Abstract
This work develops the dynamics of a perfectly elastic solid model for application to the outer crust of a magnetised neutron star. Particular attention is given to the Noether identities responsible for energy-momentum conservation, using a formulation that is fully covariant, not only (as is usual) in a fully relativistic treatment but also (sacrificing accuracy and elegance for economy of degrees of gravitational freedom) in the technically more complicated case of the Newtonian limit. The results are used to obtain explicit (relativistic and Newtonian) formulae for the propagation speeds of generalised (Alfven type) magneto-elastic perturbation modes.
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Carter, B., Khalatnikov, I.M.: Momentum, vorticity, and helicity in covariant superfluid dynamics. Ann. Phys. 219, 243–265 (1992)
Langlois, D., Sedrakian, D., Carter, B.: Differential rotation of relativistic superfluid in neutron stars. Mon. Not. R. Astr. Soc. 297, 1198–1201 (1998) [astro-ph/9711042]
Kunzle, H.P.: Lagrangian formalism for adiabatic fluids on five-dimensional space-time. Canad. J. Phys. 64, 185–189 (1986)
Duval, C., Gibbons, G., Horvathy, P.: Celestial mechanics, conformal structures, and gravitational waves. Phys. Rev. D 43, 3907–3902 (1991)
Carter, B., Khalatnikov, I.M.: Canonically covariant formulation of Landau's Newtonian superfluid dynamic. Rev. Math. Phys. 6, 277–304 (1994)
Cartan, E.: Sur les varits connexion affine et la thorie de la relativit generalise. Ann. Sci. Ecole Norm. Sup. 40, 325–412 (1923); Ann. Sci. Ecole Norm. Sup. 41, 1–25 (1924); Ann. Sci. Ecole Norm. Sup. 42, 17–88 (1925)
Carter, B., Chamel, N.: Covariant analysis of Newtonian multi-fluid models for neutron stars: I Milne – Cartan structure and variational formulation. Int. J. Mod. Phys. D 13, 291–325 (2004) [astro-ph/0305186]
Carter, B., Chamel, N.: Covariant analysis of Newtonian multi-fluid models for neutron stars: II Stress - energy tensors and virial theorems. Int. J. Mod. Phys. D14, 717 (2005) [astro–ph/0312414]
Carter, B., Chamel, N.: Covariant analysis of Newtonian multi-fluid models for neutron stars: III Transvective, viscous, and superfluid drag dissipation. Int. J. Mod. Phys. D14, 749 (2005) [astro-ph/0410660]
Battye, R.A., Carter, B., Chachoua, E., Moss, A.: Rigidity and stability of cold dark solid universe model. to appear in Phys. Rev. D (2005) [hep-th/0501244]
Palmer, D.M. et al.: A Giant γ-ray flare from the magnetar SGR 1806–20. Nature 434, 1107–1109 (2005) [astro-ph/0503030]
Eichler, D.: Waiting for the big one: a new class of soft gamma ray repeater outbursts. Mon. Not. R. Astr. Soc. 335, 883–886 (2002) [astro-ph/0204512]
Hurley, K. et al.: An exceptionally bright flare from SGR 1806–20 and the origins of short-duration γ-ray bursts. Nature 434, 1098–1103 (2005) [astro-ph/0502329]
Trautman, A.: Invariance properties and conservation laws. In: Trautman, A., Pirani, F.A.E., Bondi, H. (eds.), Brandeis Lectures on General Relativity, pp. 158–200. Prentice Hall, New Jersey (1965)
Carter, B.: Elastic perturbation theory in general relativity and a variational principle for a rotating solid star. Commun. Math. Phys. 30, 261–286 (1973)
Carter, B.: Covariant Theory of Conductivity in Ideal Fluid or Solid Media. In: Carter, B., Anile, A.M., Choquet-Bruhat, Y. (eds.), Relativistic Fluid Dynamics C.I.M.E., Noto. (1987) Lecture Notes in Mathematics, vol. 1385, pp. 1–64. Springer-Verlag, Heidelberg (1989)
Schutz, B.: Perfect fluids in general relativity: velocity potentials and a variational principle. Phys. Rev. D 2, 2762–2773 (1970)
Landau, L.D., Lifshitz, E.M.: Theory of Elasticity. Pergamon, London (1959)
Souriau, J.M.: Géometrie et Relativité. Herman, Paris (1965)
DeWitt, B.: The quantisation of geometry. In: Witten, L. (ed.), Gravitation: An Introduction to Current Research, pp. 266–381. Wiley, New York (1962)
Carter, B., Quintana, H.: Foundations of general relativistic high pressure elasticity theory. Proc. Roy. Soc. Lond. A 331, 57–83 (1972)
Carter, B.: Interaction of Gravitational Waves with an Elastic Solid Medium. In: Deruelle, N., Piran, T. (eds.), Gravitational Radiation (Proc. 1982 Les Houches Summer School), pp. 455–464. North Holland, Amsterdam (1983) [gr-qc/0102113]
Carter, B.: Kalb-Ramond vorticity variational formulation of relativistic perfectly conducting fluid theory. Int. J. Mod. Phys. D 3, 15–21 (1974)
Carter, B.: Vortex dynamics in superfluids. In: Bunkov, Yu.M., Godfrin, H. (eds.), Topological Defects and Non-equilibrium Dynamics of Symmetry Breaking Phase Transitions. (NATO ASI vol. C549. Les Houches 1999), pp. 267–301. Kluwer, Dordrecht (2000) [gr-qc/9907039]
Carter, B.: Speed of sound in a high pressure general relativistic solid. Phys. Rev. D 7, 1590–1593 (1973)
Carter, B., Gaffet, B.: Standard covariant formulation for perfect fluid dynamics. J. Fluid. Mech. 186, 1–24 (1988)
Bekenstein, J.D., Oron, A.: Conservation of circulation in magnetohydrodynamics. Phys. Rev. E 62, 5594–5603 (2000) [astro-ph/0002045]
Bekenstein, J.D., Oron, E.: New conservation laws in general-relativistic magnetohydrodynamics. Phys. Rev. D 18, 1809–1819 (1978)
Woltjer, L.: A theorem on force-free magnetic fields. Proc. Nat. Acad. Sci. USA 44, 489–491 (1958)
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Carter, B., Chachoua, E. & Chamel, N. Covariant Newtonian and relativistic dynamics of (magneto)-elastic solid model for neutron star crust. Gen Relativ Gravit 38, 83–119 (2006). https://doi.org/10.1007/s10714-005-0210-0
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DOI: https://doi.org/10.1007/s10714-005-0210-0