Symmetry operators and decoupled equations for linear fields on black hole spacetimes

Abstract

In the class of vacuum Petrov type D spacetimes with cosmological constant, which includes the Kerr-(A)dS black hole as a particular case, we find a set of four-dimensional operators that, when composed {\em off shell} with the Dirac, Maxwell and linearized gravity equations, give a system of equations for spin weighted scalars associated to the linear fields, that decouple on shell. Using these operator relations we give compact reconstruction formulae for solutions of the original spinor and tensor field equations in terms of solutions of the decoupled scalar equations. We also analyze the role of Killing spinors and Killing-Yano tensors in the spin weight zero equations and, in the case of spherical symmetry, we compare our four-dimensional formulation with the standard $2+2$ decomposition and particularize to the Schwarzschild-(A)dS black hole. Our results uncover a pattern that generalizes a number of previous results on Teukolsky-like equations and Debye potentials for higher spin fields.

arXiv comment57 pages, no figures; v3: minor changes, matches published version

References in corpus (10)

• SUSY in the sky
• Conformal Yano-Killing tensor for the Kerr metric and conserved quantities
• Killing-Yano tensors and some applications
• Teukolsky Master Equation: De Rham wave equation for the gravitational and electromagnetic fields in vacuum
• Linearized gravity and gauge conditions
• Non-modal linear stability of the Schwarzschild black hole
• Linear perturbations of self-gravitating spherically symmetric configurations
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• Two-dimensional twistor manifolds and Teukolsky operators
• Linear Stability of Black Holes and Naked Singularities
• Conserved quantities and integrability for massless spinning particles in general relativity
• First Order Symmetry Operators for the Linearized Field Equation of Metric Perturbations
• Numerical investigation of the late-time tails of the solutions of the Fackerell-Ipser equation
• Generalized wave operators, weighted Killing fields, and perturbations of higher dimensional spacetimes
• Spin-2 Green's Functions on Kerr in Radiation Gauge
• On First Order Symmetry Operators for the Field Equations of Differential Forms