Gravity and the Poincaré group.

Abstract

We discuss gravity as a gauge theory of the Poincar\'e group in three and four dimensions, i.e., in a metric-independent fashion. The fundamental fields of the theory are the gauge potentials, the matter fields, and the so-called Poincar\'e coordinates ${q}^{a}(x)$ a set of fields that are defined on the space-time manifold, but that transform as Poincar\'e vectors under gauge transformations. The presence of such coordinates is necessary in order to construct a gauge theory of the Poincar\'e group. We discuss the procedure needed to connect this theory with the Einsteinian formulation of gravity, and we show that the field equations for the gauge potentials, for pointlike sources, and for scalar and spinor matter fields reproduce the Einstein equations, the geodesics equations, and the Klein-Gordon and the Dirac equations in curved space-time, respectively. In 2+1 dimensions and in the presence of pointlike sources this gauge-theoretical approach can be further developed: the gauge potentials can be written almost everywhere as pure gauge, and a solution of the field equations provides, at the same time, the space-time metric and the set of coordinates that globally flatten the metric.