Halliday, Resnick & Walker — Relativity: Problem 1

1 Relativistic reversal of events. Figures 37-15a and b show the (usual) situation in which a primed reference frame passes an unprimed reference frame, in the common positive direction of the \(x\) and \(x'\) axes, at a constant relative velocity of magnitude \(v\). We are at rest in the unprimed frame; Bullwinkle, an astute student of relativity in spite of his cartoon upbringing, is at rest in the primed frame. The figures also indicate events \(A\) and \(B\) that occur at the following spacetime coordinates as measured in our unprimed frame and in Bullwinkle’s primed frame: Event | Unprimed | Primed --- | --- | --- \(A\) | \((x_A, t_A)\) | \((x'_A, t'_A)\) \(B\) | \((x_B, t_B)\) | \((x'_B, t'_B)\) In our frame, event \(A\) occurs before event \(B\), with temporal separation \(\Delta t = t_B - t_A = 1.00\text{ }\mu\text{s}\) and spatial separation \(\Delta x = x_B - x_A = 400\text{ m}\). Let \(\Delta t'\) be the temporal separation of the events according to Bullwinkle. (a) Find an expression for \(\Delta t'\) in terms of the speed parameter \(\beta (= v/c)\) and the given data. Graph \(\Delta t'\) versus \(\beta\) for the following 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.