The previous nine chapters have been mainly devoted to various black hole space-times and other solutions that at least contain stationary regions. The present chapter, and the following four, will concentrate on some of the most important exact solutions which represent gravitational waves in general relativity. It is convenient to start here with the simplest case of non-expanding waves, known as pp-waves, which are also important in the context of higher-dimensional theories, and their subclass of plane waves. General classes of solutions with non-expanding and expanding waves will be described, respectively, in Chapters 18 and 19, and both types of impulsive waves will be reviewed in Chapter 20. The collision and interaction of plane waves will be covered in Chapter 21. These may all be associated with other forms of pure radiation. Cylindrical gravitational waves will be more briefly described in Section 22.3.

The class of pp-waves describes plane-fronted waves with parallel rays. They are defined geometrically by the property that they admit a covariantly constant null vector field k, and may represent gravitational waves, electromagnetic waves, some other forms of matter, or any combination of these.

Using (2.15) and the properties described in Section 2.1.3, it follows from the defining property kμ;v = 0 that the vector field k is tangent to an expansion-free, shear-free and twist-free null geodesic congruence. Since this congruence is non-twisting, there exists a family of 2-surfaces orthogonal to k which may be considered as wave surfaces (see Kundt, 1961). It is also implied by the fact that k is covariantly constant that these wave surfaces are indeed planar and that rays orthogonal to them are parallel.