Pp-wave spacetime


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:
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 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. More research is indeed on its way.

Mathematical definition

A pp-wave spacetime is any Lorentzian manifold whose metric tensor can be described, with respect to Brinkmann coordinates, in the form
where is any smooth function. This was the original definition of Brinkmann, and it has the virtue of being easy to understand.
The definition which is now standard in the literature is more sophisticated.
It makes no reference to any coordinate chart, so it is a coordinate-free definition.
It states that any Lorentzian manifold which admits a covariantly constant null vector field is called a pp-wave spacetime. That is, the covariant derivative of must vanish identically:
This definition was introduced by Ehlers and Kundt in 1962. To relate Brinkmann's definition to this one, take, the coordinate vector orthogonal to the hypersurfaces. In the index-gymnastics notation for tensor equations, the condition on can be written.
Neither of these definitions make any mention of any field equation; in fact, they are entirely independent of physics. The vacuum Einstein equations are very simple for pp waves, and in fact linear: the metric obeys these equations if and only if. 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. In the next section we turn to physical interpretations of pp-wave spacetimes.
Ehlers and Kundt gave several more coordinate-free characterizations, including:
  • A Lorentzian manifold is a pp-wave if and only if it admits a one-parameter subgroup of isometries having null orbits, and whose curvature tensor has vanishing eigenvalues.
  • A Lorentzian manifold with nonvanishing curvature is a pp-wave if and only if it admits a covariantly constant bivector.

    Physical interpretation

It is a purely mathematical fact that the characteristic polynomial of the Einstein tensor of any pp-wave spacetime vanishes identically. Equivalently, we can find a Newman–Penrose complex null tetrad such that the Ricci-NP scalars and the Weyl-NP scalars each have only one nonvanishing component.
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. Also, the Weyl tensor always has Petrov type N as may be verified by using the Bel criteria.
In other 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.
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 is a vacuum solution if and only if is a harmonic function. 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. 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.
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. Hans-Jürgen Schmidt has reformulated the theory of 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, 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.