= Parallel photon beams in linearized gravity
{title2=$h_{ik}=H(u,y,z)\ell_i\ell_k,\quad\Delta_\perp H=16\pi G\rho(u)$}
For signature $(+---)$ and the curvature convention giving $\Box\bar h_{ik}=-16\pi G\epsilon^{-1}T_{ik}$, take $u=t-x$, $\ell_i=(1,-1,0,0)$ and $T_{ik}=\epsilon\rho(u)\ell_i\ell_k$. A trace-free Lorenz-gauge particular solution has the displayed form. The vector $\ell^i=(1,1,0,0)$ is null, and $\Gamma^i{}_{jk}\ell^j\ell^k=0$, so co-propagating test photons are undeflected by this beam field. For a transversely uniform source, $H=4\pi G\rho(u)(y^2+z^2)$ is one particular solution. A function of $u$ alone has vanishing wave operator and cannot source nonzero $\rho$; homogeneous gravitational perturbations and boundary conditions remain separate data.
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