Axial gravitational perturbations of an infinite static line source

Abstract

In this paper we study axial gravitational perturbations of an infinite static line source, represented by a form of the Levi-Civita metric. The perturbations are restricted to axial symmetry but break the cylindrical symmetry of the background metric. We analyze the gauge issues that arise in setting up the appropriate form of the perturbed metric and show that it is possible to restrict to diagonal terms, but that this does not fix the gauge completely. We derive the perturbation equations and show that they can be solved by solving a third order ordinary differential equation for an appropriately chosen function of the perturbed metric coefficients. The set of solutions of this equation contains gauge trivial parts, and we show how to extract the gauge non trivial components. We introduce appropriate boundary conditions on the solutions and show that these lead to a boundary value problem that determines the allowed functional forms of the perturbation modes. The associated eigenvalues determine a sort of "dispersion relation" for the frequencies and corresponding "wave vector" components. The central result of this analysis is that the spectrum of allowed frequencies contains one unstable (imaginary frequency) mode for every possible choice of the background metric. The completeness of the mode expansion in relation to the initial value problem and to the gauge problem is discussed in detail, and we show that the perturbations contain an unstable component for generic initial data, and, therefore, that the Levi-Civita space times are gravitationally unstable. We also include, for completeness, a set of approximate eigenvalues, and examples of the functional form of the solutions.