2 The Possibilities of Generalizing General Relativity: A Brief Overview

As a rule, the point of departure for unified field theory was general relativity. The additional task then was to “geometrize” the electromagnetic field. In this review, we will encounter essentially five different ways to include the electromagnetic field into a geometric setting:

• by connecting an additional linear form to the metric through the concept of “gauging” (Weyl);
• by introducing an additional space dimension (Kaluza);
• by choosing an asymmetric Ricci tensor (Eddington);
• by adding an antisymmetric tensor to the metric (Bach, Einstein);
• by replacing the metric by a 4-bein field (Einstein).

In order to bring some order into the wealth of these attempts towards “unified field theory,” I shall distinguish four main avenues extending general relativity, according to their mathematical direction: generalisation of

• geometry,
• dynamics (Lagrangians, field equations),
• number field, and
• dimension of space,

as well as their possible combinations. In the period considered, all four directions were followed as well as combinations between them like e.g., five-dimensional theories with quadratic curvature terms in the Lagrangian. Nevertheless, we will almost exclusively be dealing with the extension of geometry and of the number of space dimensions.