Newman-Penrose-Debye formalism for fields of various spins in pp-wave backgrounds

TL;DR

This paper develops a formalism using Newman-Penrose and Wald's methods to analyze massless fields of various spins in pp-wave backgrounds, providing explicit solutions and potential generalizations to massive fields relevant for supergravity.

Contribution

It introduces a novel approach combining Newman-Penrose formalism and adjoint operators to derive Debye potentials and explicit solutions for fields of various spins in pp-wave spacetimes.

Findings

Explicit solutions for Debye potentials in shock wave backgrounds

Derivation of inverse projection operators for multicomponent fields

Discussion on extending methods to massive fields like Dirac and Proca equations

Abstract

Using Newman-Penrose formalism in tetrad and spinor notation, we perform separation of variables in the wave equations for massless fields of various spins s=1/2, 1, 3/2, 2 on the background of exact plane-fronted gravitational wave metrics. Then, applying Wald's method of adjoint operators, we derive equations for Debye potentials generating these fields and find inverse projection operators expressing multicomponet fields in terms of scalar potentials. For a number of shock wave backgrounds, as a special case of non-vacuum pp-waves, the exact solutions for Debye potentials are constructed explicitly. The possibility of generalization to the case of massive fields, in particular, construction of exact solutions to the Dirac and Proca equations are discussed. These results can be used in supergravity models on pp-wave backgrounds.