Unlike earlier exact black-hole solutions based on the Wu–Yang magnetic ansatz, the new configurations contain a genuinely dynamical radial gauge amplitude, a function w(r) that must be solved for simultaneously with the metric. Both the derivative of this function and the combination 1 − w² contribute to the gauge-field invariant, so the Einstein and nonlinear Yang–Mills equations form a fully coupled boundary-value problem rather than a system solvable by quadrature. The resulting solutions are not analytic continuations of the Wu–Yang black holes: they approach a pure-gauge vacuum at spatial infinity, carry colored hair, and possess no independently specifiable magnetic charge.

The logarithmic model is defined by a Lagrangian of the form −β² ln(1 + X/β²), where X is the Yang–Mills invariant and β sets the nonlinear gauge-field scale. When the field is weak compared with β, the Lagrangian reduces to ordinary Yang–Mills theory, and the standard Einstein–Yang–Mills (EYM) equations are recovered as β tends to infinity. Unlike ordinary Yang–Mills theory, the logarithmic model has a nonvanishing stress-energy trace, which explicitly breaks the classical scale invariance of the gauge sector; the trace vanishes continuously as the linear limit is approached.

Before any numerical work, the authors derived an integral identity with far-reaching consequences. Multiplying the gauge equation by suitable factors and integrating from the horizon to infinity shows that a nontrivial, magnetically neutral solution cannot remain sign-definite throughout the black-hole exterior while approaching a Yang–Mills vacuum. Every nontrivial colored black hole must therefore possess at least one zero of the gauge function, and these nodes are necessarily simple. The study focuses on the fundamental one-node branch, for which the gauge function starts positive at the horizon, crosses zero exactly once, and settles at the opposite vacuum value at infinity.

The solutions were obtained by shooting outward from a nonextremal event horizon, with regular initial data supplied by a near-horizon expansion. For the representative calculations the horizon radius was fixed at unity, leaving the horizon value of the gauge function as the only shooting parameter, adjusted until the required vacuum was reached at a large outer boundary. Convergence tests with outer boundaries at 40, 80 and 120 showed maximum absolute variations below 1.2 × 10⁻⁴ in the shooting parameter and below 6.8 × 10⁻⁵ in the mass. As an independent validation, the authors solved the ordinary EYM equations directly by setting the nonlinear correction to unity, reproducing the published horizon value of approximately 0.6322 for the fundamental EYM colored black hole, and the b = 100 solution of the logarithmic model agreed with this benchmark to better than about 2.1 × 10⁻⁵ in relative terms.

The asymptotic structure of the solutions carries a distinctive signature of the hair. The metric is Schwarzschild only at leading order: the first gravitational imprint of the colored field appears at order r⁻⁴, rather than through a Reissner–Nordström-type r⁻² term, while the explicitly logarithmic contribution is far more strongly suppressed, entering the metric only at order 1/(b²r¹⁰). The coefficient of the 1/r gauge-field tail is not an independently conserved charge but secondary hair, fixed once the horizon radius, the nonlinear parameter, the node number and the asymptotic vacuum are specified.

Strengthening the logarithmic nonlinearity, corresponding to smaller values of the dimensionless parameter b, produces substantial changes. At fixed horizon radius, decreasing b lowers the ADM mass, raises the Hawking temperature and shifts the gauge-field node outward. Relative to the near-linear b = 100 case, the strongly nonlinear b = 0.25 solution shows a mass lower by roughly 9.1 percent, a temperature higher by roughly 96.3 percent, and a node position larger by roughly 48.7 percent. The authors continued the fundamental branch down to b = 0.002 without detecting any termination within the parameter and radial ranges investigated. As b decreases, the exterior geometry approaches Schwarzschild on fixed radial domains while the colored field develops an increasingly extended tail, a nonuniform decoupling in which the gravitational geometry becomes locally Schwarzschild-like even as the non-Abelian structure is displaced to ever larger radii.

The analysis also bounded the dynamical interpretation of the static solutions. The null, weak and dominant energy conditions are satisfied throughout the static exterior, but the strong energy condition can be violated in the strongest-field region. A characteristic analysis of the gauge equations shows that for b of 0.5 or larger both characteristic factors remain positive across the numerical exterior, whereas for b = 0.25 and b = 0.1 the transverse characteristic factor becomes negative near the event horizon, crossing zero at radii of about 1.605 and 2.620 horizon radii respectively. These configurations remain valid solutions of the static boundary-value problem, but the loss of hyperbolicity restricts their interpretation as dynamically admissible backgrounds.

Inside the event horizon, a numerically stable reformulation in terms of a flux variable allowed the authors to continue the representative solutions down to radii of 10⁻⁸ horizon radii. For all cases examined, no additional zero of the metric function was detected, providing no evidence of an inner Cauchy horizon. Instead, the mass function approaches a finite positive limit, the gauge amplitude remains finite, and the logarithmic invariant diverges as 1 + X/β² ∼ Kr⁻³/². The logarithmic growth of the effective energy density remains integrable near the center, so the logarithmic nonlinearity suppresses the oscillatory mass-inflation behavior characteristic of ordinary EYM interiors. This suppression does not render the center regular: the Kretschmann scalar grows as 48m₀²/r⁶, and the interior terminates at a Schwarzschild-type spacelike curvature singularity. Notably, the limits of taking the nonlinear parameter to infinity and taking the radius to zero do not commute, since every finite nonlinear deformation eventually enters the strongly logarithmic regime sufficiently close to the center.

Thermodynamically, the one-node families display turning points of the Hawking temperature where the heat capacity at fixed nonlinear parameter diverges and changes sign, marking Davies-type changes of local thermodynamic stability. For b = 1 and b = 2, two turning points separate three branches with alternating signs of the heat capacity, while the near-EYM case b = 100 shows a single temperature maximum, reproducing the branch structure known for ordinary EYM colored black holes. The authors emphasize that these turning points do not establish a global phase transition: an asymptotically flat black hole does not define a globally stable canonical ensemble without an additional prescription such as a finite cavity, and the calculated Helmholtz free energy shows no unambiguous crossing between globally stable competing phases. The study leaves the dynamical stability of the new branch open, noting that since the solutions approach the known unstable EYM colored black holes in the weakly nonlinear limit, the corresponding gravitational and sphaleronic instabilities are expected to persist at least in that regime, while any modification of the unstable spectrum at stronger nonlinearity would require a dedicated coupled perturbation analysis.

Subject of Research: Theoretical construction of colored black holes in logarithmic nonlinear Yang–Mills theory

Article Title: Colored black holes in logarithmic nonlinear Yang–Mills theory

Article References: Muniz, C. R. (2026). Colored black holes in logarithmic nonlinear Yang–Mills theory. The European Physical Journal C, 86(10), Article 1229. https://doi.org/10.1140/epjc/s10052-026-16495-1

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DOI: 10.1140/epjc/s10052-026-16495-1

Keywords: black holes, Yang–Mills theory, non-Abelian gauge fields, nonlinear electrodynamics, general relativity, event horizon, Hawking temperature, spacetime singularity, thermodynamics, numerical relativity, SU(2) gauge theory, theoretical physics