Ashtekar variables

In the ADM formulation of general relativity one splits spacetime into spatial slices and time, the basic variables are taken to be the induced metric, ${\displaystyle q_{ab}(x)}$, on the spatial slice (the metric induced on the spatial slice by the spacetime metric and its conjugate momentum variable is related to the extrinsic curvature, ${\displaystyle K^{ab}(x)}$ (this tells us how the spatial slice curves with respect to spacetime 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.

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