3 research outputs found
Revisiting special relativity: A natural algebraic alternative to Minkowski spacetime
Minkowski famously introduced the concept of a space-time continuum in 1908,
merging the three dimensions of space with an imaginary time dimension , with the unit imaginary producing the correct spacetime distance , 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 space-time
of two space dimensions and one time dimension, we replace the unit imaginary , with the Clifford bivector 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 and . 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 a simple formulation 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.Comment: 29 pages, 2 figure