Skip to main content
SpringerLink
Log in

The Global Structure of Spherically Symmetric Charged Scalar Field Spacetimes

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the spherical collapse of self-gravitating charged scalar fields. The main result gives a complete characterization of the future boundary of spacetime, providing a starting point for studying the cosmic censorship conjectures. In general, the boundary includes two null components, one emanating from the center of symmetry and the other from the future limit point of null infinity, joined by an achronal component to which the area-radius function r extends continuously to zero. Various components of the boundary a priori may be empty and establishing such generic emptiness would suffice to prove formulations of weak or strong cosmic censorship. As a simple corollary of the boundary characterization, the present paper rules out scenarios of ‘naked singularity’ formation by means of ‘super-charging’ (near-)extremal Reissner-Nordström black holes. The main difficulty in delimiting the boundary is isolated in proving a suitable global extension principle that effectively excludes a broad class of singularity formation. This suggests a new notion of ‘strongly tame’ matter models, which we introduce in this paper. The boundary characterization proven here extends to any such ‘strongly tame’ Einstein-matter system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aretakis S.: Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations I. Commun. Math. Phys. 307(1), 17–63 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Aretakis S.: Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations II. Ann. Henri Poincaré 12(8), 1491–1538 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Barack L.: Late time dynamics of scalar perturbations outside black holes. II. Schwarzschild geometry. Phys. Rev. D 59, 044016 (1999)

    Article  MathSciNet  Google Scholar 

  4. Barack L., Ori A.: Late-time decay of scalar perturbations outside rotating black holes. Phys. Rev. Lett. 82, 4388–4391 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Barausse E., Cardoso V., Khanna G.: Test bodies and naked singularities: is the self-force the cosmic censor? Phys. Rev. Lett. 105, 26 (2010)

    Article  Google Scholar 

  6. Burko L., Khanna G.: Universality of massive scalar field late-time tails in black-hole spacetimes. Phys. Rev. D 70, 044018 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  7. Burko L., Ori A.: Late-time evolution of non-linear gravitational collapse. Phys. Rev. D 56, 7828–7832 (1997)

    Article  ADS  Google Scholar 

  8. Chae D.: Global existence of solutions to the coupled Einstein and Maxwell-Higgs system in the spherical symmetry. Ann. Henri Poincaré 4(1), 35–62 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Challis J.: On the velocity of sound. Phil. Mag. 32(III), 494–499 (1848)

    Google Scholar 

  10. Chirco G., Liberati S., Sotiriou T.: Gedanken experiments on nearly extremal black holes and the third law. Phys. Rev. D 82, 104015 (2010)

    Article  ADS  Google Scholar 

  11. Choquet-Bruhat Y.: Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. Acta Math. 88, 141–225 (1952)

    Article  MathSciNet  Google Scholar 

  12. Choquet-Bruhat Y.: Problème de Cauchy pour le système intégro-différential d’Einstein-Liouville. Ann. Inst. Fourier 21, 181–201 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  13. Choquet-Bruhat Y., Geroch R.: Global aspects of the Cauchy problem in general relativity. Commun. Math. Phys. 14, 329–335 (1969)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Christodoulou D.: Violation of cosmic censorship in the gravitational collapse of a dust cloud. Commun. Math. Phys. 93, 171–195 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  15. Christodoulou D.: A mathematical theory of gravitational collapse. Commun. Math. Phys. 109, 613–647 (1987)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Christodoulou D.: The formation of black holes and singularities in spherically symmetric gravitational collapse. Commun. Pure Appl. Math. 44(3), 339–373 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Christodoulou D.: Bounded variation solutions of the spherically symmetric Einstein-scalar field equations. Commun. Pure Appl. Math. 46(8), 1093–1220 (1993)

    Article  MathSciNet  Google Scholar 

  18. Christodoulou D.: Examples of naked singularity formation in the gravitational collapse of a scalar field. Ann. Math. 140, 607–653 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Christodoulou D.: Self-gravitating relativistic fluids: a two-phase model. Arch. Rat. Mech. Anal. 130, 343–400 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Christodoulou D.: Self-gravitating relativistic fluids: the continuation and termination of a free phase boundary. Arch. Rat. Mech. Anal. 133, 333–398 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Christodoulou D.: Self-gravitating relativistic fluids: the formation of a free phase boundary in the phase transition from soft to hard. Arch. Ration. Mech. Anal. 134, 97–154 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Christodoulou D.: The instability of naked singularities in the gravitational collapse of a scalar field. Ann. Math. 149, 183–217 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Christodoulou D.: On the global initial value problem and the issue of singularities. Class. Quantum Grav. 16, A23–A35 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Christodoulou, D.: The formation of shocks in 3-dimensional fluids. Zürich: European Mathematical Society Publishing House, 2007

