Relativistic perfect fluids in local thermal equilibrium

Abstract

Every evolution of a fluid is uniquely described by an energy tensor. But the converse is not true: an energy tensor may describe the evolution of different fluids. The problem of determining them is called here the inverse problem. This problem may admit unphysical or non-deterministic solutions. This paper is devoted to solve the inverse problem for perfect energy tensors in the class of perfect fluids evolving in local thermal equilibrium (l.t.e.). The starting point is a previous result (Coll and Ferrando in J Math Phys 30: 2918-2922, 1989 showing that thermodynamic fluids evolving in l.t.e. admit a purely hydrodynamic characterization. This characterization allows solving this inverse problem in a very compact form. The paradigmatic case of perfect energy tensors representing the evolution of ideal gases is studied in detail and some applications and examples are outlined.

Relativistic kinematic approach to the classical ideal gas

“…We have already shown [1] (see also the recent paper [2]) that the notion of l.t.e admits a purely hydrodynamic formulation: a perfect energy tensor T evolves in l.t.e if, and only if, the hydrodynamic quantities {u, ρ, p} fulfill the hydrodynamic sonic condition (ρdṗ −ṗdρ) ∧ dρ ∧ dp = 0 .…”

“…Elsewhere [1] [2] we have shown that the system (1) admits a conditional system for the hydrodynamic quantities {u, ρ, p}:…”

“…For each class, we obtain the necessary and sufficient conditions in u to ensure that it is the velocity of some CIG. Furthermore, for each class we solve the inverse problem by obtaining the pairs (ρ, p) that complete a solution to the system (2).…”

Relativistic hydrodynamic approach to the classical ideal gas

“…the determination of the set T F of all perfect energy tensors corresponding to all possible evolutions of the set F of all perfect fluids. In [6] we have also solved the generic inverse problem for a perfect energy tensor T , i. e. the determination of the set F T of all perfect fluids for which T is the energetic description of a particular evolution.…”

“…Nevertheless, because of its practical applications, solving a specific direct problem for a family of fluids may be more interesting than to solve the generic one. We have studied in [6] the specific direct problem for the family of the generic ideal gases, i.e. those defined by the equation of state p = knΘ.…”