Linear stability of the Linet - Tian solution with positive cosmological constant

Abstract

The purpose of this paper is to extend the analysis of gravitational instability of the Linet - Tian solution to the case $\Lambda > 0$. A fundamental difference brought about by $\Lambda >0$, as compared to $\Lambda<0$ is in the structure of the resulting space time. Associated with each of the two commuting Killing vectors $\partial_{\phi}$, and $\partial_{z}$, there is a curvature singularity that has the same characteristics as that associated to $\partial_{\phi}$ in the Levi - Civita metric, and we show that there is an isometry relating these singularities that reduces the effective parameter space of the metrics. In attempting to set up and solve the linearized perturbation equations we are confronted with the problem of a gauge ambiguity that leads to the introduction of a gauge invariant function, $W_1$, that is shown to be also a {\em master function}, that satisfies a second order ODE, and in terms of which one can express all the perturbation functions. Unfortunately, the equation satisfied by $W_1$ contains singular coefficients, and, although {\em all} its solutions are regular, because of the presence of these singularities one cannot, as in the case of negative $\Lambda$, set up an associated self adjoint problem that provides a complete set of solutions for $W_1$. We are thus restricted to solving numerically the perturbation equations, and using those solutions for constructing $W_1$, for particular values of the parameters. In all the cases analyzed we find unstable modes, which strongly suggests that all the Linet - Tian space times with $\Lambda > 0$ are linearly unstable under gravitational perturbations. The problem of determining the time evolution of arbitrary initial data in terms of the $W_1$, or something equivalent, remains open.