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