Hyperbolic motion ($x^2-c^2{t}^2=c^2{t}_0^2$)

Hyperbolic motion in special relativity is rectilinear motion with constant proper acceleration. A representative positive-$x$ world line is $x=c{t}_0\cosh (\tau /t_0)$, $t=t_0\sinh (\tau /t_0)$, with proper acceleration $c/t_0$. This describes a hyperbola in spacetime, not a hyperbolic Kepler orbit.

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 $x$.

Replace instantaneous acceleration and braking by legs of constant proper-acceleration magnitude $a$, retaining the same cruise rapidity $\eta$ and fixed endpoint separation. Each leg takes proper time $c\eta /a$ and displaces $(c^2/a)(\cosh \eta -1)$ in the endpoint rest frame. Subtracting the proper cruise time displaced by both legs gives the displayed correction. A cruise segment exists only if the total endpoint separation exceeds twice this acceleration distance.