Hyperbolic motion from an initial event and rapidity ($\eta (\tau )={\eta }_0+a\tau$)

In units $c=1$, constant signed proper acceleration $a$ givesPosition data determine $t_0,x_0$ but leave the initial rapidity ${\eta }_0$ free. Proper-acceleration magnitude alone also leaves its orientation unspecified. The familiar centred hyperbola $x^2-t^2=a^{-2}$ follows for the additional initial data $t_0=0$, $x_0=1/a$, ${\eta }_0=0$ and acceleration toward increasing $x$.