Two dimensional Sen connections and quasi-local energy-momentum

TL;DR

This paper applies the two-dimensional Sen connection to define a quasi-local energy-momentum in general relativity, linking it to Penrose's charge integral and Dougan-Mason's mass, and characterizing pp-wave geometries via spinor fields.

Contribution

It demonstrates that the two-dimensional Sen connection naturally defines a quasi-local energy-momentum matching Dougan and Mason's formulation, and characterizes pp-wave geometries through spinor fields on a 2-sphere.

Findings

Penrose's charge integral expressed as a Nester--Witten integral

Holomorphy operators define acceptable spinor propagation laws

Zero quasi-local mass implies pp-wave geometry with pure radiation

Abstract

The recently constructed two dimensional Sen connection is applied in the problem of quasi-local energy-momentum in general relativity. First it is shown that, because of one of the two 2 dimensional Sen--Witten identities, Penrose's quasi-local charge integral can be expressed as a Nester--Witten integral.Then, to find the appropriate spinor propagation laws to the Nester--Witten integral, all the possible first order linear differential operators that can be constructed only from the irreducible chiral parts of the Sen operator alone are determined and examined. It is only the holomorphy or anti-holomorphy operator that can define acceptable propagation laws. The 2 dimensional Sen connection thus naturally defines a quasi-local energy-momentum, which is precisely that of Dougan and Mason. Then provided the dominant energy condition holds and the 2-sphere S is convex we show that the next statements are equivalent: i. the quasi-local mass (energy-momentum) associated with S is zero; ii.the Cauchy development $D(\Sigma)$ is a pp-wave geometry with pure radiation ($D(\Sigma)$ is flat), where $\Sigma$ is a spacelike hypersurface whose boundary is S; iii. there exist a Sen--constant spinor field (two spinor fields) on S. Thus the pp-wave Cauchy developments can be characterized by the geometry of a two rather than a three dimensional submanifold.