What is 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

d s 2 = + 2 b x y ] d u 2 + 2 d u d v + d x 2 + d y 2 {\displaystyle ds^{2}=+2bxy]du^{2}+2dudv+dx^{2}+dy^{2}}

Here, a , b {\displaystyle a,b} 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.