The decoupling problem of the Proca equation; and treatment of Dirac, Maxwell and Proca fields on the resulting pp-wave spacetimes

TL;DR

This paper investigates the decoupling of Proca equations in curved spacetimes, showing it is possible only in pp-wave vacuum backgrounds, and examines the separability of Proca, Maxwell, and Dirac equations in such spacetimes.

Contribution

It demonstrates that Proca equations can be decoupled only in pp-wave vacuum spacetimes using NP formalism, and analyzes their separability alongside Maxwell and Dirac equations.

Findings

Decoupling of Proca equations requires a covariantly constant null vector.

Proca, Maxwell, and Dirac equations are separable in pp-wave backgrounds.

Decoupling is exclusive to pp-wave vacuum spacetimes.

Abstract

In this work we take a formal approach to the problem of decoupling Proca equations in curved space-times. We use Newman-Penrose (NP) two-spinor formalism to represent the Proca vector by one complex and two real scalars. We show that a decoupled second order differential equation for one of the real scalars can be derived if and only if the background space-time admits a covariantly constant null vector. Thus, the background space-time must be a pp-wave vacuum. We evaluate the separability of Proca, Maxwell and Dirac equations on the resulting pp-wave background.