Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
We use Einstein’s summation convention. Unless stated otherwise, each pair of indices denotes the sum over the range of its values, \(D^i{_j}v^j=D^i{_1}v^1+D^i{_2}v^2+\dots +D^i{_d}v^d\).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 The Author(s)
About this chapter
Cite this chapter
Dragon, N. (2012). The Lorentz Group. In: The Geometry of Special Relativity - a Concise Course. SpringerBriefs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28329-1_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-28329-1_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-28328-4
Online ISBN: 978-3-642-28329-1
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)