Math 2.0

Relativity and gravitational theory Special relativity: kinematics Problem

Twin paradox

Builds on Minkowski spaceWorld lineProper timeLorentz factorTime dilation

Problem

Twin AA stays at rest in an inertial frame. Twin BB travels at constant speed vv, with 0<v<c0 < v < c, to a star at distance LL in that frame. Then BB turns around and comes back at the same speed. By time dilation each twin, while moving uniformly, judges the clock of the other to run slow by the Lorentz factor γ=1/1−v2/c2\gamma = 1/\sqrt{1 - v^2/c^2}. When they meet again, which twin is younger, and why does the symmetric argument fail?

Remarks

The travelling twin BB is younger. The time each twin ages is the proper time along its world line in Minkowski space. For AA it is 2L/v2L/v, and for BB it is 2L/(γv)2L/(\gamma v), which is smaller. The situation is not symmetric: the world line of AA is a straight line. The world line of BB has a corner at the turnaround, where BB changes inertial frames. Among all timelike world lines between two events, the straight one has the longest proper time; this is the reversed triangle inequality of Minkowski geometry. At the turnaround the notion of simultaneity of BB jumps, which accounts for the missing time in the naive argument. Einstein derived in 1905 that a clock moved away and back lags behind one that stayed. Langevin gave the round trip of a traveller in 1911.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Lorentz factor, concept, no evidence levelLorentz factorconceptLorentz transformation, concept, no evidence levelLorentz transformationconceptMinkowski space, concept, no evidence levelMinkowski spaceconceptProper time, concept, no evidence levelProper timeconceptWorld line, concept, no evidence levelWorld lineconceptTime dilation, statement, no evidence levelTime dilationstatementTwin paradox, problem, no evidence levelTwin paradoxproblem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Lorentz factor concept
  2. Lorentz transformation concept
  3. Minkowski space concept
  4. Proper time concept
  5. World line concept
  6. Time dilation statement
  7. Twin paradox problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Twin_paradox
About this page
TagDA3S8B613F
URIm2://DA3S8B613F
Kindproblem
WritingW0 stub
MSC83A05
StatusThis copy of the site has no build data. Run m2 status DA3S8B613F.
AboutProper time
UsesMinkowski space, World line, Proper time, Lorentz factor, Time dilation
Same as
Changed3 November 2026

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