Lecture III: Ashtekar variables for general relativity (Courses in canonical gravity)

The Palatini action for general relativity, is simply the Einstein-Hilbert ac-tion rewritten so that it is not a function of metric, but instead a function of a \connection " ! and a \frame eld " e: S[e;!] = dx4 det g R[!]: This formalism provides rst-order eld equations for general relativity. 1.1 The tetrad formalism A frame eld (triad and tetrad) provides a way to specify geometries alter-native but equivalent to metrics or line elements. Suppose that, M is an oriented n-dimensional manifold dieomorphic to Rn. Physically, we can think of M as a small open set of space-time (since every manifold can be covered with charts dieomorphic to Rn). Since the tangent bundle of Rn is trivial, so is TM. A trivialization of TM, recall, is a vector bundle isomorphism e: M Rn ! TM fpg Rn 7! TpM (1.1) The trivialization e is also called a frame eld, since for each p 2M it sends the standard basis of Rn (the Minkowski space) to a basis of tangent vectors at p, or frame: If M is a 3-dimensional manifold, then, a frame eld on M is called a triad; and if M is a 4-dimensional manifold, a frame eld on M is called a tetrad. The idea of Palatini formalism is to do a lot of work on the trivial bundle M Rn, which serves as a kind of substitute for the tangent bundle. We 1 Canonical General Relativity 2 can pass and forth between M Rn and TM by using the frame eld e and its inverse e1: TM ! M Rn: (1.2) Let us suppose that we are in the n-dimensional Lorentzian case (other case works similarly). A section of M Rn is just an Rn-valued function on M, so, for any point p 2M, there is a natural basis of sections 0; ; n