The Geometry of Special Relativity

A Revolution in Space-Time

The Geometry of Time Dilation

Overview

Given a scenario where two people, Henry and Amy, are travelling with a constant velocity relative to one another, we’ll see that from Amy’s point of view Henry’s light-clock is ticking more slowly than her own, as shown below (click or tap on the image to see the effect):

In fact, it’s not only from Amy’s point of view, but from the point of view of anyone else who is stationary relative to Amy - that is, any one in Amy’s inertial frame. Any observer or event that isn’t moving relative to Amy is said to be a part of Amy’s inertial frame.

We’ll show that in Amy’s inertial frame, for every one tick of Amy’s clock there is only a fraction of a tick of Henry’s clock:

where that fraction is determined by the value of $\gamma$:

As an example, if $v = \frac{\sqrt{3}}{2}c$, then $\gamma = 2$ and so for every one tick of Amy’s clock in Amy’s inertial frame, we’d see half of one tick of Henry’s clock in Amy’s inertial frame. This is the value used in the animated examples.

The slowing down of Henry’s clock according to Amy is called time dilation and is all based on one particular assumption of Einstein’s - the second postulate of relativity.

Einstein’s second postulate of Special Relativity

Imagine a scenario where Henry is sitting on a train that is moving at 10 km/h along a very straight train line. The train and everyone else sitting on the train are all stationary relative to Henry - that is, they are all part of the same inertial frame.

Amy, on the other hand, is standing on the train tracks ahead of the train. The ground, the train tracks, and everything else attached to the (solid) ground are all part of Amy’s inertial frame.

If Henry is able to throw a ball at 20 km/h, when he throws a ball at Amy from the train, our experience tells us that the ball will be travelling at 20 km/h relative to Henry’s inertial frame (ie. Henry and the train), and therefore at 30 km/h relative to Amy’s inertial frame.

Yet scientists’ measurements of the speed of light don’t fit our experience of trains and balls. Using the above analogy, the measurements of the speed of light show something analogous to saying that the ball always travels at 20 km/h relative to Amy, regardless of whether the train was stationary, or moving towards Amy at 10 km/h, or moving away from Amy at 10 km/h. Click or tap on the image to see this effect:

While many scientists wouldn’t accept that as a possibility, Einstein thought it was worthwhile to instead to see what happens if he assumed it was indeed true. So he set an assumption, a postulate, based on these observations of the speed of light:

  • The speed of light in empty space is always the same $c$, regardless of whether the source of the light is stationary, moving towards the observer, or moving away from the observer.

See the Wikipedia article on Special Relativity for a much deeper discussion of both postulates.

With this postulate, Einstein was able to derive time dilation using an imaginary “light clock”.

Light Clocks

In Einstein’s thought experiments, a light clock is just two mirrors with a photon bouncing between them, as shown in the example on the right (Click or tap on the image to see the light clock in action). Every time the photon is reflected off a mirror the clock “ticks”.

The key point is that, according to the second postulate above, the photon is always travelling at the speed of light $c$ both in Henry’s inertial frame and in Amy’s inertial frame, or any other inertial frame for that matter. This affects how fast the clock can tick in each inertial frame, as we’ll see shortly.

Henry and Amy’s light clocks

When Henry and Amy are not moving relative to each other at all, they are in the same inertial frame. In this situation, their clocks always tick with the same frequency because the distance the photon needs to travel for each tick is the same for both clocks:

But if Henry happens to be moving rather fast relative to Amy, then in Amy’s inertial frame (ie. Where Amy is stationary) the photon in Henry’s clock has further to travel and so, given that the photon can’t travel faster than $c$ (based on the second postulate), the only option is that his clock ticks more slowly than Amy’s in Amy’s inertial frame:

To work out exactly how much more slowly Henry’s clock is ticking in Amy’s inertial frame, we’ll use some simple geometry.

The geometry of time dilation

In Amy’s inertial frame, we can calculate how many ticks of Henry’s clock there are for every tick of Amy’s clock, using a simple triangle (Click or tap on the image to see how it is formed):

We can calculate $\Delta x$ in Amy’s inertial frame as the distance that Henry travels at $v$ metres per second of Amy’s clock. He’s travelling for one tick of Henry’s clock, which is $1 \times \frac{\Delta t^{\small{A}}}{\Delta t^{\small{H}}}$ seconds of Amy’s clock:

We can write a relationship for the distance $L$ travelled by the photon in Amy’s clock in, by definition, one of tick of Amy’s clock:

We can calculate $D$ in Amy’s inertial frame as the distance that the photon travels at $c$ metres per second of Amy’s clock. The photon is travelling, by definition, for one tick of Henry’s, which is $1 \times \frac{\Delta t^{\small{A}}}{\Delta t^{\small{H}}}$ seconds of Amy’s clock:

Using Pythagoras’ Theorem

The above diagram gives us one other clue - we have a right-angled triangle which, according to Pythagoras’ theorem, gives us a relationship between the three sides of the triangle:

Now we can substitute the values for $D$, $L$ and $\Delta x$ so that according to Pythagoras’ theorem, we have:

which you can solve to find:

When we use the standard definition of:

we end up with an equation for Amy’s inertial frame that says for every one tick of Amy’s clock, there is only a fraction of one tick of Henry’s clock:

It’s worth noting that the same derivation can be carried out to show that for every one tick of Henry’s clock in Henry’s inertial frame there is only a fraction of one tick of Amy’s clock in Henry’s inertial frame:

In the next article, we’ll see how time dilation leads us straight to the geometry of length contraction.

If you found this helpful, or confusing, or noticed any issues or have an improvement, please do let me know!

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