This not quite true for more general pp-waves. Plane waves are important for many reasons; to mention just one, they are essential for the beautiful topic of colliding plane waves. A more general subclass consists of the axisymmetric pp-waves, which in general have a two-dimensional Abelian Lie algebra of Killing vector fields. These are also called SG2 plane waves, because they are the second type in the symmetry classification of Sippel and Gönner. A limiting case of certain axisymmetric pp-waves yields the Aichelburg/Sexl ultraboost modeling an ultrarelativistic encounter with an isolated spherically symmetric object. (See also the article on plane wave spacetimes for a discussion of physically important special cases of plane waves.) J. D. Steele has introduced the notion of generalised pp-wave spacetimes. These are nonflat Lorentzian spacetimes which admit a self-dual covariantly constant null bivector field. The name is potentially misleading, since as Steele points out, these are nominally a special case of nonflat pp-waves in the sense defined above. They are only a generalization in the sense that although the Brinkmann metric form is preserved, they are not necessarily the vacuum solutions studied by Ehlers and Kundt, Sippel and Gönner, etc. Another important special class of pp-waves are the sandwich waves. These have vanishing curvature except on some range , and represent a gravitational wave moving through a Minkowski spacetime background. Relation to other theories Since they constitute a very simple and natural class of Lorentzian manifolds, defined in terms of a null congruence, it is not very surprising that they are also important in other relativistic classical field theories of gravitation. In particular, pp-waves are exact solutions in the Brans–Dicke theory, various higher curvature theories and Kaluza–Klein theories, and certain gravitation theories of J. W. Moffat. Indeed, B. O. J. Tupper has shown that the common vacuum solutions in general relativity and in the Brans/Dicke theory are precisely the vacuum pp-waves (but the Brans/Dicke theory admits further wavelike solutions). Hans-Jürgen Schmidt has reformulated the theory of (four-dimensional) pp-waves in terms of a two-dimensional metric-dilaton theory of gravity. Pp-waves also play an important role in the search for quantum gravity, because as Gary Gibbons has pointed out, all loop term quantum corrections vanish identically for any pp-wave spacetime. This means that studying tree-level quantizations of pp-wave spacetimes offers a glimpse into the yet unknown world of quantum gravity. It is natural to generalize pp-waves to higher dimensions,