Hostname: page-component-7bb8b95d7b-lvwk9 Total loading time: 0 Render date: 2024-09-06T05:22:34.599Z Has data issue: false hasContentIssue false
  • English
  • Français

Static solutions of the spherically symmetric Vlasov–Einstein system

Published online by Cambridge University Press:  24 October 2008

Gerhard Rein
Gerhard Rein
Affiliation:
Mathematisches Institut der Universität München, Theresienstr. 39, 80333 München, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the Vlasov—Einstein system in a spherically symmetric setting and prove the existence of static solutions which are asymptotically flat and have finite total mass and finite extension of the matter. Among these there are smooth, singularity-free solutions, which have a regular centre and have isotropic or anisotropic pressure, and solutions which have a Schwarzschild-singularity at the centre.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 3 , May 1994 , pp. 559 - 570
Copyright
Copyright © Cambridge Philosophical Society 1994

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]Batt, J.. Steady state solutions of the relativistic Vlasov–Poisson system. In Proceedings of the Fifth Marcel Grossmann Meeting on General Relativity, Part A (World Scientific 1989), pp. 1235–1247.Google Scholar
[2]Batt, J., Faltenbacher, W. and Horst, E.. Stationary spherically symmetric models in stellar dynamics. Arch. Rational Mech. Anal. 93 (1986), 159–183.CrossRefGoogle Scholar
[3]Batt, J. and Pfaffelmoser, K.. On the radius continuity of the models of polytropic gas spheres which correspond to the positive solutions of the generalized Emden–Fowler equation. Math. Meth. in the Appl. Sci. 10 (1988), 499–516.CrossRefGoogle Scholar
[4]Christodoulou, D.. Violation of cosmic censorship in the gravitational collapse of a dust cloud. Commun. Math. Phys. 93 (1984), 171–195.CrossRefGoogle Scholar
[5]Kijowski, J. and Magli, G.. A generalization of the relativistic equilibrium equation for a nonrotating star. General Relativity and Gravitation 24 (1992), 139–158.Google Scholar
[6]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.CrossRefGoogle Scholar
[7]Rein, G. and Rendall, A.. The Newtonian limit of the spherically symmetric Vlasov–Einstein system. Commun. Math. Phys. 150 (1992), 585–591.CrossRefGoogle Scholar
[8]Rein, G. and Rendall, A.. Smooth static solutions of the spherically symmetric Vlasov–Einstein system. To appear in Ann. de l'Inst. H. Poincaré, Physique Théorique.Google Scholar
[9]Rein, G., Rendall, A. and Schaeffer, J.. A regularity theorem for solutions of the spherically symmetric Vlasov–Einstein system. In preparation.Google Scholar
[10]Rendall, A. and Schmidt, B.. Existence and properties of spherically symmetric static fluid bodies with a given equation of state. Class. Quantum Gray. 8 (1991), 985–1000.CrossRefGoogle Scholar
[11]Shapiro, S. L. and Teukolsey, S. A.. Relativistic stellar dynamics on the computer, I. Motivation and numerical method. Astrophysical Journal 298 (1985), 34–57.CrossRefGoogle Scholar
[12]Sideris, T.. Formation of singularities in three-dimensional compressible fluids. Comniun. Math. Phys. 101 (1985), 475–485.CrossRefGoogle Scholar
[13]Smoller, J., Wasserman, A., Yau, S.-T. and Mcleod, J.. Smooth static solutions of the Einstein/Yang-Mills equations. Commun. Math. Phys. 143 (1991), 115–147.CrossRefGoogle Scholar
[14]Wald, R. M.. General Relativity. (University of Chicago Press, 1984).CrossRefGoogle Scholar
[15]Yodzis, P., Seifert, H.-J. and Müller Zum Hagen, H.. On the occurrence of naked singularities in general relativity. Commun. Math. Phys. 34 (1973), 135–148.CrossRefGoogle Scholar