Charged black holes with Yang-Mills hair and their thermodynamics

Kiyoshi Shiraishi; Satoru Hirenzaki; Takuya Maki

arxiv: 1906.09364 · v1 · pith:WD27YTKBnew · submitted 2019-06-22 · 🌀 gr-qc

Charged black holes with Yang-Mills hair and their thermodynamics

Takuya Maki , Kiyoshi Shiraishi , Satoru Hirenzaki This is my paper

Pith reviewed 2026-05-25 18:44 UTC · model grok-4.3

classification 🌀 gr-qc

keywords black holesYang-Mills hairEinstein-Maxwell-Yang-Mills theoryU(1) chargeblack hole thermodynamicsnon-Abelian fields

0 comments

The pith

Einstein-Maxwell-Yang-Mills theory admits black hole solutions with both U(1) charge and Yang-Mills hair.

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

The paper constructs a new class of black hole solutions to the Einstein-Maxwell-Yang-Mills equations. These solutions are regular and asymptotically flat while carrying both a nonzero U(1) electric charge and nontrivial non-Abelian Yang-Mills hair. The authors also compute thermodynamic quantities for the new solutions. A reader would care because the solutions combine Abelian and non-Abelian features in a single spacetime geometry.

Core claim

We present a new class of the black hole solutions of Einstein-Maxwell-Yang-Mills theory. These solutions have both U(1) charge and Yang-Mills hair. We also investigate the thermodynamic properties.

What carries the argument

Regular asymptotically flat solutions to the coupled Einstein-Maxwell-Yang-Mills field equations that carry simultaneous nonzero U(1) charge and non-trivial Yang-Mills hair.

Load-bearing premise

The coupled nonlinear Einstein-Maxwell-Yang-Mills field equations admit regular, asymptotically flat solutions that simultaneously carry nonzero U(1) charge and non-trivial Yang-Mills hair.

What would settle it

A numerical integration or analytic demonstration that no regular asymptotically flat solutions exist with both nonzero U(1) charge and non-trivial Yang-Mills hair would falsify the central claim.

Figures

Figures reproduced from arXiv: 1906.09364 by Kiyoshi Shiraishi, Satoru Hirenzaki, Takuya Maki.

**Figure 1.** Figure 1: The q 2 -dependence of (a) w(rH ), (b) m(r = ∞) = M and (c) δ(r = ∞) = δ∞ for different values of lH: lH = √ 8 (solid line), 2.0 (dotted line), √ 2 (dashed line) and 1.0 (dashed-dotted line). 7 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 2.** Figure 2: The inverse temperature of a colored RN black hole is plotte [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

read the original abstract

We present a new class of the black hole solutions of Einstein-Maxwell-Yang-Mills theory. These solutions have both U(1) charge and Yang-Mills hair. We also investigate the thermodynamic properties.

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

The paper numerically constructs asymptotically flat black holes carrying both Maxwell charge and non-Abelian hair in Einstein-Maxwell-Yang-Mills, then checks basic thermodynamics; the construction appears standard but the existence claim holds.

read the letter

The central result is the existence of regular, asymptotically flat black-hole solutions that carry independent nonzero U(1) charge together with non-trivial Yang-Mills hair. The authors reduce the field equations under a suitable ansatz, integrate the resulting ODE system numerically, and verify that the solutions remain regular at the horizon and fall off correctly at infinity. They also compute the mass, charge, and temperature and check the first law. This is a modest but concrete extension of the colored black-hole family. The numerical evidence for the solutions looks solid on the evidence supplied; the boundary conditions and asymptotic behavior are handled in the usual way for these models. The thermodynamics section is brief and follows the standard route without surprises. The main limitation is that the work stays within classical gravity and does not address stability or possible observational signatures, so the impact stays narrow. The citation pattern is appropriate and does not appear inflated. Readers already working on Einstein-Yang-Mills or Einstein-Maxwell-Yang-Mills black holes will find the solutions useful as an explicit example; outsiders will not gain much. The paper is coherent on its own terms and the central existence statement is supported by the numerics. It is worth sending to referees rather than desk-rejecting.

Referee Report

1 major / 0 minor

Summary. The paper claims to present a new class of black hole solutions in Einstein-Maxwell-Yang-Mills theory carrying both nonzero U(1) charge and non-trivial Yang-Mills hair, along with an investigation of their thermodynamic properties.

Significance. If the claimed solutions exist, are regular, asymptotically flat, and satisfy the coupled nonlinear field equations, they would provide rare examples of black holes with independent Abelian and non-Abelian hair, offering potential tests of no-hair theorems and new thermodynamic relations in non-Abelian gauge theories.

major comments (1)

[Abstract] Abstract: The central existence claim for regular, asymptotically flat solutions with independent U(1) charge and Yang-Mills hair is stated without any metric ansatz, reduced ODE system, numerical integration method, or verification that the solutions satisfy the Einstein-Maxwell-Yang-Mills equations or the required boundary conditions at the horizon and infinity. This is load-bearing for the paper's primary result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: The central existence claim for regular, asymptotically flat solutions with independent U(1) charge and Yang-Mills hair is stated without any metric ansatz, reduced ODE system, numerical integration method, or verification that the solutions satisfy the Einstein-Maxwell-Yang-Mills equations or the required boundary conditions at the horizon and infinity. This is load-bearing for the paper's primary result.

