Rapidity form of one-dimensional proper acceleration ($a=cd\eta /d\tau$)

For rectilinear motion write the four-velocity in an inertial frame as $U=(c\cosh \eta ,c\sinh \eta )$, where $\eta$ is the rapidity and $\tau$ is proper time. Differentiation gives $A=c\dot{\eta }(\sinh \eta ,\cosh \eta )$, with invariant $A\cdot A=-c^2{\dot{\eta }}^2$ in the $+-$ metric signature. The signed proper acceleration is therefore $a=c\dot{\eta }$. Constant signed acceleration gives $\eta ={\eta }_0+a\tau /c$; specifying initial position alone leaves the initial rapidity ${\eta }_0$ undetermined.