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Thermodynamic perfect fluid. Its Rainich theory
1.In pertinent units.
2.G. Y. Rainich, Trans. Am. Math. Soc. 27, 106 (1925).
3.A. Einstein, Ann. Phys. 49, 769 (1916).
4.At least, we have not been able to find it.
5.The Taub conditions for a (1,1) tensor to be the energy tensor for a perfect fluid [A. H. Taub, “Relativistic Hydrodynamics,” in Lectures in Applied Mathematics (Am. Math. Soc. Providence, RI), 1967, Vol. 8, p. 170] though not explicitly stated, apply only in absence of a given metric; see our Sec. III A.
6.J. A. Morales, Ph.D. thesis, València, 1988; see also C. Bona, B. Coll, and J. A. Morales, “Caracterización algebraica de un 2‐tensor simétrico,” in Actas de los E.R.E.86 (Pub. Univ. València, Valéncia, to be published).
7.Including the Maxwell equations as linear approximation.
8.C. Marle, Ann. Inst. H. Poincaré 10, 67 (1969);
8.C. Marle, 10, 127 (1969)., Ann. Inst. Henri Poincare, Sect. A
9.The Eckart and the Landau‐Lifchitz ones, among others.
10.Marle considers the relativistic versions of the Chapman‐Enskog and the Grad classical, methods.
11.Here, “generically” means “for almost all the versions that have been proposed in the literature.” Of course, there are always some exceptions; for example, the Arzeliés fluids [H. Arzeliés, Fluides Relativistes (Masson, Paris, 1971)].
12.For example, the Catteneo fluids [C. Catteneo, Rend. Accad. Naz. dei Lincei 46, Ser. VIII, 699 (1969)].
13.Rainich called it the skeleton of the electromagnetic field.
14.Here, this volume element is nothing but the unit velocity of the fluid.
15.This task is not easy. Restricted to the barotropic fluid, it induces an eightfold classification of the unit velocity (see B. Coll and J. J. Ferrando, “On the velocities of the barotropic perfect fluids,” to be published).
16.C. W. Misner and J. A. Wheeler, Ann. Phys. 2, 525 (1957).
17.Usually, the corresponding differential equations are presented in the form of Cauchy or underdetermined systems for the metric coefficients and some other additional unknowns (pression, electromagnetic field, etc.).
18.This is the case for Lichnerowicz’s conjecture on spherical symmetry under appropriate asymptotic conditions [see H. P. Kunzle, Commun. Math. Phys. 20, 85 (1971) and references there in],
18.or the Treciokas‐Ellis conjecture on vorticity‐free or expansion‐free consequences under distortion‐free conditions [see R. Treciokas and G. F. R. Ellis, Commun. Math. Phys. 23, 1 (1971),
18.or the more recent analysis by C. B. Collins, J. Math. Phys. 26, 2009 (1985)].
19.As an application to the Maxwell case, see, for example, B. Coll, F. Fayos, and J. J. Ferrando, J. Math. Phys. 28, 1075 (1987).
20.Fortunately, the development of field theory began, historically, with force field variables and not with energy field variables. Otherwise Maxwell equations should remain undiscovered; to think so, a glance on the nonlinear Rainich complexion equations is largely sufficient.
21.B. Coll and J. J. Ferrando, “Fluido perfecto termodinamico. Su teoria ‘à la Rainich’,” in Actas de los E.R.E.87 (Pub. Inst. Astrof. de Canarias, La Laguna, Spain, 1988).
22. denote, respectively, the divergence, interior product, normal projection, covariant derivative, exterior differentiation, and Hodge dual operators. Newton’s notation is used for timelike derivatives: being any tensorial quantity.
23.Of course, u is the proper unit velocity of the fluid, p the pression, and p the total energy density.
24.C. B. Collins, J. Math. Phys. 26, 2009 (1985).
25.Also called rest mass density, proper mass density, baryonic (average) mass density or, simply density of the fluid.
26.The definition of r as a mass balance of the baryonic number allows us to include in this scheme the study of the propagation of chemical reactions fronts; see B. Coll, Ann. Inst. H. Poincaré 25, 363 (1976).
27.See A. Lichnerowicz, Relativistic Hydrodynamics and Magnetohydrodynamics (Benjamin, New York, 1967).
28.Molecular, atomic, or baryonic mass balances.
29.See the paper quoted in Ref. 26.
30.In such a case, one does not have necessarily and every γ‐law, may be interpreted as a polytropic perfect gas [see B. Coll, C. R. Acad. Sci. Paris A 273, 1185 (1971)].
31.The symbol × denotes the cross product; contraction of the adjacent spaces of the tensor product. Of course, the operator selects the antisymmetric part of
32.J. A. Morales, Ref. 6.
33.See J. Plebański, Acta Phys. Pol. 26, 963 (1964), especially his prudent analysis (pages 1011 and 1012) on the validity of his two conditions. In The large scale structure of space‐time (Cambridge U.P., Cambridge, 1973), S. W. Hawking and G. F. R. Ellis call them the weak and dominant energy conditions, seeming to be unaware of Plebański’s work.
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