Extra-Dimensional Bi-Inflation

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.