In this article we review some basic results connnected with the use of the ‘Sparling 3-form’, a 3-form defined on the spin bundle of space-time. It is closed if and only if the vacuum equations are satisfied. This structure unifies, at least partially, some of the major spinorial developments in GR.

Ashtekar's spinorial variables for general relativity are extended to covariant variables for space-time and the Einstein-Hilbert action is given in terms of these. The reduction to the canonical formalism yields the Sparling 3-form as the gravitational Hamiltonian derived from the Einstein-Hilbert action. It is the gravitational part of Witten's integrand for the ADM energy. We derive Penrose's quasi-local mass formula from the associated boundary term when the form is restricted to appropriate sections of the spin bundle. The components of the angular momentum twistor for a 2-surface S arise as Hamiltonians generating motions of a spanning 3-surface that tend to motions generated by quasi-Killing vectors obtained from solutions of the 2-surface twistor equation on S. Connections with super-gravity are discussed.

We present some related ideas. The 3-form extends to one of a collection of 3-forms on the bundle of general frames. When pulled back to spacetime using a coordinate section of the frame bundle, this collection gives the pseudo-energy-momentum tensor of the gravitational field.