Generating Static Fluid Spheres by Conformal Transformations

TL;DR

This paper introduces a method to generate numerous exact perfect fluid solutions to Einstein's equations using conformal transformations, including notable cases like Schwarzschild interior and Einstein static universe, by solving coupled Riccati equations.

Contribution

It presents a novel approach to generate a wide class of physically acceptable solutions through conformal transformations and Riccati equations, expanding the solution space for Einstein's equations.

Findings

Generated an infinite family of exact solutions

Included well-known solutions like Schwarzschild interior

Provided a new method involving coupled Riccati equations

Abstract

We generate an explicit four-fold infinity of physically acceptable exact perfect fluid solutions of Einstein's equations by way of conformal transformations of physically unacceptable solutions (one way to view the use of isotropic coordinates). Special cases include the Schwarzschild interior solution and the Einstein static universe. The process we consider involves solving two equations of the Riccati type coupled by a single generating function rather than a specification of one of the two metric functions.