Stability, causality, and hyperbolicity in Carter's "regular" theory of relativistic heat-conducting fluids

Timothy S. Olson; William A. Hiscock

doi:10.1103/PhysRevD.41.3687

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Stability, causality, and hyperbolicity in Carter's "regular" theory of relativistic heat-conducting fluids

Timothy S. Olson and William A. Hiscock

Department of Physics, Montana State University, Bozeman, Montana 59717

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Phys. Rev. D 41, 3687 – Published 15 June, 1990

DOI: https://doi.org/10.1103/PhysRevD.41.3687

Abstract

Stability and causality are studied for linear perturbations about equilibrium in Carter's "regular" theory of relativistic heat-conducting fluids. The "regular" theory, when linearized around an equilibrium state having vanishing expansion and shear, is shown to be equivalent to the inviscid limit of the linearized Israel-Stewart theory of relativistic dissipative fluids for a particular choice of the second-order coefficients $β_{1}$ and $γ_{2}$ . A set of stability conditions is determined for linear perturbations of a general inviscid Israel-Stewart fluid using a monotonically decreasing energy functional. It is shown that, as in the viscous case, stability implies that the characteristic velocities are subluminal and that perturbations obey hyperbolic equations. The converse theorem is also true. We then apply this analysis to a nonrelativistic Boltzmann gas and to a strongly degenerate free Fermi gas in the "regular" theory. Carter's "regular" theory is shown to be incapable of correctly describing the nonrelativistic Boltzmann gas and the degenerate Fermi gas (at all temperatures).

References (22)

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C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), pp. 567 and 568
S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972), pp. 53-57
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W. A. Hiscock and L. Lindblom, Phys. Rev. D 31, 725 (1985)
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J. M. Stewart, Proc. R. Soc. London A357, 59 (1977)
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W. Israel and J. M. Stewart, Ann. Phys. (N.Y.) 118, 341 (1979)
W. A. Hiscock and L. Lindblom, Ann. Phys. (N.Y.) 151, 466 (1983)
W. A. Hiscock and L. Lindblom, Contemp. Math. 71, 181 (1988)
T. S. Olson and W. A. Hiscock, Phys. Rev. C 39, 1818 (1989)
B. Carter, in Journées Relativistes 1976, edited by M. Cahen, R. Debever, and J. Geheniau (Université Libre De Bruxelles, Bruxelles, 1976), pp. 12-27 in A Random Walk in Relativity and Cosmology, edited by N. Dadhich, J. Krishna Rao, J. V. Narlikar, and C. V. Vishveshwara (Wiley Eastern, Bombay, 1985)
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[c1] C. Eckart, Phys. Rev. 58, 919 (1940)

[c2] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), pp. 567 and 568

[c3] S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972), pp. 53-57

[c4] L. Landau and E. M. Lifshitz, Fluid Mechanics (Addison-Wesley, Reading, Mass., 1959), Sec. 127

[c5] W. A. Hiscock and L. Lindblom, Phys. Rev. D 31, 725 (1985)

[c6] W. A. Hiscock and L. Lindblom, Phys. Rev. D 35, 3723 (1987)

[c7] W. Israel, Ann. Phys. (N.Y.) 100, 310 (1976)

[c8] J. M. Stewart, Proc. R. Soc. London A357, 59 (1977)

[c9] W. Israel and J. M. Stewart, Proc. R. Soc. London A365, 43 (1979)

[c10] W. Israel and J. M. Stewart, Ann. Phys. (N.Y.) 118, 341 (1979)

[c11] W. A. Hiscock and L. Lindblom, Ann. Phys. (N.Y.) 151, 466 (1983)

[c12] W. A. Hiscock and L. Lindblom, Contemp. Math. 71, 181 (1988)

[c13] T. S. Olson and W. A. Hiscock, Phys. Rev. C 39, 1818 (1989)

[c14] B. Carter, in Journées Relativistes 1976, edited by M. Cahen, R. Debever, and J. Geheniau (Université Libre De Bruxelles, Bruxelles, 1976), pp. 12-27 in A Random Walk in Relativity and Cosmology, edited by N. Dadhich, J. Krishna Rao, J. V. Narlikar, and C. V. Vishveshwara (Wiley Eastern, Bombay, 1985)

[c15] B. Carter, in Relativistic Fluid Dynamics, edited by A. M. Anile (Springer, New York, 1989)

[c16] B. Carter, in Highlights in Gravitation and Cosmology, proceedings of the Conference, Goa, India, 1987, edited by B. R. Iyer, A. Kembhavi, J. V. Narlikar, and C. V. Vishveshwara (Cambridge University Press, Cambridge, England, 1988)

[c17] K. S. Thorne, in High Energy Astrophysics, edited by C. DeWitt, E. Schutzman, and P. Vernon (Gordon and Breach, New York, 1967), Vol. 3, p. 307

[c18] R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. 2

[c19] N. Ashcroft and N. Mermin, Solid State Physics (Holt, Rinehart, and Winston, New York, 1976), Chap. 2

[c20] S. Shapiro and S. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars (Wiley, New York, 1983), Chaps. 3 and 9

[c21] C. Cattaneo, C. R. Acad. Sci. Paris 247, 431 (1958)

[c22] J. C. Maxwell, Philos. Trans. R. Soc. London 157, 49 (1867)

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Stability, causality, and hyperbolicity in Carter's "regular" theory of relativistic heat-conducting fluids

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