Exact solutions of Einstein's equations with ideal gas sources

TL;DR

This paper presents new exact solutions to Einstein's equations with ideal gas sources, describing viscous fluids with specific thermodynamic properties, and discusses their physical and mathematical consistency.

Contribution

It introduces a new class of exact solutions with Szekeres-Szafron metrics for viscous ideal gases, linking Einstein's equations to elementary and elliptic integrals, and explores their thermodynamic and matching conditions.

Findings

Solutions are integrable in elementary functions or elliptic integrals.

Conditions for thermodynamic consistency with Extended Irreversible Thermodynamics are established.

Smooth matching with cosmological models like FLRW and Vaidya is demonstrated.

Abstract

We derive a new class of exact solutions characterized by the Szekeres-Szafron metrics (of class I), admitting in general no isometries. The source is a fluid with viscosity but zero heat flux (adiabatic but irreversible evolution) whose equilibrium state variables satisfy the equations of state of: (a) ultra-relativistic ideal gas, (b) non-relativistic ideal gas, (c) a mixture of (a) and (b). Einstein's field equations reduce to a quadrature that is integrable in terms of elementary functions (cases (a) and (c)) and elliptic integrals (case (b)). Necessary and sufficient conditions are provided for the viscous dissipative stress and equilibrium variables to be consistent with the theoretical framework of Extended Irreversible Thermodynamics and Kinetic Theory of the Maxwell-Boltzmann and radiative gases. Energy and regularity conditions are discussed. We prove that a smooth matching can be performed along a spherical boundary with a FLRW cosmology or with a Vaidya exterior solution. Possible applications are briefly outlined.