= Timelike geodesics of an inverse-time conformal metric
{title2=$ds^2=t^{-2}(dx^2-dt^2)$}
In a positive-time patch, proper-time first integrals give $\dot x=pt^2$ and $\dot t^2=t^2+p^2t^4$. If $p\ne0$, the unparametrized <timelike geodesics> are $t^2=(x-x_c)^2-p^{-2}$ on an appropriate branch; in the expanded quadratic form their discriminant is positive. If $p=0$, there is the separate family $x=x_c$, $t=t_*e^{\pm\tau}$. A description using only the nonvertical quadratic family omits genuine timelike geodesics.
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