Tuesday, February 7, 2012

Resolving the “Twin Paradox”

Fans of science-fiction space travel know that if someone goes on a rocket at close to the speed of light, he will age more slowly than someone back home. Returning to Earth and reuniting with a twin, he would find that the twin had aged more, perhaps by years. If the trip is long enough and gets really close to the speed of light, the returning traveler could find an Earth that’s millions of years in the future, maybe even devoid of human life. For many people, that’s the “twin paradox” — how could something this strange possibly happen?

Actually, twins aging differently isn’t the issue; no paradox there. The “paradox” lies in the fact that according to relativity (the very effect that causes the twins to age differently), the ideas of motion and rest are relative. Suppose rather than leaving from Earth, the experiment is done in deep space. Twin A takes off and leaves Twin B behind. But, once the twins are separated, who’s to say which twin is moving and which is at rest? After all, if you consider the picture from the perspective of either twin, it’s the other one that’s moving. Twin A sees Twin B receding rapidly in the rear-view mirror, just as if Twin B was the one who had taken off. In relativity, the question of moving vs. stationary depends upon the perspective of the observer. Shouldn’t this mean that the aging will be equal for both twins when they finally reunite? That’s the “twin paradox.”

The twin paradox isn’t really a paradox, because it can be resolved in several ways. For one thing, the situation isn’t symmetrical. One of the twins, Twin A, has to turn around at some point, whereas Twin B can just cool his heels. Physicists say that Twin B remains in an inertial reference frame — a state of constant motion (or rest), without any change in speed. Twin A, though, spends portions of his trip in two different inertial reference frames, one on the way out, and one on the way back, and in between he has to slow down, stop, and accelerate in the other direction. (Or, he could make a circular U-turn, but that still counts as an acceleration.) It’s clear how changing velocity affects the local passage of time, from the formula known as the Lorentz transformation: Higher velocities mean slower clocks, so as Twin A slows down, his onboard clock begins speeding up. Twin B’s velocity never changes, though. Therefore, it’s wrong to say the twins are moving in identical ways relative to the other.

Thinking this through, the situation is a bit peculiar. How does Twin A and his mechanical onboard clock “know” that they had turned around? Suppose Twin B, rather than staying put, secretly takes off in the other direction, goes even faster, turns around, and races back to the starting point, just in time to meet the returning Twin A. In that case, Twin B would age slower than Twin A — who, having just gone on a fast trip that included turning around, expects to meet an older Twin B. Instead he finds a younger Twin B. How does Twin B (and his clock) know that he had gone farther and faster than Twin A?

The solution all comes down to the turning-around part: acceleration. When you’re in an inertial reference frame, you can’t tell whether you’re moving or not. You could be sitting in your living room, or you could be on a rocket ship going 99% the speed of light. Close your eyes and the situations are indistinguishable. However, when you’re slowing down or speeding up, or making a turn, you can definitely tell that something’s going on. You feel a pull toward one direction, just as you do in a car with the brakes applied. This acceleration* is what causes a clock (biological or otherwise) to slow down. For Twin B who secretly went on a really fast trip, his larger accelerations slowed down his biological clock even more than Twin A’s, who now needs a facelift to look as young as his twin again.

When you add gravity into the mix, it gets even stranger. Twin A — rather than turning on the reverse-thrust engine to slow down and turn around — could instead have a close call with a massive star, and like Halley’s comet, take a tight orbit around and be fired back toward his starting point. In such a situation, the ship is in free fall with respect to the star, and counter-intuitively, doesn’t feel the acceleration as it gets slung back toward Twin B. Our traveler could continue his game of three-dimensional billiards in the weightless environment of his ship, even as it hooks sharply around the star. If there were no windows, he might not even know when he was passing behind and starting to head back. Yet incredibly, the curvature of space from the star’s gravity would slow down time aboard the ship. While the traveling twin works on his weightless billiards game, the stay-at-home twin would have the time to master not only billiards but also croquet and miniature golf, much to Twin A’s later envy.

The “twin paradox” is one of those cases where the universe just works out perfectly right, out of sheer mathematical consistency. Relative velocity determines the rate at which clocks tick; Einstein showed that this malleability of time is necessary in a universe where the speed of light is measured the same by all observers. Therefore, changing velocity (accelerating) changes the local rate at which time passes. All of this can be calculated from the Lorentz transformation. But a star’s gravity will also slow down time for Twin A and his ship, by exactly the same amount as if he had used his engines to slow down and turn around. The curvature of space due to gravity — by having the ability to sling a spaceship around and back in the other direction, on momentum alone — simply has to make an adjustment to the ship’s onboard clocks. Otherwise the math wouldn’t work out. And then we’d have a real paradox.

Anything in our universe with mass slows down clocks in its wake. For the Earth, the effect is not only measurable, it needs to be built into your GPS to avoid large, cumulative errors that would render it useless. Thanks to Einstein’s discovery, you can reliably arrive at your destination. Isn’t it nice when things work out?

* The term “acceleration” refers to both slowing down and speeding up. Slowing down is simply accelerating in the opposite direction.


  1. hello, I am puzzled about why you mention acceleration.
    1- there is no acceleration term in the Einstein formulation. Only velocity. I don't think there is acceleration in the Lorentz equations either.

    2- the synchronization of clocks can be done after the acceleration, and comparison of time elapsed can be done while speed is constant.

    for instance, a person on a train moving at constant speed passes a platform with stationary observer. They synchronize clocks as the train passes.

    They agree that after 1 minute by the train clock a flare will be sent from the train. The distance between train and platform can be measured ahead of time, and the expected delay in seeing the flare can be accounted for.

    How much time will have passed for the observer on the platform?

    There was no acceleration involved. This is such an obvious example that I am surprised it is never mentioned by the so called PhD experts.

  2. 1. Acceleration comes into play in general relativity, but not special relativity. By the equivalence principle, acceleration is equivalent to spacetime curvature, which is in Einstein's GR equations. The situations between a rocketship sitting on the launchpad on a planet can be interpreted the same as a rocketship accelerating "upward" far from any gravitational mass.
    2. If you're talking about a simple example of special relativity, such as two trains passing each other while going the same speed, then yes, acceleration is totally out of the picture.