

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.

