Advertisement

Black Hole Formation from a Complete Past for the Einstein–Vlasov System

  • Håkan AndréassonEmail author
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 157)

Abstract

A natural question in general relativity is to find initial data for the Einstein equations whose past evolution is regular and whose future evolution contains a black hole. In [1] initial data of this kind is constructed for the spherically symmetric Einstein–Vlasov system. One consequence of the result is that there exists a class of initial data for which the ratio of the Hawking mass \({\mathring{m}}={\mathring{m}}(r)\) and the area radius \(r\) is arbitrarily small everywhere, such that a black hole forms in the evolution. This result is analogous to the result [2] for a scalar field. Another consequence is that there exist black hole initial data such that the solutions exist for all Schwarzschild time \(t\in (-\infty ,\infty )\). In the present article we review the results in [1].

Keywords

Black Hole Initial Data Scalar Field Global Existence 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.
    Andréasson, H.: Black hole formation from a complete regular past for collisionless matter. Ann. Henri Poincaré 13, 1511 (2012). doi: 10.1007/s00023-012-0164-1 ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Christodoulou, D.: The formation of black holes and singularities in spherically symmetric gravitational collapse. Commun. Pure Appl. Math. 44, 339 (1991). doi: 10.1002/cpa.3160440305 CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Oppenheimer, J., Snyder, H.: On continued gravitational contraction. Phys. Rev. 56, 455 (1939). doi: 10.1103/PhysRev.56.455 ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Dafermos, M.: Black hole formation from a complete regular past. Commun. Math. Phys. 289, 579 (2009). doi: 10.1007/s00220-009-0775-7 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Christodoulou, D.: Bounded variation solutions of the spherically symmetric Einstein-scalar field equations. Commun. Pure Appl. Math. 46, 1131 (1993). doi: 10.1002/cpa.3160460803 CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Andréasson, H.: The Einstein-Vlasov system/kinetic theory. Living Rev. Relativ. 14(4), lrr-2011-4 (2011). http://www.livingreviews.org/lrr-2011-4
  7. 7.
    Binney, J., Tremaine, S.: Galactic Dynamics. Princeton Series in Astrophysics. Princeton University Press, Princeton (1987)Google Scholar
  8. 8.
    Andréasson, H.: Regularity results for the spherically symmetric Einstein-Vlasov system. Ann. Henri Poincaré 11, 781 (2010). doi: 10.1007/s00023-010-0039-2 ADSCrossRefzbMATHGoogle Scholar
  9. 9.
    Andréasson, H., Kunze, M., Rein, G.: Global existence for the spherically symmetric Einstein-Vlasov system with outgoing matter. Commun. Part. Diff. Equ. 33, 656 (2008). doi: 10.1080/03605300701454883 CrossRefzbMATHGoogle Scholar
  10. 10.
    Andréasson, H., Kunze, M., Rein, G.: The formation of black holes in spherically symmetric gravitational collapse. Math. Ann. 350, 683 (2011). doi: 10.1007/s00208-010-0578-3 CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Rendall, A.: An introduction to the Einstein-Vlasov system. In: Chruściel, P. (ed.) Mathematics of Gravitation, Part I: Lorentzian Geometry and Einstein Equations, Banach Center Publications, vol. 41, pp. 35–68. Polish Academy of Sciences, Institute of Mathematics, Warsaw (1997)Google Scholar
  12. 12.
    Rein, G., Rendall, A.: Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data. Commun. Math. Phys. 150(561), 1996 (1992). doi: 10.1007/BF02096962.Erratum:ibid.176,475-478 Google Scholar
  13. 13.
    Andréasson, H.: Sharp bounds on \(2m/r\) of general spherically symmetric static objects. J. Differ. Equ. 245, 2243 (2008). doi:  10.1016/j.jde.2008.05.010 ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Andréasson, H.: On static shells and the Buchdahl inequality for the spherically symmetric Einstein-Vlasov system. Commun. Math. Phys. 274, 409 (2007). doi: 10.1007/s00220-007-0285-4 ADSCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Gothenburg and Chalmers University of TechnologyGöteborgSweden

Personalised recommendations