James M. Chappell, Nicolangelo Iannella, Azhar Iqbal, Derek Abbott
Following the development of the special theory of relativity in 1905, Minkowski sought to provide a physical basis for Einstein's two fundamental postulates of special relativity, proposing a four dimensional spacetime structure consisting of three space and one time dimension, with the relativistic effects then being straightforward consequences of this spacetime geometry. As an alternative to Minkowski's approach, we produce the results of special relativity directly from three space ($ \Re_3 $) without the addition of an extra dimension, through identifying the local time with the three rotational degrees of freedom of this space. The natural mathematical formalism within which to describe this definition of spacetime is found to be Clifford's geometric algebra, and specifically a three-dimensional multivector. With time now identified with the three rotational degrees of freedom of space, time becomes three-dimensional, which provides a natural symmetry between space and time in the form of a complex-type number $ \mathbf{x} + i \mathbf{t} $, where $ \mathbf{x} $ and $ \mathbf{t} $ are three-vectors and $ i $ is the trivector. Similar to Minkowski spacetime, this description also immediately produces the results of special relativity, but we also show that our description can lead to explanations for quantum mechanical phenomena by virtue of the extra time dimensions. This representation also suggests a resolution of some key questions regarding the nature of time, as we can now for the first time utilize a precise mathematical definition of time in terms of spatial rotations.
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http://arxiv.org/abs/1205.5195
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