Exact solutions of Einstein's equations with ideal gas sources

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) an ultra-relativistic ideal gas; (b) a 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 Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology or with a Vaidya exterior solution. Possible applications are briefly outlined.