Two-dimensional Sen connections in general relativity

Abstract

The two dimensional version of the Sen connection for spinors and tensors on spacelike 2-surfaces is constructed. A complex metric γAB on the spin spaces is found which characterizes both the algebraic and extrinsic geometrical properties of the 2-surface $. The curvature of the two dimensional Sen operator ∆e is the pull back to $ of the anti-self-dual part of the spacetime curvature while its ‘torsion ’ is a boost gauge invariant expression of the extrinsic curvatures of $. The difference of the 2 dimensional Sen and the induced spin connections is the anti-self-dual part of the ‘torsion’. The irreducible parts of ∆e are shown to be the familiar 2-surface twistor and the Weyl–Sen– Witten operators. Two Sen–Witten type identities are derived, the first is an identity between the 2 dimensional twistor and the Weyl–Sen–Witten operators and the integrand of Penrose’s charge integral, while the second contains the ‘torsion ’ as well. For spinor fields satisfying the 2-surface twistor equation the first reduces to Tod’s formula for the kinematical twistor.