No Cauchy Horizon Theorem for Nonlinear Electrodynamics Black Holes with Charged Scalar Hairs

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We prove a no Cauchy horizon theorem for general nonlinear electrodynamics black holes with charged scalar hairs.

• By constructing a radially conserved charged, we show that there is no inner Cauchy horizon for both spherical and planar symmetric cases, independent of the form of scalar potential and nonlinear electrodynamics.
• After imposing the null energy condition, we are also able to rule out the existence of the Cauchy horizon for the hyperbolic black holes.
• We take the Born-Infeld black hole as a concrete example to study the interior dynamics beyond the event horizon.