James. M. Chappell Nicolangelo Iannella, Azhar Iqbal, Derek Abbott
Minkowski famously introduced the concept of the space-time continuum in
1908, merging the three dimensions of space with an imaginary time dimension $
i c t $, naturally producing the correct spacetime distance $ x^2 - c^2 t^2 $,
and the results of Einstein's then recently developed theory of special
relativity. As an alternative to a planar Minkowski space-time 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 two-dimensional 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 simple derivations of Dirac's and Maxwell's equations. Based on
the mathematical structure of the multivector, we produce a semi-classical
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.
View original:
http://arxiv.org/abs/1106.3748
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