Gravitational collapse of charged scalar fields

  • Jose M. TorresEmail author
  • Miguel Alcubierre
Research Article


In order to study the gravitational collapse of charged matter we analyze the simple model of an self-gravitating massless scalar field coupled to the electromagnetic field in spherical symmetry. The evolution equations for the Maxwell–Klein–Gordon sector are derived in the \(3+1\) formalism, and coupled to gravity by means of the stress–energy tensor of these fields. To solve consistently the full system we employ a generalized Baumgarte–Shapiro–Shibata–Nakamura formulation of General Relativity that is adapted to spherical symmetry. We consider two sets of initial data that represent a time symmetric spherical thick shell of charged scalar field, and differ by the fact that one set has zero global electrical charge while the other has non-zero global charge. For compact enough initial shells we find that the configuration doesn’t disperse and approaches a final state corresponding to a sub-extremal Reissner–Nördstrom black hole with \(|Q|<M\). By increasing the fundamental charge of the scalar field \(q\) we find that the final black hole tends to become more and more neutral. Our results support the cosmic censorship conjecture for the case of charged matter.


Gravitational collapse Eintein–Maxwell–Klein–Gordon system Cosmic censorship Charged scalar fields 



The authors wish to thank Dario Núñez and Marcelo Salgado for many useful discussions and comments. This work was supported in part by Dirección General de Estudios de Posgrado (DGEP-UNAM), by CONACyT through Grant 82787, and by DGAPA-UNAM through Grants IN113907 and IN115310. J.M.T. also acknowledges a CONACyT postgraduate scholarship.


  1. 1.
    Penrose, R.: J. Astrophys. Astron. 20, 233 (1999)ADSCrossRefGoogle Scholar
  2. 2.
    Shapiro, S.L., Teukolsky, S.A.: Phys. Rev. Lett. 66, 994 (1991)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Choptuik, M.W.: Phys. Rev. Lett. 70, 9 (1993)ADSCrossRefGoogle Scholar
  4. 4.
    Christodoulou, D.: Ann. Math. 140, 607 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Liebling, S.L., Palenzuela, C.: Living Rev. Rel. 15, 6 (2012)Google Scholar
  6. 6.
    Jetzer, P., Bij, J.V.D.: Phys. Lett. B 227, 341 (1989)ADSCrossRefGoogle Scholar
  7. 7.
    Petryk, R.: Ph.D. thesis, University of British Columbia (2005)Google Scholar
  8. 8.
    Oren, Y., Piran, T.: Phys. Rev. D 68, 044013 (2003)ADSCrossRefMathSciNetGoogle Scholar
  9. 9.
    Arnowitt, R., Deser, S., Misner, C.W.: In: Witten, L. (ed.) Gravitation: An Introduction to Current Research, pp. 227–265. Wiley, New York (1962)Google Scholar
  10. 10.
    York, J.: In: Smarr, L. (ed.) Sources of Gravitational Radiation. Cambridge University Press, Cambridge (1979)Google Scholar
  11. 11.
    Alcubierre, M.: Introduction to 3+1 Numerical Relativity. Oxford University Press, New York (2008)CrossRefzbMATHGoogle Scholar
  12. 12.
    Shibata, M., Nakamura, T.: Phys. Rev. D52, 5428 (1995)ADSMathSciNetGoogle Scholar
  13. 13.
    Baumgarte, T.W., Shapiro, S.L.: Phys. Rev. D59, 024007 (1998)ADSMathSciNetGoogle Scholar
  14. 14.
    Brown, J.D.: Phys. Rev. D79, 104029 (2009)ADSGoogle Scholar
  15. 15.
    Alcubierre, M., Mendez, M.D.: Gen. Relativ. Gravit. 43, 2769 (2011)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Alcubierre, M., Degollado, J.C., Salgado, M.: Phys. Rev. D80, 104022 (2009)ADSMathSciNetGoogle Scholar
  17. 17.
    Heusler, M.: Black Hole Uniqueness Theorems. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  18. 18.
    Reissner, H.: Ann. Phys. 50, 106 (1916)CrossRefGoogle Scholar
  19. 19.
    Nordström, G.: Proc. Kon. Ned. Akad. Wet. 20, 1238 (1918)ADSGoogle Scholar
  20. 20.
    Alcubierre, M., et al.: Phys. Rev. D81, 124018 (2010)ADSMathSciNetGoogle Scholar
  21. 21.
    Ruiz, M., et al.: Phys. Rev. D86, 104044 (2012)ADSGoogle Scholar
  22. 22.
    Alcubierre, M., González, J.A.: Comput. Phys. Comm. 167, 76 (2005)ADSCrossRefzbMATHGoogle Scholar
  23. 23.
    Montero, P.J., Cordero-Carrion, I.: Phys. Rev. D85, 124037 (2012)ADSGoogle Scholar
  24. 24.
    Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Spacetime. Cambridge University Press, Cambridge (1973)CrossRefGoogle Scholar
  25. 25.
    Alcubierre, M., Torres, J.M: Constraint preserving boundary conditions for the Baumgarte-Shapiro-Shibata-Nakamura formulation in spherical symmetry. Clas. Quan. Grav (submitted). arXiv:1407.8529

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Instituto de Ciencias NuclearesUniversidad Nacional Autónoma de MéxicoMexicoMexico

Personalised recommendations