Ashtekar variables

In the ADM formulation of general relativity, spacetime is split into spatial slices and a time axis. The basic variables are taken to be the induced metric on the spatial slice and the metric's conjugate momentum , which is related to the extrinsic curvature and is a measure of how the induced metric evolves in time. These are the metric canonical coordinates. In 1986 Abhay Ashtekar introduced a new set of canonical variables, Ashtekar (new) variables to represent an unusual way of rewriting the metric canonical variables on the three-dimensional spatial slices in terms of an SU(2) gauge field and its complementary variable. Ashtekar variables provide what is called the connection representation of canonical general relativity, which led to the loop representation of quantum general relativity and in turn loop quantum gravity and quantum holonomy theory. Let us introduce a set of three vector fields , that are orthogonal, that is, The are called a triad or drei-bein (German literal translation, "three-leg"). There are now two different types of indices, "space" indices that behave like regular indices in a curved space, and "internal" indices which behave like indices of flat-space (the corresponding "metric" which raises and lowers internal indices is simply ). Define the dual drei-bein as We then have the two orthogonality relationships where is the inverse matrix of the metric (this comes from substituting the formula for the dual drei-bein in terms of the drei-bein into and using the orthogonality of the drei-beins). and (this comes about from contracting with and using the linear independence of the ). It is then easy to verify from the first orthogonality relation (employing ) that we have obtained a formula for the inverse metric in terms of the drei-beins - the drei-beins may be thought of as the "square-root" of the metric (the physical meaning to this is that the metric , when written in terms of a basis , is locally flat). Actually what is really considered is which involves the densitized drei-bein instead (densitized as ).