= Solution
Take $\alpha>0$ and work in a connected patch with $x\ne0$. Setting the line element to zero gives
$$
\frac{dx}{dt}=\pm\alpha x,\qquad
\boxed{x(t)=C e^{\pm\alpha t},\quad C\ne0.}
$$
These are the two families of <Rindler null geodesics>. They remain on the same side of $x=0$ and approach it only at infinite coordinate time. $x=0$ itself is not an additional <geodesic> within this chart, since the metric is degenerate there.
The paths can also be written $t=\pm\alpha^{-1}\log|x/C|$. In two dimensions a null direction has only one independent orthogonal null direction, namely itself, so each smooth <null curve> is a <geodesic> after reparametrization. More explicitly, the transformation in part (d) makes these paths straight null lines. The coordinate $t$ is generally not an <affine parameter>.
\Image[/past-exam-of-the-relativity-course-of-the-university-of-cambridge/2021-q6-null-rays-and-reception.png]
{title=Rindler light rays and the reception horizon of a uniformly accelerated observer}
{description=Original diagrams. The left panel shows the two exponential null families on both sides of the excluded $x=0$ chart boundary. The right panel uses inertial coordinates: a hyperbolic observer receives a signal emitted within its horizon, but cannot receive the horizon ray or a ray emitted farther behind. Length and time are normalized by $R=c^2/a$.}
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