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On the covariant equations of the relativistic electrodynamics of continua. II. Fluids
1.G. A. Maugin, Part I, J. Math. Phys. 19, 1198 (1978).
2.S. R. de Groot and L. G. Suttorp, Physica 37, 284, 297 (1967);
2.S. R. de Groot and L. G. Suttorp, 39, 28, 4161, 84 (1968);
2.Foundations of Electrodynamics, (North‐Holland, Amsterdam, 1972).
3.I. Brevik, Mat. Fys. Medd. Dans. Vidensk. Selsk. 37, No. 13 (1970).
4.The notation is that of Part I to which we refer. The equations of Part I are referred to by I followed by their number, e.g., Eq. (I.5.27).
5.These fields are “objective” in the following sense. They obey the so‐called principle of objectivity or principle of material frame indifference in general relativity as formulated by the author [G. A. Maugin, in Ondes et Radiations Gravitationnelles (Editions C.N.R.S., Paris, 1974), pp. 331–8].
5.Their tetrad components are form invariant under proper time dependent rotation of the tetrad—this invariance is related to spinorial algebra. They are also “rheologically invariant” according to the invariance principle set forth by J. G. Oldroyd, Proc. R. Soc. London Ser. A 316, 1 (1970).
6.Cf. Ref. 3.
7.Compare Eq. (2.20) in Ref. 3. For the experimental validity of Helmholtz’ force we refer to the comments of Brevik in Ref. 3, p. 13.
8.Cf. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), Sec. 15.
9.This expression was deduced from a variational principle by Korteweg and Helmholtz;
9.cf. J. A. Stratton, Electromagnetic Theory (McGraw‐Hill, New York, 1941), pp. 137–40. It is also referred to as Minkowski’s force density by Brevik [cf. Ref. 3, Eq. (2.19)].
10.Cf. Eq. (2.15) in Ref. 3, and also Lord Kelvin (Sir W. Thomson), Reprints of Papers on Electrostatics and Magnetism (MacMillan, London, 1884), 2nd ed.
11.R. V. Jones and J. C. S. Richards, Proc. R. Soc. London Ser. A 221, 480 (1954).
12.H. Minkowski, Nach. Ges. Wiss. Göttingen 53 (1908);
12.Math. Ann. 68, 472 (1910).
13.We know that the oversimplified constitutive equation (4.7b) violates the principle of relativistic causality in that it yields a parabolic equation of heat. The solution to this paradox requires considerably more involved relativistic heat conduction laws. A review of such laws (there is no unique solution) is given in G. A. Maugin, J. Phys. A. Math. Nucl. Gen. 7, 465 (1974).
14.Cf A. Lichnerowicz, Relativistic Hydrodynamics and Magnetohydrodynamics (Benjamin, New York, 1967).
15.Up to the signature chosen for the space—time metric, Eq. (5.8) is the same as that on which several authors have based their mathematical studies of relativistic magnetohydrodynamics; e.g., in Ref. 14 and in K. O. Friedrichs, Commun. Pure Appl. Math. 27, 749 (1974).
15.Equation (5.8) and the corresponding Einstein field equations have been deduced simultaneously from various variational formulations; see A. H. Taub, in Relativistic Fluid Dynamics, edited by C. Cattaneo (Cremonese, Rome, 1971), and G. A. Maugin, C. R. Acad. Sci. Paris 274A, 602 (1972);
15.A. H. Taub, in Relativistic Fluid Dynamics, edited by C. Cattaneo (Cremonese, Rome, 1971), and G. A. Maugin, Ann. Inst. H. Poincaré 16, 133 (1972).
16.Cf. M. Cissoko, C. R. Acad. Sci. Ser. A 278, 463 (1974);
16.Thèse de Doctorat ès. Sci. Math. (Université de Paris VI, 1975), and other references quoted by this author.
17.P. G. De Gennes, Liquid Crystals (Oxford U.P., London, 1974).
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