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 ∂ 2 Ψz/∂t 2 + HΨz = 0, where H = −∂ 2 /∂x 2 + V (x) is the Zerilli "Hamiltonian", x 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 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,Ψ, that is a regular function of the metric perturbation for any value of M . The relation ofΨ to Ψz is provided by an intertwiner operator. The spatial pieces of the 1 + 1 wave equations thatΨ and Ψz obey are related as a supersymmetric pair of quantum hamiltonians H andĤ. For M < 0,Ĥ has a regular potential and a unique self-adjoint extension in a domain D defined by a physically motivated boundary condition at r = 0. This allows 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 ofĤ in D, and thus is excitable by generic initial data. This is further illustrated by numerically solving the linearized equations for suitably chosen initial data.

We show that a spacetime satisfying the linearized vacuum Einstein equations around a type D background is generically of type I, and that the splittings of the Principal Null Directions (PNDs) and of the degenerate eigenvalue of the Weyl tensor are non analytic functions of the perturbation parameter of the metric. This provides a gauge invariant characterization of the effect of the perturbation on the underlying geometry, without appealing to differential curvature invariants. This is of particular interest for the Schwarzschild solution, for which there are no signatures of the even perturbations on the algebraic curvature invariants. We also show that, unlike the general case, the unstable even modes of the Schwarzschild naked singularity deforms the Weyl tensor into a type II one.