The initial value problem for linearized gravitational perturbations of the Schwarzschild naked singularity

Abstract

The coupled equations for the scalar modes of the linearized Einstein equations around Schwarzschild's spacetime were reduced by Zerilli to a (1+1) wave equation , where is the Zerilli ‘Hamiltonian’ and x is the tortoise radial coordinate. From its definition, for smooth metric perturbations the field Ψz is singular at rs = −6M/(ℓ − 1)(ℓ +2), with ℓ being the mode harmonic number. The equation Ψz obeys is also singular, since V has a second-order pole at rs. This is irrelevant to the black hole exterior stability problem, where r > 2M > 0, and rs < 0, but it introduces a non-trivial problem in the naked singular case where M < 0, then rs > 0, and the singularity appears in the relevant range of r (0 < r < ∞). We solve this problem by developing a new approach to the evolution of the even mode, based on a new gauge invariant function, , that is a regular function of the metric perturbation for any value of M. The relation of to Ψz is provided by an intertwiner operator. The spatial pieces of the (1 + 1) wave equations that and Ψz obey are related as a supersymmetric pair of quantum Hamiltonians and . For has a regular potential and a unique self-adjoint extension in a domain defined by a physically motivated boundary condition at r = 0. This allows us to address the issue of evolution of gravitational perturbations in this non-globally hyperbolic background. This formulation is used to complete the proof of the linear instability of the Schwarzschild naked singularity, by showing that a previously found unstable mode belongs to a complete basis of in , and thus is excitable by generic initial data. This is further illustrated by numerically solving the linearized equations for suitably chosen initial data.