Physical interpretation of spherically symmetric perfect fluid solutions to Einstein's equations

TL;DR

This paper investigates the physical viability of spherically symmetric perfect fluid solutions to Einstein's equations, extending analysis to various symmetries and developing tools for their assessment.

Contribution

It introduces a hydrodynamic framework for interpreting perfect fluid solutions and provides a Mathematica package for analyzing their physical properties.

Findings

Wide regions of solutions are physically admissible.

Compatibility with ideal gas equations of state is established.

Methods for approximating ultrarelativistic gases are developed.

Abstract

Einstein's equations of General Relativity form a highly nonlinear system, so most exact solutions rely on symmetry assumptions. Spherically symmetric spacetimes have been particularly important, providing a tractable yet physically rich setting. Despite extensive study, many open questions remain, especially regarding the physical interpretation of perfect fluid solutions. Many such solutions were derived without a specified equation of state or under restrictive or non-physical assumptions, limiting their physical relevance. The aim of this thesis is to study the physical viability of spherically symmetric perfect fluid solutions, with extensions to plane and hyperbolic symmetries. The first part reviews the hydrodynamic approach, which interprets a perfect fluid energy-momentum tensor as a fluid in local thermal equilibrium. Interpretations as a generic ideal gas, a classical ideal gas, and fluids with transport coefficients are analysed. The framework is extended to the ultrarelativistic Synge gas, and methods to approximate its equation of state are developed. These results are applied to three families of solutions: T-models, geodesic R-models with flat synchronisation, and thermodynamic Stephani universes. For each family, general expressions for the fluid flow, energy density, pressure, speed of sound, and admissible thermodynamic schemes are obtained. Physical viability is assessed using standard energy, positivity, and compressibility conditions, with emphasis on compatibility with a generic ideal gas. In all cases, wide spacetime regions are found where the solutions represent physically admissible perfect fluids. The thesis concludes with xIdeal, a Mathematica package implementing IDEAL algorithms for the analysis of exact solutions, including spacetime characterisations, a metric database, and examples.