Editing Pp-wave spacetime (section)

==Mathematical definition==
A ''pp-wave spacetime'' is any [[Lorentzian manifold]] whose [[metric tensor]] can be described, with respect to [[Brinkmann coordinates]], in the form

:<math> ds^2 = H(u,x,y) \, du^2 + 2 \, du \, dv + dx^2 + dy^2</math>

where <math>H</math> 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 <math>k</math> is called a pp-wave spacetime.  That is, the [[covariant derivative]] of <math>k</math> must vanish identically:

:<math>\nabla k = 0.</math>

This definition was introduced by Ehlers and Kundt in 1962.  To relate Brinkmann's definition to this one, take <math>k = \partial_v</math>, the [[coordinate vector]] orthogonal to the hypersurfaces <math>v=v_0</math>.  In the ''index-gymnastics'' notation for tensor equations, the condition on <math>k</math> can be written <math>k_{a ;b} = 0</math>.

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 <math> ds^2 = H(u,x,y) \, du^2 + 2 \, du \, dv + dx^2 + dy^2</math> obeys these equations if and only if <math> H_{xx} + H_{yy} = 0</math>.   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 manifold|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 (nontrivial) pp-wave if and only if it admits a covariantly constant [[bivector]].  (If so, this bivector is a null bivector.)