The Lorentz Group
Rotations in higher dimensional spaces define one- and two dimensional subspaces in which they act just as in the Euclidean plane. Similarly Lorentz transformations in higher dimensional spaces act, up to rotations, in two dimensional subspaces as boosts. Each Lorentz transformation \(\varLambda\) of the four-dimensional spacetime corresponds uniquely to a pair \(\pm M\) of linear transformations of a complex two-dimensional space, the space of spinors. Their inspection reveals that aberration, the Lorentz transformation of the directions of light rays, acts as a Möbius transformation of the Riemann sphere.