Variational techniques in general relativity: A metric-affine approach to Kaluza's theory

TL;DR

This paper introduces a new variational principle in general relativity that treats the metric and connection as independent variables, with applications to unifying gravity and electromagnetism in Kaluza's theory.

Contribution

It presents a novel unconstrained variational approach involving both metric and connection, extending the framework of general relativity and applying it to Kaluza's unification theory.

Findings

Extremals correspond to Ricci-flat metrics with compatible Riemannian connections.

The variational principle allows independent variation of metric and connection.

Application to Kaluza's theory unifies gravitational and electromagnetic interactions.

Abstract

A new variational principle for General Relativity, based on an action functional $I\/(\Phi,\nabla)\/$ involving both the metric $\Phi\/$ and the connection $\nabla\/$ as independent, \emph{unconstrained\/} degrees of freedom is presented. The extremals of $I\/$ are seen to be pairs $\/(\Phi,\nabla)\/$ in which $\Phi\/$ is a Ricci flat metric, and $\nabla\/$ is the associated Riemannian connection. An application to Kaluza's theory of interacting gravitational and electromagnetic fields is discussed.