Solution of the Helmholtz equation for spin-2 fields

TL;DR

This paper presents a method to solve the Helmholtz equation for spin-2 tensor fields in flat space using separation of variables and spin-weighted harmonics, with applications to linearized Einstein theory.

Contribution

It introduces explicit solutions for symmetric, traceless, divergenceless tensor fields in spherical and cylindrical coordinates, expressing them via scalar potentials satisfying the Helmholtz equation.

Findings

Solutions expressed in terms of scalar potentials

Applicable to spherical and cylindrical coordinates

Relevance to linearized Einstein theory

Abstract

The Helmholtz equation for symmetric, traceless, second-rank tensor fields in three-dimensional flat space is solved in spherical and cylindrical coordinates by separation of variables making use of the corresponding spin-weighted harmonics. It is shown that any symmetric, traceless, divergenceless second-rank tensor field that satisfies the Helmholtz equation can be expressed in terms of two scalar potentials that satisfy the Helmholtz equation. Two such expressions are given, which are adapted to the spherical or cylindrical coordinates. The application to the linearized Einstein theory is discussed.