(a) Alice travels at constant speed $v$ to Alpha Centauri, which is at distance $d$ from Earth. She then turns around (taking very little time to do so), and returns at speed $v$. Bob stays at home. By how much has Bob aged during the journey? By how much has Alice aged? [No justification is required.]

Briefly explain what is meant by the twin paradox in this context. Why is it not a paradox?

(b) Suppose instead that Alice's world line is given by

$$-c^{2} t^{2}+x^{2}=c^{2} t_{0}^{2},$$

where $t_{0}$ is a positive constant. Bob stays at home, at $x=\alpha c t_{0}$, where $\alpha>1$. Alice and Bob compare their ages on both occasions when they meet. By how much does Bob age? Show that Alice ages by $2 t_{0} \cosh ^{-1} \alpha$.