Stability of magnetic black holes in general nonlinear electrodynamics

We study the perturbative stability of magnetic black holes in a general class of nonlinear electrodynamics, where the Lagrangian is given by a general function of the field strength of electromagnetic field $F_{\mu \nu }$ and its Hodge dual ${\tilde{F}}_{\mu \nu }$. We derive sufficient conditions for the stability of the black holes. We apply the stability conditions to Bardeen’s regular black holes, black holes in Euler–Heisenberg theory, and black holes in Born–Infeld theory. As a result, we obtain a sufficient condition for the stability of Bardeen’s black holes, which restricts $F_{\mu \nu }{\tilde{F}}^{\mu \nu }$ dependence of the Lagrangian. We also show that black holes in Euler–Heisenberg theory are stable for a sufficiently small magnetic charge. Moreover, we prove the stability of black holes in the Born–Infeld electrodynamics even when including $F_{\mu \nu }{\tilde{F}}^{\mu \nu }$ dependence.

Figure 1

A red line indicates the behavior of $y_1(x)$ defined by Eq. (5.14). A dashed gray line at 0.3551 divides whether the horizons exist or not.

Figure 2

A red and a blue line indicate the behaviors of the functions (5.19) and (5.20), respectively.