The high-frequency limit in general relativity

Abstract

A precise characterization of the high-frequency limit in general relativity (vacuum space-times) is presented. The averaging schemes used in earlier works are avoided by the use of weak limits, giving the characterization the advantage that the high-frequency limit can be rigorously analyzed. Using this characterization, a theorem to the effect that a certain ‘‘effective stress-energy’’ tensor acts as a source of curvature of the smooth background space-time is proved. It is shown in the special case corresponding to the presence of a single wave that this tensor has the same form as the stress-energy tensor of a null fluid. Also, the extension of this characterization of the high-frequency limit to other fields, such as electromagnetic fields, and to nonvacuum space-times is briefly discussed.