Gravity, Nonlinear Gauge Fields and Charge Confinement/Deconfinement
Eduardo Guendelman, Alexander Kaganovich, Emil Nissimov, Svetlana Pacheva
aa r X i v : . [ h e p - t h ] N ov Gravity, Nonlinear Gauge Fields and ChargeConfinement/Deconfinement ∗ Eduardo Guendelman and Alexander Kaganovich † Physics Department, Ben Gurion University of the NegevBeer Sheva, Israel
Emil Nissimov and Svetlana Pacheva ‡ Institute for Nuclear Research and Nuclear EnergyBulgarian Academy of Sciences, Sofia, Bulgaria
Abstract
We discuss in some detail the properties of gravity (including f ( R )-gravity)coupled to non-standard nonlinear gauge field system containing a square root ofthe usual Maxwell Lagrangian − f √− F . The latter is known to produce in flatspacetime a QCD-like confinement. Inclusion of gravity triggers various physi-cally interesting effects: new mechanism for dynamical generation of cosmologicalconstant; non-standard black hole solutions with constant vacuum electric fieldand with “hedge-hog”-type spacetime asymptotics, which are shown to obey thefirst law of black hole thermodynamics; new “tubelike” solutions of Levi-Civita-Bertotti-Robinson type; charge-”hiding” and charge-confining “thin-shell” worm-hole solutions; dynamical effective gauge couplings and confinement-deconfinementtransition effect when coupled to quadratic R -gravity.
1. Introduction
We consider gravity, including f ( R )-gravity [1], coupled to non-standard nonlinear gauge field system containing a square root of the ordinary MaxwellLagrangian − f √− F . In flat spacetime the latter model has been shown[2] to produce a QCD-like confinement .We exhibit several interesting features of the above system (see alsoRefs.[3, 4]) : • New mechanism for dynamical generation of cosmological constantdue to nonlinear gauge field dynamics: Λ eff = Λ + 2 πf (Λ – barecosmological constant, may be absent at all). ∗ Work supported in part by Bulgarian National Science Foundation grant DO 02-257 † e-mail address: [email protected], [email protected] ‡ e-mail address: [email protected], [email protected] E. Guendelman, A. Kaganovich, E. Nissimov, S. Pacheva • Non-standard black hole solutions of Reissner-Nordstr¨om-(anti-)de-Sitter type containing a constant radial vacuum electric field (in addi-tion to the Coulomb one), in particular, in electrically neutral blackholes of Schwarzschild-(anti-)de-Sitter type. It is shown that thesenon-standard black holes obey the first law of black hole thermody-namics. • In case of vanishing effective cosmological constant Λ eff ( i.e. , Λ < , | Λ | = 2 πf ) the resulting Reissner-Nordstr¨om-type black hole,apart from carrying an additional constant vacuum electric field, turnsout to be non-asymptotically flat – a feature resembling the gravita-tional effect of a hedgehog [6]. • Appearance of confining-type effective potential in charged test particledynamics in the above black hole backgrounds. • New “tubelike” solutions of Levi-Civita-Bertotti-Robinson [7] type, i.e. , with spacetime geometry of the form M × S , where M is atwo-dimensional anti-de Sitter, Rindler or de Sitter space dependingon the relative strength of the electric field w.r.t. the coupling f ofthe square-root gauge field term.When in addition one or more lightlike branes are self-consistently cou-pled to the above gravity/nonlinear-gauge-field system (as matter and chargesources) they produce (“thin-shell”) wormhole solutions displaying twonovel physically interesting effects [4]: • “Charge-hiding” effect - a genuinely charged matter source of grav-ity and electromagnetism may appear electrically neutral to