= Ricci tensor of a Brinkmann plane wave
{c}
{title2=$R_{uu}=H_{xx}+H_{yy}$}
For $ds^2=-2du\,dv+dx^2+dy^2-2H(u,x,y)du^2$, the nonzero connections are $\Gamma^v_{uu}=H_u$, $\Gamma^v_{ux}=H_x$, $\Gamma^v_{uy}=H_y$, $\Gamma^x_{uu}=H_x$, and $\Gamma^y_{uu}=H_y$, with symmetric lower indices. Their traces and quadratic Ricci products vanish, leaving only the displayed Ricci component. Hence the vacuum equations require a harmonic transverse profile. The sign is tied to the metric's $-2Hdu^2$ convention.
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