Abstract
The necessary and sufficient conditions for a unit time-like vector field to be the unit velocity of a classical ideal gas (CIG) are obtained. In a recent paper (Coll et al 2019 Phys. Rev. D 99 084035) we have offered a purely hydrodynamic description of a CIG. 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 CIG.
Highlights
In Relativity, a conservative energy tensor of the form T = (ρ + p)u ⊗ u + pg represents the energetic description of the evolution of a perfect fluid
If we want to describe the evolution of a perfect fluid in local thermal equilibrium we must add to the hydrodynamic quantities a set of thermodynamic quantities constrained by the usual thermodynamic laws
Elsewhere [1] [2] we have shown that the system (1) admits a conditional system for the hydrodynamic quantities {u, ρ, p}: D(u, ρ, p) = 0
Summary
We have solved the restricted direct problem by obtaining the conditional system in the hydrodynamic quantities (2) associated with the fundamental system of the classical ideal gas hydrodynamics (1). The main goal of this paper is to show that the answer is affirmative: by starting from the hydrodynamic characterization (2) we obtain a conditional system in the kinematic quantity u: D(u) = 0 , This result solves the restricted direct problem and offers a purely kinematic description of the CIG. We offer the necessary and sufficient conditions for u to be the velocity of a CIG, and we explain the full set of pairs (ρ, p) which solve the inverse problem. The symbols ∇, ∇·, d and ∗ denote, respectively, the covariant derivative, the divergence operator, the exterior derivative and the Hodge dual operator, and i(x)t denotes the interior product of a vector field x and a p-form t
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