The Geometry of Special Relativity

A Revolution in Space-Time

The Geometry of the Lorentz Transformation

Overview

Given a scenario where two people, Henry and Amy, are travelling with a constant velocity relative to one another, we’ll use only the second postulate of relativity - that the speed of light is the same in all inertial frames - together with some geometry to derive the standard Lorentz transformation (using $c = 1$ for simplicity):

where,

The geometry of the derivation is shown below - choose any point below with your mouse pointer and check it yourself using the Lorentz transformation:

This derivation of the Lorentz transformation can be summarised as follows: for any given photon emission event $(x_{\small{e}}, t_{\small{e}})$ from a stationary object (let’s say a torch) in Amy’s inertial frame:

  • calculate where the photon from that event, travelling at the speed of light relative to Amy, intercepts Henry’s path,
  • calculate where, in Henry’s time-dilated and Lorentz-contracted coordinates, a photon observed at the same point must have originated from the torch given that the photon observed by Henry must be travelling at the speed of light relative to Henry.

The result is the well known Lorentz transformation although the understanding of the equations is quite different.

Let’s break the derivation down into small steps and take a look.

Amy’s inertial frame

Let’s assume that Amy is positioned at $x = 0$ in her inertial frame and has a stationary light-source (a torch) positioned some distance away.

Try pointing at different positions on the visual below to demonstrate the obvious - that the torch always remains the same distance from Amy as time moves forwards or backwards (ie. it is stationary):

If we use $(x_{\small{e}}, t_{\small{e}})$ for the event where the photon is emitted from the torch, we can also write an equation to describe the photon that travels towards Amy in Amy’s inertial frame (ie. a line with a gradient of -1, as shown in the above visualisation):

which we can rearrange to define:

For example, for the photon emitted at $(x_{\small{e}}, t_{\small{e}}) = (50, -50)$, the photon will be defined by $x = -t$, so at $t = 0$, it’s at the origin. Try moving the photon emission event to $(50, 0)$ in the visualisation above and check the equation for the photon.

Henry in Amy’s inertial frame

We now add Henry to the scene such that Henry is moving relative to Amy at $v$ metres per second to the right (ie. along the x-axis in the positive direction). We can write an equation that will tell us Henry’s position in Amy’s inertial frame at any point in time:

Play with the visualisation below where $v=0.3c$ to demonstrate that according to Amy, for every 10 seconds Henry moves 3 metres, or for every 50 seconds he moves 15 metres.

We can then work out where the photon intercepts Henry’s path in Amy’s inertial frame, by equating the equation for Henry and the equation for the Photon in Amy’s inertial frame:

so that we end up with the time when Henry meets the photon in Amy’s inertial frame, defined as:

Check this in the above visualisation. For example, if you move the photon event to $(50, 0)$ then the photon should intersect Henry when $ t = \frac{50}{1 + 0.3} = 38.5 $, at the point $(11.5, 38.5)$.

Time dilation and length contraction

In previous articles we saw how Einstein’s second postulate, that light always travels at the same speed in all inertial frames, leads directly to time dilation and therefore length contraction.

Specifically, for every one tick of Amy’s clock there’s only a fraction of a tick of Henry’s clock:

and for every metre rod-length of Amy’s (along the direction of the relative velocity) the rod is only a fraction of a metre for Henry:

We then saw a simple geometry of space-time which enables us to visualise the velocity as:

while still preserving time dilation and length contraction:

The Lorentz transformation

We’re now ready for the last step - deriving the Lorentz transformation.

So far, given a photon emission event from a stationary torch in Amy’s inertial frame at $(x_{\small{e}}, t_{\small{e}})$, we have a method for calculating where that photon will cross Henry’s path in Amy’s inertial frame:

at which point, according to time-dilation, Henry’s clock shows a fraction of that time:

We can define a photon travelling at the speed of light relative to Henry, that passes through this point (ie. a line with a gradient of -1 relative to Henry’s axis):

so that,

We can also define an equation for the torch relative to Henry, with the knowledge that it will be moving with a velocity of $-v$ and at $t’ = 0$, the distance to the torch will be the contracted $\frac{x_{\small{e}}}{\gamma}$:

Solving those two equations to find the point when this photon must have been emitted from the torch if it was indeed travelling at the speed of light relative to Henry, results in the Lorentz transformation for time:

And substituting this equation for $t’$ into the equation for the torch relative to Henry results in the Lorentz transformation for space:

Pulling it all together

We now have a geometric derivation of the Lorentz transformation such that for any given photon emission event $(x_{\small{e}}, t_{\small{e}})$ from a stationary torch in Amy’s inertial frame, if we:

  • calculate where the photon from that event, travelling at the speed of light relative to Amy, intercepts Henry’s path, and then
  • calculate where, in Henry’s time-dilated and Lorentz-contracted coordinates, a photon observed at the same point must have originated from the torch given that the photon observed by Henry must be travelling at the speed of light relative to Henry,

then the result is the Lorentz transformation:

This geometry is visualised as:

The transformation is a combination of

  • a physical transformation of space-time due to the relative velocity, including the Lorentz contraction and time dilation for Henry with respect to Amy.
  • an observational distortion between the event which a stationary observer will see (that is, an observer in the same inertial frame as the source of events, the torch in the examples), and the different event which a moving observer (Henry) at the same point in space-time will see.

Both are the direct result of the second postulate of relativity. The former is testable and consistent with experimental results. The latter is contrary to the idea that the Lorentz transformation transforms a single event from one inertial frame to another.

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