= Hyperbolic motion from an initial event and rapidity
{title2=$\eta(\tau)=\eta_0+a\tau$}
In units $c=1$, constant signed <proper acceleration> $a$ gives
$$
t-t_0=\frac{\sinh(\eta_0+a\tau)-\sinh\eta_0}{a},\qquad
x-x_0=\frac{\cosh(\eta_0+a\tau)-\cosh\eta_0}{a}.
$$
Position 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 $x$.
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