Gauge invariant perturbations of self-similar Lemaître-Tolman-Bondi spacetime: Even parity modes with $l\ge 2$

In this paper we consider gauge invariant linear perturbations of the metric and matter tensors describing the self-similar Lemaître-Tolman-Bondi (timelike dust) spacetime containing a naked singularity. We decompose the angular part of the perturbation in terms of spherical harmonics and perform a Mellin transform to reduce the perturbation equations to a set of ordinary differential equations with singular points. We fix initial data so the perturbation is finite on the axis and the past null cone of the singularity, and follow the perturbation modes up to the Cauchy horizon. There we argue that certain scalars formed from the modes of the perturbation remain finite, indicating linear stability of the Cauchy horizon.

Figure 1

Conformal diagram for the self-similar LTB admitting a globally naked singularity. There are three similarity horizons at which the similarity coordinate $y$ is null: $y=y_p$ the past null cone, $y=y_c$ shown dashed, and $y=y_e$ shown as a double line. We identify $y=y_c$ as the Cauchy horizon, and will call $y=y_e$ the second future similarity horizon (SFSH). The apparent horizon is shown as a bold curve.

Figure 2

The lines $\sigma =1$ and $\overline{\sigma }=1$ plotted in the $\mathrm{Re}(s)$, $\mu$ parameter space for $0<\mu <{\mu }_{*}$.

Figure 3

The lines $\sigma =1,2$ and $\overline{\sigma }=1,2$ plotted in the $\mathrm{Re}(s)$, $\mu$ parameter space for $0<\mu <{\mu }_{*}$.

Figure 4

Integrating over a contour in the complex plane of $s$.