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Title: Charged massive particle at rest in the field of a Reissner-Nordstroem black hole

Abstract

The interaction of a Reissner-Nordstroem black hole and a charged massive particle is studied in the framework of perturbation theory. The particle backreaction is taken into account, studying the effect of general static perturbations of the hole following the approach of Zerilli. The solutions of the combined Einstein-Maxwell equations for both perturbed gravitational and electromagnetic fields to first order of the perturbation are exactly reconstructed by summing all multipoles, and are given explicit closed form expressions. The existence of a singularity-free solution of the Einstein-Maxwell system requires that the charge-to-mass ratios of the black hole and of the particle satisfy an equilibrium condition which is in general dependent on the separation between the two bodies. If the black hole is undercritically charged (i.e. its charge-to-mass ratio is less than one), the particle must be overcritically charged, in the sense that the particle must have a charge-to-mass ratio greater than one. If the charge-to-mass ratios of the black hole and of the particle are both equal to one (so that they are both critically charged, or 'extreme'), the equilibrium can exist for any separation distance, and the solution we find coincides with the linearization in the present context of the well-knownmore » Majumdar-Papapetrou solution for two extreme Reissner-Nordstroem black holes. In addition to these singularity-free solutions, we also analyze the corresponding solution for the problem of a massive particle at rest near a Schwarzschild black hole, exhibiting a strut singularity on the axis between the two bodies. The relations between our perturbative solutions and the corresponding exact two-body solutions belonging to the Weyl class are also discussed.« less

Authors:
; ;  [1];  [2];  [3]
  1. Istituto per le Applicazioni del Calcolo 'M. Picone', CNR I-00161 Rome (Italy) and ICRA, University of Rome 'La Sapienza', I-00185 Rome (Italy)
  2. (Italy)
  3. (Italy) and ICRANet, I-65100 Pescara (Italy)
Publication Date:
OSTI Identifier:
21011077
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 4; Other Information: DOI: 10.1103/PhysRevD.75.044012; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; BLACK HOLES; COSMOLOGY; DISTURBANCES; EINSTEIN-MAXWELL EQUATIONS; ELECTROMAGNETIC FIELDS; MATHEMATICAL SOLUTIONS; MULTIPOLES; PARTICLE INTERACTIONS; PERTURBATION THEORY; QUANTUM FIELD THEORY; SCHWARZSCHILD METRIC; SINGULARITY; TWO-BODY PROBLEM

