Free The Elegant Universe by Brian Greene

travel apart, each claims that the other’s clock is running slower. To confront this disagreement head on, George and Gracie must rejoin each other and directly compare the time elapsed on their clocks. But how can they do this? Well, George has a jetpack that he can use, from his perspective, to catch up with Gracie. But if he does this, the symmetry of their two perspectives, which is the cause of the apparent paradox, is broken since George will have undergone accelerated, non-force-free motion. When they rejoin in this manner, less time will indeed have elapsed on George’s clock as he can now definitively say that he was in motion, since he could feel it. No longer are George’s and Gracie’s perspectives on equal footing. By turning on the jet-pack, George relinquishes his claim to being at rest.
    If George chases after Gracie in this manner, the time difference that their clocks will show depends on their relative velocity and the details of how George uses his jet-pack. As is by now familiar, if the speeds involved are small, the difference will be minuscule. But if substantial fractions of light speed are involved, the differences can be minutes, days, years, centuries, or more. As one concrete example, imagine that the relative speed of George and Gracie when they pass and are moving apart is 99.5 percent of light speed. Further, let’s say that George waits 3 years, according to his clock, before firing up his jet-pack for a momentary blast that sends him closing in on Gracie at the same speed that they were previously moving apart, 99.5 percent of light speed. When he reaches Gracie, 6 years will have elapsed on his clock since it will take him 3 years to catch her. However, the mathematics of special relativity shows that 60 years will have elapsed on her clock. This is no sleight of hand: Gracie will have to search her distant memory, some 60 years before, to recall passing George in space. For George, on the other hand, it was a mere 6 years ago. In a real sense, George’s motion has made him a time traveler, albeit in a very precise sense: He has traveled into Gracie’s future.
    Getting the two clocks back together for direct comparison might seem to be merely a logistical nuisance, but it is really at the heart of the matter. We can imagine a variety of tricks to circumvent this chink in the paradox armor, but all ultimately fail. For instance, rather than bringing the clocks back together, what if George and Gracie compare their clocks by cellular telephone communication? If such communication were instantaneous, we would be faced with an insurmountable inconsistency: reasoning from Gracie’s perspective, George’s clock is running slow and hence he must communicate less elapsed time; reasoning from George’s perspective, Gracie’s clock is running slow and hence she must communicate less elapsed time. They both can’t be right, and we would be sunk. The key point of course is that cell phones, like all forms of communication, do not transmit their signals instantaneously. Cell phones operate with radio waves, a form of light, and the signal they transmit therefore travels at light speed. This means that it takes time for the signals to be received—just enough time delay, in fact, to make each perspective compatible with the other.
    Let’s see this, first, from George’s perspective. Imagine that every hour, on the hour, George recites into his cell phone, “It’s twelve o’clock and all is well,” “It’s one o’clock and all is well,” and so forth. Since from his perspective Gracie’s clock runs slow, at first blush he thinks that Gracie will receive these messages prior to her clock’s reaching the appointed hour. In this way, he concludes, Gracie will have to agree that hers is the slow clock. But then he rethinks it: “Since Gracie is receding from me, the signal I send to her by cell phone must travel ever longer distances to reach her. Maybe this additional travel time