The Twin Paradox

A number of paradoxes arise in the context of special relativity, due to the fact that the theory doesn’t fit our commonsense intuitions about time and space. Common sense is basically right so long as we’re talking about things that move much slower than the speed of light (relative to us) and that aren’t too far away. At small speeds and distances, relativistic time dilation and length contraction are negligible—so small we never notice them. However, when dealing with things that are very fast or very far away, our intuitions aren’t so reliable.

One of the most famous paradoxes of special relativity is the twin paradox. In the cow abduction scenario discussed earlier, imagine that the farmer has an identical twin who hitches a ride on the alien spaceship. We’ll call the two brothers Farmer Fred and Traveling Ted. As the spaceship whizzes away from Earth at nearly the speed of light, Fred and Ted each regard themselves as being at rest, and they regard each other as moving close to the speed of light. From Farmer Fred’s perspective (regarding the earth at rest), his space-traveling twin brother is experiencing time dilation: time goes by more slowly on the spaceship than it does on Earth. Let’s suppose the spaceship travels so fast that for every minute that ticks by on Fred’s own wristwatch, he reckons that only one second has elapsed on Ted’s wristwatch. The spaceship would have to be traveling very close to the speed of light for the time dilation to be that extreme. No human technology can propel a rocket anywhere close to that speed, but we can imagine that alien spaceships have such technology. From Traveling Ted’s perspective, on the other hand, Farmer Fred is the one experiencing time dilation! Ted sees the earth moving away from his spaceship at nearly the speed of light, so from his perspective physical processes on Earth occur more slowly. In Ted’s frame of reference (regarding the spaceship at rest), Farmer Fred experiences just one second for every minute that goes by on Ted’s wristwatch.

Each of the two men regards his twin as aging more slowly than himself. How can they both be right? Our commonsense intuitions tell us that this is impossible; yet the special theory of relativity says that both perspectives are equally valid.

But wait a minute.Or wait a second, depending on whose point of view we’re talking about. What if the alien spaceship returns to Earth, so that Fred and Ted can compare their ages? As the alien spaceship zooms back toward earth, Fred and Ted again regard each other as traveling close to the speed of light, and as before, each regards the other as aging more slowly. When the spaceship nears Earth, it slows down and stops to drop off Ted at his brother’s farm. Now the two men share the same reference frame, and they can’t both be younger than each other! Doesn’t this prove that Einstein’s theory is incorrect?

No, it doesn’t. The paradox is easily resolved if we remember that special relativity only applies to inertial frames of reference. So long as the spaceship is moving with constant velocity, Ted is allowed to consider himself at rest, and he can legitimately infer that his brother on Earth is aging more slowly. But in order for the spaceship to return to Earth, it must turn around. It may slow down and then speed up again in the opposite direction, or it may simply turn in a half-circle while maintaining full speed. Either way, it must be accelerated. In the special theory of relativity, accelerated observers are not allowed to consider themselves at rest.

Therefore, during the brief period of time when the spaceship is turning around in order to return back to Earth, Ted can’t infer that his brother is aging more slowly. To the contrary, he will regard his brother as aging extremely quickly—so quickly, in fact, that he’ll expect his brother Fred to be much older than himself when the spaceship finally arrives back on Earth, even though he regarded Fred as aging more slowly for most of the trip. 

To see why, let’s represent the situation in a spacetime diagram. We’ll have to use Fred’s spacetime diagram, since Ted doesn’t know how to draw the part of his own diagram where his spaceship accelerated. (He’s not allowed to consider the spaceship as being at rest during that time.) Technically, Fred’s frame of reference isn’t completely inertial either, since the earth is constantly accelerating around the sun. But Earth’s acceleration is small compared to that of the spaceship, so we can ignore it for present purposes. In Fred’s frame of reference, the spaceship zooms away from Earth, turns around, and flies home again. During the first half of the trip, Ted’s planes of simultaneity are slanted upward to the right (the direction in which his spaceship is traveling). But during the return trip, the spaceship is moving to the left, so his planes of simultaneity are slanted upward in that direction instead. And at the precise moment when the spaceship isn’t moving to the right or left relative to Earth, Ted’s plane of simultaneity coincides with Fred’s: it is horizontal. (See figure below.)

spacetime diagram illustrating the twin paradox
Fred’s Spacetime Diagram

During the brief period when Ted’s spaceship accelerates (at the curved part of his worldline), his planes of simultaneity pivot, sweeping past a huge portion of Fred’s time axis.

