All static spherically symmetric anisotropic solutions of Einstein's equations

TL;DR

This paper extends Lake's algorithm to generate all static spherically symmetric solutions of Einstein's equations for anisotropic fluids, requiring two functions, and recovers known solutions to validate the method.

Contribution

It introduces a new formalism for anisotropic fluids, expanding the solution space beyond perfect fluids by requiring two functions.

Findings

Successfully recovers known anisotropic solutions

Provides a systematic method for generating new solutions

Extends previous algorithms to anisotropic cases

Abstract

An algorithm recently presented by Lake to obtain all static spherically symmetric perfect fluid solutions, is extended to the case of locally anisotropic fluids (principal stresses unequal). As expected, the new formalism requires the knowledge of two functions (instead of one) to generate all possible solutions. To illustrate the method some known cases are recovered.