Past exam of the mathematics course of the University of Cambridge / 2016 / ia / Paper 4 / 11B / b / Solution

The meetings occur where Bob's constant position $x=\alpha c{t}_0$ intersects Alice's positive-$x$ branch of the hyperbolic motion:Bob is at rest in this inertial frame, so his elapsed proper time is

Parametrize Alice's world line bySince $t$ increases with $\eta$, the proper time element isAt the meetings, $\cosh \eta =\alpha$, giving endpoints $\eta =\pm \mathrm{arcosh}\alpha$. Thus Alice's elapsed proper time isHere $\mathrm{arcosh}$ is the nonnegative real inverse hyperbolic cosine. As a check, if $\alpha =\cosh {\eta }_{\ast }$ with ${\eta }_{\ast }>0$, then Bob ages by $2t_0\sinh {\eta }_{\ast }$ and Alice by $2t_0{\eta }_{\ast }$; $\sinh {\eta }_{\ast }>{\eta }_{\ast }$ confirms that Bob again ages more.