Citation Formats

Bini, D., Geralico, A., Ruffini, R., Physics Department and ICRA, University of Rome 'La Sapienza', I-00185 Rome, and Physics Department and ICRA, University of Rome 'La Sapienza', I-00185 Rome. Charged massive particle at rest in the field of a Reissner-Nordstroem black hole. United States: N. p., 2007. Web. doi:10.1103/PHYSREVD.75.044012.
Bini, D., Geralico, A., Ruffini, R., Physics Department and ICRA, University of Rome 'La Sapienza', I-00185 Rome, & Physics Department and ICRA, University of Rome 'La Sapienza', I-00185 Rome. Charged massive particle at rest in the field of a Reissner-Nordstroem black hole. United States. doi:10.1103/PHYSREVD.75.044012.
Bini, D., Geralico, A., Ruffini, R., Physics Department and ICRA, University of Rome 'La Sapienza', I-00185 Rome, and Physics Department and ICRA, University of Rome 'La Sapienza', I-00185 Rome. Thu . "Charged massive particle at rest in the field of a Reissner-Nordstroem black hole". United States. doi:10.1103/PHYSREVD.75.044012.
@article{osti_21011077,
title = {Charged massive particle at rest in the field of a Reissner-Nordstroem black hole},
author = {Bini, D. and Geralico, A. and Ruffini, R. and Physics Department and ICRA, University of Rome 'La Sapienza', I-00185 Rome and Physics Department and ICRA, University of Rome 'La Sapienza', I-00185 Rome},
abstractNote = {The interaction of a Reissner-Nordstroem black hole and a charged massive particle is studied in the framework of perturbation theory. The particle backreaction is taken into account, studying the effect of general static perturbations of the hole following the approach of Zerilli. The solutions of the combined Einstein-Maxwell equations for both perturbed gravitational and electromagnetic fields to first order of the perturbation are exactly reconstructed by summing all multipoles, and are given explicit closed form expressions. The existence of a singularity-free solution of the Einstein-Maxwell system requires that the charge-to-mass ratios of the black hole and of the particle satisfy an equilibrium condition which is in general dependent on the separation between the two bodies. If the black hole is undercritically charged (i.e. its charge-to-mass ratio is less than one), the particle must be overcritically charged, in the sense that the particle must have a charge-to-mass ratio greater than one. If the charge-to-mass ratios of the black hole and of the particle are both equal to one (so that they are both critically charged, or 'extreme'), the equilibrium can exist for any separation distance, and the solution we find coincides with the linearization in the present context of the well-known Majumdar-Papapetrou solution for two extreme Reissner-Nordstroem black holes. In addition to these singularity-free solutions, we also analyze the corresponding solution for the problem of a massive particle at rest near a Schwarzschild black hole, exhibiting a strut singularity on the axis between the two bodies. The relations between our perturbative solutions and the corresponding exact two-body solutions belonging to the Weyl class are also discussed.},
doi = {10.1103/PHYSREVD.75.044012},
journal = {Physical Review. D, Particles Fields},
number = 4,
volume = 75,
place = {United States},
year = {Thu Feb 15 00:00:00 EST 2007},
month = {Thu Feb 15 00:00:00 EST 2007}
}
  • The properties of the electric field of a two-body system consisting of a Reissner-Nordstroem black hole and a charged massive particle at rest have recently been analyzed in the framework of first order perturbation theory following the standard approach of Regge, Wheeler, and Zerilli. In the present paper we complete this analysis by numerically constructing and discussing the lines of force of the ''effective'' electric field of the sole particle with the subtraction of the dominant contribution of the black hole. We also give the total field due to the black hole and the particle. As the black hole becomesmore » extreme an effect analogous to the Meissner effect arises for the electric field, with the ''effective field'' lines of the point charge being expelled by the outer horizon of the black hole. This effect existing at the level of test field approximation, i.e. by neglecting the backreaction on the background metric and electromagnetic field due to the particle's mass and charge, is here found also at the complete perturbative level. We point out analogies with similar considerations for magnetic fields by Bicak and Dvorak. We also explicitly show that the linearization of the recently obtained Belinski-Alekseev exact solution coincides with our solution in the Regge-Wheeler gauge. Our solution thus represents a bridge between the test field solution, which neglects all the feedback terms, and the exact two-body solution, which takes into account all the nonlinearity of the interaction.« less
  • We investigate quasinormal modes of a massless charged scalar field on a small Reissner-Nordstroem-anti-de Sitter (RN-AdS) black hole both with analytical and numerical approaches. In the analytical approach, by using the small black hole approximation (r{sub +}<<L), we obtain the quasinormal mode frequencies in the limit of r{sub +}/L{yields}0, where r{sub +} and L stand for the black hole event horizon radius and the AdS scale, respectively. We then show that the small RN-AdS black hole is unstable if its quasinormal modes satisfy the superradiance condition and that the instability condition of the RN-AdS black hole in the limit ofmore » r{sub +}/L{yields}0 is given by Q>(3/eL)Q{sub c}, where Q, Q{sub c}, and e are the charge of the black hole, the critical (maximum) charge of the black hole, and the charge of the scalar field, respectively. In the numerical approach, we calculate the quasinormal modes for the small RN-AdS black holes with r{sub +}<<L and confirm that the RN-AdS black hole is unstable if its quasinormal modes satisfy the superradiance condition. Our numerical results show that the RN-AdS black holes with r{sub +}=0.2L, 0.1L, and 0.01L become unstable against scalar perturbations with eL=4 when the charge of the black hole satisfies Q > or approx. 0.8Q{sub c}, 0.78Q{sub c}, and 0.76Q{sub c}, respectively.« less
  • We study the dynamical evolution of a scalar field coupling to Einstein's tensor in the background of a Reissner-Nordstroem black hole. Our results show that the coupling constant {eta} imprints in the wave dynamics of a scalar perturbation. In the weak coupling, we find that with the increase of the coupling constant {eta} the real parts of the fundamental quasinormal frequencies decrease and the absolute values of imaginary parts increase for fixed charge q and multipole number l. In the strong coupling, we find that for l{ne}0 the instability occurs when {eta} is larger than a certain threshold value {eta}{submore » c} which deceases with the multipole number l and charge q. However, for the lowest l=0, we find that there does not exist such a threshold value and the scalar field always decays for arbitrary coupling constant.« less
  • Complex frequencies of the normal modes of the Reissner-Nordstroem black hole are computed by two independent methods. The first is a high-order WKB approach devised by Schutz and Will and extended by Iyer and Will for the Schwarzschild case. The second is direct numerical integration using a method developed by Chandrasekhar and Detweiler, thereby extending earlier results of Gunter. The WKB results agree with the numerical ones with an error less than 1% for the lowest-order modes. For somewhat higher orders, the numerical techniques fail but the WKB method continues to give eigenfrequencies that should be reasonably accurate.
  • The stationary multipole solutions of the Einstein-Maxwell equations representing coupled electromagnetic and gravitational perturbations of the extreme Reissner-Nordstroem black hole are simple rational functions of the radial coordinate. These solutions are used to construct the axially symmetric field of a current loop in the equatorial plane and of a small current loop on the polar axis. The magnetic lines of force are calculated numerically and exhibited graphically. The field representing a general Reissner-Nordstroem black hole in an asymptotically uniform magnetic field is also given, and its relation to the exact solutions of Ernst describing Kerr-Newman black holes in Melvin's magneticmore » universe is discussed.« less