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

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.

15 pages, Plain Tex, no figures

References in corpus (5)

• The World as a Hologram
• Thermodynamics of Spacetime: The Einstein Equation of State
• Two dimensional Sen connections in general relativity
• Two dimensional Sen connections and quasi-local energy-momentum
• Total Mass-Momentum of Arbitrary Initial-Data Sets in General Relativity

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