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The plane gravitational waves may be defined in terms of Brinkmann coordinates by the line element:

${\displaystyle ds^{2}=du^{2}+2dudv+dx^{2}+dy^{2}}$

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.