Relativistic hydrodynamic approach to the classical ideal gas

Abstract

The necessary and sufficient condition for a conservative perfect fluid energy tensor to be the energetic evolution of a classical ideal gas is obtained. This condition forces the square of the speed of sound to have the form ${c}_{s}^{2}=\frac{\ensuremath{\gamma}p}{\ensuremath{\rho}+p}$ in terms of the hydrodynamic quantities, energy density $\ensuremath{\rho}$ and pressure $p$, $\ensuremath{\gamma}$ being the (constant) adiabatic index. The inverse problem for this case is also solved, that is, the determination of all the fluids whose evolutions are represented by a conservative energy tensor endowed with the above expression of ${c}_{s}^{2}$, and it shows that these fluids are, and only are, those fulfilling a Poisson law. The relativistic compressibility conditions for the classical ideal gases and the Poisson gases are analyzed in depth and the values for the adiabatic index $\ensuremath{\gamma}$ for which the compressibility conditions hold in physically relevant ranges of the hydrodynamic quantities $\ensuremath{\rho}$, $p$ are obtained. Some scenarios that model isothermal or isentropic evolutions of a classical ideal gas are revisited, and preliminary results are presented in applying our hydrodynamic approach to looking for perfect fluid solutions that model the evolution of a classical ideal gas or of a Poisson gas.