18 - Kundt solutions

Summary

Solutions are said to belong to Kundt's class if they admit a null geodesic congruence, generated by a vector field k, which is shear-free, twist-free and expansion-free. They include the pp-waves and plane wave space-times that have been described in the previous chapter. The whole class, initially investigated by Kundt (1961, 1962) in the case of vacuum or with an aligned pure radiation field, is however much wider. Because the null vector field k is not in general covariantly constant, the rays of the corresponding non-expanding waves are not necessarily parallel, as in the case of pp-waves, and the wave surfaces need not be planar. This greater freedom permits, for example, the presence of a cosmological constant Λ or aligned electromagnetic fields, both null and non-null.

For vacuum or some specific matter content, generalised Goldberg—Sachs theorems imply that the Kundt space-times must be algebraically special, that is of Petrov type II, D, III or N (or conformally flat), with k being a repeated principal null direction of the Weyl tensor. Any Einstein—Maxwell and pure radiation fields must also be aligned: that is, k is the common eigendirection of the Weyl and Ricci tensor.

A number of physically interesting space-times of this class are explicitly known. These will be briefly described in this chapter, with emphasis given to those that describe exact non-expanding gravitational waves of various kinds.