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Colored Black Holes in Logarithmic Nonlinear Yang--Mills Theory

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Pith's one-line read Colored black holes in logarithmic nonlinear Yang-Mills theory are constructed and shown to have no inner horizon.

desk verdict A solid and honest numerical construction of a new one-node colored black-hole family in logarithmic Yang–Mills theory; the exterior is well validated, and the interior claims are plausible, though stronger with shipped code and deeper precision. read the letter →

arxiv 2608.07150 v1 pith:YM7RJOGU submitted 2026-08-07 gr-qc

Celio R. Muniz This is my paper · ORCID
classification gr-qc MSC 83C5781T13 PACS 04.70.-s
keywords coloredblackholesEinstein-Yang-Millstheorylogarithmicnonlineargaugenon-AbelianhairnodalprofilesholeinteriorsHawkingtemperaturemassinflation
topics Yang-Mills Mass Gap
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper constructs a new family of static, spherically symmetric, asymptotically flat colored black holes in four-dimensional Einstein gravity coupled to a logarithmic nonlinear SU(2) Yang-Mills field. Unlike earlier Wu-Yang based solutions, the gauge field keeps a dynamical radial profile $w(r)$, so the Einstein and gauge equations form a genuine coupled boundary-value problem. The paper proves an integral identity forcing any nontrivial magnetically neutral colored solution to have at least one node, and then numerically builds the fundamental one-node branch. The central result is that strong logarithmic nonlinearity changes both the exterior and the interior: it lowers the mass, raises the Hawking temperature, displaces the node outward, and, inside the horizon, removes the inner Cauchy horizon and the oscillatory mass inflation of ordinary Einstein-Yang-Mills black holes while leaving a Schwarzschild-type curvature singularity.

What carries the argument

The load-bearing object is the logarithmic gauge Lagrangian $L(X)=-\beta^2\ln(1+X/\beta^2)$ with positive function $P=1/(1+X/\beta^2)$; it reduces to ordinary Yang-Mills as $\beta\to\infty$ and sets the nonlinear scale $b$. Substituting the magnetic SU(2) ansatz yields the coupled radial system for mass, redshift, and gauge amplitude, and an integration-by-parts identity using $P>0$ rules out sign-definite nodeless solutions, forcing nodes. The interior analysis is carried by a flux-variable reformulation, $H=N P w'$, together with the algebraic root $\Xi=2B/(1+\sqrt{1-4AB})$, which avoids catastrophic cancellation and allows integration down to $r/r_h=10^{-8}$, leading to the near-center asymptotic forms used to conclude no Cauchy horizon and finite limiting mass.

What would settle it

An independent high-precision integration of the original second-order gauge equation, or a spectral method, that reaches $r/r_h<10^{-8}$ and either finds a second zero of $N(r)$ or sees $m(r)$ oscillate or diverge before the center would refute the central interior claim. Similarly, a reliable solver that locates a lower critical $b$ below which no one-node solution exists would refute the claim of branch continuation.

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Extended reading notes

Core claim

The central claim is that the one-node colored black-hole branch of the logarithmic model exists for all values of the nonlinearity parameter explored (down to $b=0.002$), converges smoothly to the ordinary Einstein-Yang-Mills colored black hole as $b\to\infty$, and has an interior that differs qualitatively from the linear theory. Representative solutions have a single event horizon with no inner Cauchy horizon; the mass function tends to a finite positive limit $m_0$ near $r=0$, while $1+X/b^2 \sim K r^{-3/2}$, so the curvature invariant diverges as $48 m_0^2/r^6$ and the center is a Schwarzschild-type spacelike singularity. The paper presents this as a nonuniform recovery of the linear theory: for every finite $b$ the deep interior enters the strongly logarithmic regime, and the $b\to\infty$ and $r\to0$ limits do not commute. The thermodynamic analysis also identifies temperature turning points that produce divergences and sign changes of the heat capacity, interpreted as local stability transitions.

