### 2.2 Dynamics

Within a particular geometry, usually various options for the dynamics of the fields (field equations, in
particular as following from a Lagrangian) exist as well as different possibilities for the identification of
physical observables with the mathematical objects of the formalism. Thus, in general relativity, the field
equations are derived from the Lagrangian
where is the Ricci scalar, the cosmological constant, and the matter
Lagrangian depending on the metric, its first derivatives, and the matter variables. This Lagrangian leads to
the well-known field equations of general relativity,
with the energy-momentum(-stress) tensor of matter
and , where is Newton’s gravitational constant. is called the Einstein
tensor. In empty space, i.e., for , Equation (92) reduces to
If only an electromagnetic field derived from the 4-vector potential is present in the
energy-momentum tensor, then the Einstein–Maxwell equations follow:
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

implied by it.
For point particles, due to the singularities appearing, in general this is a tricky task, up to now solved only
approximately. However, the world lines for point particles falling freely in the gravitational field are, by
definition, the geodesics of the Riemannian metric. This definition is consistent with the rigourous
derivation of the geodesic equation for non-interacting dust particles in a fluid matter description. It is also
consistent with all observations.
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]).