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

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.

57 pages

References in corpus (8)

• Black holes, hidden symmetries, and complete integrability
• Kerr-CFT and gravitational perturbations
• Generalization of the Geroch-Held-Penrose formalism to higher dimensions
• Maxwell's Equations in the Myers-Perry Geometry
• Perturbations of near-horizon geometries and instabilities of Myers-Perry black holes
• Perturbations of higher-dimensional spacetimes
• Symmetry operators and decoupled equations for linear fields on black hole spacetimes
• Instability in near-horizon geometries of even-dimensional Myers-Perry black holes

Cited by in corpus (4)

• Separation of variables in Maxwell equations in Plebanski-Demianski spacetime
• Two-dimensional twistor manifolds and Teukolsky operators
• Linear Stability of Black Holes and Naked Singularities
• First Order Symmetry Operators for the Linearized Field Equation of Metric Perturbations