Load-bearing premise

The interior conclusions rest on numerical integration only down to $r/r_h=10^{-8}$ together with an assumed near-center asymptotic form; if the chosen root of the algebraic equation is not the physical continuation all the way to $r=0$, or the matched asymptotics fail, the claims of no Cauchy horizon and finite limiting mass could be wrong.

Editorial extensions

If this is right

  • If the fundamental branch exists for all explored $b$ down to at least $b=0.002$, then the logarithmic model contains a continuum of colored black holes parametrized by $b$ and $r_h$, with ordinary EYM as the $b\to\infty$ limit.
  • The absence of an inner Cauchy horizon for finite $b$ implies that the interior structure of these solutions is qualitatively simpler than in the linear EYM case, with the singularity behaving like Schwarzschild rather than exhibiting oscillatory mass inflation.
  • Since the outer geometry is Schwarzschild to leading order with hair entering at $r^{-4}$, any probe of the asymptotic tail would see the colored hair only as a subleading correction.
  • The temperature turning points and heat-capacity divergences define alternating locally stable and unstable branches, so the thermodynamic phase structure of colored black holes persists and is modified by the nonlinear scale.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An implication beyond the paper: since the nodal identity uses only $P=-L_X>0$, the same argument should force nodes for any monotonic nonlinear Yang-Mills deformation, so a power-law or exponential gauge Lagrangian is a cheap test case.
  • Because the small-$b$ behavior pushes the node outward while the exterior geometry approaches Schwarzschild, a matched-asymptotic treatment might turn the apparent decoupling limit into a rigorous statement about hair being expelled to infinity.
  • The noncommuting $b\to\infty$ and $r\to0$ limits warn that any effective-field-theory truncation of the Lagrangian would miss the deep interior behavior, so interior predictions from truncated actions should be checked against the full logarithmic model.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper constructs static, spherically symmetric, asymptotically flat black-hole solutions of Einstein gravity minimally coupled to a logarithmic nonlinear SU(2) Yang–Mills field. It reduces the field equations to ODEs, derives horizon and asymptotic expansions, proves a nodal restriction, solves the boundary-value problem by shooting for the fundamental one-node branch, and studies parameter dependence, interior continuation, and thermodynamics. The main claimed results are a one-node colored branch that reduces to ordinary EYM as b tends to infinity, interiors with no Cauchy horizon, finite limiting mass, and no mass inflation, and temperature turning points with sign changes of the heat capacity.

Significance. If the interior claims hold, the paper makes a useful contribution: it constructs colored black holes with a genuine dynamical radial gauge amplitude in a logarithmic nonlinear Yang–Mills theory, exhibiting a nodal structure and a nonuniform b→∞ and r→0 limit. The exterior construction is well supported: the shooting method is validated against a direct EYM integration and a published horizon value, and convergence checks with respect to outer boundary and integration tolerances are reported. The nodal integral identity is simple and robust. However, the advertised interior conclusions are less firmly established than the exterior ones, and a horizon-derivative formula used for varying horizon radii appears to be inconsistent as printed. The result is therefore significant and publishable after revision, but the interior evidence and several equations need attention.

