General Relativity and Gravitation

, Volume 45, Issue 12, pp 2483–2492 | Cite as

Stationary scalar configurations around extremal charged black holes

  • Juan Carlos Degollado
  • Carlos A. R. Herdeiro
Research Article


We consider the minimally coupled Klein-Gordon equation for a charged, massive scalar field in the non-extremal Reissner-Nordström background. Performing a frequency domain analysis, using a continued fraction method, we compute the frequencies \(\omega \) for quasi-bound states. We observe that, as the extremal limit for both the background and the field is approached, the real part of the quasi-bound states frequencies \(\mathcal{R }(\omega )\) tends to the mass of the field and the imaginary part \(\mathcal{I }(\omega )\) tends to zero, for any angular momentum quantum number \(\ell \). The limiting frequencies in this double extremal limit are shown to correspond to a distribution of extremal scalar particles, at stationary positions, in no-force equilibrium configurations with the background. Thus, generically, these stationary scalar configurations are regular at the event horizon. If, on the other hand, the distribution contains scalar particles at the horizon, the configuration becomes irregular therein, in agreement with no hair theorems for the corresponding Einstein-Maxwell-scalar field system.


Charged black holes Scalar fields Accretion 



We would like to thank João Rosa for discussions and comments on this draft. JCD Acknowledges CONACyT-México support. This work was also supported by the NRHEP–295189 FP7-PEOPLE-2011-IRSES Grant, and by FCT – Portugal through the project PTDC/FIS/116625/2010.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Juan Carlos Degollado
    • 1
  • Carlos A. R. Herdeiro
    • 1
  1. 1.Departamento de Física daUniversidade de Aveiro and I3N AveiroPortugal

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