Generating Spherically Symmetric Static Anisotropic Fluid Solutions of Einstein's Equations from Hydrostatic Equilibrium

TL;DR

This paper develops solution-generating techniques for static anisotropic fluid spheres in Einstein's equations, enabling the construction of new, physically relevant solutions with the same density profile and analyzing their properties.

Contribution

It introduces methods to generate new solutions from existing ones using Riccati and Bernoulli equations, expanding the set of exact solutions for anisotropic fluid spheres.

Findings

Solution-generating techniques based on Riccati and Bernoulli equations.

New three-parameter family of exact solutions satisfying physical conditions.

Comparison of equilibrium configurations with identical spatial metrics.

Abstract

For static fluid spheres, the condition of hydrostatic equilibrium is given by the generalized Tolman--Oppenheimer--Volkoff (TOV) equation, a Riccati equation in the radial pressure. For a perfect fluid source, it is known that finding a new solution from an existing solution requires solving a Bernoulli equation, if the density profile is kept the same. In this paper, we consider maps between static (an)isotropic fluid spheres with the same (arbitrary) density profile and present solution-generating techniques to find new solutions from existing ones. The maps, in general, require solving an associated Riccati equation, which, unlike the Bernoulli equation, cannot be solved by quadrature. In any case, it can be shown that the output solution is not, in general, regular for a given regular input solution. However, if pressure anisotropy is kept the same, the new solution is both regular and can be found by solving a Bernoulli equation. We give a few examples where the generalized TOV equation, under algebraic constraints, can be converted into a Bernoulli equation and thus, solved exactly. We discuss the physical significance of these Bernoulli equations. Since the density profile remains the same in our approach, the spatial line element is identical for all solutions, which facilitates direct comparison between various equilibrium configurations using fluid variables as functions of the radial coordinate. Finally, combining with the previous study on generation algorithms, we show how this study leads us to a new three-parameter family of exact solutions that satisfy all desirable physical conditions.