Stationary scalar configurations around extremal charged black holes
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Stationary scalar configurations around extremal charged black holes Juan Carlos Degollado · Carlos A. R. Herdeiro
Received: 18 July 2013 / Accepted: 2 September 2013 / Published online: 17 September 2013 © Springer Science+Business Media New York 2013
Abstract 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 ω 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 R(ω) tends to the mass of the field and the imaginary part I(ω) tends to zero, for any angular momentum quantum number . 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. Keywords
Charged black holes · Scalar fields · Accretion
1 Introduction Scalar test fields in black hole (BH) geometries do not admit, generically, stationary configurations with an asymptotic decay and with real frequencies, i.e bound states. This follows from the physical requirement that only ingoing waves can exist at the
J. C. Degollado (B) · C. A. R. Herdeiro Departamento de Física da, Universidade de Aveiro and I3N, Campus de Santiago, 3810-183 Aveiro, Portugal e-mail: [email protected] C. A. R. Herdeiro e-mail: [email protected]
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horizon, therefore preventing a real equilibrium configuration between the field and the BH. Consequently, the configurations allowed in BH backgrounds are quasi-bound states [1–3], for which the frequencies are complex, with the imaginary part revealing a time dependence for the states, signaling either their absorption or, in the case of superradiant instabilities, their amplification by the BH [4]. The profile and some of the physical properties of quasi-bound states diverge at the horizon. This is intimately related to the inability that BH backgrounds have to accommodate, in a regular fashion, the scalar field, as an exact stationary solution, a property established by no-hair theorems [5,6]. But even with this caveat such quasibound states are informative. For instance in [7], performing numerical simulations and starting with regular initial data for a scalar field around a Schwarzschild BH, there were found damped oscillating solutions with frequency and decay rate described by the real and imaginary parts of quasi-bound state frequencies. These decay rates can be very small [8] and thus long lived scalar field configurations could
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