= Finite-acceleration correction to a relativistic cruise
{title2=$\Delta\tau=(2c/a)[\eta-(\cosh\eta-1)/\sinh\eta]$}
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.
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