Form Geometry and the 'tHooft-Plebanski Action

TL;DR

This paper explores a novel formulation of four-dimensional Riemannian geometry and Einstein's equations using a connection that acts on two-forms, with a new approach to the conformal factor as a variational field.

Contribution

It introduces a new perspective on the conformal factor in the 'tHooft-Plebanski action, treating it as a dynamical field rather than a fixed component.

Findings

Reformulation of Riemannian geometry using two-form connections.

New treatment of the conformal factor as a variational field.

Insights into the geometric structure underlying Einstein's equations.

Abstract

Riemannian geometry in four dimensions, including Einstein's equations, can be described by means of a connection that annihilates a triad of two-forms (rather than a tetrad of vector fields). Our treatment of the conformal factor of the metric differs from the original presentation of this result, due to 'tHooft. In the action the conformal factor now appears as a field to be varied.