Cauchy-horizon singularity inside perturbed Kerr black holes

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 ${\psi }_0(u=\text{const},v)$, where $u$, $v$ are advanced and retarded times, respectively. We do find, however, that the exponential growth rate of $R_{\alpha \beta \gamma \delta }{R}^{\alpha \beta \gamma \delta }(u=\text{const},v)$ approaching the singularity is dramatically slower than that found in perturbation analysis, and that the angular frequency is in excellent agreement.

Figure 1

The Penrose diagram for the domain of integration and the coordinates. The solid (dashed) thin curves are hypersurfaces $\tau =\text{const}$ ($\rho =\text{const}$). Notice that the $\tau =\text{const}$ surfaces intersect on the diagram with ${ℐ}^{+}$, and penetrate through the event horizon (EH) and the outgoing inner horizon (OIH), which is the inner boundary for our computational domain. The earliest of these surfaces is our partial Cauchy surface $\mathrm{\Sigma }$. The dotted thin line is a constant advanced-time ray ($u=\text{const}$) that intersects with the Cauchy horizon (CH).

Figure 2

Sketch of the numerical scheme for Cauchy data with two timelike coordinate ($t$, $x$). Cauchy data are specified for $t_0$ at $x_{-1}$ and at $x_1$. With one spacelike $x$ coordinate one normally finds the fields at $(t_1,x_0)$. Here, that event is outside the domain of influence of the initial data. One may, however, find the fields at, say, $(t_{1/2},x_0)$, which is inside the domain of influence. Shown also are retarded and advanced times.

Figure 3

The left hand-side of inequality (1) as a function of $\rho$ for various values of the Courant factor, for a Kerr BH with $a/M=0.8$. The horizons are at ${\rho }_{+}=1.479$ and ${\rho }_{-}=0.392$ ($S=19.6$). When the Courant factor is smaller than $\sim 0.8$ the CFL condition is globally satisfied. The condition on advanced time is satisfied globally for all Courant factors $\le 1$.

Figure 4

The Teukolsky fields ${\mathrm{\Delta }}^{-2}{\mathrm{\Psi }}_4$ [panels (a) and (b)] and ${\mathrm{\Delta }}^2{\mathrm{\Psi }}_0$ [panels (c) and (d)] along the event horizon on the equatorial plane as functions of retarded time for $a/M=0.8$ and $m=2$ in horizon regularized coordinates.

Figure 5

The Weyl scalars ${\mathrm{\Delta }}^2{\psi }_0$ in regularized coordinates (upper panel) and ${\psi }_4$ in ingoing Kerr coordinates (lower panel) as functions of $v/M$ along an outgoing null ray that intersects with the early part of the Cauchy horizon (as $v/M\to \infty$). For ${\mathrm{\Delta }}^2{\psi }_0$ (${\psi }_4$) we present in addition to our numerical results in a solid curve also the prediction of perturbation analysis [Ansatz (3)] in a dashed curve. Here, $a/M=0.8$ and $m=2$ and the fields are on the equatorial plane.

Figure 6

The curvature scalar $R_{\alpha \beta \gamma \delta }{R}^{\alpha \beta \gamma \delta }$ as a function of $v/M$ along an outgoing null ray on the equatorial plane approaching the Cauchy horizon. Here, $a/M=0.8$ and $m=2$, and data are on the equatorial plane. The numerical simulation is shown with a solid curve, and the prediction of [11] with a dashed curve.