Almost Symmetric Spaces and Gravitational Radiation

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Abstract
A generalization of the idea of Killing fields to spaces which are not symmetric is given. The field so defined specifies coordinate lines along which the variation of the metric tensor is the slowest possible in a global sense. Thus it generalizes the Killing fields in spaces with a symmetry where the metric tensor does not change along Killing trajectories. Several examples are given, and the method is then applied to spaces containing gravitational radiation of the type considered by Issacson. For spaces containing radiation, it is shown that a real functional λ[ξ], associated with every vector field ξ, measures some parameters associated with the radiation. In the simplest case this parameter is the ``energy density'' of the radiation, but, if a sufficient number of vector fields can be invariantly defined in the background, the average gravitational ``stress'' associated with the wave may also be measured. We conclude with some conjectures about further application of these ideas to the theory of gravitational radiation.