Petrov type of linearly perturbed type D spacetimes

TL;DR

This paper investigates how linear perturbations affect the Petrov classification of type D spacetimes, revealing that generic perturbations lead to type I, with non-analytic changes in principal null directions, especially impacting Schwarzschild solutions.

Contribution

It provides a gauge-invariant method to analyze the algebraic type changes of spacetimes under linear perturbations, highlighting the transition from type D to type I or II.

Findings

Perturbed type D spacetimes are generally of type I.

Splittings of PNDs are non-analytic functions of perturbations.

Even modes can deform Schwarzschild into a type II spacetime.

Abstract

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.