From the perspective of an inertial reference frame (like Fred’s), Ted’s planes of simultaneity pivot like a seesaw during the period of acceleration. As his spaceship begins to slow down, the planes of simultaneity gradually level off to horizontal. And as he speeds up again in the opposite direction, Ted’s planes of simultaneity begin to tilt the other way. Since the spaceship is far away from Earth when this acceleration occurs, Ted’s pivoting planes of simultaneity sweep past a huge portion of his brother Fred’s time axis.

For this reason, when Traveling Ted arrives back on Earth he will be much younger than his twin brother. Perhaps sixty years have gone by on Earth, and Farmer Fred is now a feeble old man. Yet Ted is only a year older than he was when he left! Ted has essentially travelled into the future by speeding away from Earth at close to the speed of light and then accelerating in the opposite direction to return home again.

The fanciful tale of Fred and Ted reveals a couple of interesting insights about the nature of time and space. One point worth noting is that it is the acceleration of the spaceship, not its speed, which allows Ted to return home younger than his twin brother. When a reference frame is accelerated, it experiences time dilation in an absolute (non-relative) way. We’ll return to this point in the context of Einstein’s second theory, general relativity.

The Hafele–Keating Experiment

The time dilation caused by acceleration has been measured experimentally, using extremely precise atomic clocks. In 1971, physicist Joseph Hafele and astronomer Richard Keating took four atomic clocks aboard commercial airliners, flew eastward around the world, and returned to compare their clocks with ones that had remained at the United States Naval Observatory. They then flew the clocks westward around the world, and again compared the times.

Since the earth rotates eastward, the clocks aboard the airliners experienced greater acceleration when flying around the world eastward than when flying westward. Just as Einstein’s theory predicted, Hafele and Keating found that their clocks ran slightly slower (compared to the Naval Observatory clocks) when flown eastward. When flown westward, on the other hand, their clocks ran slightly faster than the ground-based clocks, because the westbound aircraft stayed in an almost inertial reference frame while the Earth rotated beneath it.

A second observation that we can make in light of the above story is that the relativity of time is more pronounced at long distances. If you pace back and forth in a room, you are accelerating very slightly, and this acceleration makes your planes of simultaneity pivot one way or another, just as Ted’s did when his spaceship accelerated. Your planes of simultaneity don’t pivot very much, obviously, since you’re only accelerating a little. But even the slightest pivoting of your planes of simultaneity could make a significant difference at long distances. Physicist Paul Davies estimates that the small acceleration you experience just from pacing back and forth in a room is enough to change what day it is “now” (according to your frame of reference) in the Andromeda Galaxy.Paul Davies, About Time: Einstein’s Unfinished Revolution (New York: Simon & Schuster, 1995), 70.

Don’t misunderstand: your motion doesn’t make any difference to what happens in the Andromeda Galaxy. You can’t change the events that occur there, nor can you change the order in which they happen. What you can change is your own perspective about what day it is “right now” in that distant place. If you first regard the earth as being at rest, and then change your mind and regard yourself as being at rest while you pace across the room (and regard the Earth as moving slowly beneath your feet), that change in perspective makes a tiny difference in the slope of your present plane of simultaneity—the things you regard as happening “right now.” And that miniscule shift in your present plane of simultaneity is enough to make a difference of more than 24 hours at a distant planet in Andromeda. When you walk in the direction towards Andromeda (and regard yourself as being at rest), your plane of simultaneity tilts upward along the alien planet’s time axis; when you walk the other way, it tilts downward along that time axis. And the degree of tilt is enough to change which day (on that planet) you regard as simultaneous with today on Earth.