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 $I\text{/}(\mathrm{\Phi },\mathrm{\nabla })\text{/}$ involving both the metric $\mathrm{\Phi }\text{/}$ and the connection $\mathrm{\nabla }\text{/}$ as independent, \emph{unconstrained/} degrees of freedom is presented. The extremals of $I\text{/}$ are seen to be pairs $\text{/}(\mathrm{\Phi },\mathrm{\nabla })\text{/}$ in which $\mathrm{\Phi }\text{/}$ is a Ricci flat metric, and $\mathrm{\nabla }\text{/}$ is the associated Riemannian connection. An application to Kaluza’s theory of interacting gravitational and electromagnetic fields is discussed.