Generating static spherically symmetric anisotropic solutions of Einstein's equations from isotropic Newtonian solutions

TL;DR

This paper presents a method to generate exact anisotropic solutions of Einstein's equations from Newtonian isotropic solutions, expanding the set of known solutions and including those satisfying energy conditions.

Contribution

It introduces a novel approach to derive Einstein's solutions using Newtonian hydrostatic equilibrium and constructs multiple regular anisotropic models.

Findings

Generated infinite regular anisotropic solutions

Some solutions satisfy all standard energy conditions

Two classes generalize Newtonian polytropes of index 0 and 1

Abstract

I use the Newtonian equation of hydrostatic equilibrium for an isotropic fluid sphere to generate exact anisotropic solutions of Einstein's equations. The input function is simply the density. An infinite number of regular solutions are constructed, some of which satisfy all the standard energy conditions. Two classes of these solutions generalize the Newtonian polytropes of index 0 and 1.