Halliday, Resnick & Walker — Relativity: Problem 2

2 For the passing reference frames in Fig. 37-15, events A and B occur at the following spacetime coordinates: according to the unprimed frame, $(x_A, t_A)$ and $(x_B, t_B)$; according to the primed frame, $(x'_A, t'_A)$ and $(x'_B, t'_B)$. In the unprimed frame, $\Delta t = t_B - t_A = 1.00\ \mu\text{s}$ and $\Delta x = x_B - x_A = 400\text{ m}$. (a) Find an expression for $\Delta t'$ in terms of the speed parameter $\beta$ and the given data. Graph $\Delta t'$ versus $\beta$ for two ranges of $\beta$: (b) 0 to 0.01 and (c) 0.1 to 1. (d) At what value of $\beta$ is $\Delta t' = 0$? For what range of $\beta$ is the sequence of events A and B according to Bullwinkle (e) the same as ours and (f) the reverse of ours? (g) Can event A cause event B, or vice versa? Explain.