  25. Chruściel P., Cortier J.: Maximal analytic extensions of the Emparan-Reall black ring. J. Diff. Geom. 85, 425–459 (2010)

    MATH  Google Scholar 

  26. Dafermos M.: Stability and instability of the Cauchy horizon for the spherically symmetric Einstein-Maxwell-scalar field equations. Ann. Math. 158, 875–928 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Dafermos M.: The interior of charged black holes and the problem of uniqueness in general relativity. Commun. Pure Appl. Math. LVIII, 0445–0504 (2005)

    Article  MathSciNet  Google Scholar 

  28. Dafermos M.: On naked singularities and the collapse of self-gravitating Higgs fields. Adv. Theor. Math. Phys. 9(4), 575–591 (2005)

    MathSciNet  MATH  Google Scholar 

  29. Dafermos M.: Spherically symmetric spacetimes with a trapped surface. Class. Quantum Grav. 22, 2221–2232 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Dafermos, M.: Black holes without spacelike singularities. http://arxiv.org/abs/1201.1797v1 [gr-qc], 2012

  31. Dafermos M., Holzegel G.: On the nonlinear stability of higher-dimensional triaxial Bianchi IX black holes. Adv. Theor. Math. Phys. 10, 503–523 (2006)

    MathSciNet  MATH  Google Scholar 

  32. Dafermos M., Rendall A.: An extension principle for the Einstein-Vlasov system in spherical symmetry. Ann. Henri Poincaré 6, 1137–1155 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Dafermos M., Rendall A.: Inextendibility of expanding cosmological models with symmetry. Class. Quantum Grav. 22, L143–L147 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Dafermos, M., Rendall, A.: Strong cosmic censorship for surface-symmetric cosmological spacetimes with collisionless matter. http://arxiv.org/abs/gr-qc/0701034v1, 2007

  35. Dafermos M., Rodnianski I.: A proof of Price’s law for the collapse of a self-gravitating scalar field. Invent. Math. 162, 381–457 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. de Felice F., Yunqiang Y.: Turning a black hole into a naked singularity. Class. Quantum Grav. 18, 1235–1244 (2001)

    Article  ADS  MATH  Google Scholar 

  37. Frankel, T.:The Geometry of Physics: An Introduction. Cambridge: Cambridge University Press, 1997

  38. Gundlach C., Price R., Pullin J.: Late-time behavior of stellar collapse and explosions. I: Linearized perturbations. Phys. Rev. D 49, 883–889 (1994)

    Article  ADS  Google Scholar 

  39. Gundlach C., Price R., Pullin J.: Late-time behavior of stellar collapse and explosions. II: Nonlinear evolution. Phys. Rev. D 49, 890–899 (1994)

    Article  ADS  Google Scholar 

  40. Helfer A.: Null infinity does not carry massive fields. J. Math. Phys. 34(8), 3478–3480 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. Heusler, M.:Black hole uniqueness theorems. Cambridge: Cambridge University Press, 1996

  42. Hod S., Piran T.: Late-time evolution of charged gravitational collapse and decay of charged scalar hair. II. Phys. Rev. D 58, 024018 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  43. Hod S., Piran T.: Late-time evolution of charged gravitational collapse and decay of charged scalar hair. III. Nonlinear analysis. Phys. Rev. D 58, 024019 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  44. Hod S., Piran T.: Mass inflation in dynamic gravitational collapse of a charged scalar field. Phys. Rev. Lett. 81, 8 (1998)

    Google Scholar 

  45. Holzegel, G., Smulevici, J.: Self-gravitating Klein-Gordon fields in asymptotically Anti-de-Sitter spacetimes. http://arxiv.org/abs/1103.0712v1 [gr-qc], 2011

  46. Holzegel, G., Smulevici, J.:Stability of Schwarzschild-AdS for the spherically symmetric Einstein-Klein-Gordon system. http://arxiv.org/abs/1103.3672v1 [gr-qc], 2011

  47. Hubeny V.: Overcharging a black hole and cosmic censorship. Phys. Rev. D 59, 064013 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  48. Jacobson T., Sotiriou T.: Overspinning a black hole with a test body. Phys. Rev. Lett. 103, 14 (2009)

    MathSciNet  Google Scholar 

  49. Jetzer P., van der Bij J.: Charged boson stars. Phys. Lett. B 227, 341–346 (1989)

    Article  ADS  Google Scholar 

  50. Klainerman S.: The null condition and global existence to nonlinear wave equations. Lect. Appl. Math. 23, 293–326 (1986)

