Neighbours of Einstein's Equations: Connections and Curvatures

TL;DR

This paper explores the relationship between the Riemann tensor of a metric derived from Einstein's equations reformulated with an SO(3,C) connection and the connection's curvature tensor, highlighting special cases where metrics coincide.

Contribution

It analyzes the complex relation between the Riemann tensor and connection curvature in a reformulation of Einstein's equations, identifying conditions for metric coincidence.

Findings

In Einstein's case, two natural SO(3) metrics on fibers coincide.

The general case involves a bimetric structure on fibers.

The relation between Riemann and connection curvature is generally complex.

Abstract

Once the action for Einstein's equations is rewritten as a functional of an SO(3,C) connection and a conformal factor of the metric, it admits a family of ``neighbours'' having the same number of degrees of freedom and a precisely defined metric tensor. This paper analyzes the relation between the Riemann tensor of that metric and the curvature tensor of the SO(3) connection. The relation is in general very complicated. The Einstein case is distinguished by the fact that two natural SO(3) metrics on the GL(3) fibers coincide. In the general case the theory is bimetric on the fibers.