Einstein-Cartan Theory

In the last section, several generalizations of GRT have been mentioned. Einstein stated in his book ''Grundzüge der Relativitätstheorie'' [10] (in English ''The Meaning of Relativity'' [11]): There is no proof that nature can be always described by the simplest theory, but one should use the simplest theory unless there are physical reasons to use a more complicated one^1.7. As discussed in the previous sections GRT is no longer sufficient for our purpose. This means one should use a generalization of GRT which makes only the necessary generalizations needed for constitutive theory. A theory of gravitation that allows non-symmetric energy-momentum tensors with non-vanishing divergence and that also has a geometric quantity to that the spin can couple directly is needed.
For the purpose of constitutive theory the Einstein-Cartan theory is the most suitable generalization of GRT. ECT contains geometric quantities to which both, energy-momentum and spin, can couple. Energy-momentum is coupled to curvature and spin is coupled to torsion. The fundamental quantity is no longer only the metric, but the metric and the (non-symmetric) connection, whose skew-symmetric part defines the torsion tensor. From this follows that there is a reason for including the connection (or the Ricci tensor) in the state space. It is possible to construct covariant state spaces which are compatible with the concept of extended thermodynamics.
The notation is mainly the one that is used by J.A. Schouten [40,41], with the exception that torsion will be denoted by $\mathcal{T}$.
Einstein-Cartan theory is formulated in Riemann-Cartan space.