The Bach equations in Spin-Coefficient form

TL;DR

This paper reformulates the Bach equations, which are central to conformal gravity theories, in spin-coefficient form for more efficient analysis and demonstrates their application to specific spacetime solutions.

Contribution

It introduces a compacted spin-coefficient formalism for the Bach equations, providing an efficient alternative to tensor form and applying it to pp-wave and spherically symmetric spacetimes.

Findings

Bach equations expressed in spin-coefficient form

Solutions obtained for pp-wave spacetimes

Solutions obtained for static spherically symmetric spacetimes

Abstract

Conformal gravity theories are defined by field equations that determine only the conformal structure of the spacetime manifold. The Bach equations represent an early example of such a theory, we present them here in component form in terms of spin- and boost-weighted spin-coefficients using the compacted spin-coefficient formalism. These equations can be used as an efficient alternative to the standard tensor form. As a simple application we solve the Bach equations for pp-wave and static spherically symmetric spacetimes.