2013/07/02 by Patrick L. Nash, Patrick Lee Nash, Nash, Patrick Lee
Physics and Astronomy · #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Relativity and Gravitational Theory #physics.gen-ph
paper · pdf · doi:10.48550/arxiv.1307.0741
replaced by "Possible consistent extra time dimensions in the early universe," http://arxiv.org/abs/1310.0697
arxiv created 2015/04/14 · arxiv updated 2015/04/15
A mathematically interesting hyperbolic solution to the Einstein field equations is studied on an eight-dimensional pseudo-Riemannian manifold \mathbbX4,4 that is a spacetime of four space dimensions and four time dimensions. [The signature and dimension of \mathbbX4,4 are chosen because its tangent spaces satisfy a triality principle \citeNash2010 (vectors and spinors are equivalent).] This solution exhibits temporal hyperbolic inflation of three of the four space dimensions and temporal hyperbolic deflation of three of the four time dimensions. Comoving coordinates for the unscaled dimensions are chosen to be (x4 ↔ \textrmtime, x8 ↔ \textrmspace), where the x4 coordinate corresponds to our universe's observed physical time dimension and the x8 coordinate corresponds to a predicted new physical spatial dimension. This solution of the field equations manifests temporal hyperbolic inflation \cosh(13 H x4) of the scale factor associated with three of the four space dimensions, and temporal deflation \textrmsech(13 H x4) of the scale factor associated with three of the four time dimensions. (Here H is the Hubble parameter.) The scale factors possess a more complicated dependence on the spatial x8 coordinate, which, however, turn out to be periodic in x8 with period ∝ 1 / H. After "inflation" the observed physical macroscopic world has three space dimensions and one time dimension.