Multidimensional integrable vacuum cosmology with two curvatures

TL;DR

This paper analyzes multidimensional vacuum cosmological models with two curvatures, deriving exact solutions, studying their behavior near singularities, and identifying special non-singular solutions with specific topologies.

Contribution

It provides explicit solutions for vacuum cosmologies with two Einstein spaces, including Abel equation reductions, Kasner-like behavior analysis, and non-singular solutions with specific topologies.

Findings

Solutions for n=2 reduced to Abel equations

Kasner-like behavior near singularities

Existence of non-singular solutions with R^7 x M_2 topology

Abstract

The vacuum cosmological model on the manifold $R \times M_1 \times \ldots \times M_n$ describing the evolution of $n$ Einstein spaces of non-zero curvatures is considered. For $n = 2$ the Einstein equations are reduced to the Abel (ordinary differential) equation and solved, when $(N_1 = $dim $ M_1, N_2 = $ dim$ M_2) = (6,3), (5,5), (8,2)$. The Kasner-like behaviour of the solutions near the singularity $t_s \to +0$ is considered ($t_s$ is synchronous time). The exceptional ("Milne-type") solutions are obtained for arbitrary $n$. For $n=2$ these solutions are attractors for other ones, when $t_s \to + \infty$. For dim $ M = 10, 11$ and $3 \leq n \leq 5$ certain two-parametric families of solutions are obtained from $n=2$ ones using "curvature-splitting" trick. In the case $n=2$, $(N_1, N_2)= (6,3)$ a family of non-singular solutions with the topology $R^7 \times M_2$ is found.