= Rindler null geodesics
{c}
{title2=$x=C e^{\pm\alpha t}$}
For $ds^2=(\alpha x)^2dt^2-dx^2$ in a patch with $x\ne0$, the null paths solve $dx/dt=\pm\alpha x$, giving $x=C e^{\pm\alpha t}$ with $C\ne0$. They are geodesics after choosing an <affine parameter>. The <Rindler coordinates> $X=x\cosh(\alpha t)$, $cT=x\sinh(\alpha t)$ give $ds^2=c^2dT^2-dX^2$ 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.
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