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Annales Henri Poincaré

, Volume 13, Issue 7, pp 1511–1536 | Cite as

Black Hole Formation from a Complete Regular Past for Collisionless Matter

  • Håkan AndréassonEmail author
Article

Abstract

Initial data for the spherically symmetric Einstein–Vlasov system is constructed whose past evolution is regular and whose future evolution contains a black hole. This is the first example of initial data with these properties for the Einstein-matter system with a “realistic” matter model. One consequence of the result is that there exists a class of initial data for which the ratio of the Hawking mass m̊= m̊ (r) and the area radius r is arbitrarily small everywhere, such that a black hole forms in the evolution. This result is in a sense analogous to the result (Christodoulou Commun Pure Appl Math 44:339–373, 1991) for a scalar field. Another consequence is that there exist black hole initial data such that the solutions exist for all Schwarzschild time \({t \in (-\infty,\infty)}\) .

Keywords

Black Hole Initial Data Global Existence Vlasov Equation Black Hole Formation 
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.
    Andréasson H.: The Einstein–Vlasov system/kinetic theory. Living Rev. Relat. 14, 4 (2011)ADSGoogle Scholar
  2. 2.
    Andréasson H.: Regularity results for the spherically symmetric Einstein–Vlasov system. Ann. Henri Poincaré 11, 781–803 (2010)ADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Andréasson H.: Sharp bounds on 2m/r of general spherically symmetric static objects. J. Differ. Equ. 245, 2243–2266 (2008)zbMATHCrossRefGoogle Scholar
  4. 4.
    Andréasson H.: On static shells and the Buchdahl inequality for the spherically symmetric Einstein–Vlasov system. Commun. Math. Phys. 274, 409–425 (2007)ADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Andréasson H., Kunze M., Rein G.: Global existence for the spherically symmetric Einstein–Vlasov system with outgoing matter. Commun. Partial Differ. Equ. 33, 656–668 (2008)zbMATHCrossRefGoogle Scholar
  6. 6.
    Andréasson H., Kunze M., Rein G.: The formation of black holes in spherically symmetric gravitational collapse. Math. Ann. 350, 683–705 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Andréasson H., Kunze M., Rein G.: Gravitational collapse and the formation of black holes for the spherically symmetric Einstein–Vlasov system. Q. Appl. Math. 68, 17–42 (2010)zbMATHGoogle Scholar
  8. 8.
    Andréasson H., Kunze M., Rein G.: Existence of axially symmetric static solutions of the Einstein–Vlasov system. Commun. Math. Phys. 308, 23–47 (2011)ADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Andréasson H., Rein G.: The asymptotic behaviour in Schwarzschild time of Vlasov matter in spherically symmetric gravitational collapse. Math. Proc. Camb. Phil. Soc. 149, 173–188 (2010)zbMATHCrossRefGoogle Scholar
  10. 10.
    Andréasson H., 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, 3659–3677 (2006)ADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Andréasson H., Rein G.: On the steady states of the spherically symmetric Einstein–Vlasov system. Class. Quantum Grav. 24, 1809–1832 (2007)ADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Binney J., Tremaine S.: Galactic Dynamics. Princeton University Press, Princeton (1987)zbMATHGoogle Scholar
  13. 13.
    Christodoulou D.: On the global initial value problem and the issue of singularities. Class. Quantum Grav. 16, A23–A35 (1999)MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Christodoulou D.: Bounded variation solutions of the spherically symmetric Einstein-scalar field equations. Commun. Pure Appl. Math. 46, 1131–1220 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Christodoulou D.: The formation of black holes and singularities in spherically symmetric gravitational collapse. Commun. Pure Appl. Math. 44, 339–373 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Christodoulou D., Klainerman S.: The Global Nonlinear Stability of Minskowski Space. Princeton Mathematical Series, vol. 41. Princeton University Press, Princeton (1993)Google Scholar
  17. 17.
    Dafermos M.: Black hole formation from a complete regular past. Commun. Math. Phys. 289, 579–596 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 18.
    Olabarrieta I., Choptuik M.W.: Critical phenomena at the threshold of black hole formation for collisionless matter in spherical symmetry. Phys. Rev. D 65(024007), 110 (2001)MathSciNetGoogle Scholar
  19. 19.
    Oppenheimer J.R., Snyder H.: On continued gravitational contraction. Phys. Rev. 56, 455–459 (1939)ADSzbMATHCrossRefGoogle Scholar
  20. 20.
    Rein G.: The Vlasov–Einstein System with Surface Symmetry. Habilitationsschrift, Munich (1995)Google Scholar
  21. 21.
    Rein G.: Collisionless kinetic equations from astrophysics. In: Dafermos, C.M., Feireisl, E. (eds) The Vlasov Poisson System. Handbook of Differential Equations: Evolutionary Equations, vol. 3, pp. 383–476. Elsevier/North-Holland, Amsterdam (2006)Google Scholar
  22. 22.
    Rein, G., Rendall, A.D.: Global existence of solutions of the spherically symmetric Vlasov–Einstein system with small initial data. Commun. Math. Phys. 150, 561–583 (1992). Erratum: Commun. Math. Phys. 176, 475–478 (1996)Google Scholar
  23. 23.
    Rein G., Rendall J., Schaeffer A.D.: A regularity theorem for solutions of the spherically symmetric Vlasov–Einstein system. Commun. Math. Phys. 168, 467–478 (1995)MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Rein G., Rendall A.D., Schaeffer J.: Critical collapse of collisionless matter: a numerical investigation. Phys. Rev. D 58(044007), 18 (1998)Google Scholar
  25. 25.
    Rendall A.D.: An introduction to the Einstein–Vlasov system. Banach Center Publ. 41, 35–68 (1997)MathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Mathematical SciencesUniversity of Gothenburg, Chalmers University of TechnologyGöteborgSweden

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