The Geometry of Special Relativity

A Revolution in Space-Time

Overview

Special Relativity is underpinned by the Lorentz Transformation, which is understood to transform a single event from the space-time coordinates of one reference frame $(x,\ t)$ to the space-time coordinates of the same event in another reference frame $(x’,\ t’)$. This requires a geometry of space-time which is quite different from our everyday experience and as a result quite difficult to think about - which is of course no reason to question its validity. Nonetheless, if there was a simpler geometry which also provided an understanding of the Lorentz Transformation, time dilation and the other properties of Special Relativity, that was also experimentally verifiable it would be worth considering.

This site exists to demonstrate how a simpler coordinate transformation - a simple rotation of space-time - can also lead to a geometric derivation of the Lorentz transformation formulas with a different understanding of their meaning, while still preserving all the normal experimentally verifiable properties of Special Relativity such as time dilation.

Yet the geometry does predict a small but testable experimental difference, and although the probability that the test confirms this thought experiment in geometry may be very small, the implications for our understanding of the universe we live in would be huge.

The Geometry of Time Dilation

We start with one assumption - Einstein’s second postulate of relativity that the speed of light is the same in all reference frames - and follow a very standard geometrical derivation of time dilation. This article introduces inertial frames, the second postulate and light clocks.

The Geometry of length contraction

Length contraction is here derived as a consequence of time dilation, in the sense that you can not have time dilation without length contraction.

The geometry of Space-Time

If we start without any previous assumptions of space-time geometry (like Minkowski Space-Time) then the simplest way to visualise the relative velocity between inertial frames is a rotation.

In this thought experiment, a normal geometric rotation and scale is used to define a coordinate transform that preserves time dilation and length contraction. It’s not the Lorentz transformation, but…

The Geometry of the Lorentz Transformation

Using the simple geometry of space-time defined above we ask a question:

Suppose a photon is emitted at $(x_{\small{e}}, t_{\small{e}})$ from a stationary object (let’s say a torch) in one inertial frame and travelling at $c$ in that inertial frame intersects a moving observer at a particular point in space-time, let’s say the point of observation. Then switching to the second inertial frame, where that same observer is stationary, where must a photon from the same torch be emitted from so as to reach that same point of observation when travelling at $c$ in that inertial frame?

The result as you probably guessed is the exact formula of the Lorentz transformation (Don’t worry, there are visualisations the whole way).

A prediction to test: Seasonal redshifts

Still to come. The slightly different understanding of the Lorentz transformation that comes from the above derivation predicts a different result for the change in redshift that we observe as the Earth’s relative velocity to the source changes. This section will derive that expected difference.

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