    MathSciNet  Google Scholar 

  51. Kommemi, J.: On Cauchy horizon stability for spherically symmetric Einstein-Maxwell-Klein-Gordon black holes. Preprint, 2013

  52. Kommemi, J.:Trapped surface formation in the collapse of spherically symmetric charged scalar fields. Preprint, 2013

  53. Langfelder P., Mann R.: A note on spherically symmetric naked singularities in general dimension. Class. Quantum Grav. 22, 1917–1932 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  54. Lemaître G.: L’universe en expansion. Ann. Soc. Scient. Bruxelles 53A, 51–85 (1933)

    ADS  Google Scholar 

  55. Matsas G., da Silva A.: Overspinning a nearly extreme charged black hole via a quantum tunneling process. Phys. Rev. Lett. 99, 181301 (2007)

    Article  ADS  Google Scholar 

  56. Mayo A., Bekenstein J.: No hair for spherical black holes: charged and nonminimally coupled scalar field with self-interaction. Phys. Rev. D 54, 5059–5069 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  57. Naber, G.:Topology, Geometry, and Gauge Fields. Interactions. Berlin-Heidelberg-New York: Springer-Verlag, 2000

  58. Narita, M.: On collapse of spherically symmetric wave maps coupled to gravity. In: Nonlinear phenomena with energy dissipation (2008), Vol. 29 of GAKUTO Internat. Ser. Math. Sci. Appl., Tokyo: Gakkotosha, 2008, pp. 313–327

  59. Narita, M.: On spherically symmetric gravitational collapse in the Einstein-Gauss-Bonnet theory. In: Physics and mathematics of gravitation: Proceedings of the Spanish relativity meeting 2008 (2009), Vol. 1122, AIP Conference Proceedings, Melville, VY: Amer. Inst. Phys., 2009, pp. 356–359

  60. Oppenheimer J., Snyder H.: On continued gravitational collapse. Phys. Rev. 56, 455–459 (1939)

    Article  ADS  MATH  Google Scholar 

  61. Papapetrou A., Hamoui A.: Surfaces coustiques dégénérés dans la solutions de Tolman. La singularité physique en relativité générale. Ann. Inst. Henri Poincaré 6, 343–364 (1967)

    Google Scholar 

  62. Poisson E., Israel W.: Internal structure of black holes. Phys. Rev. D 3(6), 1796–1809 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  63. Price R.: Nonspherical perturbations of relativistic gravitational collapse. I. scalar and gravitational perturbations. Phys. Rev. D 5, 2419–2438 (1972)

    Article  MathSciNet  ADS  Google Scholar 

  64. Rendall A., Ståhl F.: Shock waves in plane symmetric spacetimes. Commun. PDE 33, 2020–2039 (2008)

    Article  MATH  Google Scholar 

  65. Richartz M., Saa A.: Overspinning a nearly extreme black hole and the weak cosmic censorship conjecture. Phys. Rev. D 78, 081503 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  66. Saa A., Santarelli R.: Destroying a near-extremal Kerr-Newman black hole. Phys. Rev. D 84, 027501 (2011)

    Article  ADS  Google Scholar 

  67. Schunck F., Mielke E.: General relativistic boson stars. Class. Quantum Grav. 20, R301 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  68. Sideris T.: Formation of singularities in three-dimensional compressible fluids. Commun. Math. Phys. 101, 475–485 (1985)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  69. Tolman R.: Effect of inhomogeneity on cosmological models. Proc. Nat. Acad. Sci. U.S. 20, 169–176 (1934)

    Article  ADS  Google Scholar 

  70. Wald R.: Gedanken experiments to destroy a black hole. Ann. Phys. 83, 548–556 (1974)

    ADS  Google Scholar 

  71. Williams C.: Asymptotic behavior of spherically symmetric marginally trapped tubes. Ann. Henri Poincaré. 9, 1029–1067 (20088)

    Article  ADS  MATH  Google Scholar 

  72. Yodzis P., Seifert H.-J., Müllerzum Hagen H.: On the occurrence of naked singularities in general relativity. Commun. Math. Phys. 34, 135–148 (1973)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK

    Jonathan Kommemi

Authors
  1. Jonathan Kommemi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonathan Kommemi.

Additional information

Communicated by P. T. Chruściel

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kommemi, J. The Global Structure of Spherically Symmetric Charged Scalar Field Spacetimes. Commun. Math. Phys. 323, 35–106 (2013). https://doi.org/10.1007/s00220-013-1759-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-013-1759-1

Keywords

Search

Navigation