Stable colored black holes with quartic self-interactions

Gabriel Gomez; Jose F. Rodriguez-Ruiz

arxiv: 2605.16005 · v2 · pith:DD22STZRnew · submitted 2026-05-15 · 🌀 gr-qc · hep-th

Stable colored black holes with quartic self-interactions

Jose F. Rodriguez-Ruiz , Gabriel Gomez This is my paper

Pith reviewed 2026-05-22 10:16 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords black holesstabilitynon-AbelianProca fieldhairy black holesgeneral relativitylinear perturbationsWu-Yang ansatz

0 comments

The pith

Non-Abelian Proca black holes are linearly stable in one branch of the exact solutions.

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

The paper shows that black holes sourced by a non-Abelian Proca field with quartic self-interactions can be linearly stable. The solutions are built from the Wu-Yang magnetic monopole ansatz and fall into two branches distinguished by the value of a single coupling parameter. In the odd perturbation sector stability holds for both branches. In the even sector the effective potential is positive on branch I, which remains stable for all regular solutions free of naked singularities, while branch II develops negative regions and is expected to be unstable. A reader would care because most known hairy black holes in four-dimensional gravity are unstable, so concrete examples of stable ones constrain how matter fields can persist outside event horizons.

Core claim

Using the Wu-Yang magnetic monopole ansatz we construct exact asymptotically flat black-hole solutions carrying non-Abelian magnetic charge controlled by the coupling χ. The solutions admit two branches. Linear radial perturbations decouple into odd and even sectors. The odd sector is always stable. In the even sector the effective potential is positive for branch I and negative for branch II, establishing linear stability for branch I throughout the physical domain of χ where the solutions are regular and free of naked singularities. This proves the existence of the first linearly stable asymptotically flat hairy black holes in four dimensions with a minimally coupled non-Abelian Proca self

What carries the argument

The Wu-Yang magnetic monopole ansatz, which produces an exact black-hole background whose linear radial perturbations reduce to a Schrödinger-like equation whose effective potential sign fixes stability in each parity sector.

If this is right

The odd sector remains stable for every regular solution in both branches.
Branch I is linearly stable for the full range of χ that yields regular black holes without naked singularities.
Branch II is expected to be unstable because its effective potential is negative in places and it connects to the known unstable Einstein-Yang-Mills Reissner-Nordström solution.
These solutions supply explicit examples of asymptotically flat hairy black holes that survive linear radial perturbations.

Where Pith is reading between the lines

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

The same ansatz and stability analysis could be applied to related quartic or higher-order self-interactions to search for additional stable branches.
Nonlinear time evolution of the stable branch I solutions would test whether the linear stability survives finite perturbations.
These configurations may serve as benchmarks for numerical studies of black-hole uniqueness theorems when non-Abelian fields are present.

Load-bearing premise

The Wu-Yang ansatz captures the relevant solutions and linear stability is completely determined by the sign of the effective potential obtained from radial perturbations.

What would settle it

An explicit calculation of the even-sector effective potential for branch I that finds a negative region at some allowed value of χ would show unstable modes and falsify the stability result.

Figures

Figures reproduced from arXiv: 2605.16005 by Gabriel Gomez, Jose F. Rodriguez-Ruiz.

read the original abstract

We analytically prove the linear radial stability of non-Abelian black holes with quartic self-interactions. The background, constructed from the Wu--Yang magnetic monopole ansatz, is an exact black-hole solution carrying a non-Abelian magnetic charge $Q_{\rm NA}^2$ controlled by a single coupling parameter $\chi$, and admits two distinct branches. The odd sector is always stable, while in the even sector the effective potential is positive for branch~I and negative for branch~II, establishing stability and potential instability, respectively. The potential instability of branch~II is consistent with its connection to the perturbatively unstable Einstein--Yang--Mills Reissner--Nordstr\"{o}m solution. Branch~I remains linearly stable throughout the physical domain of $\chi$ where the solutions are regular and free of naked singularities. Our results prove the existence of the first linearly stable asymptotically flat hairy black holes in four dimensions with a minimally coupled non-Abelian Proca self-interaction.

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.

