Asymptotics of Solutions of a Perfect Fluid Coupled with a Cosmological Constant in Four-Dimensional Spacetime with Toroidal Symmetry

TL;DR

This paper investigates the asymptotic behavior of perfect fluid solutions coupled with a cosmological constant in four-dimensional spacetime with toroidal symmetry, deriving solutions for specific equations of state and analyzing their properties.

Contribution

It reduces the problem of self-similar solutions to a master equation and explicitly solves it for particular values of the equation of state parameter k, exploring their characteristics.

Findings

Explicit solutions for k=0 and k=-1/3 are obtained.

Main local and global properties of these solutions are analyzed.

The problem is reduced to solving a master differential equation.

Abstract

Asymptotics of solutions of a perfect fluid when coupled with a cosmological constant in four-dimensional spacetime with toroidal symmetry are studied. In particular, it is found that the problem of self-similar solutions of the first kind for a fluid with the equation of state, $p = k \rho$, can be reduced to solving a master equation of the form, $$ 2 F(q, k)\frac{q''(\xi)}{q'(\xi)} - G(q,k) q'(\xi) = \frac{4}{\xi}. $$ For $k = 0$ and $k = -1/3$ the general solutions are obtained and their main local and global properties are studied in detail.