Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T11:37:52.340Z Has data issue: false hasContentIssue false
  • English
  • Français

On the small redshift limit of steady states of the spherically symmetric Einstein–Vlasov system and their stability

Published online by Cambridge University Press:  05 October 2015

MAHIR HADŽIĆ  and
GERHARD REIN
MAHIR HADŽIĆ
Affiliation:
King's College London, Department of Mathematics Strand, London, WC2R 2LS. e-mail: [email protected]
GERHARD REIN
Affiliation:
Mathematisches Institut der Universität Bayreuth D-95440 Bayreuth, Germany. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Families of steady states of the spherically symmetric Einstein–Vlasov system are constructed, which are parametrised by the central redshift. It is shown that as the central redshift tends to zero, the states in such a family are well approximated by a steady state of the Vlasov–Poisson system, i.e., a Newtonian limit is established where the speed of light is kept constant as it should be and the limiting behavior is analysed in terms of a parameter which is tied to the physical properties of the individual solutions. This result is then used to investigate the stability properties of the relativistic steady states with small redshift parameter in the spirit of recent work by the same authors, i.e., the second variation of the ADM mass about such a steady state is shown to be positive definite on a suitable class of states.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 3 , November 2015 , pp. 529 - 546
Copyright
Copyright © Cambridge Philosophical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Andréasson, H., Kunze, M. and Rein, G. Gravitational collapse and the formation of black holes for the spherically symmetric Einstein–Vlasov system. Quarterly Appl. Math. LXVIII (2010), 17–42.Google Scholar
[2] Andréasson, H., Kunze, M. and Rein, G. The formation of black holes in spherically symmetric gravitational collapse. Math. Ann. 350 (2011), 683–705.Google Scholar
[3] Andréasson, H. and Rein, G. A numerical investigation of the stability of steady states and critical phenomena for the spherically symmetric Einstein–Vlasov system. Class. Quantum Grav. 23 (2006), 3659–3677.Google Scholar
[4] Andréasson, H. and Rein, G. Formation of trapped surfaces for the spherically symmetric Einstein–Vlasov system. J. Hyperbolic Differential Eqns. 7 (2010), 707–731.Google Scholar
[5] Dafermos, M. and Rendall, A. An extension principle for the Einstein–Vlasov system under spherical symmetry. Ann. Henri Poincaré 6 (2005), 1137–1155.Google Scholar
[6] Guo, Y. and Rein, G. A non-variational approach to nonlinear stability in stellar dynamics applied to the King model. Commun. Math. Phys. 271 (2007), 489–509.Google Scholar
[7] Hadžić, M. and Rein, G. Stability for the spherically symmetric Einstein–Vlasov system–-a coercivity estimate. Math. Proc. Camb. Phil. Soc. 155 (2013), 529–556.Google Scholar
[8] Ipser, J. and Thorne, K. Relativistic, spherically symmetric star clusters I. Stability theory for radial perturbations. Astrophys. J. 154 (1968), 251–270.Google Scholar
[9] Ipser, J. Relativistic, Spherically Symmetric Star Clusters. II. Sufficient Conditions for Stability against Radial Perturbations. Astrophys. J. 156 (1969), 509–527.Google Scholar
[10] Ipser, J. Relativistic, Spherically Symmetric Star Clusters. III. Stability of Compact Isotropic Models. Astrophys. J. 158 (1969), 17–43.Google Scholar
[11] Kandrup, H. and Morrison, P. Hamiltonian structure of the Vlasov–Einstein system and the problem of stability for spherical relativistic star clusters. Annals of Physics 225 (1993), 114–166.Google Scholar
[12] Kandrup, H. and Sygnet, J. F. A simple proof of dynamical stability for a class of spherical clusters. Astrophys. J. 298 (1985), 27–33.Google Scholar
[13] Lemou, M., Mehats, F. and Raphaël, P. Orbital stability of spherical systems. Inventiones Math. 187 (2012), 145–194.Google Scholar
[14] Mouhot, C. Stabilité orbitale pour le système de Vlasov–Poisson gravitationnel, d'après Lemou–Méhats–Raphaël, Guo, Lin, Rein et al. Séminaire Nicolas Bourbaki Nov. 2011, arXiv:1201.2275 (2012).Google Scholar
[15] Ramming, T. and Rein, G. Spherically symmetric equilibria for self-gravitating kinetic or fluid models in the non-relativistic and relativistic case–-A simple proof for finite extension. SIAM Journal on Mathematical Analysis 45 (2013), 900–914.Google Scholar
[16] Rein, G. Static solutions of the spherically symmetric Vlasov–Einstein system. Math. Proc. Camb. Phil. Soc. 115 (1994), 559–570.Google Scholar
[17] Rein, G. Collisionless kinetic equations from astrophysics–-The Vlasov–Poisson system. In Handbook of Differential Equations, Evolutionary Equations, vol. 3, edited by Dafermos, C. M. and Feireisl, E. (Elsevier, 2007).Google Scholar
[18] Rein, G. and Rendall, A. Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data. Commun. Math. Phys. 150 (1992), 561–583. Erratum: Commun. Math. Phys. 176 (1996), 475–478.Google Scholar
[19] Rein, G. and Rendall, A. The Newtomian limit of the spherically symmetric Vlasov-Einstein system. Commun. Math. Phys. 150 (1992), 585–591.Google Scholar
[20] Rein, G. and Rendall, A. Smooth static solutions of the spherically symmetric Vlasov–Einstein system. Ann. Inst. H. Poincaré Physique Théorique 59 (1993), 383–397.Google Scholar
[21] Rein, G. and Rendall, A. Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics. Math. Proc. Camb. Phil. Soc. 128 (2000), 363–380.Google Scholar
[22] Wolansky, G. Static solutions of the Vlasov-Einstein system. Arch. Rational Mech. Anal. 156 (2001), 205–230.Google Scholar
[23] Zel'dovich, Ya. B. and Novikov, I. D. Relativistic Astrophysics, vol. 1 (Chicago University Press, 1971).Google Scholar
[24] Zel'dovich, Ya. B. and Podurets, M. A. The evolution of a system of gravitationally interacting point masses. Soviet Astronomy–AJ 9 (1965), 742–749. Translated from Astronomicheskii Zhurnal 42.Google Scholar