Desk Editor's Note private letter to a colleague

This paper gives an analytical stability proof for one branch of non-Abelian black holes with a quartic Proca term, which would be the first such stable hairy solutions in flat 4D if the reduction holds.

read the letter

The main point is that these authors construct exact black-hole solutions carrying non-Abelian magnetic charge via the Wu-Yang ansatz plus a quartic self-interaction, then show that branch I stays linearly stable across its regular domain while branch II does not. The odd sector is stable by inspection, and the even sector reduces to a Schrödinger problem whose effective potential stays positive on branch I. That analytical step is what is new here; most prior hairy solutions either lack a full stability analysis or turn out unstable under perturbations. The consistency check with the known unstable Einstein-Yang-Mills-Reissner-Nordström limit for branch II is a nice touch and strengthens the overall picture. The background solutions themselves are straightforward once the ansatz is fixed, and the single free parameter χ controls the hair in a clean way. The work is careful about regularity and the absence of naked singularities, which keeps the claim focused. The soft spot is the perturbation reduction itself. The abstract states that the even-sector potential is positive, but without the explicit form, the tortoise coordinate choice, or the boundary conditions written out, it is hard to rule out missed cross terms from the non-Abelian structure or the quartic interaction. If those terms are absent or cancel cleanly, the conclusion stands; if a negative dip appears for some χ, the stability claim narrows. The stress-test concern about whether the reduction captures all radial degrees of freedom is therefore the right one to check in the full text. This paper is for people working on no-hair theorems, non-Abelian gauge fields on curved backgrounds, or extensions of Einstein-Proca theory. A reader already familiar with the Wu-Yang ansatz and effective-potential methods will get the most out of it. It is worth sending to a serious referee because the claim is concrete, the methods are standard, and the result directly addresses an open question even if the proof needs one more round of verification.

Referee Report

1 major / 0 minor

Summary. The manuscript constructs exact asymptotically flat black-hole solutions in Einstein gravity minimally coupled to a non-Abelian Proca field with quartic self-interaction, employing the Wu-Yang magnetic monopole ansatz. The solutions are parameterized by a single coupling χ that controls the non-Abelian magnetic charge and fall into two branches. Linear radial perturbations are reduced to a Schrödinger-like equation; the odd sector is stable for both branches, while the even sector has a positive effective potential for branch I (implying linear stability) and a negative effective potential for branch II (implying potential instability). The latter is consistent with the known instability of the Einstein-Yang-Mills-Reissner-Nordström limit. The authors conclude that branch I furnishes the first example of linearly stable asymptotically flat hairy black holes in four dimensions with this interaction.

Significance. If the reduction to a Schrödinger equation and the positivity of the effective potential are rigorously established, the result would be significant: it supplies an explicit, analytically tractable counter-example to the expectation that non-Abelian hair necessarily leads to instability in asymptotically flat spacetime, and it may serve as a benchmark for numerical or perturbative studies of hairy black holes in related theories.

major comments (1)

[stability analysis / effective potential paragraph] The central stability claim for branch I rests on the assertion that the effective potential in the even sector is positive throughout the regular domain of χ. The manuscript must supply the explicit linearized perturbation equations, the gauge choice, the definition of the tortoise coordinate, the resulting Schrödinger operator, and the algebraic or numerical verification that this potential has no negative regions for all admissible χ (including the limiting values where the solutions remain regular and free of naked singularities). Without these steps the sign-based stability conclusion cannot be independently checked.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive suggestion to improve the transparency of the stability analysis. We address the major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

Referee: [stability analysis / effective potential paragraph] The central stability claim for branch I rests on the assertion that the effective potential in the even sector is positive throughout the regular domain of χ. The manuscript must supply the explicit linearized perturbation equations, the gauge choice, the definition of the tortoise coordinate, the resulting Schrödinger operator, and the algebraic or numerical verification that this potential has no negative regions for all admissible χ (including the limiting values where the solutions remain regular and free of naked singularities). Without these steps the sign-based stability conclusion cannot be independently checked.

Authors: We appreciate the referee's request for explicit intermediate steps that would allow independent verification of the positivity of the effective potential. The current manuscript derives the Schrödinger-like equation from the linearized even-sector perturbations around the exact background solution and states that the resulting effective potential is positive definite for branch I throughout the regular domain of χ. To address the comment, the revised version will explicitly present: the full linearized perturbation equations obtained by varying the action, the gauge choice (Regge-Wheeler-type gauge with the non-Abelian field perturbations decomposed into even parity), the definition of the tortoise coordinate r* = ∫ dr / f(r) where f(r) is the metric lapse function, the explicit form of the Schrödinger operator, and both the algebraic demonstration that the potential has no negative regions (via factorization or completion of squares for the quartic interaction term) together with numerical plots confirming V_eff(r) > 0 for representative values of χ, including the endpoints of the regular interval. These additions will make the linear stability proof fully self-contained while leaving the physical conclusions unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; stability follows from derived effective potential sign

full rationale

The paper derives the background solution from the Wu-Yang ansatz and reduces linear radial perturbations to a Schrödinger-like equation whose effective potential sign determines stability for each branch. This is a direct analytical step with no reduction of the stability conclusion to a fitted parameter, self-definitional loop, or load-bearing self-citation chain. The result remains independent of the input ansatz once the potential is explicitly constructed and its positivity verified in the physical domain of χ for branch I.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The construction rests on the standard Einstein-Proca action with quartic term, the Wu-Yang ansatz, and the assumption that positivity of the effective potential implies linear stability. χ is the single free coupling that sets the non-Abelian charge.

