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.

•• Like time dilation, the Lorentz contraction cannot be seen directly (that is, perceived by the normal process of vision). To understand this claim, consider a rod of proper length l0 moving relative to S. Careful measurements made by observers in S [as in Fig. 1.(a), for example] will show that the rod has the contracted length l = l0/γ. But now consider what is seen by observer Q in Fig. 1(b) (with Q to the right of points A and B). What Q sees at any one instant is determined by the light entering her eyes at that instant. Now, consider the light reaching Q at one instant from the front and back of the rod. (a) Explain why these two rays must have left the rod (from points A and B) at different times. If the x axis has a graduated scale as shown, Q sees (and a photograph would record) a rod extending from A to B; that is, Q sees a rod of length AB. (b) Prove that Q sees a rod that is longer than l. (In fact, at certain speeds it is even seen to be longer than l0, and the Lorentz contraction is distorted into an expansion.) (c) Prove that once it has passed her, Q will see the rod to be shorter than l.

FIGURE 1

(a) One can measure the Lorentz-contracted length l = l0/γ using two observers to record the positions of the front and back at the same instant. (b) What a single observer sees is determined by light that left the rod at different times.