major comments (3)
  1. [§4.1, Eq. (38)] Equation (38) is inconsistent with the definition N=1−2m/r. Differentiating gives N1=(1−2m1)/r_h, not 1−2m1/r_h; Eq. (41) effectively uses the corrected form. For r_h=1 the two expressions coincide, but the paper uses r_h≠1 in Fig. 3 and Table 3. Please correct Eq. (38) and verify that the numerical families with varying r_h were initialized with the correct horizon derivative; otherwise the varying-r_h results are suspect.
  2. [§8, Eqs. (123)–(129)] The conclusions that the representative interiors have no Cauchy horizon, a finite limiting mass m0, and no mass inflation rest on integrating the flux-variable system only down to r/r_h=10^-8 and matching to the assumed near-center asymptotics, with agreement at the 3×10^-4 level. This does not exclude further zeros of N below the cutoff or a different exponent than the assumed r^{-3/2}. Please provide an independent check, such as inward/outward matched shooting, rigorous asymptotic error estimates, or a substantially deeper high-precision integration, or alternatively restrict the abstract and conclusions to the numerically resolved range.
  3. [§8, Eq. (117)] As printed, Eq. (117) reads S'/S = 2 w'^2 Ξ/r, but combining Eqs. (24) and (112) gives S'/S = 2P w'^2/r = 2 w'^2/(Ξ r), or equivalently 2H^2Ξ/(N^2 r). The subsequent asymptotic coefficient 4γ^2/K matches the corrected form, so this is likely a typographical error, but it must be fixed because the interior integration is central to the paper's main claims.
minor comments (5)
  1. [Abstract] There are several typos in the abstract, including 'oscillary mass inflation' and 'near the cetonter'; these should be corrected.
  2. [§4.1, Eqs. (37)–(41)] The nonextremality inequality in Eq. (41) follows from the corrected expression N1=(1−2m1)/r_h, not from Eq. (38) as printed; the derivation should be made explicit.
  3. [§6] The symbol R is used for the outer boundary while the curvature scalar also appears later; this is not confusing in context, but a distinct symbol for the outer boundary would improve readability.
  4. [§7 and Table 1] The claim that no lower critical value of b exists is only tested down to b=0.002 and the text already notes the reduced precision for b<0.1; the abstract's 'branch termination is not detected over the full range' should retain this qualification explicitly.
  5. [§8] The paper does not mention any code or data release. Given that the interior claims depend on a nonstandard flux-variable integration, making the code or representative data available would substantially strengthen reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning found: the colored black holes are constructed by an outward shooting method with boundary conditions at the horizon and infinity, and the EYM limit is validated against an external published benchmark.

full rationale

The paper's central construction is not circular. The shooting parameter w_h is fixed by the exterior boundary condition w(infinity) = -1, not by any target value of the claimed outputs (ADM mass, Hawking temperature, node position, heat capacity). The parameter b is a model scale, not a fitted constant, and the b -> infinity comparison uses the published EYM horizon value w_h ~ 0.6322 from reference [27] as an external benchmark; the agreement with the b = 100 solution is a validation, not an input. The integral identity restricting nodeless solutions is proved directly from the field equations. The thermodynamic quantities are evaluated from the constructed solution families rather than imposed. The only potentially fragile step is the interior continuation: the no-Cauchy-horizon, finite-limiting-mass, and no-mass-inflation claims rely on integrating down to r/r_h = 10^-8 and matching the assumed near-center asymptotic forms (123)-(127). That is a numerical and physical robustness concern, not a circularity: the asymptotic form is not used to define the solution or to select the shooting parameter, and the paper explicitly declines to assert global regularity at the central singularity. Since every circular pattern requires a specific reduction of a claim to its own input or to a self-citation, and none is present, the appropriate score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new particles, forces, dimensions, or conserved quantities are introduced. The logarithmic Yang-Mills Lagrangian is taken from prior literature (Soleng [24] and related work). The listed free parameters are model scales and boundary labels, not hidden fits to an external target.

free parameters (3)
  • b = Scanned from 0.002 to 100; representative values 0.25, 0.5, 1, 2, 100
    Dimensionless logarithmic nonlinearity scale. It is a model parameter scanned by hand, not fitted to external data, but the central claims are statements about its variation.
  • r_h = 1 for representative solutions; 0.85 to 3 for thermodynamic families
    Dimensionless horizon radius used as a boundary condition and family label.
  • w_h = 0.421258 (b=0.1) to 0.632206 (b=100) at r_h=1
    Horizon gauge amplitude; the shooting parameter adjusted to enforce w(infinity)=-1. It is determined by the boundary-value problem, not by fitting to a target observable.
assumptions (4)
  • domain assumption The logarithmic Lagrangian L(X) = -beta^2 ln(1+X/beta^2) with P = -L_X > 0 is the matter model.
    Defines the theory; positivity of P is used in Eq. (22) and in Proposition 1. The reality condition 1+X/beta^2>0 is assumed.
  • domain assumption The static, spherically symmetric, purely magnetic SU(2) ansatz (14) captures all relevant colored configurations.
    The paper only analyzes this ansatz; no argument excludes other spherically symmetric gauge configurations.
  • standard math Standard ODE existence and uniqueness apply to the reduced system with regular horizon data.
    Used in Sec. 5 to show nodes are simple and in the shooting construction.
  • domain assumption The numerical integrations with the stated tolerances and finite outer boundaries faithfully approximate the continuum solutions.
    Convergence checks are reported, but no machine-checked proof or shipped code is provided.