free parameters (1)

χ
Single coupling parameter that controls the non-Abelian magnetic charge Q_NA^2 on the background solution.

axioms (2)

domain assumption The Wu-Yang magnetic monopole ansatz produces an exact black-hole solution of the Einstein-Proca system with quartic self-interaction.
Invoked to construct the background geometry and charge.
domain assumption Linear radial stability is determined by the sign of the effective potential in the Schrödinger-like perturbation equation.
Standard assumption in black-hole perturbation theory used to conclude stability or instability.

pith-pipeline@v0.9.0 · 5696 in / 1419 out tokens · 47061 ms · 2026-05-22T10:16:09.624708+00:00 · methodology

Review history (2 revisions) →

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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unclear
Relation between the paper passage and the cited Recognition theorem.

Branch I remains linearly stable throughout the physical domain of χ where the solutions are regular

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 8 internal anchors

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work page internal anchor Pith review Pith/arXiv arXiv 2016
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G´ omez and J

G. G´ omez and J. F. Rodr´ ıguez, New non-Abelian Reissner-Nordstr¨ om black hole solutions in the gen- eralized SU(2) Proca theory and some astrophysi- cal implications, Phys. Rev. D108, 024069 (2023), arXiv:2301.05222 [gr-qc]

work page arXiv 2023
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Generalized Proca action for an Abelian vector field

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work page internal anchor Pith review Pith/arXiv arXiv
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Gallego Cadavid, Y

A. Gallego Cadavid, Y. Rodriguez, and L. G. G´ omez, Generalized SU(2) Proca theory reconstructed and beyond, Phys. Rev. D102, 104066 (2020), arXiv:2009.03241 [hep-th]

work page arXiv 2020
[16]

G. W. Horndeski, Second-order scalar-tensor field equa- tions in a four-dimensional space, Int. J. Theor. Phys. 10, 363 (1974)

work page 1974
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Witten, Some Exact Multi - Instanton Solutions of Classical Yang-Mills Theory, Phys

E. Witten, Some Exact Multi - Instanton Solutions of Classical Yang-Mills Theory, Phys. Rev. Lett.38, 121 (1977)

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Gomez and J

G. Gomez and J. F. Rodriguez, Internal symmetry to the rescue: well-posed 1 + 1 evolution of self- interacting vector fields, Eur. Phys. J. C85, 921 (2025), arXiv:2503.09757 [gr-qc]

work page arXiv 2025
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R. M. Wald, Note on the stability of the Schwarzschild metric, J. Math. Phys.20, 1056 (1979)

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Gravitating Solitons and Hairy Black Holes

P. Bizon, Gravitating solitons and hairy black holes, Acta Phys. Polon. B25, 877 (1994), arXiv:gr-qc/9402016

work page internal anchor Pith review Pith/arXiv arXiv 1994
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Bartnik and J

R. Bartnik and J. McKinnon, Particlelike solutions of the Einstein-Yang-Mills equations, Phys. Rev. Lett.61, 141 (1988)

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J. N. Martinez, J. F. Rodriguez, Y. Rodriguez, and G. Gomez, Particle-like solutions in the generalized SU(2) Proca theory, JCAP04, 032, arXiv:2212.13832 [gr-qc]

work page arXiv
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J. N. Martinez, J. F. Rodriguez, L. M. Becerra, Y. Ro- driguez, and G. Gomez, Neutron stars in the generalized SU(2) Proca theory, Phys. Rev. D110, 104070 (2024), arXiv:2408.07674 [gr-qc]

work page arXiv 2024

[1] [1]

Israel, Event horizons in static vacuum space-times, Phys

W. Israel, Event horizons in static vacuum space-times, Phys. Rev.164, 1776 (1967)

work page 1967

[2] [2]

Carter, Axisymmetric Black Hole Has Only Two De- grees of Freedom, Phys

B. Carter, Axisymmetric Black Hole Has Only Two De- grees of Freedom, Phys. Rev. Lett.26, 331 (1971)

work page 1971

[3] [3]

D. C. Robinson, Uniqueness of the Kerr black hole, Phys. Rev. Lett.34, 905 (1975)

work page 1975

[4] [4]

M. S. Volkov and D. V. Gal’tsov, Gravitating nonAbelian solitons and black holes with Yang-Mills fields, Phys. Rept.319, 1 (1999), arXiv:hep-th/9810070

work page internal anchor Pith review Pith/arXiv arXiv 1999

[5] [5]

Bizon, Colored black holes, Phys

P. Bizon, Colored black holes, Phys. Rev. Lett.64, 2844 (1990)

work page 1990

[6] [6]

M. S. Volkov and D. V. Galtsov, NonAbelian Einstein Yang-Mills black holes, JETP Lett.50, 346 (1989)

work page 1989

[7] [7]

