Scalar torsion and a new symmetry of general relativity

TL;DR

This paper reformulates general relativity within Riemann-Cartan geometry, introducing scalar torsion and a new symmetry called Cartan transformations, which reveal different perspectives of the same gravitational phenomena.

Contribution

It introduces a scalar torsion formulation of general relativity with a new invariance group, expanding the geometric and symmetry understanding of the theory.

Findings

General relativity can be expressed as a scalar-tensor theory in Riemann-Cartan geometry.

Cartan transformations relate different gauges, providing alternative descriptions of gravitational phenomena.

Conservation laws for auto-parallel motion are derived in a static spherically symmetric vacuum space-time.

Abstract

We reformulate the general theory of relativity in the language of Riemann-Cartan geometry. We start from the assumption that the space-time can be described as a non-Riemannian manifold, which, in addition to the metric field, is endowed with torsion. In this new framework, the gravitational field is represented not only by the metric, but also by the torsion, which is completely determined by a geometric scalar field. We show that in this formulation general relativity has a new kind of invariance, whose invariance group consists of a set of conformal and gauge transformations, called Cartan transformations. These involve both the metric tensor and the torsion vector field, and are similar to the well known Weyl gauge transformations. By making use of the concept of Cartan gauges, we show that, under Cartan transformations, the new formalism leads to different pictures of the same gravitational phenomena. We show that in an arbitrary Cartan gauge general relativity has the form of a scalar-tensor theory. In this approach, the Riemann-Cartan geometry appears as the natural geometrical setting of the general relativity theory when the latter is viewed in an arbitrary Cartan gauge. We illustrate this fact by looking at the one of the classical tests of general relativity theory, namely the gravitational spectral shift. Finally, we extend the concept of space-time symmetry to the more general case of Riemann-Cartan space-times endowed with scalar torsion. As an example, we obtain the conservation laws for auto-parallel motion in a static spherically symmetric vacuum space-time in a Cartan gauge, whose orbits are identical to Schwarzschild orbits in general relativity.