On a relation between the Bach equation and the equation of geometrodynamics

TL;DR

This paper explores the relationship between the Bach equation and the equation of geometrodynamics, revealing that solutions to the latter are encompassed by a more general conformally invariant framework.

Contribution

It demonstrates that a conformally invariant Lagrangian generalizes the Bach equation and includes solutions to the equation of geometrodynamics.

Findings

Solutions of geometrodynamics are more general than those of the Bach equation.

A conformally invariant Lagrangian encompasses solutions to both equations.

The relation between the two equations is clarified through this generalized framework.

Abstract

The Bach equation and the equation of geometrodynamics are based on two quite different physical motivations, but in both approaches, the conformal properties of gravitation plays the key role. In this paper we present an analysis of the relation between these two equations and show that the solutions of the equation of geometrodynamics are of a more general nature. We show the following non-trivial result: there exists a conformally invariant Lagrangian, whose field equation generalizes the Bach equation and has as solutions those Ricci tensors which are solutions to the equation of geometrodynamics.