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 ${F}_{\ensuremath{\mu}\ensuremath{\nu}}$ and its Hodge dual ${\stackrel{\texttildelow{}}{F}}_{\ensuremath{\mu}\ensuremath{\nu}}$. 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 ${F}_{\ensuremath{\mu}\ensuremath{\nu}}{\stackrel{\texttildelow{}}{F}}^{\ensuremath{\mu}\ensuremath{\nu}}$ 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 ${F}_{\ensuremath{\mu}\ensuremath{\nu}}{\stackrel{\texttildelow{}}{F}}^{\ensuremath{\mu}\ensuremath{\nu}}$ dependence.

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