Electrostatic spherically symmetric configurations in gravitating nonlinear electrodynamics

J. Diaz-Alonso and D. Rubiera-Garcia

Phys. Rev. D 81, 064021 – Published 18 March 2010

Abstract

We perform a study of the gravitating electrostatic spherically symmetric (G-ESS) solutions of Einstein field equations minimally coupled to generalized nonlinear Abelian gauge models in three space dimensions. These models are defined by Lagrangian densities which are general functions of the gauge field invariants, restricted by some physical conditions of admissibility. They include the class of nonlinear electrodynamics supporting electrostatic spherically symmetric (ESS) nontopological soliton solutions in absence of gravity. We establish that the qualitative structure of the G-ESS solutions of admissible models is fully characterized by the asymptotic and central-field behaviors of their ESS solutions in flat space (or, equivalently, by the behavior of the Lagrangian densities in vacuum and on the point of the boundary of their domain of definition, where the second gauge invariant vanishes). The structure of these G-ESS configurations for admissible models supporting divergent-energy ESS solutions in flat space is qualitatively the same as in the Reissner-Nordström case. In contrast, the G-ESS configurations of the models supporting finite-energy ESS solutions in flat space exhibit new qualitative features, which are discussed in terms of the Arnowitt-Deser-Misner mass, the charge, and the soliton energy. Most of the results concerning well-known models, such as the electrodynamics of Maxwell, Born-Infeld, and the Euler-Heisenberg effective Lagrangian of QED, minimally coupled to gravitation, are shown to be corollaries of general statements of this analysis.

Received 21 September 2009

DOI:https://doi.org/10.1103/PhysRevD.81.064021

Authors & Affiliations

J. Diaz-Alonso^* and D. Rubiera-Garcia^†

LUTH, Observatoire de Paris, CNRS, , USAUniversité Paris Diderot, 5 Place Jules Janssen, 92190 Meudon, France and Departamento de Física, Universidad de Oviedo, Avda. Calvo Sotelo 18, E-33007 Oviedo, Asturias, Spain

^*joaquin.diaz@obspm.fr
^†rubieradiego@uniovi.es

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Vol. 81, Iss. 6 — 15 March 2010

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Figure 1
Behaviors of the Lagrangian densities of admissible NEDs for $Y = 0$ . Around the vacuum ( $X = 0$ ) they determine the asymptotic behaviors of the ESS solutions, necessary for the integral of energy to converge at infinity (cases B1, B2, and B3). The central behavior of the ESS solutions characterizes the structure of the corresponding G-ESS configurations. This behavior is governed by the form of the Lagrangian densities on the boundary of their domain of definition. The cases A1 and A2 correspond to ESS solutions for which the integral of energy converges at the center. In case A2 the boundary is reached at finite $X$ and the Lagrangian densities exhibit there a vertical asymptote (if $σ \leq 2$ ) or an absolute maximum with divergent slope (if $σ > 2$ ). In case A1 the boundary is at infinity and the Lagrangian densities diverge asymptotically as $φ (X \to \infty) \sim X^{γ}$ with $γ > 3 / 2$ , exhibiting a vertical parabolic branch there. Lagrangian densities with similar asymptotic form and $1 / 2 < γ \leq 3 / 2$ ( $p \leq - 1$ ), leading to ESS fields with a divergent integral of energy (at the center), have also been drawn.Reuse & Permissions
Figure 2
Exterior integral of energy $ε_{ex} (r)$ for NEDs with (flat-space) energy-divergent ESS solutions. The straight lines correspond to different values of the ADM mass $M$ . The cut points give the horizons of the black hole solutions: (1) tangency point corresponding to single-horizon extreme black holes ( $M = M_{extr} (q)$ ); (2) naked singularities ( $M < M_{extr} (q)$ ); and (3) black holes with two horizons ( $M > M_{extr} (q))$ .Reuse & Permissions
Figure 3
Exterior integral of energy $ε_{ex} (r, q)$ for the two different cases of NEDs with (flat-space) finite-energy ESS solutions. The bottom curve corresponds to models with ESS fields divergent (but integrable) at the center (case A1). In this case the slope of the curve at the origin diverges. The remaining curves are representative of ESS solutions of NEDs leading to a finite value of the fields at the center (A2 cases). They correspond to the slopes at the center associated to different values of the electric charge [ $ε_{ex}^{'} (r = 0) = - 8 π a q ⋛ 1 / 2$ , see Eq. (44)]. The cut points with the straight (dashed) lines $y = (M - r / 2)$ give the horizons of the different black hole solutions (see the discussion in the text). For the sake of simplicity curves corresponding to different values of the charge $q$ have been superimposed on the same figure in such a way that all of them begin at the same point $ε (q)$ . Consequently, each curve should be viewed as if it were represented in a different figure. Note that the slope of the beam of straight lines is not affected by this trick.Reuse & Permissions
Figure 4
Qualitative behavior of the metric function $λ (r)$ for the critical configurations ( $M = ε (q)$ ) in the cases A1 ( $λ (0) \to - \infty$ , case A1-4 in the text), A2a ( $λ (0) = 1 - 16 π a q > 0$ , case A2a-2 in the text), A2b ( $λ (0) = 1 - 16 π a q < 0$ , case A2b-4 in the text), and A2c ( $λ (0) = 1 - 16 π a q = 0$ , case A2c-1 in the text). The three curves associated to each (A2) case correspond to the three ranges of the parameter $σ ⪋ 1$ , which determine their slopes at $r = 0$ [see Eq. (47)]. The small frame displays the metric function for different values of the ADM mass in the case A2b with $σ > 1$ . The dashed line in this frame corresponds to the critical configuration ( $M = ε (q)$ , case A2b-4 in the text). The upper curves ( $M < ε (q)$ ) correspond to naked singularities (A), extreme black holes (B), and two-horizon black holes (C and D). The lower curves (E and F, with $M > ε (q)$ ) correspond to the single-horizon black holes. Curves D and E show the approximation to the critical configuration (for $r > 0$ ) in the limits $M - ε (q) \to 0^{\mp}$ , respectively, illustrating the formation of the isolated discontinuities of the limit configurations at $r = 0$ . The horizon radii for the extreme ( $r_{hextr}$ ) and critical ( $r_{hcrit}$ ) black holes are indicated. Similar figures to the one of the small frame can be straightforwardly displayed for the other values of $σ$ and for the cases A1, A2a, and A2c, illustrating the discussion of Sec. 4.Reuse & Permissions

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