# Gravitational plane wave

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