Gravitational plane wave

In general relativity, a gravitational plane wave is a special class of a vacuum pp-wave spacetime, and may be defined in terms of Brinkmann coordinates by

${\displaystyle ds^{2}=[a(u)(x^{2}-y^{2})+2b(u)xy]du^{2}+2dudv+dx^{2}+dy^{2}}$

Here, ${\displaystyle a(u),b(u)}$ can be any smooth functions; they control the waveform of the two possible polarization modes of gravitational radiation. In this context, these two modes are usually called the plus mode and cross mode, respectively.