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

A new variational principle for General Relativity, based on an action functional $I\/(Φ,\nabla)\/$ involving both the metric $Φ\/$ and the connection $\nabla\/$ as independent, \emph{unconstrained\/} degrees of freedom is presented. The extremals of $I\/$ are seen to be pairs $\/(Φ,\nabla)\/$ in which $Φ\/$ 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.