5 - The ‘Chiral’ Formulation of General Relativity

Summary

This is the longest Chapter of the book introducing the "chiral" formulations of 4D GR. The most important concept here is that of self-duality. We describe the associated decomposition of the Riemann tensor, and then the chiral version of Einstein-Cartan theory, together with its Yang-Mills analog. The geometry that is necessary to understand the fact that the knowledge of the Hodge star is equivalent to the knowledge of the conformal metric is explained in some detail. We also describe the different signature pseudo-orthogonal groups in 4 dimensions, and in particular explain that it is natural to put coordinates of a 4D space into a 2x2 matrix, the fact that is going to play central role in the later description of twistors. The notions of the chiral part of the spin connection, as well as the chiral soldering form are introduced. We give an example of a computation of Riemann curvature using the chiral formalism, to illustrate its power. We then describe the Plebanski formulation, as well as its linearisation. This allows to derive the linearisation of the chiral pure connection action, to be studied in full in the following Chapter. We describe coupling to matter in Plebanski formalism, and then some alternative descriptions related to Plebanski formalism.