Hyperbolic Inflation

Abstract

A mathematically interesting hyperbolic solution to the Einstein field equations is studied on an eight-dimensional pseudo-Riemannian manifold $\mathbb{X}_{4,4}$ that is a spacetime of four space dimensions and four time dimensions. [The signature and dimension of $\mathbb{X}_{4,4}$ are chosen because its tangent spaces satisfy a triality principle \cite{Nash2010} (vectors and spinors are equivalent).] This solution exhibits temporal hyperbolic inflation of three of the four space dimensions and temporal hyperbolic \textbf{deflation} of three of the four time dimensions. Comoving coordinates for the \textbf{unscaled} dimensions are chosen to be $(x^4 \leftrightarrow \textrm{time}, x^8 \leftrightarrow \textrm{space})$, where the $x^4$ coordinate corresponds to our universe's observed physical time dimension and the $x^8$ coordinate corresponds to a predicted new physical spatial dimension. This solution of the field equations manifests temporal hyperbolic inflation $\cosh{({1}{3}\,H\,x^4)}$ of the scale factor associated with three of the four space dimensions, and temporal deflation $\textrm{sech}({{{1}{3}\,H\,x^4}})$ 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 $x^8$ coordinate, which, however, turn out to be \textbf{periodic} in $x^8$ with period $\propto 1 / H$. \textbf{After "inflation" the observed physical macroscopic world has three space dimensions and one time dimension.}