Editing Pp-wave spacetime (section)

==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 <math>k</math> always has identically vanishing [[optical scalars]].  Therefore, pp-waves belong to the [[Kundt spacetime|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 <math>H</math> is a [[harmonic function]] (with respect to the spatial coordinates <math>x,y</math>).  Physically, these represent purely gravitational radiation propagating along the null rays <math>\partial_v</math>.

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 <math>k=\partial_v</math>.  However, for various special forms of <math>H</math>, 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 <math>H</math> is quadratic, and can hence be transformed to the simple form

:<math>H(u,x,y)=a(u) \, (x^2-y^2) + 2 \, b(u) \, xy + c(u) \, (x^2+y^2)</math>

Here, <math>a,b,c</math> are arbitrary smooth functions of <math>u</math>.
Physically speaking, <math>a,b</math> describe the wave profiles of the two linearly independent [[polarization modes]] of gravitational radiation which may be present, while <math>c</math> describes the wave profile of any nongravitational radiation.
If <math>c = 0</math>, 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 <math>X</math>, including <math>X = \partial_v</math> and four more which have the form

:<math> X = \frac{\partial}{\partial u}(p x + q y) \, \partial_v 
+ p \, \partial_x + q \, \partial_y </math>

where

:<math> \ddot{p} = -a p + b q - c p </math>
:<math> \ddot{q} = a q - b p - c q. </math>

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 [[Lie algebra#Abelian.2C nilpotent.2C and solvable|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 <math>u_1 < u < u_2</math>, and represent a gravitational wave moving through a [[Minkowski spacetime]] background.