Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

••• Time dilation implies that if a clock moves relative to a frame S, careful measurements made by observers in S [as in Fig. 1(a), for example] will find that the clock runs slow. This is not at all the same thing as saying that a single observer in S will see the clock running slow; and the latter statement is, in fact, not always true. To understand this, remember that what we see is determined by the light as it arrives at our eyes. Consider the observer Q in Fig. 1(b) and suppose that as the clock moves from A to B, it registers the passage of a time τ0. As measured in S, the time between these two events (“clock at A” and “clock at B”) is of course τ = γτ0. However, B is closer to Q than A is; thus light from the clock when at B will reach Q in a shorter time than will light from the clock when at A. Therefore, the time τsee between Q’s seeing the clock at A and seeing it at B is less than r. (a) Prove that in fact

(Prove both equalities.) Since τsee is less than τ0, the observer Q actually sees the clock running fast. (b) What will Q see once the clock has passed her? That is, find the new value of τsee when the clock is moving away from Q.

Your answers here are closely related to the Doppler effect discussed in Section 1.14. The moral of this problem is that one must be very careful how one states (and thinks about) time dilation. It is safe to say “moving clocks are observed to run slow” [where to “observe” means to “measure carefully” as in Fig. 1(a)], but it is certainly wrong to say “moving clocks are seen to run slow.”

FIGURE 1

(a) Two observers at rest in frame S at A and B time the moving clock as it passes them; they find the dilated time τ = γτ0. (b) The single observer Q sees the moving clock at A and B by means of light that has traveled different distances, AQ and BQ.