= Timelike geodesics of a quadratic-time conformal metric
{title2=$ds^2=(t^2/\alpha)(dx^2-c^2dt^2)$}
For $\alpha>0$ and $t\ne0$, 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 $ds^2=-c^2d\tau^2$ this becomes
$$
P=\frac{|t|}{\sqrt\alpha}\frac{v}{\sqrt{1-v^2/c^2}}.
$$
Thus $v/\sqrt{1-v^2/c^2}$ alone is not constant. Set $b=\sqrt\alpha P/c$. Every unparametrized <timelike geodesic> in a connected nonzero-$t$ patch has the form
$$
x(t)=x_c+cb\operatorname{arsinh}(t/|b|)\quad(b\ne0),\qquad x(t)=x_c\quad(b=0).
$$
The nonvertical paths have $v=cb/\sqrt{t^2+b^2}$ and proper-time parametrization
$$
\tau-\tau_c=\frac{t\sqrt{t^2+b^2}-b^2\operatorname{arsinh}(t/|b|)}{2\sqrt\alpha}.
$$
The chart excludes $t=0$, where its metric is degenerate; a vertical family must not be lost by dividing by the conserved momentum.
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