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

TL;DR

This paper proves that nonlinear electrodynamics black holes with charged scalar hairs do not have an inner Cauchy horizon, revealing universal interior dynamics and linking scalar hair to horizon instability.

Contribution

It establishes a general no Cauchy horizon theorem for such black holes, independent of scalar potential and electrodynamics form, and analyzes interior dynamics beyond the horizon.

Findings

No inner Cauchy horizon exists in these black holes.

Interior near-singularity exhibits universal Kasner behavior.

Scalar hairs induce instability of the inner Cauchy horizon.

Abstract

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. When the contribution from the scalar potential can be neglected, the asymptotic near-singularity takes a universal Kasner form. We also confirm that the intricate interior dynamics is closely associated with the instability of the inner Cauchy horizon triggered by scalar hairs.