Extra-Dimensional Bi-Inflation

TL;DR

This paper explores a novel eight-dimensional spacetime model with inflationary behavior in both space and time dimensions, proposing a geometric mechanism for particle and field evolution in a higher-dimensional universe.

Contribution

It introduces a new eight-dimensional pseudo-Riemannian spacetime solution exhibiting simultaneous inflation and deflation of space and time dimensions, with implications for understanding physical time and particle dynamics.

Findings

Inflation of space and deflation of time dimensions demonstrated

Particle/field evolution is geometrically entrapped and entrained by higher-dimensional structure

New insights into the nature of physical time are proposed

Abstract

We study a solution to the Einstein field equations on an eight-dimensional pseudo-Riemannian manifold (a spacetime of four space dimensions and four time dimensions) that exhibits inflation of three of the four space dimensions and deflation of three of the four time dimensions. Hubble parameters $(\mathbb{H}_4, \mathbb{H}_8)$ exist for each of the unscaled dimensions $(\textrm{time}x^4, \textrm{space}x^8)$. The scale factor for the three ordinary space dimensions $(x^1, x^2, x^3)$ is $a = a(x^4, x^8)$, and during inflation is functionally a product of exponentials $a = e^{\mathbb{H}_4\, x^4\,/\,c} \; e^{\mathbb{H}_8\, x^8\,/\,c}$ of each Hubble parameter times its respective unscaled coordinate ($c$ is the physical speed of gravitational waves in vacuum in the observed universe). We investigate the conjecture that every particle/field in our universe is carried along the $x^8$-axis in the same way that it is carried along the $x^4$-axis, that is, in a manner that is independent of its energy, and with no freedom to change either its rate of passage along the $x^8$-axis or its location within the $x^8$-dimension. The evolution of the particle/field through the unscaled dimensions $(x^4, x^8) \in \mathbb{R}^{1,1}$ in our universe is entrapped and entrained by geometry. We employ a line element of the form $ {ds}^2 = -|g_{4\,4}| \; {(c_4 \, d x^4)}^2 + g_{8\,8} \; {(d x^8)}^2 \;+\;...$ to define a physical time $t = x^0$ through the relation % $ |g_{t\,t}| \; {(c \, d t)}^2 = |g_{4\,4}| \; {(c_4 \, d x^4)}^2 - g_{8\,8} \; {(d x^8)}^2 $. Here $c_4$ is a speed related parameter (in this paper $g_{t\,t} = g_{4\,4} = -1 $). This model offers new insight into physical time.