K.-M. Lee, V. P. Nair, and E. J. Weinberg, A Classical instability of Reissner-Nordstrom solutions and the fate of magnetically charged black holes, Phys. Rev. Lett.68, 5 1100 (1992), arXiv:hep-th/9111045

work page internal anchor Pith review Pith/arXiv arXiv 1992

[8] [8]

Straumann and Z

N. Straumann and Z. H. Zhou, Instability of a colored black hole solution, Phys. Lett. B243, 33 (1990)

work page 1990

[9] [9]

Existence of stable hairy black holes in su(2) Einstein-Yang-Mills theory with a negative cosmological constant

E. Winstanley, Existence of stable hairy black holes in SU(2) Einstein Yang-Mills theory with a negative cos- mological constant, Class. Quant. Grav.16, 1963 (1999), arXiv:gr-qc/9812064

work page internal anchor Pith review Pith/arXiv arXiv 1963

[10] [10]

Stable black hole solutions with non-Abelian fields

E. Radu and D. H. Tchrakian, Stable black hole solutions with non-Abelian fields, Phys. Rev. D85, 084022 (2012), arXiv:1111.0418 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2012

[11] [11]

B. R. Greene, S. D. Mathur, and C. M. O’Neill, Elud- ing the no hair conjecture: Black holes in spontaneously broken gauge theories, Phys. Rev. D47, 2242 (1993), arXiv:hep-th/9211007

work page internal anchor Pith review Pith/arXiv arXiv 1993

[12] [12]

Solitons and hairy black holes in Einstein-non-Abelian-Proca theory in anti-de Sitter space-time

S. Ponglertsakul and E. Winstanley, Solitons and hairy black holes in Einstein–non-Abelian–Proca theory in anti–de Sitter spacetime, Phys. Rev. D94, 044048 (2016), arXiv:1606.04644 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2016

[13] [13]

G´ omez and J

G. G´ omez and J. F. Rodr´ ıguez, New non-Abelian Reissner-Nordstr¨ om black hole solutions in the gen- eralized SU(2) Proca theory and some astrophysi- cal implications, Phys. Rev. D108, 024069 (2023), arXiv:2301.05222 [gr-qc]

work page arXiv 2023

[14] [14]

Generalized Proca action for an Abelian vector field

E. Allys, P. Peter, and Y. Rodriguez, Generalized Proca action for an Abelian vector field, JCAP02, 004, arXiv:1511.03101 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv

[15] [15]

Gallego Cadavid, Y

A. Gallego Cadavid, Y. Rodriguez, and L. G. G´ omez, Generalized SU(2) Proca theory reconstructed and beyond, Phys. Rev. D102, 104066 (2020), arXiv:2009.03241 [hep-th]

work page arXiv 2020

[16] [16]

G. W. Horndeski, Second-order scalar-tensor field equa- tions in a four-dimensional space, Int. J. Theor. Phys. 10, 363 (1974)

work page 1974

[17] [17]

Witten, Some Exact Multi - Instanton Solutions of Classical Yang-Mills Theory, Phys

E. Witten, Some Exact Multi - Instanton Solutions of Classical Yang-Mills Theory, Phys. Rev. Lett.38, 121 (1977)

work page 1977

[18] [18]

Gomez and J

G. Gomez and J. F. Rodriguez, Internal symmetry to the rescue: well-posed 1 + 1 evolution of self- interacting vector fields, Eur. Phys. J. C85, 921 (2025), arXiv:2503.09757 [gr-qc]

work page arXiv 2025

[19] [19]

R. M. Wald, Note on the stability of the Schwarzschild metric, J. Math. Phys.20, 1056 (1979)

work page 1979

[20] [20]

Gravitating Solitons and Hairy Black Holes

P. Bizon, Gravitating solitons and hairy black holes, Acta Phys. Polon. B25, 877 (1994), arXiv:gr-qc/9402016

work page internal anchor Pith review Pith/arXiv arXiv 1994

[21] [21]

Bartnik and J

R. Bartnik and J. McKinnon, Particlelike solutions of the Einstein-Yang-Mills equations, Phys. Rev. Lett.61, 141 (1988)

work page 1988

[22] [22]

J. N. Martinez, J. F. Rodriguez, Y. Rodriguez, and G. Gomez, Particle-like solutions in the generalized SU(2) Proca theory, JCAP04, 032, arXiv:2212.13832 [gr-qc]

work page arXiv

[23] [23]

J. N. Martinez, J. F. Rodriguez, L. M. Becerra, Y. Ro- driguez, and G. Gomez, Neutron stars in the generalized SU(2) Proca theory, Phys. Rev. D110, 104070 (2024), arXiv:2408.07674 [gr-qc]

work page arXiv 2024