Exact lambdavacuum solutions in higher dimensions

Abstract

In this work, we obtain exact solutions to the $(n+2)$-dimensional Einstein Field Equations with a non-zero cosmological constant for $n>1$. These solutions depend on a set $\{A_a,a=1,2,\dots ,m\}$ of pairwise commuting constant matrices in $\mathfrak{sl}(n,\mathbb{R})$ and on a constant matrix $g_0$ in $\mathcal{I}(\{A_a,a=1,\dots ,m\})$, determined in previous work. Different choices of $\{A_a,a=1,\dots ,m\}$ and $g_0$ correspond to different solutions. As examples, we show how to obtain the de Sitter metric, the Anti-de Sitter metric, the Birmingham metric, the Nariai metric and the Anti-Nariai metric in higher dimensions. The generalized Nariai and Anti-Nariai solutions are direct topological products of $Ad{S}_{\frac{n}{2}+1}\times H^{\frac{n}{2}+1}$, $d{S}_{\frac{n}{2}+1}\times S^{\frac{n}{2}+1}$, $Ad{S}_2\times H^n$, $Ad{S}_n\times H^2$, $d{S}_2\times S^n$ and $d{S}_n\times S^2$. In addition, we study a solution in the context of cosmology.