Following Ashtekar's (1987) recently revised version of the standard canonical theory, the construction of a new variables canonical formalism for Einstein's theory of gravity is investigated when the time parameter has level sets which are null hypersurfaces. The configuration space variables are the components of a tetrad and the self-dual components of a connection. Because a null time parameter is used, the Hamiltonian formalism has second-class constraints as well as the first-class constraints which are associated with the invariance of the theory under diffeomorphisms and local gauge transformations. The first-class constraint algebra is discussed and reality conditions which relate the complex formalism to real general relativity are displayed.

We use the canonical formalism developed together with David Robinson to study the Einstein equations on a null surface. Coordinate and gauge conditions are introduced to fix the triad and the coordinates on the null surface. Together with the previously found constraints, these form a sufficient number of second class constraints so that the phase space is reduced to one pair of canonically conjugate variables: A 3 2 and Σ 3 2 . The formalism is related to both the Bondi-Sachs and the Newman-Penrose methods of studying the gravitational field at null infinity. Asymptotic solutions in the vicinity of null infinity which exclude logarithmic behavior require the connection to fall off like 1/r 3 after the Minkowski limit. This, of course, gives the previous results of Bondi-Sachs and Newman-Penrose. Introducing terms which fall off more slowly leads to logarithmic behavior which leaves null infinity intact, allows for meaningful gravitational radiation, but the peeling theorem does not extend to Ψ 1 in the terminology of Newman-Penrose. The conclusions are in agreement with those of Chrusciel, MacCallum, and Singleton. This work was begun as a preliminary study of a reduced phase space for quantization of general relativity.

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The constraint equations for vacuum general relativity, formulated in terms of the new Hamiltonian variables, are investigated. A simple local coordinate presentation of the equations is used.