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On the locally rotationally symmetric Einstein-Maxwell perfect fluid. (English) Zbl 1386.83031
Summary: We examine the stability of Einstein-Maxwell perfect fluid configurations with a privileged radial direction by means of a \(1+1+2\)-tetrad formalism. We use this formalism to cast in a quasilinear symmetric hyperbolic form the equations describing the evolution of the system. This hyperbolic reduction is used to discuss the stability of linear perturbations in some special cases. By restricting the analysis to isotropic fluid configurations, we assume a constant electrical conductivity coefficient for the fluid. As a result of this analysis we provide a complete classification and characterization of various stable and unstable configurations. We find, in particular, that in many cases the stability conditions are strongly determined by the constitutive equations and the electric conductivity. A threshold for the emergence of the instability appears in both contracting and expanding systems.
83C15 Exact solutions to problems in general relativity and gravitational theory
83C22 Einstein-Maxwell equations
83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)
53Z05 Applications of differential geometry to physics
76W05 Magnetohydrodynamics and electrohydrodynamics
Full Text: DOI
[1] Etienne, ZB; Liu, YT; Shapiro, SL, Relativistic magnetohydrodynamics in dynamical spacetimes: a new AMR implementation, Phys. Rev. D, 82, 084031, (2010)
[2] Font, J.A.: Numerical hydrodynamics in general relativity. Living Rev. Relativ. 6, 4 (2003) · Zbl 1068.83501
[3] Font, JA, Numerical hydrodynamics and magnetohydrodynamics in general relativity, Living Rev. Relat., 11, 7, (2007) · Zbl 1166.83003
[4] Alcubierre, M.: Introduction to \(3+1\) Numerical Relativity. Oxford University Press, Oxford (2008) · Zbl 1140.83002
[5] Giacommazo, B; Rezzolla, L, Whiskeymhd: a new numerical code for general relativistic MHD, Class. Quantum Gravity, 24, s235, (2007) · Zbl 1117.83002
[6] Lichnerowicz, A.: Relativistic Hydrodynamics and Magnetohydrodynamics. Benjamin, New York (1967) · Zbl 0193.55401
[7] Anton, L; Zanotti, O; Miralles, JA; Marti, JM; Ibanez, JM; Font, JA; Pons, JA, Numerical 3 + 1 general relativistic magnetohydrodynamics: a local characteristic approach, Astrophys. J., 637, 296, (2006)
[8] Radice, D; Rezzolla, L; Galeazzi, F, High-order fully general-relativistic hydrodynamics: new approaches and tests, Class. Quantum Gravity, 31, 075012, (2014) · Zbl 1291.83092
[9] Witek, H, Numerical relativity in higher-dimensional space-times, Int. J. Mod. Phys. A, 28, 1340017, (2013) · Zbl 1277.83061
[10] Pugliese, D; Valiente Kroon, JA, On the evolution equations for ideal magnetohydrodynamics in curved spacetime, Gen. Relat. Gravit., 44, 2785, (2012) · Zbl 1253.83011
[11] Shibata, M; Sekiguchi, Y, Magnetohydrodynamics in full general relativity: formulations and tests, Phys. Rev. D, 72, 044014, (2005)
[12] Baumgarte, TW; Shapiro, SL, General relativistic magnetohydrodynamics for the numerical construction of dynamical spacetimes, Astrophys. J., 585, 921, (2003)
[13] Palenzuela, C; Garrett, D; Lehner, L; Liebling, S, Magnetospheres of black hole systems in force-free plasma, Phys. Rev. D, 82, 044045, (2010)
[14] Anile, A.M.: Relativistic fluids and magneto-fluids: with applications in astrophysics and plasma physics. Cambridge UniversityPress, Cambridge (1989) · Zbl 0701.76003
[15] Disconzi, MM, On the well-posedness of relativistic viscous fluids, Nonlinearity, 27, 1915, (2014) · Zbl 1296.83012
[16] Lubbe, C; Valiente Kroon, JA, A conformal approach for the analysis of the non-linear stability of pure radiation cosmologies, Ann. Phys., 328, 1, (2013) · Zbl 1263.83188
[17] Clarkson, C, A covariant approach for perturbations of rotationally symmetric spacetimes, Phys. Rev. D, 76, 104034, (2007)
[18] Horst, E, Symmetric plasmas and their decay, Commun. Math. Phys., 126, 613-633, (1990) · Zbl 0694.76049
[19] Hsu, SC; Awe, TJ; Brockington, S; Case, A; Cassibry, JT; Kagan, G; Messer, SJ; Stanic, M; Tang, X; Welch, DR; Witherspoon, FD, Spherically imploding plasma liners as a standoff driver for magnetoinertial fusion, IEEE Trans. Plasma Sci., 40, 5, (2012)
[20] Tan, Tai-Ho; Borovsky, Joseph E, Spherically symmetric high-velocity plasma expansions into background gases, J. Plasma Phys., 35, 02-239, (1986)
[21] Laskyand, PD; Lun, AWC, Gravitational collapse of spherically symmetric plasmas in Einstein-Maxwell spacetimes, Phys. Rev. D, 75, 104010, (2007)
[22] Viana, RL; Clemente, RA; Lopes, SR, Spherically symmetric stationary MHD equilibria with azimuthal rotation, Plasma Phys. Control. Fus., 39, 197, (1997)
