A causal magnetic black hole with finite self-energy

Abstract

We construct a nonlinear electrodynamics (NLED) model for a static magnetic black hole. Rather than imposing a regular center, we require a Maxwell weak-field limit, finite magnetic self-energy, a positive and subluminal electromagnetic cone, the standard energy conditions, and a static exterior that satisfies known sufficient stability criteria. A positive mixture of power kernels restricts the exponent to $1/4<γ\leq1/2$. For the minimal two-kernel model, with $γ_1=1/3$ and $γ_2=1/2$, the field equations admit an exact solution in terms of incomplete beta functions. For the branches with nonnegative Schwarzschild mass parameter $M_0$, the metric function is strictly increasing, so there is at most one positive-radius horizon and no inner Cauchy horizon. We construct the maximal extension and show that the black-hole, horizonless, and critical branches have spacelike, timelike, and null singularities, respectively. The representative black-hole families have positive temperature and negative fixed-charge heat capacity. Electromagnetic waves split into ordinary and extraordinary optical branches. We calculate their photon spheres, shadow radii, instability rates, and illustrative Event Horizon Telescope size bands. We also construct ISCO-truncated thin-disk images. The full images remain close, but jointly normalized residual maps and radial profiles reveal a coherent branch-dependent shift near the lensed inner edge and the critical region. The extraordinary branch moves the critical curve outward and partly compensates the shadow reduction caused by magnetic charge.