On the geometry of Petrov type II spacetimes

Steffen Aksteiner,Bernard Whiting,Lars Andersson,Bernardo Araneda

doi:10.1088/1361-6382/abf542

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.

Highlights

A remarkable property of vacuum spacetimes of Petrov type D is the existence of ‘hidden symmetries’, namely, appropriate generalizations of Killing vectors such as Killing tensors and conformal Killing-Yano tensors
Penrose and Walker have shown [30] the existence of a valence 2 Killing spinor, from which one can obtain the above mentioned symmetries
Because of the Goldberg-Sachs theorem, the vacuum type D condition is equivalent to the existence of two independent null geodesic congruences that are shear-free

Summary

Introduction

A remarkable property of vacuum spacetimes of Petrov type D is the existence of ‘hidden symmetries’, namely, appropriate generalizations of Killing vectors such as Killing tensors and conformal Killing-Yano tensors. In this paper we focus on the spin-2 case, and consider the construction of solutions to the linearized Einstein equations on backgrounds of Petrov type II, in terms of Debye potentials. For vacuum type D spacetimes, it is sometimes assumed that, up to gauge, all real solutions of the linearized Einstein vacuum equations can be obtained, locally, as the real part of a metric generated by a Debye potential; for recent advances in the Schwarzschild and Kerr cases see respectively [20] and [11]. We will give a geometric interpretation to the origin of (1.6) in terms of spinors that are parallel under a suitable connection especially adapted to the geometry, which is the conformal-GHP connection This allows us to generalize the result (1.4)-(1.6) to non-vacuum spacetimes in the real-analytic case, and to derive conservation laws associated to projected Killing spinors. Most computations were performed with Spinframes [2], based on the symbolic computer algebra package xAct for Mathematica

Preliminaries

Special geometry and parallel spinors

Perturbation theory in terms of a Debye potential