an exter-nal observer – a phenomenon opposite to the famous Misner-Wheeler“charge without charge” effect [5]; • Charge-confining “tubelike” wormhole with two “throats” occupied bytwo oppositely charged lightlike branes – the whole electric flux isconfined within the finite-extent “middle universe” of generalized Levi-Civita-Bertotti-Robinson type – no flux is escaping into the outer non-compact “universes”.Additional interesting features appear when we couple the “square-root” confining nonlinear gauge field system to f ( R )-gravity with f ( R ) = R + αR and a dilaton. Reformulating the model in the physical “Einstein”frame we find (cf. second Ref.[3]): • “Confinement-deconfinement” transition due to appearance of “flat”region in the effective dilaton potential; • The effective gauge couplings as well as the induced cosmological con-stant become dynamical depending on the dilaton v.e.v. In particular,a conventional Maxwell kinetic term for the gauge field is dynamicallygenerated even if absent in the original theory; • Regular black hole solution ( no singularity at r = 0) with confiningvacuum electric field : the bulk spacetime consist of two regions –an interior de Sitter and an exterior Reissner-Nordstr¨om-type (with“hedgehog asymptotics”) glued together along their common horizon ravity, Nonlinear Gauge Fields and Charge Confinement/Deconfinement 3 occupied by a charged lightlike brane. The latter also dynamically de-termines the non-zero cosmological constant in the interior de-Sitterregion. This result is analogous to the regular black hole solution inthe case of ordinary Einstein gravity presented in Ref.[8] and will bediscussed in more detail in a subsequent paper.Concluding the introductory remarks, let us briefly mention the princi-pal motivation for studying non-standard gauge field models with √− F .G. ‘t Hooft has shown [9] that in any effective quantum gauge theory,which is able to describe linear confinement phenomena, the energy den-sity of electrostatic field configurations should be a linear function of theelectric displacement field in the infrared region (the latter appearing as an“infrared counterterm”).The simplest way to realize these ideas in flat spacetime was proposedin Refs.[2]: S = Z d xL ( F ) , L ( F ) = − F − f p − F , (1) F ≡ F µν F µν , F µν = ∂ µ A ν − ∂ ν A µ , The square root of the Maxwell term naturally arises as a result of sponta-neous breakdown of scale symmetry of the original scale-invariant Maxwellaction with f appearing as an integration constant responsible for thelatter spontaneous breakdown. For static field configurations the model(1) yields an electric displacement field ~D = ~E − f √ ~E | ~E | and the corre-sponding energy density turns out to be ~E = | ~D | + f √ | ~D | + f ,so that it indeed contains a term linear w.r.t. | ~D | . The model (1) pro-duces, when coupled to quantized fermions, a confining effective potential V ( r ) = − βr + γr (Coulomb plus linear one with γ ∼ f ) which is of the formof the well-known “Cornell” potential in the phenomenological descriptionof quarkonium systems in QCD [10].
2. Einstein Gravity Coupled to Confining Nonlinear GaugeField
The pertinent action is given by ( R -scalar curvature; Λ - bare cosmologicalconstant, might be absent): S = Z d x √− G h R − π + L ( F ) i , L ( F ) = − F − f p − F , (2) F ≡ F κλ F µν G κµ G λν , F µν = ∂ µ A ν − ∂ ν A µ . Remark.
One could start with the non-Abelian version of the gauge fieldaction in (2). Since we will be interested in static spherically symmetricsolutions, the non-Abelian gauge theory effectively reduces to an Abelianone.