[23] Guo, Yan; Shadi Tahvildar-Zadeh, A, Formation of singularities in relativistic fluid dynamics and in spherically symmetric plasma dynamics, Contem. Math., 238, 151-161, (1999) · Zbl 0973.76100
[24] Clarkson, CA; Marklund, M; Betschart, G; Dunsby, PKS, The electromagnetic signature of black hole ring-down, Astrophys. J., 613, 492, (2004)
[25] Ellis, G F R; van Elst, H, Cosmological models: cargese lectures 1998, NATO Adv. Study Inst. Ser. C Math. Phys. Sci., 541, 1, (1998)
[26] Burston, RB, 1 \(+\) 1 \(+\) 2 gravitational perturbations on LRS class II spacetimes: decoupling gravito-electromagnetic tensor harmonic amplitudes, Class. Quantum Gravity, 25, 075004, (2008) · Zbl 1155.83323
[27] Clarkson, CA; Barrett, RK, Covariant perturbations of Schwarzschild black holes, Class. Quantum Gravity, 20, 3855, (2003) · Zbl 1054.83018
[28] Elsty, H; Ellis, GFR, The covariant approach to LRS perfect fluid spacetime geometries, Class. Quantum Gravity, 13, 1099, (1996) · Zbl 0855.53052
[29] Burston, RB; Lun, AWC, 1 \(+\) 1 \(+\) 2 electromagnetic perturbations on general LRS space-times: Regge-Wheeler and Bardeen-press equations, Class. Quantum Gravity, 25, 075003, (2008) · Zbl 1195.83029
[30] Burston, RB, 1 \(+\) 1 \(+\) 2 electromagnetic perturbations on non-vacuum LRS class II space-times: decoupling scalar and 2-vector harmonic amplitudes, Class. Quantum Gravity, 25, 075002, (2008) · Zbl 1195.83028
[31] Zanotti, O; Pugliese, D, Von zeipel’s theorem for a magnetized circular flow around a compact object, Gen. Relat. Gravit., 47, 4-44, (2015) · Zbl 1317.83015
[32] Komissarov, SS, Magnetized tori around Kerr black holes: analytic solutions with a toroidal magnetic field, Mon. Not. R. Astron. Soc., 368, 993-1000, (2006)
[33] Bekenstein, JD; Oron, E, New conservation laws in general-relativistic magnetohydrodynamics, Phys. Rev. D, 18, 1809, (1978)
[34] Bekenstein, JD; Oron, E, Interior magnetohydrodynamic structure of a rotating relativistic star, Phys. Rev. D, 19, 2827-72837, (1979)
[35] Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations. Cambridge Monographs on Mathematical Physics, Paperback, Cambridge (2009) · Zbl 1179.83005
[36] Betschart, G; Clarkson, CA, Scalar field and electromagnetic perturbations on locally rotationally symmetric spacetimes, Class. Quantum Gravity, 21, 5587, (2004) · Zbl 1065.83019
[37] Reula, O, Hyperbolic methods for einstein’s equations, Living Rev. Relat., 3, 1, (1998) · Zbl 1024.83004
[38] Reula, O, Exponential decay for small nonlinear perturbations of expanding flat homogeneous cosmologies, Phys. Rev. D, 60, 083507, (1999)
[39] Alho, A. Mena, F. C., & Valiente Kroon, J. A.: The Einstein-Friedrich-nonlinear scalar field system and the stability of scalar field Cosmologies. In arXiv:1006.3778 (2010) · Zbl 1386.83005
[40] Kreiss, HO; Lorenz, J, Stability for time-dependent differential equations, Acta Numer., 7, 203, (1998) · Zbl 0909.35017
[41] Kreiss, HO; Ortiz, OE; Reula, OA, Stability of quasi-linear hyperbolic dissipative systems, J. Differ. Equ., 142, 78, (1998) · Zbl 0932.35024
[42] Kreiss, HO; Nagy, GB; Ortiz, OE; Reula, OA, Global existence and exponential decay for hyperbolic dissipative relativistic fluid theories, J. Math. Phys., 38, 5272, (1997) · Zbl 0943.76097
[43] Ortiz, OE, Stability of nonconservative hyperbolic systems and relativistic dissipative fluids, J. Math. Phys., 42, 1426, (2001) · Zbl 1053.35089
[44] Marklund, M; Clarkson, C, The general relativistic MHD dynamo, Mon. Not. R. Astron. Soc., 358, 892, (2005)
[45] Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials: Critical Points. Zeros and Extremal Properties. London Mathematical Society Monographs, Clarendon Press, New York (2002)
[46] Barnett, S, A new formulation of the theorems of Hurwitz, Routh and Sturm, J. Inst. Math. Appl., 7, 240, (1971) · Zbl 0226.65037
[47] Choquet-Bruhat, Y.: General Relativity and the Einstein Equations. Oxford University Press, Oxford (2009) · Zbl 1157.83002
[48] Rezzolla, L., Zanotti, O.: Relativistic Hydrodynamics. Oxford University Press, New York (2013) · Zbl 1297.76002
[49] Pugliese, D; Montani, G; Bernardini, MG, On the Polish doughnut accretion disc via the effective potential approach, Mon. Not. R. Astron. Soc., 428, 952, (2012)
[50] Camenzind, M.: Compact Objects in Astrophysics White Dwarfs, Neutron Stars and Black Holes. Springer, Berlin (2007)
[51] Frank, J., King, A., Raine, D.: Accretion Power in Astrophysics. Cambridge University Press, Cambridge (2002)
[52] Collins, G.W.: The Fundamentals of Stellar Astrophysics. W H Freeman & Co, New York (1989)
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