Toda chains with type A_m Lie algebra for multidimensional m-component perfect fluid cosmology

TL;DR

This paper models multidimensional cosmology with perfect fluids using Toda chains associated with Lie algebra A_m, providing exact solutions to Einstein's equations in Kasner-like form.

Contribution

It establishes a connection between multidimensional cosmological models with perfect fluids and classical Toda chains related to Lie algebra A_m, enabling exact solutions.

Findings

Reduction to classical open m-body Toda chain

Explicit integration of Einstein equations

Kasner-like metric solutions

Abstract

We consider a D-dimensional cosmological model describing an evolution of Ricci-flat factor spaces, M_1,...M_n (n > 2), in the presence of an m-component perfect fluid source (n > m > 1). We find characteristic vectors, related to the matter constants in the barotropic equations of state for fluid components of all factor spaces. We show that, in the case where we can interpret these vectors as the root vectors of a Lie algebra of Cartan type A_m=sl(m+1,C), the model reduces to the classical open m-body Toda chain. Using an elegant technique by Anderson (J. Math. Phys. 37 (1996) 1349) for solving this system, we integrate the Einstein equations for the model and present the metric in a Kasner-like form.