The components of the metrical tensor are
identified with gravitational potentials. Consequently, the
components of the (Levi-Civita) connection correspond
to the gravitational “field strength”, and the components of the
curvature tensor to the gradients of the gravitational field. The
equations of motion of material particles should follow, in
principle, from Equation (92) through the relation
For most of the unified field theories to be
discussed in the following, such identifications were made on
internal, structural reasons, as no link-up to empirical data was
possible. Due to the inherent wealth of constructive possibilities,
unified field theory never would have come off the ground proper as
a physical theory even if all the necessary formal requirements
could have been satisfied. As an example, we take the
identification of the electromagnetic field tensor with either the
skew part of the metric, in a “mixed geometry” with metric
compatible connection, or the skew part of the Ricci tensor in
metric-affine theory, to list only two possibilities. The latter
choice obtains likewise in a purely affine theory in which the
metric is a derived secondary concept. In this case, among the many
possible choices for the metric, one may take it proportional to
the variational derivative of the Lagrangian with respect to the
symmetric part of the Ricci tensor. This does neither guarantee the
proper signature of the metric nor its full rank. Several
identifications for the electromagnetic 4-potential and the
electric current vector density have also been suggested
(cf. below and [143]).