Teukolsky Master Equation: De Rham wave equation for the gravitational and electromagnetic fields in vacuum

TL;DR

This paper introduces a new form of the Teukolsky Master Equation that links curvature perturbation theory with wave equations for gravitational and electromagnetic fields in vacuum, offering a recursive scheme for higher-order analysis.

Contribution

It presents a novel wave equation form of the Teukolsky Master Equation incorporating curvature terms, connecting it with de Rham-Lichnerowicz Laplacian equations in vacuum spacetimes.

Findings

Establishes a relation between curvature perturbation theory and exact wave equations.

Provides a recursive scheme for higher-order perturbative expansions.

Clarifies the origins of perturbative results within the exact theory.

Abstract

A new version of the Teukolksy Master Equation, describing any massless field of different spin $s=1/2,1,3/2,2$ in the Kerr black hole, is presented here in the form of a wave equation containing additional curvature terms. These results suggest a relation between curvature perturbation theory in general relativity and the exact wave equations satisfied by the Weyl and the Maxwell tensors, known in the literature as the de Rham-Lichnerowicz Laplacian equations. We discuss these Laplacians both in the Newman-Penrose formalism and in the Geroch-Held-Penrose variant for an arbitrary vacuum spacetime. Perturbative expansion of these wave equations results in a recursive scheme valid for higher orders. This approach, apart from the obvious implications for the gravitational and electromagnetic wave propagation on a curved spacetime, explains and extends the results in the literature for perturbative analysis by clarifying their true origins in the exact theory.