Collins in Wonderland

Abstract

What is the asymptotic future of a scalar-field model if the assumption of isotropy is relaxed in generic, homogeneous space-times with general relativity? This paper is a continuation of our previous work on Bianchi cosmologies with a $p$-form field (where $p\,\in\,\{1,3\}$)---or equivalently: an inhomogeneous, mass-less scalar gauge field with a homogeneous gradient. In this work we investigate such matter sector in General Relativity, and restrict to space-times of the particular Bianchi types VI$_0$ and VI$_{\tilde{h}}$, where $\tilde{h}=h<0\,\cap\,\neq\,-1/9\,\cup\,-1$. We show that the previously found fabric of exact solutions named Wonderland are future attractors in $\mathcal{B}$(VI$_0$) and $\mathcal{B}$(VI$_{\tilde{h}}$), extending the Collins perfect-fluid equilibrium set to include a $p$-form (with $p\,\in\,\{1,3\}$). We also write down the line-element corresponding to Wonderland in VI$_{\tilde{h}}$ and give explicit expressions for the underling gauge-potential $\phi(t,\mathbf{x})$ corresponding to this solution. Simulation of a path approaching Wonderland in Bianchi type I is also given.