A Radiation Scalar for Numerical Relativity

This Letter describes a scalar curvature invariant for general relativity with a certain, distinctive feature. While many such invariants exist, this one vanishes in regions of space-time which can be said unambiguously to contain no gravitational radiation. In more general regions which incontrovertibly support nontrivial radiation fields, it can be used to extract local, coordinate-independent information partially characterizing that radiation. While a clear, physical interpretation is possible only in such radiation zones, a simple algorithm can be given to extend the definition smoothly to generic regions of space-time.

Figure 1

The Riemann surface ${\mathcal{Z}}_{\mathcal{S}}$ is a triple cover of the Riemann sphere. The values of $Z(\mathcal{S})$ on its three sheets correspond to the three values of $\xi$. The labels in the figure indicate contiguous regions of ${\mathcal{Z}}_{\mathcal{S}}$ pictured on different sheets.