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

TL;DR

This paper demonstrates that pp-wave spacetimes can be characterized by data on a two-surface boundary, linking quasi-local mass and matter fields to the geometry of spacetime in a novel way.

Contribution

It shows that pp-wave metrics and matter fields are fully determined by boundary data on a two-surface, extending the understanding of initial data in general relativity.

Findings

Vanishing Dougan--Mason quasi-local mass implies pp-wave geometry.

Matter and metric are determined by boundary zero-rest-mass-field and Sen--geometry.

Ludvigsen--Vickers angular momentum properties hold for axially symmetric pp-waves.

Abstract

In the present paper the determination of the {\it pp}-wave metric form the geometry of certain spacelike two-surfaces is considered. It has been shown that the vanishing of the Dougan--Mason quasi-local mass $m_{\$}$, associated with the smooth boundary $\$:=\partial\Sigma\approx S^2$ of a spacelike hypersurface $\Sigma$, is equivalent to the statement that the Cauchy development $D(\Sigma)$ is of a {\it pp}-wave type geometry with pure radiation, provided the ingoing null normals are not diverging on $\$ $ and the dominant energy condition holds on $D(\Sigma)$. The metric on $D(\Sigma)$ 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 {\it pp}-wave metric of $D(\Sigma)$ 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 {\it pp}-waves can be characterized not only by the usual Cauchy data on a {\it three} dimensional $\Sigma$ but by data on its {\it two} dimensional boundary $\$ $ too. In addition, it is shown that the Ludvigsen--Vickers quasi-local angular momentum of axially symmetric {\it pp}-wave geometries has the familiar properties known for pure (matter) radiation.