On the geometry of Petrov type II spacetimes

TL;DR

This paper explores the geometric properties of Petrov type II spacetimes, introducing a weaker symmetry condition, and analyzes perturbations leading to solutions of the linearized Einstein equations with specific algebraic properties.

Contribution

It introduces a weaker form of Killing equations for Petrov type II spacetimes and studies their properties and implications for perturbations and conservation laws.

Findings

Existence of a generalized Killing spinor in Petrov type II spacetimes.

Perturbations yield complex solutions to linearized Einstein equations.

Linearized Weyl tensor becomes half Petrov type N under perturbations.

Abstract

In general, geometries of Petrov type II do not admit symmetries in terms of Killing vectors or spinors. We introduce a weaker form of Killing equations which do admit solutions. In particular, there is an analog of the Penrose-Walker Killing spinor. Some of its properties, including associated conservation laws, are discussed. Perturbations of Petrov type II Einstein geometries in terms of a complex scalar Debye potential yield complex solutions to the linearized Einstein equations. The complex linearized Weyl tensor is shown to be half Petrov type N. The remaining curvature component on the algebraically special side is reduced to a first order differential operator acting on the potential.