The Geometry of Relativity

Both relativity and quantum mechanics are often hard to grasp for our minds because they describe physical situations beyond the extremes of our every-day experience.

The mathematical tools that we use to describe Special Relativity can be marvelously helpful, but may also unintentionally hide from view a geometric understanding of the Lorentz transformation.

Geometric Algebra is a relatively recent branch of Mathematics which simplifies many physical processes involving vectors. This site is my attempt to understand and communicate the beauty of geometric algebra as simply as possible while having fun investigating a derivation of the Lorentz transformation as a geometric rotation in space-time, as well as looking at the implications of that geometry.

  • Intro to Geometric Algebra
  • Geometric Algebra is a line of Mathematics which treats vectors as first-class citizens, rather than arrows defined by coordinates on a coordinate system.

    For a flat Euclidean space, Geometric Algebra can be derived from a single rule that the square of the magnitude of any vector \(\v{a}\) is equal to the square of the vector itself.

    With this one rule we can introduce addition and multiplication of two-dimensional vectors, as well as a simple rotation.

  • Rotations in Space
  • At the end of the previous section we saw that we can rotate an arbitrary vector by 90 degrees, \(\frac{\pi}{2}\) radians, simply by multiplying that vector by \(\mathbf{xy}\). In this section we will see that this is because \(\mathbf{xy} = e^{\mathbf{xy}\frac{\pi}{2}}\) and that more generally we can rotate the same vector by any arbitrary angle \(\theta\) simply by multiplying by \(e^{\mathbf{xy}\theta}\).

    But to get to that point we'll first need to take a closer look at Euler's formula and its application to geometric algebra - where we no longer need to define the imaginary unit \(i\).

  • Rotations in Space-Time
  • Building on rotations using geometric algebra, in this section we define a two-dimensional space-time multivector and then investigate a rotation in space as well as a rotation in space-time, the result of which is equivalent to the Lorentz transformation.

  • Coordinate-free rotations
  • The previous section derived the Lorentz Transformation as a geometric rotation in two-dimensional Spacetime. Rather than simply generalising this to three dimensions, we will now take the opportunity to further our understanding of Geometric Algebra so that we can remove some of the cognitive load of dealing with lots of coordinates.

    One of the most important aspects of Geometric Algebra is that we can stop thinking in terms of coordinates of a specific reference frame, and rather represent everything as vectors.

    We will begin by revisiting the two-dimensional rotation in space and move towards the Lorentz Transformation with three spatial and one temporal dimension.