Time Dilation

Vermilion and Cerulean construct identical clocks, consisting of a light beam which bounces off a mirror. Tick, the light beam hits the mirror, tock, the beam returns to its owner. As long as Vermilion and Cerulean remain at rest relative to each other, both agree that each other's clock tick-tocks at the same rate as their own.

But now suppose Cerulean goes off at velocity \(v\) relative to Vermilion, in a direction perpendicular to the direction of the mirror.

A far as Cerulean is concerned, his clock tick-tocks at the same rate as before, a tick at the mirror, a tock on return.

But from Vermilion's point of view, although the distance between Cerulean and his mirror at any instant remains the same as before, the light has further to go. And since the speed of light is constant, Vermilion thinks it takes longer for Cerulean's clock to tick-tock than her own.

Thus Vermilion thinks Cerulean's clock runs slow relative to her own.

How much slower does Cerulean's clock run, from Vermilion's point of view? In special relativity the factor is called \(\gamma\) the Lorentz gamma factor, introduced by the Dutch physicist Hendrik A. Lorentz in 1904, one year before Einstein proposed his theory of special relativity. Let us see how the Lorentz gamma factor \(\gamma\) is related to Cerulean's velocity \(v\).

In units where the speed of light is one, \(c = 1\), Vermilion's mirror is one tick away from her, and from her point of view the vertical distance between Cerulean and his mirror is the same, one tick. But Vermilion thinks that the distance travelled by the light beam between Cerulean and his mirror is \(\gamma\) ticks. Cerulean is moving at speed \(v\), so Vermilion thinks he moves a distance of \(\gamma v\) ticks during the \(\gamma\) ticks of time taken by the light to travel from Cerulean to his mirror. Thus, from Vermilion's point of view, the vertical line from Cerulean to his mirror, Cerulean's light beam, and Cerulean's path form a triangle with sides \(1\), \(\gamma\) and \(\gamma v\) as illustrated. Pythogoras' theorem implies that \[ 1^2 + ( \gamma v )^2 = \gamma^2 \] from which it follows that the Lorentz gamma factor \(\gamma\) is related to Cerulean's velocity \(v\) by \[ \gamma = {1 \over \sqrt{1 - v^2}} \] which is Lorentz's famous formula.

Vermilion thinks Cerulean's clock runs slow. But of course from Cerulean's perspective it is Vermilion who is moving, and Vermilion whose clock runs slow. How can both think the other's clock runs slow? Paradox!

The resolution of the paradox, as usual in special relativity, involves simultaneity, and as usual it helps to draw a spacetime diagram, such as this one from the Centre of the Lightcone page.

While Vermilion thinks events happen simultaneously along horizontal planes in this diagram, Cerulean thinks events occur simultaneously along skewed planes. Thus Vermilion thinks her clock ticks when Cerulean is at the point , before Cerulean's clock ticks. Conversely, Cerulean thinks his clock ticks when Vermilion is at the point , before Vermilion's clock ticks.

Where do the two light beams in Vermilion's and Cerulean's clocks go in this spacetime diagram? Here.