E. Guendelman, A. Kaganovich, E. Nissimov, S. Pacheva
The corresponding equations of motion read accordingly – Einsteinequations: R µν − G µν R + Λ G µν = 8 πT ( F ) µν , (3) T ( F ) µν = (cid:16) − f √− F (cid:17) F µκ F νλ G κλ − (cid:16) F + 2 f p − F (cid:17) G µν , (4)and nonlinear gauge field equations: ∂ ν (cid:18) √− G (cid:16) − f √− F (cid:17) F κλ G µκ G νλ (cid:19) = 0 . (5) Important remark . Note the non-zero value of the trace of energy-momentum tensor unlike ordinary Maxwell theory: T ( F ) ≡ T ( F ) µν G µν = − f p − F . Solving Eqs.(3)–(5) we find new non-standard
Reissner-Nordstr¨om-(anti-)de-Sitter-type black holes depending on the sign of a dynamically gener-ated cosmological constant Λ eff : ds = − A ( r ) dt + dr A ( r ) + r ( dθ + sin θdϕ ) , (6) A ( r ) = 1 − √ π | Q | f − mr + Q r − Λ eff r , Λ eff = 2 πf + Λ , (7)with static spherically symmetric electric field containing apart from theCoulomb term an additional constant “vacuum” piece: F r = ε F f √ Q √ π r , ε F ≡ sign( F r ) = sign( Q ) . (8)The latter corresponds to a confining “Cornell”-type [10] potential A = − ε F f √ r + Q √ π r . When Λ eff = 0, A ( r ) → − √ π | Q | f for r → ∞ , i.e. , theblack hole exhibits “hedgehog” [6] non-flat-spacetime asymptotics.Furthermore, we find three distinct types of static solutions of “tube-like” Levi-Civita-Bertotti-Robinson [7] type with spacetime geometry of theform M × S , where M is some 2-dimensional manifold ((anti-)de Sitter( A ) dS , Rindler Rind ): ds = − A ( η ) dt + dη A ( η ) + r ( dθ + sin θdϕ ) , −∞ < η < ∞ , (9) F η = c F = const , r = 4 πc F + Λ (= const) . (10) ravity, Nonlinear Gauge Fields and Charge Confinement/Deconfinement 5 (i) AdS × S with constant vacuum electric field | F η | ≡ | ~E | = | c F | : A ( η ) = 4 π (cid:20) c F − √ f | c F | − Λ π (cid:21) η ( η − Poincare patch coordinate) , (11)provided either | c F | > f √ (cid:16) q Λ πf (cid:17) for Λ ≥ − πf or | c F | > q π | Λ | for Λ < , | Λ | > πf .(ii) Rind × S with constant vacuum electric field | F η | = | c F | , where Rind is the flat 2-dimensional Rindler spacetime with: A ( η ) = η for 0 < η < ∞ or A ( η ) = − η for − ∞ < η < | c F | = f √ (cid:16) q Λ πf (cid:17) for Λ > − πf .(iii) dS × S with weak const vacuum electric field | F η | = | c F | ,where dS is the 2-dimensional de Sitter space with: A ( η ) = 1 − π (cid:20) √ f | c F | − c F + Λ π (cid:21) ) η , (13)when | c F | < f √ (cid:16) q Λ πf (cid:17) for Λ > − πf . Note that dS has twohorizons at η = ± η ≡ ± h π (cid:0) √ f | c F | − c F (cid:1) + Λ i − .
3. Bulk Gravity/Nonlinear Gauge Field Coupled to Light-like Brane Sources
In the following two Sections we will consider bulk Einstein/non-lineargauge field system (2) self-consistently coupled to N ≥ lightlike p -brane ( LL-brane ) sources (here p = 2).World-volume LL-brane actions in a reparametrization-invariant Nambu-Goto-type or in an equivalent Polyakov-type formulation were proposed inRefs.[11]: S LL [ q ] = − Z d p +1 σ T b p − √− γ h γ ab ¯ g ab − b ( p − i , (14)¯ g ab ≡ ∂ a X µ G µν ∂ b X ν − T ( ∂ a u + q A a )( ∂ b u + q A b ) , A a ≡ ∂ a X µ A µ . (15)Here and below the following notations are used: • γ ab is the intrinsic world-volume Riemannian metric; g ab = ∂ a X µ G µν ( X ) ∂ b X ν is the induced metric on the world-volume,which becomes singular on-shell (manifestation of the lightlike nature); b is world-volume “cosmological constant”. E. Guendelman, A. Kaganovich, E. Nissimov, S. Pacheva • X µ ( σ ) are the p -brane embedding coordinates in the bulk D -dimensionalspacetime with Riemannian metric G µν ( x ) ( µ, ν = 0 , , . . . , D − σ ) ≡ (cid:0) σ ≡ τ, σ i (cid:1) with i = 1 , . . . , p ; ∂ a ≡ ∂∂σ a . • u is auxiliary world-volume scalar field defining the lightlike directionof the induced