Physic Labs

Theory of relativity

The twin paradox

Compare the two worldlines of a round trip and measure the elapsed times: Earth records Δt=2D/v\Delta t=2D/v while the traveler ages only Δτ=(2D/v)1−v2/c2\Delta\tau=(2D/v)\sqrt{1-v^2/c^2}, showing that the “paradox” is resolved by the change of inertial frame at turnaround.

Undergraduate

Equipment

  • 3D spacetime diagram with the twins' two worldlines
  • Slider “Tốc độ β = v/c” (speed)
  • Slider “Tham số mô hình” (model parameter, trip distance scale)
  • Quantitative γ(β) graph with marker; “Đặt lại góc nhìn” and “Tạm dừng” buttons

Procedure

  1. Read the two worldlines

    Inspect the spacetime diagram: the stay-at-home twin follows a straight worldline, while the traveler's worldline bends at the turnaround point. Elapsed aging is the proper time along each worldline, Δτ=∫1−v2/c2 dt\Delta\tau=\int\sqrt{1-v^2/c^2}\,dt, which is shorter for the bent path.

  2. Vary the speed β

    Drag the β slider from 0 toward 0.9 and read the elapsed times for both twins. The lower graph tracks γ=1/1−β2\gamma=1/\sqrt{1-\beta^2}; as β rises, the gap Δt−Δτ\Delta t-\Delta\tau grows, and it diverges as β→1\beta\to1.

  3. Check the round-trip numbers

    Pick β and read off the trip distance scale from the model slider. Compute Δt=2D/v\Delta t=2D/v and Δτ=Δt/γ\Delta\tau=\Delta t/\gamma by hand and compare with the readouts; then predict the age gap for a faster trip before moving the slider to confirm.

Simulation

Experiment history

Einstein's 1905 paper on special relativity already noted that clocks in relative motion cannot stay synchronized: a clock transported around a closed path returns behind one that stayed put. Paul Langevin popularized the striking traveler version in 1911, framing it as a genuine puzzle rather than a mathematical footnote. Hermann Minkowski's 1908 spacetime picture gave the clean resolution: each twin ages by the proper time along his worldline, and the traveler's bent worldline simply has less proper time. The effect is not hypothetical — atomic clocks flown around the world in the Hafele–Keating experiment (1971) and daily corrections in GPS satellites confirm the predicted gains and losses of time.

Related physicists

Albert Einstein

Related library topics

The twin paradox