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Our story stars two twins, sometimes unimaginatively named A and B; we prefer the monikers Stella and Terence. Terence sits at home on Earth. Stella flies off in a spaceship at nearly the speed of light, turns around after a while, thrusters blazing, and returns. (So Terence is the terrestrial sort; Stella sets her sights on the stars.)
When our heroes meet again, what do they find? Did time slow down for Stella, making her years younger than her home-bound brother? Or can Stella declare that the Earth did the travelling, so Terence is the younger?
Not to keep anyone in suspense, Special Relativity (SR for short) plumps unequivocally for the first answer: Stella ages less than Terence between the departure and the reunion.
Perhaps we can make short work of the "travelling Earth" argument. SR does not declare that all frames of reference are equivalent, only so-called inertial frames. Stella's frame is not inertial while she is accelerating. And this is observationally detectable: Stella had to fire her thrusters midway through her trip; Terence did nothing of the sort. The Ming vase she had borrowed from Terence fell over and cracked. She struggled to maintain her balance, like the crew of Star Trek. In short, she felt the acceleration, while Terence felt nothing.
Whew! One short paragraph, and we've polished off the twin paradox. Is that really all there is to it? Well, not quite. There's nothing wrong with what we've said so far, but we've left out a lot. There are reasons for the popular confusion.
For one thing, we've been rather unfair to Stella. We've said why she can't simply adopt Terence's viewpoint, but we haven't said how things look from her perspective. It seems passing strange that Terence could age several years just because Stella engages her thrusters. The Time Gap and Distance Dependence Objections put a sharper edge on this uneasy feeling.
There are versions of the twin paradox where Stella doesn't turn on her thrusters and feels no acceleration (the Slingshot variation, where Stella whips round a distant star in free fall, and the Magellan variation, where Stella travels round a cylindrical or spherical universe). These cast doubt on how relevant the acceleration is in the usual version. (We may add FAQ entries for these variations sometime in the future, but at the moment they are left as Exercises for the Reader.)
Finally, what about the Equivalence Principle? Doesn't that say that Stella can still claim to be motionless the whole time, but that a huge pseudo-gravitational field just happened to sweep through the universe when she hit her "thrusters on" button? (For that matter, Terence experiences the Earth's gravity, but his frame can be considered to be approximately inertial.) Some people claim that the twin paradox can or even must be resolved only by invoking General Relativity (which is built on the Equivalence Principle). This is not true, but the Equivalence Principle Analysis of the twin paradox does provide some additional analysis of the subject. The EP viewpoint is nearly mandatory for understanding some of the twin paradox variations.
Let's lay out a standard version of the paradox in detail, and settle on some terminology. We'll get rid of Stella's acceleration at the start and end of the trip. Stella flashes past Terence in her spaceship both times, coasting along.
Here's the itinerary according to Terence:
How much over? Well, Terence says the turnaround took a day. Stella's speed was changing throughout the turnaround, and so her time dilation factor was changing, varying between 1 and 7. So Stella's measurement of the turnaround time will be something between 1 day and 1/7 of a day. If you work it out, it turns out to be a bit over 15 hours.