Introduction to Classical Mechanics · Problem 11.19
Relativity (Kinematics) — Morin
Physics — Mechanics · modified-twin-paradox
Problem
Consider the following variation of the twin paradox. , , and each have a clock. In 's reference frame, flies past with speed to the right. When passes , they both set their clocks to zero. Also, in 's reference frame, starts far to the right and moves to the left with speed . When and pass each other, sets his clock to read the same as 's. Finally, when passes , they compare the readings on their clocks. At this moment, let 's clock read , and let 's clock read . (a) Working in 's frame, show that , where . (b) Working in 's frame, show again that . (c) Working in 's frame, show again that .Intuitive explanation
The key to this problem is understanding that proper time (time measured by a clock moving with an observer) is invariant across reference frames, even though simultaneity is relative. Each observer ages according to their own proper time, which depends on their motion through spacetime. By carefully tracking which events are simultaneous in each frame and how moving clocks run slow, you'll see that all three frames agree on the final clock readings—even though they disagree on when certain events happened. The factor appears because and are moving relative to , so their clocks tick slower in 's frame, and this effect persists when you transform to their frames using relativity of simultaneity.
Intuitive explanation
The key to this problem is understanding that proper time (time measured by a clock moving with an observer) is invariant across reference frames, even though simultaneity is relative. Each observer ages according to their own proper time, which depends on their motion through spacetime. By carefully tracking which events are simultaneous in each frame and how moving clocks run slow, you'll see that all three frames agree on the final clock readings—even though they disagree on when certain events happened. The factor appears because and are moving relative to , so their clocks tick slower in 's frame, and this effect persists when you transform to their frames using relativity of simultaneity.