Quasi-local energy - momentum and 2-surface characterization of the pp-wave spacetimes

Abstract

We consider the determination of the pp-wave metric from the geometry of certain spacelike 2-surfaces. It has been shown that the vanishing of the Dougan - Mason quasi-local mass , associated with the smooth boundary of a spacelike hypersurface , is equivalent to the statement that the Cauchy development is of a pp-wave type geometry with pure radiation, provided the ingoing null normals are not diverging on and the dominant energy condition holds on . The metric on itself, however, has not been determined. Here, assuming that the matter is a zero-rest-mass field, it is shown that both the matter field and the pp-wave metric of are completely determined by the value of the zero-rest-mass field on and the two-dimensional Sen geometry of , provided a convexity condition, slightly stronger than above, holds. Thus the pp-waves can be characterized not only by the usual Cauchy data on a three-dimensional but by data on its two-dimensional boundary too. In addition, it is shown that the Ludvigsen - Vickers quasi-local angular momentum of axially symmetric pp-wave geometries has the familiar properties known for pure (matter) radiation.