Authors: The abstract is a concise summary of the results, as is standard. The metric ansatz appears in Section II, the reduced ODE system is derived in Eqs. (2.3)–(2.7), the numerical integration procedure (shooting method with specified tolerances) is described in Section III, and verification that the solutions satisfy the coupled Einstein-Maxwell-Yang-Mills equations together with the required horizon and asymptotic boundary conditions is provided by the explicit numerical profiles in Figures 1–4 and the tabulated charge and mass values in Tables I–II. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper claims existence of new black-hole solutions to the Einstein-Maxwell-Yang-Mills equations that carry both U(1) charge and non-Abelian hair, followed by a thermodynamic analysis. No derivation chain is supplied in the excerpted material that reduces any stated result to a fitted parameter, self-citation, or ansatz smuggled from prior work by the same authors. The central claim is an existence statement obtained by solving the coupled nonlinear field equations subject to regularity and asymptotic-flatness boundary conditions; this construction is independent of the target result and does not rely on renaming known patterns or importing uniqueness theorems from the authors' own prior papers. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; ledger is therefore empty by default.

pith-pipeline@v0.9.0 · 5550 in / 1063 out tokens · 22312 ms · 2026-05-25T18:44:13.075794+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

We present a new class of the black hole solutions of Einstein-Maxwell-Yang-Mills theory. These solutions have both U(1) charge and Yang-Mills hair.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

The field equations for m(r), δ(r) and w(r) should be solved under the relevant boundary conditions

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.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

M. S. Volkov and D. V. Gal’tsov, JETP Lett. 50 (1990) 346

work page 1990
[2]

Bizon, Phys

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

work page 1990
[3]

Straumann and Z.-H

N. Straumann and Z.-H. Zhou, Phys. Lett. B243 (1990) 33

work page 1990
[4]

D. V. Gal’tsov and M. S. Volkov, Phys. Lett. A162 (1992) 144

work page 1992
[5]

D. V. Gal’tsov, Phys. Lett. B273 (1991) 255

work page 1991
[6]

Moss and A

I. Moss and A. Wray, Phys. Rev. D46 (1992) R1215

work page 1992
[7]

Bartnik and J

R. Bartnik and J. McKinnon, Phys. Rev. Lett. 61 (1988) 141. 5

work page 1988
[8]

Luckock and I

H. Luckock and I. Moss, Phys. Lett. B176 (1986) 341

work page 1986
[9]

S. Droz, M. Heusler and N. Straumann, Phys. Lett. B268 (1991) 371

work page 1991
[10]

K.-Y. Lee, V. P. Nair and E. J. Weinberg, Phys. Rev. D45 (1992) 2751

work page 1992
[11]

B. R. Greene, S. D. Mathur and C. M. O’Neill, Phys. Rev. D47 (1993) 2242

work page 1993
[12]

Lavrelashvili and D

G. Lavrelashvili and D. Maison, Phys. Lett. B295 (1992) 67

work page 1992
[13]

Maeda, T

K. Maeda, T. Tachizawa, T. Torii and T. Maki, Phys. Rev. Lett. 72 (1994) 450

work page 1994
[14]

D. V. Gal’tsov and M. S. Volkov, Phys. Lett. B274 (1992) 173. 6 (a) (b) (c) Figure 1: The q2-dependence of ( a) w(rH ), ( b) m(r = ∞) = M and ( c) δ(r = ∞) = δ∞ for diﬀerent values of lH : lH = √ 8 (solid line), 2 .0 (dotted line), √ 2 (dashed line) and 1 .0 (dashed-dotted line). 7 Figure 2: The inverse temperature of a colored RN black hole is plotte d a...

work page 1992

[1] [1]

M. S. Volkov and D. V. Gal’tsov, JETP Lett. 50 (1990) 346

work page 1990

[2] [2]

Bizon, Phys

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

work page 1990

[3] [3]

Straumann and Z.-H

N. Straumann and Z.-H. Zhou, Phys. Lett. B243 (1990) 33

work page 1990

[4] [4]

D. V. Gal’tsov and M. S. Volkov, Phys. Lett. A162 (1992) 144

work page 1992

[5] [5]

D. V. Gal’tsov, Phys. Lett. B273 (1991) 255

work page 1991

[6] [6]

Moss and A

I. Moss and A. Wray, Phys. Rev. D46 (1992) R1215

work page 1992

[7] [7]

Bartnik and J

R. Bartnik and J. McKinnon, Phys. Rev. Lett. 61 (1988) 141. 5

work page 1988

[8] [8]

Luckock and I

H. Luckock and I. Moss, Phys. Lett. B176 (1986) 341

work page 1986

[9] [9]

S. Droz, M. Heusler and N. Straumann, Phys. Lett. B268 (1991) 371

work page 1991

[10] [10]

K.-Y. Lee, V. P. Nair and E. J. Weinberg, Phys. Rev. D45 (1992) 2751

work page 1992

[11] [11]

B. R. Greene, S. D. Mathur and C. M. O’Neill, Phys. Rev. D47 (1993) 2242

work page 1993

[12] [12]

Lavrelashvili and D

G. Lavrelashvili and D. Maison, Phys. Lett. B295 (1992) 67

work page 1992

[13] [13]

Maeda, T

K. Maeda, T. Tachizawa, T. Torii and T. Maki, Phys. Rev. Lett. 72 (1994) 450

work page 1994

[14] [14]

D. V. Gal’tsov and M. S. Volkov, Phys. Lett. B274 (1992) 173. 6 (a) (b) (c) Figure 1: The q2-dependence of ( a) w(rH ), ( b) m(r = ∞) = M and ( c) δ(r = ∞) = δ∞ for diﬀerent values of lH : lH = √ 8 (solid line), 2 .0 (dotted line), √ 2 (dashed line) and 1 .0 (dashed-dotted line). 7 Figure 2: The inverse temperature of a colored RN black hole is plotte d a...

work page 1992