Two-dimensional Sen connections and quasi-local energy--momentum

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 two-dimensional Sen--Witten identities, Penrose's quasi-local charge integral can be expressed as a Nester--Witten integral. Then, to find the spinor propagation laws appropriate to the Nester--Witten integral, all the possible first order linear differential operators that can be constructed from the irreducible chiral parts, only, 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 two-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 is convex, we show that the following statements are equivalent: (i) the quasi-local mass (energy--momentum) associated with a 2-sphere is zero; (ii) the Cauchy development is a pp-wave geometry with pure radiation ( is flat), where is a spacelike hypersurface with ; (iii) there exists a Sen-constant spinor field (two spinor fields) on . Thus the pp-wave Cauchy developments can be characterized by the geometry of a two- rather than a three-dimensional submanifold.