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.

  • A (para)Vector of space and time
  • So far we have considered operations on vectors only - linear combinations of the two basis vectors, \(\mathbf{x}\) and \(\mathbf{y}\). Here we will consider a more general combination of a vector together with a scalar component \(a_0\) to form a something called a paravector - a scalar value plus a vector value. Furthermore, we will see that this scalar component results in a amplitude-squared metric matching the metric of space-time in Special Relativity. Finally, we will see that we can rotate these space-time paravectors in space without affecting the temporal component, corresponding to changing our frame of reference within a single inertial frame.

  • The Lorentz Transformation
  • In the previous section we saw that a combination of a scalar and a vector, a paravector, \(\v{A} = a_0 + \v{a}\), has an amplitude-squared with the same metric as that of spacetime and yet can be rotated like a normal vector such that the scalar part is unchanged, while the vector part rotates in space, as \(\v{\c{R}}\v{A}\v{R}\), where the rotor \(\v{R}\) is itself comprised of a scalar and a bivector, \(\v{R} = e^{-\v{xy}\frac{\theta}{2}}\).

    We also saw that we can define a normalised paravector as \(\v{B} = e^{-\v{b}\frac{\phi}{2}}\), so that \(\v{B}\c{\v{B}} = 1\). In this section we will consider rotating the paravector \(\v{A}\) by a normalized paravector and see that the result, \(\v{B}\v{A}\v{B}\), is equivalent to the standard Lorentz transformation.