20 - Impulsive waves

Summary

Physical situations exist in which different regions of space-time have different matter contents. These can be modelled by compound space-times. (For example, in Subsection 9.5.2, a space-time was discussed in which a region represented by the Vaidya metric is sandwiched between a Minkowski and a Schwarzschild region.) In such cases, we have followed the approach of Lichnerowicz (1955) and used a global metric form that is at least C³ everywhere except on junctions, represented by hypersurfaces N, on which the metric is only C. Since the curvature of the space-time involves second derivatives of the metric, such situations give rise to discontinuities in the curvature across N. When N is null, these may represent various forms of shock waves which propagate with the speed of light.

More extreme situations may also be considered for which the metric is still (at least) C³ almost everywhere, but merely C⁰ on some hypersurface N. In such situations, some components of the curvature of the space-time will formally contain a δ-function. When N is null, these may be interpreted as impulsive waves. They are regarded as impulsive gravitational waves when the δ-function components occur in the Weyl tensor, or impulsive components of some kind of null matter when they occur in the Ricci tensor.

The geometry of impulsive waves in flat space was first described in detail by Penrose (1972). However, some particular examples of exact solutions which include impulsive gravitational waves or thin sheets of null matter were known before then. Many further examples have subsequently been obtained, and involve a variety of backgrounds.