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

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.