= Rapidity form of one-dimensional proper acceleration
{title2=$a=c\,d\eta/d\tau$}
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.
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