Spin and torsion in general relativity II: Geometry and field equations

Abstract

From physical arguments space-time is assumed to possess a connection

$\left\{ {_{ij}^k } \right\}$ is Christoffel's symbol built up from the metric g _ij and already appearing in General Relativity (GR). Cartan's torsion tensor$s_{ij} ^k = \tfrac{1}{2}\left( {\Gamma _{ij}^k - \Gamma _{ji}^k } \right)$ and the contortion tensor K _ij ^k, in contrast to the theory presented here, both vanish identically in conventional GR.

Using the connection introduced above, in this series of articles we will discuss the consequences for GR in the framework of a consistent formalism. There emerges a theory describing in a unified way gravitation and a very weakspin-spin contact interaction.

In Part I of this work† we discussed the foundations of the theory.

In this Part II we present in section 3 the geometrical apparatus necessary for the formulation of the theory. In section 4 we take the curvature scalar (or rather its density) as Lagrangian density of the field. In this way we obtain in subsection 4.1 the field equations in their explicit form. In particular it turns out that torsion is essentially proportional to spin. We then derive the angular momentum and the energy-momentum theorems (subsections 4.2-4); the latter yields a force proportional to curvature, acting on any matter with spin. In subsection 4.5 we compare the theory so far developed with GR. Torsion leads to a universal spin-spin contact