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

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 $m_{\mathrm{\$}}$, associated with the smooth boundary $\mathrm{\$}:=\mathrm{\partial }\mathrm{\Sigma }\approx S^2$ of a spacelike hypersurface $\mathrm{\Sigma }$, is equivalent to the statement that the Cauchy development $D(\mathrm{\Sigma })$ is of a {\it pp}-wave type geometry with pure radiation, provided the ingoing null normals are not diverging on $\mathrm{\$}$ and the dominant energy condition holds on $D(\mathrm{\Sigma })$. The metric on $D(\mathrm{\Sigma })$ 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 $D(\mathrm{\Sigma })$ are completely determined by the value of the zero-rest-mass-field on $\mathrm{\$}$ and the two dimensional Sen–geometry of $\mathrm{\$}$ 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 $\mathrm{\Sigma }$ but by data on its {\it two} dimensional boundary $\mathrm{\$}$ 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.