The prediction that moving clocks run slow has been amply demon-strated by experiment. Convincing evidence comes from the realm of subatomic physics, but we shall defer discussing that evidence until after we discuss atoms and elementary particles. Here we shall discuss evidence provided by extremely accurate clocks. Three such clocks were compared at rest and found to keep the same time within experimental error. Then one clock was kept at rest, while the other two clocks were put on airplanes and flown around the world, one in each direction.
The earth rotates west-to-east, as we can tell from the apparent daily movement of the sun east-to-west. If an airplane flies west-to-east, it is flying in the same direction as the earth is rotating, and so its speed is greater than the speed of rotation of the earth. Consequently, the clock on that airplane moves at a slower speed than the clock that is stationary with respect to the earth. Therefore, when the two clocks are compared after the trip, the clock on the airplane is slightly behind the clock that has remained stationary on the earth. On the other hand, the airplane traveling east-to-west moves slower than than the earth, so its clock runs slightly ahead of the stationary clock.
The experiment confirmed these predictions. The differences among the clocks were found to be only a tiny fraction of a second, owing to the fact that the speed of the airplane (about 900 km/hr) is only a
10.3. THE TWIN PARADOX 117 small fraction of the speed of light.
According to special relativity, it is only relative speeds that count.
Why then, was a difference found between the clocks in the two air-planes, when both moved at the same speed relative to the clock that was stationary on the earth? The answer is that because the earth rotates, an observer on the earth is not strictly speaking in an inertial frame, where the predictions of special relativity are valid.
Another complication arises from the earth’s gravity. Although for most purposes, the earth’s gravity is sufficiently weak that New-ton’s law of gravity is adequate, for very precise measurements it is not. In such situations, Einstein’s general theory of relativity, which is a better theory of gravity than Newton’s, must be used. According to general relativity, clocks are affected by gravity. Because the clocks in the airplanes are not at the same height as the the clock on earth, the gravity they feel is different and they do not run at the same speed as the clock on earth. After correcting for the effect of gravity on the clocks and the fact that the earth is not strictly speaking an inertial frame, the experimenters obtained agreement with the pre-diction of special relativity that clocks moving in space slow down in their measurement of time.
We now turn to the “twin paradox,” as it illustrates the impor-tance of making measurements from an inertial frame. The twin paradox is not really a paradox, but appears to be one to those who do not understand the restriction that the special theory of relativity does not necessarily give the right answer unless measurements are made from an inertial frame. The statement goes as follows:
There are twin boys on earth, each ten years old. One boy goes
off in a space ship at nearly the speed of light. (As a practical matter, there is no way for a space ship to achieve this speed, but we are doing a thought experiment.) The boy remaining on earth observes that the clock on the space ship runs slow. Not only does the clock run slow but so do the heartbeats and aging processes of the boy in the space ship. But the boy in the space ship observes that the clock of his twin on earth runs slow. After a time the boy in the space ship returns to earth. Each boy expects to be older than his twin, but each cannot be older than the other. What actually happens?
To simplify the problem, let’s neglect the fact that the earth is not strictly in an inertial frame. The difference is small. Because the twin remaining on earth remains for all practical purposes in an inertial frame during the entire journey of his brother, his observations are in accord with the predictions of special relativity. Therefore, his conclusion is correct that he will be older than his brother after his brother returns to earth. However, the twin in the space ship can-not remain in an inertial frame and get back to earth. In order to return, the space ship must turn around. While it is turning around it is accelerating. (Remember that a change in direction is an accel-eration.) While one is in an accelerating frame one cannot conclude that moving clocks run slow. It turns out that the twin in the space ship will observe that, while he is turning around, the twin on earth will age rapidly. After turning around, while heading back to earth, the twin remaining on earth will appear to age less rapidly than the twin in the space ship, but the effect will not be large enough to undo the rapid aging during the acceleration. So when the space ship gets back to earth, and the twins look at each other, both will observe that
10.4. LIGHT AS A LIMITING SPEED 119