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

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

Here, $$a(u), b(u)$$ 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.