In general relativity, the pp-wave spacetimes, or pp-waves for short, are an important family of exact solutions of Einstein's field equation. The term pp stands for plane-fronted waves with parallel propagation, and was introduced in 1962 by Jürgen Ehlers and Wolfgang Kundt. Overview The pp-waves solutions model radiation moving at the speed of light. This radiation may consist of: electromagnetic radiation, gravitational radiation, massless radiation associated with Weyl fermions, massless radiation associated with some hypothetical distinct type relativistic classical field, or any combination of these, so long as the radiation is all moving in the same direction. A special type of pp-wave spacetime, the plane wave spacetimes, provide the most general analogue in general relativity of the plane waves familiar to students of electromagnetism. In particular, in general relativity, we must take into account the gravitational effects of the energy density of the electromagnetic field itself. When we do this, purely electromagnetic plane waves provide the direct generalization of ordinary plane wave solutions in Maxwell's theory. Furthermore, in general relativity, disturbances in the gravitational field itself can propagate, at the speed of light, as "wrinkles" in the curvature of spacetime. Such gravitational radiation is the gravitational field analogue of electromagnetic radiation. In general relativity, the gravitational analogue of electromagnetic plane waves are precisely the vacuum solutions among the plane wave spacetimes. They are called gravitational plane waves. There are physically important examples of pp-wave spacetimes which are not plane wave spacetimes. In particular, the physical experience of an observer who whizzes by a gravitating object (such as a star or a black hole) at nearly the speed of light can be modelled by an impulsive pp-wave spacetime called the Aichelburg–Sexl ultraboost. The gravitational field of a beam of light is modelled, in general relativity, by a certain axi-symmetric pp-wave. An example of pp-wave given when gravity is in presence of matter is the gravitational field surrounding a neutral Weyl fermion: the system consists in a gravitational field that is a pp-wave, no electrodynamic radiation, and a massless spinor exhibiting axial symmetry. In the Weyl-Lewis-Papapetrou spacetime, there exists a complete set of exact solutions for both gravity and matter. Pp-waves were introduced by Hans Brinkmann in 1925 and have been rediscovered many times since, most notably by Albert Einstein and Nathan Rosen in 1937. Mathematical definition A pp-wave spacetime is any Lorentzian manifold whose metric tensor can be described, with respect