The problem is to reconcile this result with the principle of reciprocity (that whatever A thinks of B, B thinks of A) and with the absence of preferred frames of reference. The reconciliation can be given mathematically, but I shall instead discuss a closely analogous problem that illuminates the twin paradox and makes it seem less paradoxical.
Suppose that Antonius is standing motionless on Earth as his twin brother Bartholomeus speeds past in a train. Both note as they pass that their watches point to the same time, but, making rapid observations, each concludes that the other’s watch is running slowly. Each therefore predicts that later on the other’s watch will be behind his own. Can they both be right? Oddly enough, they can. It depends on how they go about checking up on their watches later on.
Let us suppose that these twins pass each other in this manner on two occasions. On the first occasion, Bartholomeus leaves the train at the next station, catches the next train back, and rejoins Antonius to find his watch behind Antonius’s. “Thats odd,” says Bartholomeus, “I could have sworn it was your watch that was slow. Oh, well,” he adds philosophically, “at least I havent aged as much as you.” On the second occasion, Antonius rents a fast motorcar, overtakes the train and boards it at the next station. Now it is Antonius, finding himself younger than Bartholomeus, who is surprised, but takes what comfort he can from his lower rate of aging. Each observer’s prediction has been borne out, provided that he remains patiently in his own inertial frame of reference and lets the other observer come to him. All quite simple, says an impartial observer, Clavius, who takes the human point of view that the Earth is really the preferred frame of reference. Bartholomeus’s watch was of course running slowly because he was moving, as was proved when he came back to Antonius. But if Antonius is so impatient that he travels even faster than Bartholomeus in his effort to overtake him, small wonder that Antonius’s watch slowed down even more than Bartholomeus’s.
The general rule to cover these thought experiments as well as the rocket-ship version of the twin paradox is that any observer who leaves an inertial frame, moves with respect to it, and then rejoins it, will find that his or her time is behind the time of observers who remained in the frame. There is actually nothing paradoxical about the twin paradox. Indeed, the explanation given above has even been verified with twin muons,1 not twin people. Nevertheless it raises some fundamental questions to which no satisfactory answers have yet been provided. Why are there any inertial frames at all? Given that there are inertial frames, what determines which frames of reference are inertial? We can at best define an inertial frame operationally. It is one in which Newtons first law holds true: Objects free of outside influences are unaccelerated. And any frame moving uniformly with respect to an inertial frame is inertial.
The most appealing hypothesis about the true nature of inertial frames is due to Ernst Mach,2 and is usually called Mach’s principle. Mach suggested, in the latter part of the nineteenth century,3 that the distribution of matter in the universe determines preferred frames of reference, and that any frame unaccelerated with respect to the “fixed stars”—that is, with respect to the average distribution of matter in the universe—is an inertial frame. This would mean that matter throughout the universe determines the properties of spacetime at the Earth—a challenging idea, but one that has so far defied a mathematical formulation, and that has not been fruitful in leading to the prediction of new experiments, although it may have played a role in shaping Einstein’s thinking about general relativity. Actually, the more modern view would replace Mach’s matter with matter plus energy, but since most of the energy in the universe seems to be in the form of matter, the difference is not significant.
1 Muons have a mean lifetime of 2.2 μs. If some are collected at one location and some (their “twins”) are sent at high speed on a circular path (guided by a magnetic field), those that make the circuit live longer, on average, than those that stay put.
2 Yes, the same Mach for whom speeds given in Mach numbers are named.
3 Mach apparently never wrote down this idea, but it influenced Einstein, who gave it the name “Mach’s principle.”