= Affine transverse trace in a diagonal linearized plane wave is pure gauge
For a perturbation $h_{xx}=2a(t-z)$, $h_{yy}=2b(t-z)$ with all other components zero, the linearized <Vacuum Einstein equations> give $(a+b)''=0$. Put $s=a+b$ and choose $\xi_x=sx/2$, $\xi_y=sy/2$, $\xi_t=-s'(x^2+y^2)/4$, $\xi_z=s'(x^2+y^2)/4$. Under the <linearized coordinate gauge transformation>, the mixed time/transverse and longitudinal/transverse components cancel, while the other time/longitudinal changes are proportional to $s''=0$. The remaining components are $h_{xx}=a-b$, $h_{yy}=-(a-b)$, $h_{xy}=0$, a <transverse-traceless gauge> perturbation of <plus polarization>. This explicitly removes a nonzero affine trace without assuming it vanishes by boundary conditions. If $(a-b)''=0$ too, all linearized curvature vanishes.
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