words, pp-waves model various kinds of classical and massless radiation traveling at the local speed of light. This radiation can be gravitational, electromagnetic, Weyl fermions, or some hypothetical kind of massless radiation other than these three, or any combination of these. All this radiation is traveling in the same direction, and the null vector plays the role of a wave vector. Relation to other classes of exact solutions 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 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 is a harmonic function (with respect to the spatial coordinates ). Physically, these represent purely gravitational radiation propagating along the null rays . 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 . However, for various special forms of , 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 is quadratic, and can hence be transformed to the simple form Here, are arbitrary smooth functions of . Physically speaking, describe the wave profiles of the two linearly independent polarization modes of gravitational radiation which may be present, while describes the wave profile of any nongravitational radiation. If , 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 , including and four more which have the form where Intuitively, the distinction is that the wavefronts of plane waves are truly planar; all points on a given two-dimensional wavefront are equivalent.