Special Relativity: Time Dilation and Length Contraction

A train passes a station. Switch between the two points of view, change the speed, and compare the clocks and the lengths that each observer measures.

Speed (v)0.50 c
Lorentz factor (γ)1.15
Moving clocks tick at0.87 × the rate
Moving objects measure0.87 × l0
Platform and groundTrainMeasured lengthRest length (dashed outline)

Whose point of view?
Playback

Show the formulas and the notes

Time dilation. A clock that moves relative to the observer runs slow: t = γt0. Length contraction. An object that moves relative to the observer is shorter along its direction of motion: l = l0/γ. Here γ = 1/√(1 − v2/c2).

Each side sees the other contracted. The platform sees the train short and its clocks slow. The train sees the platform short and the station clocks slow. There is no contradiction, because each observer is comparing a moving object with their own stationary rulers and synchronised clocks.

Simultaneity. Clocks that are in step in one frame are out of step in the other. Along the train, the clocks at the back are ahead of the ones at the front, as the platform sees them (by vΔx/c2 for clocks a distance Δx apart in the train). Try the readout under the picture.

What you measure is not what you see. This picture is a world-map: where things are, and what the clocks read, at the same instant, as measured by a grid of rulers and synchronised clocks, after allowing for the time light takes to travel. It is not a photograph. For a train coming towards you at 0.866c, light from its clock arrives squeezed together, so a camera would show the clock ticking about 3.7 times faster, and the train looking longer, not shorter. Time dilation and length contraction are measurements, not optical illusions.

Learn more in the Special relativity lesson and the Evidence for special relativity lesson. Original idea by Aaron (Scots College), rebuilt in the Physics High style.