Black holes and causal nonlinear electrodynamics

TL;DR

The paper analyzes how causality constraints in nonlinear electrodynamics (NLED) shape static, spherically symmetric Einstein–NLED black holes. By reducing the Einstein–NLED equations to a single radial Hamiltonian function ${\rm H}(r)$ and leveraging convexity/concavity properties of the associated energy and effective charge functions, the authors derive strong global-structure results: causal NLED admit no regular charged black holes, extremal entropy is reduced relative to Reissner–Nordström (RN) at fixed charge, and transitions between RN-type and Schwarzschild-type interiors occur with a Barriola–Vilenkin monopole geometry at the transition point. The work also shows that acausal Born-type theories can yield RN as an exact (but unstable) solution for certain dyons, while self-dual theories allow a clean mapping of charges through the duality-invariant $Q=\\sqrt{q_e^2+q_m^2}$. Collectively, these results clarify how strong-field causality constraints govern admissible black-hole geometries, horizon structures, and extremal limits in NLED, with implications for black-hole interiors and the interpretation of singularities under nonlinear electromagnetic dynamics.

Abstract

For generic theories of nonlinear electrodynamics (NLED) we investigate the restrictions imposed by causality on spherically-symmetric charged black-hole solutions of the Einstein-NLED equations. For a large class of (acausal) Born-type NLED theories, we find that the Reissner-Nordstrom (RN) metric is an exact, but unstable, solution for some dyonic black holes. For all causal NLED we show that there are no regular black holes, and that the entropy of extremal black holes is less than the RN entropy for fixed charge. We also find the conditions for a parameter-space transition between RN-type and Schwarzschild-type global structure. For the transition from Schwarzschild-type to naked singularity, which occurs at finite mass, we show that the metric at the transition point is a Barriola-Vilenkin global monopole.

Figures (5)

• Figure 1: The electric field $E_r(r)$ for Maxwell (black), Born-Infeld (red) and two other examples of causal theories with $E_r\sim 1/r$ (blue) and $E_r\to E_0$ (green). For any causal NLED, $E_r(r)$ is a monotonically decreasing function of $r$ that is less than $Q/r^2$ for all finite $r$, and hence lies entirely in the grey-shaded area.
• Figure 2: Location of horizons for Born-Infeld at the intersection of $f_L\equiv(r-GM)^2$ (green) and $f_R\equiv (MG)^2-4\pi G Q_{\rm eff}^2(r)$ (red), which is asymptotic to the horizontal dashed line: $f_R(\infty) = (MG)^2-4\pi G Q^2$. a) $Q=1$, $\mu =1.1$ (here $G=1$, $T=1/100$). b) $Q=0.6$, $\mu =1.1$. In this case there is no Cauchy horizon and the global structure of the black hole is Schwarzschild type with a space-like singularity.
• Figure 3: The figures show that $\mu<1$ for any NLED different from Maxwell theory. a) For $\mu=1$, the horizons have not yet merged (here $G=1$, $T=1/100$, $Q=0.8$). Extremality requires a lower value of $\mu$. b) Extremal solution, occurring at $\mu=0.974$ (same conventions as fig. \ref{['ddab']}).
• Figure 4: Phase diagram for Born-Infeld black hole, illustrating a $\nu=\frac{1}{2}$ case ($G=1$).
• Figure 5: Phase diagram for a charged black hole in a theory with $\nu=\frac{5}{8}$.