Cauchy-horizon singularity inside perturbed Kerr black holes

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 ${\ensuremath{\psi}}_{0}$ and ${\ensuremath{\psi}}_{4}$ and for the curvature scalar ${R}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}\ensuremath{\delta}}{R}^{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}\ensuremath{\delta}}$ 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 ${\ensuremath{\psi}}_{0}(u=\text{const},v)$, where $u$, $v$ are advanced and retarded times, respectively. We do find, however, that the exponential growth rate of ${R}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}\ensuremath{\delta}}{R}^{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}\ensuremath{\delta}}(u=\text{const},v)$ approaching the singularity is dramatically slower than that found in perturbation analysis, and that the angular frequency is in excellent agreement.