2011/12/06 by Christian Fleischhack, Fleischhack, Christian, Philipp Levermann +1
Mathematics · Physics and Astronomy · #53B05 (Primary) 53C07 #53C05 #83C05 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph) #gr-qc #math-ph #math.DG #math.MP #msc:53B05 #msc:53C05 #msc:53C07 #msc:83C05
paper · pdf · doi:10.48550/arxiv.1112.1262
11 pages, LaTeX
arxiv created 2011/12/06 · arxiv updated 2011/12/07
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 R3. Lifted to the spin bundle, one regains the well-known SU(2) Ashtekar connection. At the end, we apply our results to FRW spacetimes.