Advertisement

Existence of Axially Symmetric Static Solutions of the Einstein-Vlasov System

  • Håkan Andréasson
  • Markus Kunze
  • Gerhard ReinEmail author
Article

Abstract

We prove the existence of static, asymptotically flat non-vacuum spacetimes with axial symmetry where the matter is modeled as a collisionless gas. The axially symmetric solutions of the resulting Einstein-Vlasov system are obtained via the implicit function theorem by perturbing off a suitable spherically symmetric steady state of the Vlasov-Poisson system.

Keywords

Implicit Function Theorem Bianchi Identity Regularity Property Energy Momentum Tensor Vlasov Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Andersson L., Beig R., Schmidt B.: Static self-gravitating elastic bodies in Einstein gravity. Commun. Pure Appl. Math. 61, 988–1023 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Andersson L., Beig R., Schmidt B.: Rotating elastic bodies in Einstein gravity. Commun. Pure Appl. Math. 63, 559–589 (2009)MathSciNetGoogle Scholar
  3. 3.
    Andréasson, H.: The Einstein-Vlasov System/Kinetic Theory. Living Rev. Relativity 8 (2005), available at http://relativity.livingreviews.org/Articles/lrr-2005-z, 2005
  4. 4.
    Bardeen, J.: Rapidly rotating stars, disks, and black holes. In: Black Holes / Les Astres Occlus, ed. by C. DeWitt, B. S. DeWitt, Les Houches, 1972, London-NewYork-Paris: Gordon and Breach, 1973Google Scholar
  5. 5.
    Batt J., Faltenbacher W., Horst E.: Stationary spherically symmetric models in stellar dynamics. Arch. Rat. Mech. Anal. 93, 159–183 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Batt J., Pfaffelmoser K.: On the radius continuity of the models of polytropic gas spheres which correspond to positive solutions of the generalized Emden-Fowler equations. Math. Meth. Appl. Sci. 10, 499–516 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Deimling K.: Nonlinear Functional Analysis. Springer, Berlin-New York (1985)zbMATHGoogle Scholar
  8. 8.
    Fjällborg M., Heinzle M., Uggla C.: Self-gravitating stationary spherically symmetric systems in relativistic galactic dynamics. Math. Proc. Cambridge Philos. Soc. 143, 731–752 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Heilig U.: On Lichtenstein’s analysis of rotating Newtonian stars. Ann. de l’Inst. H. Poincaré, Physique Théorique 60, 457–487 (1994)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Heilig U.: On the existence of rotating stars in general relativity. Commun. Math. Phys. 166, 457–493 (1995)MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Jackson D.: Classical Electrodynamics. Wiley, New York (1975)zbMATHGoogle Scholar
  12. 12.
    Lichtenstein L.: Untersuchung über die Gleichgewichtsfiguren rotierender Flüssigkeiten, deren Teilchen einander nach dem Newtonschen Gesetze anziehen. Erste Abhandlung. Homogene Flüssigkeiten. Allgemeine Existenzsätze. Math. Z. 1, 229–284 (1918)MathSciNetGoogle Scholar
  13. 13.
    Lichtenstein L.: Gleichgewichtsfiguren rotierender Flüssigkeiten. Springer, Berlin (1933)CrossRefGoogle Scholar
  14. 14.
    Lieb, E., Loss, M.: Analysis, Providence, RI: Amer. Math. Soc., 1997Google Scholar
  15. 15.
    Müller C.: Spherical Harmonics. Lecture Notes in Mathematics 17. Springer, Berlin (1966)Google Scholar
  16. 16.
    Rein G.: Static solutions of the spherically symmetric Vlasov-Einstein system. Math. Proc. Camb. Phil. Soc. 115, 559–570 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Rein G.: Stationary and static stellar dynamic models with axial symmetry. Nonlinear Analysis; Theory, Methods & Applications 41, 313–344 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Rein G., Rendall A.: Smooth static solutions of the spherically symmetric Vlasov-Einstein system. Ann. de l’Inst. H. Poincaré, Physique Théorique 59, 383–397 (1993)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Rein G., Rendall A.: Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics. Math. Proc. Camb. Phil. Soc. 128, 363–380 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Schulze A.: Existence of axially symmetric solutions to the Vlasov-Poisson system depending on Jacobi’s integral. Commun. Math. Sci. 6, 711–727 (2008)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Wald, R.: General Relativity, Chicago, IL: Chicago University Press, 1984Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Håkan Andréasson
    • 1
  • Markus Kunze
    • 2
  • Gerhard Rein
    • 3
    Email author
  1. 1.Mathematical SciencesChalmers University of Technology, Göteborg UniversityGöteborgSweden
  2. 2.Fakultät für MathematikUniversität Duisburg-EssenEssenGermany
  3. 3.Fakultät für Mathematik, Physik und InformatikUniversität BayreuthBayreuthGermany

Personalised recommendations