Asymptotically de Sitter and anti-de Sitter black holes with confining electric potential

doi:10.1016/j.physletb.2011.09.003

Physics Letters B

Volume 704, Issue 3, 13 October 2011, Pages 230-233

https://doi.org/10.1016/j.physletb.2011.09.003 Get rights and content

Abstract

We study gravity interacting with a special kind of QCD-inspired nonlinear gauge field system which earlier was shown to yield confinement-type effective potential (the “Cornell potential”) between charged fermions (“quarks”) in flat space–time. We find new static spherically symmetric solutions generalizing the usual Reissner–Nordström–de Sitter and Reissner–Nordström–anti-de Sitter black holes with the following additional properties: (i) appearance of a constant radial electric field (in addition to the Coulomb one); (ii) novel mechanism of dynamical generation of cosmological constant through the non-Maxwell gauge field dynamics; (iii) appearance of confining-type effective potential in charged test particle dynamics in the above black hole backgrounds.

Introduction

It has been shown by ‘t Hooft [1] that in any effective quantum theory, which is able to describe linear confinement phenomena, the energy density of electrostatic field configurations should be a linear function of the electric displacement field in the infrared region due to appropriate infrared counterterms. The simplest way to achieve this in Minkowski space–time is by considering a square root of the field strength squared, in addition to the standard Maxwell term, leading to a very peculiar non-Maxwell nonlinear effective gauge field model [2]: $S = \int d^{4} x L (F^{2}), L (F^{2}) = - \frac{1}{4} F^{2} - \frac{f}{2} \sqrt{- F^{2}},$ $F^{2} \equiv F_{μ ν} F^{μ ν}, F_{μ ν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ},$ with f being a positive coupling constant. It has been shown in first three references in [2] that the square root of the Maxwell term naturally arises as a result of spontaneous breakdown of scale symmetry of the original scale-invariant Maxwell theory with f appearing as an integration constant responsible for the latter spontaneous breakdown. The model (1) produces a confining effective potential $V (r) = - \frac{α}{r} + β r$ (Coulomb plus linear one) which is of the form of the well-known “Cornell” potential [3] in quantum chromodynamics (QCD). For static field configurations the model (1) yields the following electric displacement field $\vec{D} = \vec{E} - \frac{f}{\sqrt{2}} \frac{\vec{E}}{| \vec{E} |}$ . The pertinent energy density turns out to be (there is no contribution from the square-root term in (1)) $\frac{1}{2} {\vec{E}}^{2} = \frac{1}{2} {| \vec{D} |}^{2} + \frac{f}{\sqrt{2}} | \vec{D} | + \frac{1}{4} f^{2}$ , so that it indeed contains a term linear w.r.t. $| \vec{D} |$ .

It is crucial to stress that the Lagrangian $L (F^{2})$ (1) contains both the usual Maxwell term as well as a non-analytic function of $F^{2}$ and thus it is a non-standard form of nonlinear electrodynamics. In this way it is significantly different from the original “square root” Lagrangian $- \frac{f}{2} \sqrt{F^{2}}$ first proposed by Nielsen and Olesen [4] to describe string dynamics. Moreover, it is important that the square-root term in (1) is in the “electrically” dominated form ( $\sqrt{- F^{2}}$ ) as opposed to the “magnetically” dominated Nielsen–Olesen form ( $\sqrt{F^{2}}$ ).

Let us remark that one could start with the non-Abelian version of the action (1). Since we will be interested in static spherically symmetric solutions, the non-Abelian theory effectively reduces to an Abelian one as pointed out in the first reference in [2].

Our main goal in the present Letter is to study possible new effects by coupling the confining potential generating nonlinear gauge field system (1) to gravity. We find:

(i)
appearance of a constant radial electric field (in addition to the Coulomb one) in charged black holes within Reissner–Nordström–de Sitter and/or Reissner–Nordström–anti-de Sitter space–times as well as in electrically neutral black holes with Schwarzschild–de Sitter and/or Schwarzschild–anti-de Sitter geometry;
(ii)
novel mechanism of dynamical generation of cosmological constant through the non-Maxwell gauge field dynamics of the nonlinear action $L (F^{2})$ (1);
(iii)
appearance of confining-type effective potential in charged test particle dynamics in the above black hole backgrounds.

