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 $q_{ab}(x)$ on the spatial slice and the metric's conjugate momentum $K^{ab}(x)$ , which is related to the extrinsic curvature and is a measure of how the induced metric evolves in time.^[1] 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.^[2]

^ Gravitation by Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, published by W. H. Freeman and company. New York.
^ Ashtekar, A (1986). "New variables for classical and quantum gravity". Physical Review Letters. 57 (18): 2244–2247. Bibcode:1986PhRvL..57.2244A. doi:10.1103/physrevlett.57.2244. PMID 10033673.

[1] Gravitation by Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, published by W. H. Freeman and company. New York.

[2] Ashtekar, A (1986). "New variables for classical and quantum gravity". Physical Review Letters. 57 (18): 2244–2247. Bibcode:1986PhRvL..57.2244A. doi:10.1103/physrevlett.57.2244. PMID 10033673.

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