Dynamics

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

√ --- ℒ = − g (R + 2Λ − 2κLM ),

where $R(g ) ik$ is the Ricci scalar, $g := det g ,Λ ik$ the cosmological constant, and $L M$ 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 $κ = 8πG4- c$ , where $G$ is Newton’s gravitational constant. $Gik := Rik − 1Rgik 2$ is called the Einstein tensor. In empty space, i.e., for $Tik = 0$ , Equation (92

) reduces to

If only an electromagnetic field $∂A ∂A Fik = -∂xki − ∂xik$ derived from the 4-vector potential $Ak$ 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 it⁴¹ . 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 ]).