Algorithmic construction of static perfect fluid spheres

TL;DR

This paper refines Lake's algorithm for constructing static perfect fluid spheres, providing explicit formulas and a clearer physical interpretation, enabling the generation of new solutions and better understanding of existing ones.

Contribution

It presents a re-cast of Lake's algorithm using physically meaningful variables, with explicit formulas for mass, pressure, and central pressure, and facilitates the discovery of new solutions.

Findings

Explicit formulas for mass and pressure profiles

Closed-form expression for central pressure

Generation of new exact solutions

Abstract

Perfect fluid spheres, both Newtonian and relativistic, have attracted considerable attention as the first step in developing realistic stellar models (or models for fluid planets). Whereas there have been some early hints on how one might find general solutions to the perfect fluid constraint in the absence of a specific equation of state, explicit and fully general solutions of the perfect fluid constraint have only very recently been developed. In this article we present a version of Lake's algorithm [Phys. Rev. D 67 (2003) 104015; gr-qc/0209104] wherein: (1) we re-cast the algorithm in terms of variables with a clear physical meaning -- the average density and the locally measured acceleration due to gravity, (2) we present explicit and fully general formulae for the mass profile and pressure profile, and (3) we present an explicit closed-form expression for the central pressure. Furthermore we can then use the formalism to easily understand the pattern of inter-relationships among many of the previously known exact solutions, and generate several new exact solutions.