Two dimensional Sen connections in general relativity

TL;DR

This paper constructs a two-dimensional version of the Sen connection in general relativity, revealing its geometric properties, relations to curvature, and deriving identities relevant to twistor theory and charge integrals.

Contribution

It introduces a novel 2D Sen connection framework, linking it to twistor operators, spacetime curvature, and gauge invariants, advancing the understanding of 2-surface geometry in GR.

Findings

The 2D Sen operator's curvature relates to spacetime's anti-self-dual curvature.

The difference between Sen and induced spin connections is the anti-self-dual torsion.

Derived identities connect twistor operators, the Sen operator, and Penrose's charge integral.

Abstract

The two dimensional version of the Sen connection for spinors and tensors on spacelike 2-surfaces is constructed. A complex metric $\gamma_{AB}$ 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 $\Delta_e$ 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 $\Delta_e$ 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.