Gravitational plane wave

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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

$d s 2 = [a (u)(x 2 - y 2) + 2 b (u) x y] d u 2 + 2 d u d v + d x 2 + d y 2$

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.