The Fourth Root of Gravity

TL;DR

This paper explores a novel approach to gravity by expressing the metric and connection as quadratic and quartic forms of basis vectors attached to matter fields, aiming for a fully covariant formulation.

Contribution

It introduces a method to render free action densities fully covariant using basis vectors, connecting the metric, connection, and tetrads through quadratic and quartic forms.

Findings

Connection and tetrads are quadratic forms in basis vectors.

Spacetime metric becomes quartic in basis vectors.

Provides a covariant formulation of gravity using basis vectors.

Abstract

By attaching basis vectors to the components of matter fields, one may render free action densities fully covariant. Both the connection and the tetrads are quadratic forms in these basis vectors. The metric of spacetime, which is quadratic in the tetrads, is then quartic in the basis vectors.