Form Connections

TL;DR

This paper explores how different subgroup choices within SL(3) connections in four-dimensional Riemannian geometry lead to various gravitational equations, including Einstein's and non-Einstein metrics, with implications for 3D space-times.

Contribution

It demonstrates how alternative subgroup selections in SL(3) connections produce a family of gravitational equations beyond Einstein's, including explicit non-Einstein solutions.

Findings

Multiple gravitational equations arise from different subgroup choices.

An explicit non-Einstein metric solution is constructed.

Remarks on three-dimensional space-times are included.

Abstract

Riemannian geometry in four dimensions naturally leads to an SL(3) connection that annihilates a basis for self-dual two-forms. Einstein's equations may be written in terms of an SO(3) connection, with SO(3) chosen as an appropriate subgroup of SL(3). We show how a set of "neighbours" of Einstein's equations arises because the subgroup may be chosen in different ways. An explicit example of a non-Einstein metric obtained in this way is given. Some remarks on three dimensional space-times are made.