Rindler null geodesics ($x=C{e}^{\pm \alpha t}$)

For $d{s}^2=(\alpha x{)}^{2}d{t}^2-d{x}^2$ in a patch with $x\ne 0$, the null paths solve $dx/dt=\pm \alpha x$, giving $x=C{e}^{\pm \alpha t}$ with $C\ne 0$. They are geodesics after choosing an affine parameter. The Rindler coordinates $X=x\cosh (\alpha t)$, $cT=x\sinh (\alpha t)$ give $d{s}^2=c^{2}d{T}^2-d{X}^2$ and $X\pm cT=x{e}^{\pm \alpha t}$. Thus each exponential path is a straight inertial null line. A literal $x=0$ curve is outside this degenerate coordinate chart, not an extra null geodesic within it.