Hyperbolic Inflation

TL;DR

This paper presents a hyperbolic inflation solution in an eight-dimensional spacetime with four space and four time dimensions, showing hyperbolic expansion of space and deflation of time dimensions, with implications for the structure of the universe.

Contribution

It introduces a novel hyperbolic inflation model on an eight-dimensional manifold with triality properties, predicting a new spatial dimension and complex scale factor behaviors.

Findings

Hyperbolic inflation of three space dimensions modeled by cosh function.

Deflation of three time dimensions modeled by sech function.

Periodic dependence of scale factors on an extra spatial coordinate.

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.}