Direct gauging of the Poincar� group. IV. Curvature, holonomy, spin, and gravity

Abstract

The minimal replacement operator of direct Poincare gauge theory converts Minkowski space-time to a new spaceU4 with curvature and torsion. Explicit representations of the connection and curvature forms ofU4 are obtained. This enable us to prove that the Ricci lemma is always satisfied (U4 is a Riemann-Cartan space) and that the holonomy group ofU4 is the component of the Lorentz group that is continuously connected to the identity. A specific form of the “free field” Lagrangian for local action of the Poincare group is studied. Although the Lagrangian is independent of torsion, spin currents are supported by a system of algebraic relations between spin current and torsion. The field equations for the translation part of theP10 gauge fields are shown to be relations between the Ricci curvature and the total momentum-energy tensors, although these equations have nontrivial skew-symmetric parts whenever torsion is present. If the spin currents vanish and the total momentum-energy tensor is symmetric, Einstein's equations of general relativity with cosmological constant obtain as exact rather than approximate results. This leads to explicit evaluations of all coupling constants for theP10 sector, and to the fact that any solution of Einstein's field equations has the proper orthochronous Lorentz group as holonomy group. Direct gauge theory for the Poincare group thus provides a simple and explicit method of introducing gravitational effects whenever an adequate description of matter and internal gauge structures is known on Minkowski space. An alternative system of field variables is shown to lead to a decomposition of the gravitational equations that is analogous to the decomposition of the Klein-Gordon equation via Dirac spinors.