Time Dilation (2D)

The same light clock seen from its own frame and from the lab, a spacetime diagram, and the cosmic-ray muon test.

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Presets

Playback

Live readout

Lorentz factor gammaLab time tClock proper time tauTicks seen in labTicks on the clockLength contraction 1/gamma

Muon survival (beta from slider)

Flight time in labMuon proper timeSurvival without dilationSurvival with dilationRatio

What to try

1. Press 0.866: gamma = 2, the lab sees half as many ticks. 2. Press Muon: survival jumps by many orders of magnitude. 3. Watch the world-line in the spacetime diagram lean toward the light cone as beta approaches 1.

Model checks

The science

In the clock's rest frame a photon travels straight up and down between mirrors a distance L apart, so one tick lasts T0 = 2L/c. In the lab the clock moves at v, so the photon's path is a zigzag: its vertical speed is sqrt(c² - v²) because the speed of light is the same in every inertial frame. The tick then lasts T = 2L / sqrt(c² - v²) = gamma T0 with gamma = 1 / sqrt(1 - beta²).

On the spacetime diagram the interval c²t² - x² = c²tau² is the same for every observer, and the clock's world-line is x = beta c t. Muons made about 15 km up live tau0 = 2.197 microseconds at rest; the surviving fraction is exp(-t / (gamma tau0)).

Honest simplifications: a single inertial frame (no acceleration or twin paradox), muons all made at the same height and moving straight down at one speed, no energy loss. Mirror gap is in light-seconds so the picture is readable; real light clocks are far smaller.