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Mathematics for Physics

207 Twin Paradox

The thought experiment in which a traveling twin returns younger than the twin who stayed on Earth, illustrating time dilation along different worldlines.

Asymmetry of the two frames. The two twins are not in equivalent situations. The stay-at-home twin remains in one inertial frame, while the traveling twin does not. An inertial frame is a coordinate system in which an isolated object moves at constant velocity. This principle is used to remove the apparent contradiction of mutual time dilation.

Maximal proper time on a geodesic. Among worldlines joining the same departure and return events, the unaccelerated path has the largest proper time. A timelike geodesic is that straight unaccelerated worldline. Proper time is the time elapsed on a clock that follows the worldline. This principle is used to prove that the stay-at-home twin is older at reunion.

Frame-switching at turnaround. Completing the round trip requires the traveler to switch from an outbound inertial frame to a return inertial frame. Frame-switching is that change of rest frame at turnaround. This principle is used to show that the traveler’s notion of simultaneous Earth time jumps during the turnaround.

Measurable proper acceleration. The traveling twin feels a proper acceleration at turnaround that the stay-at-home twin does not. Proper acceleration is the acceleration registered by an onboard accelerometer. This principle is used to identify which twin left inertial motion and is younger at reunion.

Note: Also called the clock paradox.

207.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — source for the heading explanation.
  2. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — asymmetric inertial histories; proper acceleration of the traveler.
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — non-symmetry of the twins; frame-switching at turnaround.
  4. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — maximal proper time on a straight timelike path.
  5. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — turnaround and simultaneity.
  6. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — proper acceleration of the traveling twin.