metric; • T is dynamical (variable) brane tension; • q – the coupling to bulk spacetime gauge field A µ is LL-brane surfacecharge density.The on-shell singularity of the induced metric g ab , i.e. , the lightlikeproperty, directly follows from the LL-brane equations of motion: g ab (cid:16) ¯ g bc ( ∂ c u + q A c ) (cid:17) = 0 . (16)Now, let us consider the full action of self-consistently coupled bulkEinstein/non-linear gauge field/ LL-brane system ( L ( F ) = − F − f √− F ): S = Z d x √− G h R ( G ) − π + L ( F ) i + N X k =1 S LL [ q ( k ) ] , (17)where the superscript ( k ) indicates the k -th LL-brane .The corresponding equations of motion are as follows: R µν − G µν R + Λ G µν = 8 π h T ( F ) µν + N X k =1 T ( k ) µν i , (18) ∂ ν h √− G (cid:16) − f √− F (cid:17) F κλ G µκ G νλ i + N X k =1 j µ ( k ) = 0 . (19)The energy-momentum tensor and the charge current density of k -th LL-brane are straightforwardly derived from the pertinent
LL-brane world-volume action (14): T µν ( k ) = − Z d σ δ (4) (cid:16) x − X ( k ) ( σ ) (cid:17) √− G T ( k ) q | ¯ g ( k ) | ¯ g ab ( k ) ∂ a X µ ( k ) ∂ b X ν ( k ) , (20) j µ ( k ) = − q ( k ) Z d σ δ (4) (cid:16) x − X ( k ) ( σ ) (cid:17)q | ¯ g ( k ) | ¯ g ab ( k ) ∂ a X µ ( k ) ∂ b u ( k ) + q ( k ) A ( k ) b T ( k ) . (21)Solving Eqs.(18)–(19) with (20)–(21) we find “thin-shell” wormhole so-lutions of static “spherically-symmetric” type (in Eddington-Finkelstein co-ordinates dt = dv − dηA ( η ) , F η = F vη ): ds = − A ( η ) dv + 2 dvdη + C ( η ) h ij ( θ ) dθ i dθ j , F vη = F vη ( η ) , (22) −∞ < η < ∞ , A ( η ( k )0 ) = 0 for η (1)0 < . . . < η ( N )0 . (23) ravity, Nonlinear Gauge Fields and Charge Confinement/Deconfinement 7 The derivation of these “thin-shell” wormhole solutions proceeds along thefollowing main steps:(i) Take “vacuum” solutions of (18)–(19) (without delta-function
LL-brane terms) in each spacetime region (separate “universe”) given by ( −∞ <η < η (1)0 ) , . . . , ( η ( N )0 < η < ∞ ) with common horizon(s) at η = η ( k )0 ( k =1 , . . . , N ).(ii) Each k -th LL-brane automatically locates itself on the horizon at η = η ( k )0 – intrinsic property of LL-brane dynamics [11].(iii) Match discontinuities of the derivatives of the metric and the gaugefield strength across each horizon at η = η ( k )0 using the explicit expressionsfor the LL-brane stress-energy tensor and charge current density (20)–(21).
4. Charge “Hiding”and Charge Confining Wormholes
First we will construct “one-throat” wormhole solutions to (17) with thecharged
LL-brane occupying the wormhole “throat”, which connects (i) anon-compact “universe” with Reissner-Nordstr¨om-(anti)-de-Sitter-type ge-ometry (where the cosmological constant is partially or entirely dynamically generated) to (ii) a compactified (“tubelike”) “universe” of (generalized)Levi-Civita-Bertotti-Robinson type with geometry
AdS × S or Rind × S .These wormholes possess the novel property of hiding electric chargefrom external observer in the non-compact “universe”. Namely, the wholeelectric flux produced by the charged LL-brane at the wormhole “throat”is pushed into the “tubelike” “universe”. As a result, the non-compact“universe” becomes electrically neutral with Schwarzschild-(anti-)de-Sitteror purely Schwarzschild geometry. Therefore, an external observer in thenon-compact “universe” detects a genuinely charged matter source (thecharged
LL-brane ) as electrically neutral .The explicit form ds = − A ( η ) dv + 2 dvdη + C ( η ) (cid:0) dθ + sin θdϕ (cid:1) forthe metric and the nonlinear gauge theory’s electric field F vη ( η ) read: • “Left universe” of Levi-Civita-Bertotti-Robinson (“tubelike”) type withgeometry AdS × S for η < A ( η ) = 4 π (cid:18) c F − √ f | c F | − Λ π (cid:19) η , C ( η ) ≡ r = 14 πc F + Λ , (24) | F vη | ