Scalar tensor theory of gravitation with torsion and conformally flat space-times

Full text: We develop a scalar-tensor theory of gravitation in a Riemann-Cartan space-time, where the trace of the torsion tensor is derived from a scalar potential. The theory is conformally invariant when the torsion's trace behaves as a gauge vector field. The field equations are determined from the requirement of compatibility between minimal action principle and minimal coupling procedure. Light and massive particles follow null and time-like auto-parallels curves, respectively. We develop the idea of Cartan gauges, where different configurations of the gravitational field are given by the metric and the torsion scalar gauge potential. The theory admits an Einstein gauge, where the torsion potential is null, the Riemann-Cartan space-time is transformed into a pseudo-Riemannian space-time and the equations of General Relativity are obtained. We apply this result to conformally flat space-times and show that, although the mathematics is the same, a rather new picture may arise when the pseudo-Riemannian spacetime is transformed into a Riemann-Cartan space-time with a Minkowski metric and a scalar field. We take advantage of this mathematical equivalence to examine how some conformally at spacetime configurations in General Relativity, that is, in the Einstein gauge, look when viewed from the standpoint of a Cartan gauge. (author)