Lsdiff M and the Einstein Equations

TL;DR

This paper presents a novel formulation of vacuum Einstein equations using volume-preserving vector fields on a four-manifold, generalizing Yang-Mills equations for flat spacetime.

Contribution

It introduces a new geometric approach to Einstein equations through volume-preserving vector fields, extending Yang-Mills theory.

Findings

Formulation of Einstein equations via volume-preserving vector fields

Generalization of Yang-Mills equations for flat spacetime

Potential new methods for solving Einstein equations

Abstract

We give a formulation of the vacuum Einstein equations in terms of a set of volume-preserving vector fields on a four-manifold ${\cal M}$. These vectors satisfy a set of equations which are a generalisation of the Yang-Mills equations for a constant connection on flat spacetime.