Chapter 2, Problem 120CE

In Example 2.5, we noted that Anna could go wherever she wished in as little time as desired by going fast enough to length−contract the distance to an arbitrarily small value. This overlooks a physiological limitation. Accelerations greater than about 30g are fatal, and there are serious concerns about the effects of prolonged accelerations greater than 1g. Here we see how far a person could go under a constant acceleration of 1g, producing a comfortable artificial gravity.
(a) Though traveler Anna accelerates, Bob, being on near−inertial Earth, is a reliable observer and will see less time go by on Anna’s clock $(dt\text{'})$ than on his own $(dt)$ . Thus, $dt\text{'}=(1/{\gamma }_u)dt$ , where u isAnna’s instantaneous speed relative to Bob, Using the result of Exercise 117(c), with g replacing $F/m$ , substitute for u, then integrate to show that
$t=\frac{c}{g}\sinh \frac{gt\text{'}}{c}$

(b) How much time goes by for observes on Earth as they “see” Anna age 20 years?
(c) Using the result of Exercise 119, show that when Anna has aged a time $t\text{'}$ . She is a distance fromEarth (according to Earth observers) of
$x=\frac{c^2}{g}\left(\cosh \frac{gt\text{'}}{c}-1\right)$

(d) If Anna accelerates away from Earth while aging 20 years and then slows to a stop while aging another 20, how far away from Earth will she end up, and how much time will have passed on Earth?