36.6.3 The Twin Paradox
The most well-known relativity paradox is the twin paradox. George and Helen are twins. On their 25th birthday, Helen departs on a starship voyage to a distant star. Let’s imagine, to be specific, that her starship accelerates almost instantly to a speed of 0.95c and that she travels to a star that is 9.5 light years (9.5 ly) from earth. Upon arriving, she discovers that the planets circling the star are inhabited by fierce aliens, so she immediately turns around and heads home at 0.95c.
The global positioning system (GPS), which allows you to pinpoint your location anywhere in the world to within a few meters, uses a set of orbiting satellites. Because of their motion, the atomic clocks on these satellites keep time differently from clocks on the ground. To determine an accurate position, the software in your GPS receiver must carefully correct for time-dilation effects.
A light year, abbreviated ly, is the distance that light travels in one year. A light year is vastly larger than the diameter of the solar system. The distance between two neighboring stars is typically a few light years. For our purpose, we can write the speed of light as . That is, light travels 1 light year per year.
This value for c allows us to determine how long, according to George and his fellow earthlings, it takes Helen to travel out and back. Her total distance is 19 ly and, due to her rapid acceleration and rapid turnaround, she travels essentially the entire distance at speed . Thus the time she’s away, as measured by George, is
George will be 45 years old when his sister Helen returns with tales of adventure.
While she’s away, George takes a physics class and studies Einstein’s theory of relativity. He realizes that time dilation will make Helen’s clocks run more slowly than his clocks, which are at rest relative to him. Her heart—a clock—will beat fewer times and the minute hand on her watch will go around fewer times. In other words, she’s aging more slowly than he is. Although she is his twin, she will be younger than he is when she returns.
Calculating Helen’s age is not hard. We simply have to identify Helen’s clock, because it’s always with Helen as she travels, as the clock that measures proper time . From Equation 36.9,
George will have just celebrated his 45th birthday as he welcomes home his 31-year-and-3-month-old twin sister.
This may be unsettling because it violates our commonsense notion of time, but it’s not a paradox. There’s no logical inconsistency in this outcome. So why is it called “the twin paradox”?
Helen, knowing that she had quite of bit of time to kill on her journey, brought along several physics books to read. As she learns about relativity, she begins to think about George and her friends back on earth. Relative to her, they are all moving away at 0.95c. Later they’ll come rushing toward her at 0.95c. Time dilation will cause their clocks to run more slowly than her clocks, which are at rest relative to her. In other words, as FIGURE 36.24 shows, Helen concludes that people on earth are aging more slowly than she is. Alas, she will be much older than they when she returns.
FIGURE 36.24 The twin paradox.
Finally, the big day arrives. Helen lands back on earth and steps out of the starship. George is expecting Helen to be younger than he is. Helen is expecting George to be younger than she is.
Here’s the paradox! It’s logically impossible for each to be younger than the other at the time they are reunited. Where, then, is the flaw in our reasoning? It seems to be a symmetrical situation—Helen moves relative to George and George moves relative to Helen—but symmetrical reasoning has led to a conundrum.
But are the situations really symmetrical? George goes about his business day after day without noticing anything unusual. Helen, on the other hand, experiences three distinct periods during which the starship engines fire, she’s crushed into her seat, and free dust particles that had been floating inside the starship are no longer, in the starship’s reference frame, at rest or traveling in a straight line at constant speed. In other words, George spends the entire time in an inertial reference frame, but Helen does not. The situation is not symmetrical.
The principle of relativity applies only to inertial reference frames. Our discussion of time dilation was for inertial reference frames. Thus George’s analysis and calculations are correct. Helen’s analysis and calculations are not correct because she was trying to apply an inertial reference frame result while traveling in a noninertial reference frame. (Or, alternatively, Helen was in two different inertial frames while George was only in one, and thus the situation is not symmetrical.)
Helen is younger than George when she returns. This is strange, but not a paradox. It is a consequence of the fact that time flows differently in two reference frames moving relative to each other.