 Timelike geodesics of a quadratic-time conformal metric

For $\alpha >0$ and $t\ne 0$, the conformal factor $t^2/\alpha$ makes spatial translation a Killing vector field. Using proper time, its conserved momentum is $P=(t^2/\alpha )dx/d\tau$. With $v=dx/dt$ and $d{s}^2=-c^{2}d{\tau }^2$ this becomesThus $v/\sqrt{1-v^2/c^2}$ alone is not constant. Set $b=\sqrt{\alpha }P/c$. Every unparametrized timelike geodesic in a connected nonzero-$t$ patch has the formThe nonvertical paths have $v=cb/\sqrt{t^2+b^2}$ and proper-time parametrizationThe chart excludes $t=0$, where its metric is degenerate; a vertical family must not be lost by dividing by the conserved momentum.