Metric-Affine Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invariance

Abstract

In Einstein's gravitational theory, the spacetime is Riemannian, that is, it has vanishing torsion and vanishing nonmetricity (covariant derivative of the metric). In the gauging of the general affine group ${A}(4,R)$ and of its subgroup ${GL}(4,R)$ in four dimensions, energy--momentum and hypermomentum currents of matter are canonically coupled to the one--form basis and to the connection of a metric--affine spacetime with nonvanishing torsion and nonmetricity, respectively. Fermionic matter can be described in this framework by half--integer representations of the $\overline{SL}(4,R)$ covering subgroup. --- We set up a (first--order) Lagrangian formalism and build up the corresponding Noether machinery. For an arbitrary gauge Lagrangian, the three gauge field equations come out in a suggestive Yang-Mills like form. The conservation--type differential identities for energy--momentum and hypermomentum and the corresponding complexes and superpotentials are derived. Limiting cases such as the Einstein--Cartan theory are discussed. In particular we show, how the ${A}(4,R)$ may ``break down'' to the Poincar\'e (inhomogeneous Lorentz) group. In this context, we present explicit models for a symmetry breakdown in the cases of the Weyl (or homothetic) group, the ${SL}(4,R)$, or the ${GL}(4,R)$.