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

Salvador Mengual

TL;DR

The thesis develops a comprehensive LTE-based framework to test the physical viability of spherically symmetric (and related planar/hyperbolic) perfect-fluid solutions of Einstein's equations. By recasting fluid dynamics in terms of hydrodynamic quantities and thermodynamic schemes, it derives sonic conditions, energy/positivity criteria, and compressibility constraints, then applies them to three main families: T-models (G3 acting on S2 with tangent curvature gradient), R-models (G3/S2 with orthogonal flat synchronisation), and thermodynamic Stephani universes. It provides explicit results for the fluid content (ρ,p), the speed of sound χ(ρ,p), and thermodynamic schemes (n, ε, s, Θ), identifying wide spacetime regions where these solutions model physically admissible LTE fluids, including generic ideal gases and Synge-like relativistic gases via various approximations (TM, SG, etc.). The work further delivers algorithmic tools (xIdeal) to automate IDEAL characterisations, offer a metric database, and enable rapid verification of novel exact solutions, thereby linking exact GR solutions to physical fluid models and enabling cross-checks against energy, positivity, and compressibility constraints. Overall, the study produces a catalog of physically viable exact spacetimes within these symmetry classes and furnishes practical computational machinery to classify and interpret them in terms of realistic fluid dynamics.

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.

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

TL;DR

The thesis develops a comprehensive LTE-based framework to test the physical viability of spherically symmetric (and related planar/hyperbolic) perfect-fluid solutions of Einstein's equations. By recasting fluid dynamics in terms of hydrodynamic quantities and thermodynamic schemes, it derives sonic conditions, energy/positivity criteria, and compressibility constraints, then applies them to three main families: T-models (G3 acting on S2 with tangent curvature gradient), R-models (G3/S2 with orthogonal flat synchronisation), and thermodynamic Stephani universes. It provides explicit results for the fluid content (ρ,p), the speed of sound χ(ρ,p), and thermodynamic schemes (n, ε, s, Θ), identifying wide spacetime regions where these solutions model physically admissible LTE fluids, including generic ideal gases and Synge-like relativistic gases via various approximations (TM, SG, etc.). The work further delivers algorithmic tools (xIdeal) to automate IDEAL characterisations, offer a metric database, and enable rapid verification of novel exact solutions, thereby linking exact GR solutions to physical fluid models and enabling cross-checks against energy, positivity, and compressibility constraints. Overall, the study produces a catalog of physically viable exact spacetimes within these symmetry classes and furnishes practical computational machinery to classify and interpret them in terms of realistic fluid dynamics.

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.
Paper Structure (145 sections, 9 theorems, 357 equations, 32 figures, 9 tables)

This paper contains 145 sections, 9 theorems, 357 equations, 32 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Physical interpretation of a perfect energy tensor
  3. Conditions for physical reality
  4. Local thermal equilibrium
  5. Hydrodynamic$\,$ characterisation$\,$ of$\,$ the$\,$ local thermal$\,$ equilibrium
  6. Solving the inverse problem
  7. Other necessary physical reality conditions
  8. Generic ideal gas
  9. Classical ideal gas
  10. Solutions$\,$ compatible$\,$ with$\,$ non-vanishing transport$\,$ coefficients
  11. Hydrodynamic approach to the Synge gas
  12. Hydrodynamic flow of a relativistic Synge gas
  13. Hydrodynamic characterisation of the Synge gas
  14. Behaviour of the Synge gas at low and high temperatures
  15. Rainich-like theory for the Einstein-Synge solutions
  16. ...and 130 more sections

Key Result

Lemma 1

A perfect energy tensor evolves in local thermal equilibrium if, and only if, the equation admits solutions $s(x^\mu)$ such that $\emph{d}s \wedge \emph{d}\rho \wedge \emph{d}p = 0$.

Figures (32)

  • Figure 1: Schematic diagram representing the relations among the inhomogeneous solutions considered in this thesis.
  • Figure 2: (a) $\textbf{T}_{\textit{f}}$ is the set of perfect energy tensors $T_{\textit{f}}$ corresponding to all the possible evolutions of a fluid $f$. (b) The sets $\textbf{T}_{\textit{f}}$ and $\textbf{T}_{\textit{$\bar{f}$}}$ corresponding to the possible evolutions of two different perfect fluids f and $\textit{$\bar{f}$}$ might not be disjoint, $\textbf{T}_{\textit{f}}\cap\textbf{T}_{\textit{$\bar{f}$}}\neq\emptyset$ (reproduced from Hydro-LTE).
  • Figure 3: (a) F is the set of all the perfect fluids $f$, $\textbf{T}_{\textbf{F}}$ is the set of all the perfect energy tensors corresponding to some possible evolution of a perfect fluid and T, the set of all the perfect energy tensors. Then $\textbf{T}_{\textbf{F}}\subset\textbf{T}$. (b) Every fluid in the subset $\textbf{F}_{\textit{T}}$ of F have as a possible evolution the one corresponding to T (reproduced from Hydro-LTE).
  • Figure 4: This plot shows the relative error with respect to the Synge EoS of the TM EoS and of the other proposed EoS. The P3/P2 and P3/P1 EoS are more accurate than the TM EoS, while P3 and P1/P2 EoS are not.
  • Figure 5: (a) Here, we show the behaviour of the indicatrix function $\chi(\pi)$ (\ref{['chi-T-ideal']}) defined in the whole interval $]0,1[$ for different values of the parameter $\tilde{\gamma}$. This is the domain where the energy conditions E$^{\rm G}$ hold and, therefore, the compressibility conditions H$_1^{\rm G}$ also hold in the whole domain. (b) Here, we also show the behaviour of the square of the speed of sound but in the interval $]0,1/3[$, for the values of $\tilde{\gamma}$ that we study in detail in this section, $\tilde{\gamma} = 4/3$ and $\tilde{\gamma} = 2$, and for the Synge gas. The case $\tilde{\gamma} = 4/3$ approaches a Synge gas at high temperatures. The shaded area is forbidden by the compressibility condition H$_2^{\rm G}$. Thus, the hydrodynamic variable is limited to an interval $]\pi_m,1[$.
  • ...and 27 more figures

Theorems & Definitions (9)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 1
  • Proposition 2
  • Proposition 3