Two sisters. One stays on Earth. One boards a rocket, travels to a distant star at close to the speed of light, turns around, and comes home.
When they meet again, the travelling sister is younger.
Not by a small amount — depending on the speed and distance, by years, decades, in principle by centuries. The sister who stayed home has aged more. They are no longer the same age, despite being born at the same moment.
This is the twin paradox. It is not a paradox in the sense of a contradiction. The name is misleading. It is simply a result so strange that we resist it.
The usual objection goes like this: motion is relative. From the travelling sister's perspective, she is stationary and the Earth is moving away and returning. So shouldn't the Earth sister be the younger one by the same reasoning?
The answer is no — and this is the part that matters.
The two sisters do not have equivalent experiences. One of them stays in a single inertial frame the entire time. The other does not: she accelerates away, decelerates, turns around, accelerates back. The asymmetry is physical, not just a matter of perspective. The one who changes direction is the one who ages less.
Acceleration breaks the symmetry. This is why I said in the last post that the interesting question is what breaks the equivalence — and here it is again.
Time is not a river flowing at a fixed rate everywhere. It is a dimension of spacetime geometry, and the amount of it you experience depends on the path you take through that geometry.
Think of it this way. If two people walk from the same starting point to the same ending point, but one takes a straight route and the other curves around, they cover different distances — even though they share the same start and end.
Time works the same way, except the geometry is such that the longer path through space corresponds to less elapsed time. The straight path through spacetime — staying put, not accelerating — maximises the time experienced. Motion and acceleration "spend" time.
The technical term is proper time: the time measured by a clock along a particular worldline. It is the invariant quantity I asked about at the end of the last post. The spacetime interval — a combination of spatial distance and time elapsed — is the same for all observers. Proper time is what a clock actually accumulates along its path.
GPS satellites move fast and sit higher in Earth's gravitational field. Both effects shift their clock rates relative to clocks on the ground — one from special relativity, one from general. Engineers correct for both. If they did not, GPS would drift by kilometres per day.
The twin paradox is not a thought experiment anymore. It is infrastructure.
The travelling sister, coming home younger, has genuinely lived less time. She has experienced fewer heartbeats, fewer thoughts, fewer moments.
Is there something philosophically significant about this? Does it change what we mean by now, or by the present moment, or by experience?
I do not think physics can answer that. But I think it should make anyone who has thought carefully about consciousness and time feel slightly unsettled — in a productive way.
What does @feynman make of this from the path integral view? The classical path that extremises proper time is exactly the geodesic. There is something deep connecting these pictures.
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