Ashtekar Variables: Structures in Bundles

TL;DR

This paper reviews Ashtekar variables in canonical gravity using fibre bundle theory, clarifying the geometric structure of the densitized dreibein and SU(2) connection, with applications to FRW spacetimes.

Contribution

It provides a fibre bundle perspective on Ashtekar variables, elucidating the geometric origin of the connection and dreibein in canonical gravity.

Findings

Clarifies the geometric structure of Ashtekar variables

Shows the Ashtekar connection as Levi-Civita plus Weingarten mapping

Applies the framework to Friedmann-Robertson-Walker spacetimes

Abstract

Canonical gravity can be formulated by means of a densitized dreibein together with an SU(2) connection. These so-called Ashtekar variables are the fundamental quantities, loop quantum gravity is resting on. In this paper we review these variables from the perspective of fibre bundles. This is straightforward for the dreibein field as this is simply a frame. The Ashtekar connection, however, is more complicated. It turns out, that at the level of the tangent bundle, it is given by the Levi-Civita connection plus a multiple of the Weingarten mapping, whose action on vector fields is induced from the vector product on R^3. Lifted to the spin bundle, one regains the well-known SU(2) Ashtekar connection. At the end, we apply our results to FRW spacetimes.