The term pp stands for plane-fronted waves with parallel propagation, and was introduced in 1962 by Jürgen Ehlers and Wolfgang Kundt.
In particular, in general relativity, we must take into account the gravitational effects of the energy density of the electromagnetic field itself.
Furthermore, in general relativity, disturbances in the gravitational field itself can propagate, at the speed of light, as "wrinkles" in the curvature of spacetime.
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.
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.
[1] 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.
A pp-wave spacetime is any Lorentzian manifold whose metric tensor can be described, with respect to Brinkmann coordinates, in the form where
The vacuum Einstein equations are very simple for pp waves, and in fact linear: the metric
But the definition of a pp-wave spacetime does not impose this equation, so it is entirely mathematical and belongs to the study of pseudo-Riemannian geometry.
Ehlers and Kundt gave several more coordinate-free characterizations, including: It is a purely mathematical fact that the characteristic polynomial of the Einstein tensor of any pp-wave spacetime vanishes identically.
Specifically, with respect to the NP tetrad the only nonvanishing component of the Ricci spinor is and the only nonvanishing component of the Weyl spinor is This means that any pp-wave spacetime can be interpreted, in the context of general relativity, as a null dust solution.
In other words, pp-waves model various kinds of classical and massless radiation traveling at the local speed of light.
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
It turns out that the most general pp-wave spacetime has only one Killing vector field, the null geodesic congruence
The most important class of particularly symmetric pp-waves are the plane wave spacetimes, which were first studied by Baldwin and Jeffery.
describe the wave profiles of the two linearly independent polarization modes of gravitational radiation which may be present, while
A limiting case of certain axisymmetric pp-waves yields the Aichelburg/Sexl ultraboost modeling an ultrarelativistic encounter with an isolated spherically symmetric object.
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.
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.
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, where they enjoy similar properties to those we have discussed.
C. M. Hull has shown that such higher-dimensional pp-waves are essential building blocks for eleven-dimensional supergravity.
Consider an inertial observer in Minkowski spacetime who encounters a sandwich plane wave.
If he looks into the oncoming wavefronts at distant galaxies which have already encountered the wave, he will see their images undistorted.
The effect of a passing polarized gravitational plane wave on the relative positions of a cloud of (initially static) test particles will be qualitatively very similar.
Roger Penrose has observed that near a null geodesic, every Lorentzian spacetime looks like a plane wave.
Penrose also pointed out that in a pp-wave spacetime, all the polynomial scalar invariants of the Riemann tensor vanish identically, yet the curvature is almost never zero.
Such statement does not hold in higher-dimensions since there are higher-dimensional pp-waves of algebraic type II with non-vanishing polynomial scalar invariants.