Unfortunately, the terminology concerning pp-waves, while fairly standard, is highly confusing and tends to promote misunderstanding. In any pp-wave spacetime, the covariantly constant vector field k always has identically vanishing
optical scalars. Therefore, pp-waves belong to the
Kundt class (the class of Lorentzian manifolds admitting a
null congruence with vanishing optical scalars). Going in the other direction, pp-waves include several important special cases. From the form of Ricci spinor given in the preceding section, it is immediately apparent that a pp-wave spacetime (written in the Brinkmann chart) is a
vacuum solution if and only if H is a
harmonic function (with respect to the spatial coordinates x,y). Physically, these represent purely gravitational radiation propagating along the null rays \partial_v. Ehlers and Kundt and Sippel and Gönner have classified vacuum pp-wave spacetimes by their
autometry group, or group of
self-isometries. This is always a
Lie group, and as usual it is easier to classify the underlying
Lie algebras of
Killing vector fields. It turns out that the most general pp-wave spacetime has only one Killing vector field, the null geodesic congruence k=\partial_v. However, for various special forms of H, there are additional Killing vector fields. The most important class of particularly symmetric pp-waves are the
plane wave spacetimes, which were first studied by Baldwin and Jeffery. A plane wave is a pp-wave in which H is quadratic, and can hence be transformed to the simple form :H(u,x,y)=a(u) \, (x^2-y^2) + 2 \, b(u) \, xy + c(u) \, (x^2+y^2) Here, a,b,c are arbitrary smooth functions of u. Physically speaking, a,b describe the wave profiles of the two linearly independent
polarization modes of gravitational radiation which may be present, while c describes the wave profile of any nongravitational radiation. If c = 0, we have the vacuum plane waves, which are often called
plane gravitational waves. Equivalently, a plane-wave is a pp-wave with at least a five-dimensional Lie algebra of Killing vector fields X, including X = \partial_v and four more which have the form : X = \frac{\partial}{\partial u}(p x + q y) \, \partial_v + p \, \partial_x + q \, \partial_y where : \ddot{p} = -a p + b q - c p : \ddot{q} = a q - b p - c q. Intuitively, the distinction is that the wavefronts of plane waves are truly
planar; all points on a given two-dimensional wavefront are equivalent. 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 u_1 , and represent a gravitational wave moving through a
Minkowski spacetime background. ==Relation to other theories==