Section snippets

Lagrangian formulation. Spherically symmetric solutions

We will consider the simplest coupling of the nonlinear gauge field system (1) to gravity described by the action (we use units with Newton constant $G_{N} = 1$ ): $S = \int d^{4} x \sqrt{- g} [\frac{R (g)}{16 π} - \frac{1}{4} F^{2} - \frac{f}{2} \sqrt{- F^{2}}], F^{2} \equiv F_{κ λ} F_{μ ν} g^{κ μ} g^{λ ν},$ where $R (g)$ is the scalar curvature of the space–time metric $g_{μ ν}$ and $g \equiv \det ‖ g_{μ ν} ‖$ . It is important to stress that for the time being we will not introduce any bare cosmological constant term.

The energy–momentum tensor $T_{μ ν}^{(F)}$ of the nonlinear gauge field, which appears in the pertinent equations of

Bare negative cosmological constant versus induced cosmological constant

Let us now introduce in (2) from the very beginning a negative bare cosmological constant $Λ = - | Λ |$ : $S = \int d^{4} x \sqrt{- g} [\frac{1}{16 π} (R (g) - 2 Λ) - \frac{1}{4} F^{2} - \frac{f}{2} \sqrt{- F^{2}}] .$ Then the corresponding static spherically symmetric solution is given by (9), (6) with: $A (r) = 1 - \frac{2 m}{r} + \frac{Q^{2}}{r^{2}} + \frac{1}{3} (| Λ | - 2 π f^{2}) r^{2} .$ Thus, we find also black hole solution with Reissner–Nordström–anti-de Sitter geometry (17) and with additional global constant electric field (9) provided the full effective cosmological constant (bare one plus dynamically induced one) satisfies: $Λ$

Charged test particle dynamics

Let us now briefly discuss the dynamics of a test particle with mass $m_{0}$ and electric charge $q_{0}$ in the above black hole backgrounds – (6) with (12), (9) or (6) with (17), (9). It is given by the standard reparametrization invariant point–particle action: $S_{particle} = \int d λ [\frac{1}{2 e} g_{μ ν} (x) {\dot{x}}^{μ} {\dot{x}}^{ν} - \frac{1}{2} e m_{0}^{2} - q_{0} {\dot{x}}^{μ} A_{μ} (x)],$ where e denotes the world-line “einbein”. The standard treatment, using energy $E$ and angular momentum $J$ conservation in the static spherically symmetric background under consideration and

Discussion

It is possible to rewrite the action (2) in an explicitly Weyl-conformally invariant form using the method of two volume forms (two integration measures) [6] introduced earlier in the context of gravity–matter models with primary applications in cosmology. Namely, apart from the standard reparametrization covariant integration density $\sqrt{- g}$ in terms of the intrinsic Riemannian metric $g_{μ ν}$ as in (2), one introduces an alternative reparametrization covariant integration density $Φ (φ)$ in terms of

Acknowledgements

E.N. and S.P. are supported by Bulgarian NSF grant DO 02-257. Also, all of us acknowledge support of our collaboration through the exchange agreement between the Ben-Gurion University of the Negev (Beer-Sheva, Israel) and the Bulgarian Academy of Sciences.

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Asymptotically de Sitter and anti-de Sitter black holes with confining electric potential

Abstract

Introduction

Section snippets

Lagrangian formulation. Spherically symmetric solutions

Bare negative cosmological constant versus induced cosmological constant

Charged test particle dynamics

Discussion

Acknowledgements

Nucl. Phys. B (Proc. Suppl.)

Phys. Lett. B

Phys. Lett. B

Int. J. Mod. Phys. A

Mod. Phys. Lett. A

Int. J. Mod. Phys. A

Int. J. Mod. Phys. A

Rend. R. Acad. Naz. Lincei

Phys. Rev. D

Bull. Akad. Pol.

Phys. Rev. Lett.

Ann. of Phys.