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

TL;DR

This paper develops symmetry operators and decoupled scalar equations for linear fields on black hole spacetimes, enabling simplified analysis and solution reconstruction for fields like Dirac, Maxwell, and gravity in these backgrounds.

Contribution

It introduces four-dimensional operators that decouple field equations on Petrov type D spacetimes, generalizing previous Teukolsky and Debye potential results.

Findings

Derived off-shell operators that decouple field equations

Provided reconstruction formulas for original fields

Analyzed role of Killing spinors and tensors in decoupling

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.