Einstein's Elevator

• Alice and her colleagues, living in the pre-general-relativity era, try a series of experiments in flat space-time in order to guess what a gravitational theory would look like. Having heard about Einstein’s gedanken experiment involving an accelerating frame, they try to mimic a stationary motion of constant acceleration. As the leader of the team, Alice undergoes a hyperbolic motion described by
  $z = \sqrt{t^2 + 1/g^2}$
  1. Show that her motion in terms of her proper time is given by $t(\tau) = \frac{1}{g}\sinh(g\tau)$
     $z(\tau) = \frac{1}{g} \cosh(g\tau)$

Meanwhile each of her colleagues, labelled by a number $h$, moves in such a way that at any moment $\tau$ alice can find them a distance $h$ up above her head.

2. Describe the motion of the colleague, labelled $h$. Show that if Bob is flying a distance $h$ up above Alice's head and Charlie is flying a distance $h^\prime$ up above Bob's head, then Alice finds Charlie a distance $h+h^\prime$ up above his head.