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.
• A space explorer sets off at a steady v = 0.95c to a distant star. After exploring the star for a short time he returns at the same speed and gets home after a total absence of 80 years (as measured by earthbound observers). How long do his clocks say that he was gone, and by how much has he aged? Note: This is the twin paradox discussed in Example 1.3. It is easy to get the right answer by judicious insertion of a factor γ in the right place, but to understand the result you need to recognize that it involves three inertial frames: the earthbound frame S, the frame S′ of the outward-bound rocket, and the frame S″ of the returning rocket. You can write down the time-dilation formula (1.14) for each of the two halves of the journey, and add these to give the desired relation. (Notice that the experiment is not symmetrical between the explorer and his friends who stay behind on earth — the earthbound clocks stay at rest in a single inertial frame, but the rocket’s clock and crew occupy at least two different frames. This is what allows the result to be unsymmetrical.)
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