Relativistic kinematic approach to the classical ideal gas

Abstract

The necessary and sufficient conditions for a unit time-like vector field to be the unit velocity of a classical ideal gas are obtained. In a recent paper [Coll, Ferrando and Sáez, Phys. Rev D 99 (2019)] we have offered a purely hydrodynamic description of a classical ideal gas. Here we take one more step in reducing the number of variables necessary to characterize these media by showing that a plainly kinematic description can be obtained. We apply the results to obtain test solutions to the hydrodynamic equation that model the evolution in local thermal equilibrium of a classical ideal gas. PACS numbers: 04.20.-q, 04.20.Jb D(u, ρ, p) = 0 .This means that (2) is a consequence of (1), and conversely, for any solution {u, ρ, p} of (2), a solution {u, ρ, p, n, ǫ, s, Θ} of (1) exists. In other words, (2) is the integrability condition for the system (1) to admit a solution {n, ǫ, s, Θ}.

Thermodynamic perfect fluid spheres admitting an orthogonal flat synchronization

“…It is worth remarking that the ideal Szafron models do not represent the evolution of a classical ideal gas because the equation of state (54) is not compatible with the one of a classical ideal gas, namely, χ(π) = γπ/(1+π) [31]. This result agrees with a result on the study of the velocities of the classical ideal gases [43]: a geodesic and expanding time-like unit vector is the unit velocity of a classical ideal gas if, and only if, it is vorticity-free and its expansion is homogeneous.…”

A thermodynamic approach to the T-models

“…The answer is affirmative. Indeed, in [46] we have characterized the unit velocities of the classical ideal gas solutions of the hydrodynamic equations, and we have shown the following result: a geodesic and expanding time-like unit vector u is the unit velocity of a classical ideal gas if, and only if, u is vorticity-free and its expansion is homogeneous, that is, u = −dt and θ = θ(t).…”

On the flow of perfect energy tensors

“…In other words, (2) is the integrability condition for the system (1) to admit a solution {ρ, p}. Consequently, our study shows two aspects that we will analyze in depth: (i) the direct problem, namely, the determination of the conditional system (2) from the initial one (1), and (ii) the inverse problem, namely, the obtention of the solutions of (1) associated with a given solution of (2).…”

“…They were rediscovered [32] as the conformally flat class of expanding, irrotational and shear-free perfect fluid spacetimes. The metric line element takes the expression ds 2…”

“…It should be mentioned that a similar approach was taken years ago for the system of the barotropic hydrodynamics [1], and more recently for the system of the classical ideal gas hydrodynamics [2]. In this relativistic hydrodynamics framework, the direct and inverse problems have been analyzed for the thermodynamic perfect fluids [3,4], for a generic ideal gas [4], for a classical ideal gas [5] and, not long ago, for a relativistic Synge gas [6].…”