Cauchy-horizon singularity inside perturbed Kerr black holes

TL;DR

This paper investigates the development of a curvature singularity at the Cauchy horizon inside perturbed Kerr black holes by solving the linearized Weyl scalars and curvature scalar along outgoing null rays, confirming some prior perturbation results while revealing differences in growth rates.

Contribution

The study provides a detailed numerical analysis of the Weyl scalars and curvature scalar near the Cauchy horizon, validating previous perturbation findings and highlighting differences in exponential growth rates.

Findings

Cauchy horizon becomes a null, scalar-curvature singularity.

Excellent agreement with perturbation analysis for Weyl scalar $\psi_0$.

Slower exponential growth rate of curvature scalar than previously found.

Abstract

The Cauchy horizon inside a perturbed Kerr black hole develops an instability that transforms it into a curvature singularity. We solve for the linearized Weyl scalars $\psi_0$ and $\psi_4$ and for the curvature scalar $R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}$ along outgoing null rays approaching the Cauchy horizon in the interior of perturbed Kerr black holes using the Teukolsky equation, and compare our results with those found in perturbation analysis. Our results corroborate the previous perturbation analysis result that at its early parts the Cauchy horizon evolves into a deformationally-weak, null, scalar-curvature singularity. We find excellent agreement for $\psi_0(u={\rm const},v)$, where $u,v$ are advanced and retarded times, respectively. We do find, however, that the exponential growth rate of $R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}(u={\rm const},v)$ approaching the singularity is dramatically slower than that found in perturbation analysis, and that the angular frequency is in excellent agreement.