Integrable cases of gravitating static isothermal fluid spheres

TL;DR

This paper investigates specific integrable cases of Einstein's equations for static, spherically symmetric perfect fluids with a gamma-law equation of state, identifying known solutions through Abel differential equations.

Contribution

It demonstrates that various solution approaches reduce to an Abel differential equation, revealing the only integrable cases including flat spacetime, de Sitter, Einstein static universe, and Klein-Tolman solutions.

Findings

Identifies integrable cases as flat spacetime, de Sitter, Einstein static universe, and Klein-Tolman solutions.

Shows reduction of Einstein equations to Abel differential equations for these cases.

Highlights the limited set of integrable solutions for the specified fluid spheres.

Abstract

It is shown that different approaches towards the solution of the Einstein equations for a static spherically symmetric perfect fluid with a gamma-law equation of state lead to an Abel differential equation of the second kind. Its only integrable cases at present are flat spacetime, de Sitter solution and its Buchdahl transform, Einstein static universe and the Klein-Tolman solution.