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 Vector in Spacetime
  • 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 an \(\mathbf{xy}\) component to form a "multivector". Furthermore, we will see that this extra component behaves geometrically as we expect time to behave in the space-time of Special Relativity and, by extension, in our every day experience. We will then see that we can rotate these space-time multivectors in space without affecting the temporal component, corresponding to changing your frame of reference within a single inertial frame.

  • The Lorentz Transformation
  • In the previous section we saw that we can rotate a space-time multivector \(\mathbf{A}\) in space - using a rotation from one normal vector \(\mathbf{b}\) to another normal vector \(\mathbf{c}\) to form the rotation \(\mathbf{bc} = e^{\mathbf{xy}\frac{\theta}{2}}\). We also saw that we can define a normal space-time multivector as \(\mathbf{B} = \mathbf{b}e^{\mathbf{bt}\phi}\), where \(\mathbf{b}^2 = 1\). In this section we will consider a space-time multivector \(\mathbf{A}\) being rotated in space-time - using a rotation from one normal multivector \(\mathbf{B}\) to another normal multivector \(\mathbf{C}\), and see that the result is the standard Lorentz transformation.