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

G. DOTTI; R. J. GLEISER

CLASSICAL AND QUANTUM GRAVITY

IOP PUBLISHING LTD

Lugar: Londres; Año: 2009 vol. 25 p. 215002 - 215002

0264-9381

The coupled equations for the scalar modes of the linearized Einstein equations aroundSchwarzschild´s spacetime were reduced by Zerilli to a 1+1 wave equation where + V (x) is the Zerilli Hamiltonian", x the tortoise radial coordinate. Fromits denition, for smooth metric perturbations the field  is singular at rs = 􀀀6M=(` 􀀀 1)(` + 2),with ` the mode harmonic number. The equation    obeys is also singular, since V has a secondorder 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, thenrs > 0, and the singularity appears in the relevant range of r ( 0 < r < 1). We solve this problemby developing a new approach to the evolution of the even mode, based on a new gauge invariantfunction, ^     , that is a regular function of the metric perturbation for any value of M. The relationof ^     to     z is provided by an intertwiner operator. The spatial pieces of the 1 + 1 wave equationsthat ^     and     z obey are related as a supersymmetric pair of quantum hamiltonians H and H^. ForM < 0, H^ has a regular potential and a unique self-adjoint extension in a domain D dened by aphysically motivated boundary condition at r = 0. This allows to address the issue of evolution ofgravitational perturbations in this non globally hyperbolic background. This formulation is used tocomplete the proof of the linear instability of the Schwarzschild naked singularity, by showing thata previously found unstable mode belongs to a complete basis of H^ in D, and thus is excitable bygeneric initial data. This is further illustrated by numerically solving the linearized equations for suitably chosen initial data.