A new algorithm for anisotropic solutions

TL;DR

This paper introduces a novel algorithm that constructs anisotropic solutions to Einstein's equations from isotropic seeds, providing explicit solutions that generalize known models like the isothermal sphere.

Contribution

The authors develop a new integral-based method to generate anisotropic solutions with exact forms, extending isotropic solutions and ensuring isotropic limits.

Findings

Derived explicit anisotropic solutions generalizing classical models

Provided solutions in closed form with elementary functions

Demonstrated the method with examples related to the isothermal sphere

Abstract

We establish a new algorithm that generates a new solution to the Einstein field equations, with an anisotropic matter distribution, from a seed isotropic solution. The new solution is expressed in terms of integrals of an isotropic gravitational potential; and the integration can be completed exactly for particular isotropic seed metrics. A good feature of our approach is that the anisotropic solutions necessarily have an isotropic limit. We find two examples of anisotropic solutions which generalise the isothermal sphere and the Schwarzschild interior sphere. Both examples are expressed in closed form involving elementary functions only.