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The phenomenon occurring in the special theory of relativity wherein two observers who start together with identical clocks and then undergo different motions can have different total elapsed time on their clocks when they rejoin later. This effect is a well-defined, mathematically consistent prediction of special relativity which has been verified by experiment but, historically, it has been referred to as a paradox because of erroneous reasoning in the manner in which the effect is commonly analyzed. The clock-paradox phenomenon arises because there is no notion of absolute simultaneity in the theory of special relativity.
The clock-paradox effect is illustrated by the following hypothetical example. (In the context of the example the effect usually is referred to as the twin paradox.) Two identical twins are separated as young adults. One of the twins remains on Earth and, for the purposes of the discussion, is assumed to undergo inertial motion. (In fact, in the context of general relativity, the twin on Earth would be viewed as accelerating—only freely falling observers would be viewed as inertial—but in this example the corrections made to the effect by performing a proper, general-relativistic analysis would be negligible.) The other twin is placed in a rocket ship which accelerates very rapidly away from Earth until it is receding from Earth at nearly the speed of light. After coasting for a while at this speed, the rocket ship turns around and accelerates very rapidly back toward Earth, so that it soon is traveling at nearly the speed of light. The rocket ship then lands on Earth and the twins rejoin. The twin on Earth is old (and everything else on Earth has aged considerably), but the twin in the rocket ship (and everything else therein) has barely aged at all.
The clock-paradox phenomenon has been observed directly in an experiment performed in 1971 by J. C. Hafele and R. E. Keating, who observed differences in elapsed times of atomically stabilized clocks flown in airplanes as compared with ones on the ground. In this experiment the special relativistic effect is so small, since the velocities achieved by airplanes are much smaller than c, that the tiny corrections due to general relativity cannot be neglected, so the experiment actually must be viewed as verifying the analogous clock effect in general relativity rather than purely the clock effect of special relativity. See Relativity, Space-time