Generalized wave operators, weighted Killing fields, and perturbations of higher dimensional spacetimes

TL;DR

This paper develops a framework of weighted wave operators and covariant derivatives for analyzing perturbations in higher-dimensional Einstein spacetimes, generalizing key equations and symmetries to arbitrary dimensions.

Contribution

It introduces weighted covariant derivatives, wave operators, and conformal Killing vectors for higher-dimensional spacetimes, extending the Teukolsky formalism beyond four dimensions.

Findings

Weighted wave equations for perturbations are derived.

Higher-dimensional principal null directions are weighted conformal Killing vectors.

Solutions to original field equations are constructed from wave equation solutions.

Abstract

We present weighted covariant derivatives and wave operators for perturbations of certain algebraically special Einstein spacetimes in arbitrary dimensions, under which the Teukolsky and related equations become weighted wave equations. We show that the higher dimensional generalization of the principal null directions are weighted conformal Killing vectors with respect to the modified covariant derivative. We also introduce a modified Laplace-de Rham-like operator acting on tensor-valued differential forms, and show that the wave-like equations are, at the linear level, appropriate projections {\em off shell} of this operator acting on the curvature tensor; the projection tensors being made out of weighted conformal Killing-Yano tensors. We give off shell operator identities that map the Einstein and Maxwell equations into weighted scalar equations, and using adjoint operators we construct solutions of the original field equations in a compact form from solutions of the wave-like equations. We study the extreme and zero boost weight cases; extreme boost corresponding to perturbations of Kundt spacetimes (which includes Near Horizon Geometries of extreme black holes), and zero boost to static black holes in arbitrary dimensions. In 4 dimensions our results apply to Einstein spacetimes of Petrov type D and make use of weighted Killing spinors.