Revisiting Special Relativity: A Natural Algebraic Alternative to Minkowski Spacetime
Abstract
Minkowski famously introduced the concept of a spacetime continuum in 1908, merging the three dimensions of space with an imaginary time dimension $ i c t $, with the unit imaginary producing the correct spacetime distance $ x^2  c^2 t^2 $, and the results of Einstein's then recently developed theory of special relativity, thus providing an explanation for Einstein's theory in terms of the structure of space and time. As an alternative to a planar Minkowski spacetime of two space dimensions and one time dimension, we replace the unit imaginary $ i = \sqrt{1} $, with the Clifford bivector $ \iota = e_1 e_2 $ for the plane that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis $ e_1 $ and $ e_2 $. We find that with this model of planar spacetime, using a twodimensional Clifford multivector, the spacetime metric and the Lorentz transformations follow immediately as properties of the algebra. This also leads to momentum and energy being represented as components of a multivector and we give a new efficient derivation of Compton's scattering formula, and a simple formulation of Dirac's and Maxwell's equations. Based on the mathematical structure of the multivector, we produce a semiclassical model of massive particles, which can then be viewed as the origin of the Minkowski spacetime structure and thus a deeper explanation for relativistic effects. We also find a new perspective on the nature of time, which is now given a precise mathematical definition as the bivector of the plane.
 Publication:

PLoS ONE
 Pub Date:
 December 2012
 DOI:
 10.1371/journal.pone.0051756
 arXiv:
 arXiv:1106.3748
 Bibcode:
 2012PLoSO...751756C
 Keywords:

 Physics  Classical Physics;
 Mathematical Physics
 EPrint:
 29 pages, 2 figures