Stability of the Cauchy horizon in accelerating black-hole spacetimes.

The extendibility of spacetime and the existence of weak solutions to the Einstein field equations beyond Cauchy horizons, is a crucial ingredient to examine the limits of General Relativity. Strong Cosmic Censorship serves as a firewall for gravitation by demanding inextendibility of spacetime beyond the Cauchy horizon. For asymptotically flat spacetimes, the predominance of the blueshift instability and the subsequent formation of a mass-inflation singularity at the Cauchy horizon have, so far, substantiated the conjecture. Accelerating black holes, described by the $C-$metric, are exact solutions of the field equations without a cosmological constant, which possess an acceleration horizon with similar causal properties to the cosmological horizon of de Sitter spacetime. Here, by considering linear scalar field perturbations, we provide numerical evidence for the stability of the Cauchy horizon of charged accelerating black holes. In particular, we show that the stability of Cauchy horizons in accelerating charged black holes is connected to quasinormal modes, we discuss the regularity requirement for which weak solutions to the field equations exist at the Cauchy horizon and show that Strong Cosmic Censorship may be violated near extremality.