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Cite this review

Pith. "Pith review of Colored Black Holes in Logarithmic Nonlinear Yang--Mills Theory." pith.science (2026). https://pith.science/paper/YM7RJOGU

@misc{pith2026260807150,
  author       = {Pith},
  title        = {Pith review of: Colored Black Holes in Logarithmic Nonlinear Yang--Mills Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YM7RJOGU}},
  note         = {Machine review of arXiv:2608.07150}
}
read the original abstract

We construct static, spherically symmetric, and asymptotically flat colored black holes in four-dimensional Einstein gravity coupled to a logarithmic nonlinear Yang--Mills field with gauge group SU(2). Unlike solutions based on the Wu--Yang ansatz, the gauge sector contains a dynamical radial amplitude, and the configurations arise from a genuinely coupled nonlinear boundary-value problem. An integral identity excludes nontrivial sign-definite solutions approaching a magnetically neutral Yang--Mills vacuum, implying that colored configurations must be nodal. We numerically construct the fundamental one-node branch and verify its convergence to the ordinary Einstein--Yang--Mills colored black hole in the linear limit. Increasing the logarithmic nonlinearity lowers the ADM mass, raises the Hawking temperature, and displaces the gauge-field node toward larger radii. Branch termination is not detected over the full range of parameters investigated. Instead, the exterior geometry approaches Schwarzschild on fixed radial domains, while the colored structure develops an increasingly extended tail. Although no independent asymptotic Yang--Mills charge is present, the non-Abelian hair leaves a subleading imprint on the far-field geometry. Inside the event horizon, the representative nonlinear solutions possess no Cauchy horizon and approach a Schwarzschild-type spacelike curvature singularity, with a finite limiting mass and no oscillary mass inflation. The recovery of the linear theory is therefore nonuniform near the cetonter: solutions that are nearly linear in the exterior eventually enter the strongly logarithmic regime in the deep interior. Finally, turning points of the Hawking temperature produce divergences and sign changes in heat capacity, indicating transitions of local thermodynamic stability.

Figures

Figures reproduced from arXiv: 2608.07150 by the authors.

Figure 1
Figure 1. Radial gauge-field profiles w(r) for the fundamental one-node logarithmic Yang–Mills colored black holes with rh = 1. The inset enlarges the nodal region. Decreasing b strengthens the logarithmic nonlinear effects, lowers wh, and shifts the node away from the event horizon. The b = 100 curve represents the near-EYM regime. All profiles approach w = −1 asymptotically. 2 4 6 8 10 r/rh 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0… view at source ↗
Figure 2
Figure 2. Metric functions N(r) (left panel) and S(r) (right panel) for the fundamental one-node colored black holes with rh = 1. The redshift function is normalized by S(∞) = 1. Strong logarithmic nonlinearities, corresponding to smaller b, reduce the asymptotic mass and weaken the variation of S(r) outside the horizon. The profiles converge rapidly toward the near-EYM behavior as b increases. 14 [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 3
Figure 3. Dimensionless ADM mass M, Hawking temperature ℓEYMTH, and normalized node position rnode/rh as functions of the horizon radius for the fundamental one-node colored branch. Each curve corresponds to a fixed value of b. Nonlinear effects are most pronounced for small horizons and small b. For weaker logarithmic deformation, the temperature develops turning points, while the normalized node position exhibits a b-depend… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Interior behavior of the fundamental one-node colored black-hole solutions with [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: Thermodynamic behavior of the one-node ( [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

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