AS MAXIMS GO, Time is relative may not be quite as famous as Time is money. But the notion that time speeds up or slows down depending on how fast one object is traveling relative to another surely ranks as one of Albert Einstein's most inspired insights.

The term time dilation was coined to describe the slowing of time caused by motion. And to illustrate the effect of time dilation, he proposed an example--the twin paradox--that is arguably the most famous thought experiment in relativity theory. In this supposed paradox, one of two twins travels at near the speed of light to a distant star and returns to Earth. Relativity dictates that when he comes back, he is younger than his identical twin [see How to Build a Time Machine, by Paul Davies, on page 6].

The paradox lies in the question Why is the traveling brother younger? Special relativity tells us that an observed clock, traveling at high speed past an observer, appears to run more slowly--that is, it experiences time dilation. (Many of us solved this traveling-clock problem in sophomore physics, to demonstrate one effect of the absolute nature of the speed of light.) Because special relativity says that there is no absolute motion, wouldn't the brother traveling to the star also see his brother's clock on Earth move more slowly? If this were the case, wouldn't they both be the same age?

This paradox is discussed in many books but solved in very few. It is typically explained by saying that the one who feels the acceleration is the one who is younger at the end of the trip; hence, the brother who travels to the star is younger. Although the result is correct, the explanation is misleading. Some people may falsely assume that the acceleration causes the age difference and that the general theory of relativity, which deals with noninertial or accelerating reference frames, is required to explain the paradox. But the acceleration incurred by the traveler is incidental, and the paradox can be unraveled by special relativity alone.

**A Long, Strange Space Trip**

LET US ASSUME that twin brothers, nicknamed the traveler and the homebody, live in Hanover, N.H. They differ in their wanderlust but share a common desire to build a spacecraft that can achieve 0.6 times the speed of light (0.6 *c*). After working on the spacecraft for years, they are ready to launch it, manned by the traveler, toward a star six light-years away.

His craft will quickly accelerate to 0.6 *c*. To reach that speed, it would take a little more than 100 days at an acceleration of two g's. Two g's is two times the acceleration of gravity, about what one experiences on a sharp loop on a roller coaster. If, however, the traveler were an electron, he could be accelerated to 0.6 *c* in a tiny fraction of a second. Hence, the time to reach 0.6 *c* is not central to the argument.

The traveler uses the length-contraction equation of special relativity to measure distance. So the star six light-years away to the homebody appears to be only 4.8 light-years away to the traveler at a speed of 0.6 *c*. Therefore, to the traveler, the trip to the star takes only eight years (4.8/0.6), whereas the homebody calculates it taking 10 years (6.0/0.6). To solve the twin paradox, we need to consider how each twin would view his and the other's clocks during the trip. Let us assume that each twin has a very powerful telescope that permits such observation. Surprisingly, by focusing on the time it takes light to travel between the two, the paradox can be explained.

Both the traveler and homebody set their clocks at zero when the traveler leaves Earth for the star [*see box at left*]. When the traveler reaches the star, his clock reads eight years. But when the homebody sees the traveler reach the star, the homebody's clock reads 16 years. Why 16 years? Because, to the homebody, the craft takes 10 years to make it to the star, and the light takes six additional years to come back to Earth showing the traveler at the star. So, viewed through the homebody's telescope, the traveler's clock appears to be running at half the speed of his clock (8/16).

As the traveler reaches the star, he reads his clock at eight years as mentioned, but he sees the homebody's clock as it was six years ago (the amount of time it takes for the light from Earth to reach him), or at four years (10 6). So the traveler also views the homebody's clock as running at half the speed of his clock (4/8).

**From Twin to Younger Brother**

ON THE TRIP BACK, the homebody views the traveler's clock going from eight years to 16 years in only four years time, because his clock was at 16 years when he saw the traveler leave the star and will be at 20 years when the traveler arrives back home. So the homebody now sees the traveler's clock advance eight years in four years of his time; it is now twice as fast as his clock.

As the traveler returns home, he sees the homebody's clock advance from four to 20 years in eight years of his time. Therefore, he also sees his brother's clock advancing at twice the speed of his. They both agree, however, that at the end of the trip the traveler's clock reads 16 years and the homebody's 20 years. So the traveler is four years younger.

The asymmetry in the paradox is that the traveler leaves Earth's reference frame and comes back, whereas the homebody never leaves Earth. It is also an asymmetry that the traveler and the homebody agree with the reading on the traveler's clock at each event, but they don't agree about the reading on the homebody's clock at each event. The traveler's actions define the events.

The Doppler effect and relativity together explain this effect mathematically at any instant. The reader should also note that the speed that an observed clock appears to run depends on whether it is traveling away from or toward the observer.

Finally, we should point out that the twin paradox today is more than a theory, because its fundamentals have been exhaustively confirmed experimentally. In one such experiment, the lifetime of muon decay verifies the existence of time dilation. Stationary muons have a lifetime of about 2.2 microseconds. When traveling past an observer at 0.9994 *c*, their lifetime stretches to 63.5 microseconds, just as predicted by special relativity. Experiments in which atomic clocks are transported at varying speeds have also produced results that confirm both special relativity and the twin paradox.

**THE AUTHOR**

*RONALD C. LASKY* is a visiting professor at Dartmouth College and a senior technologist at Indium Corporation. He has edited several books on electronic packaging and optoelectronics and written numerous technical papers and patent disclosures. One of his hobbies is studying the subtleties of relativity and quantum theory so that he can explain it to laypeople.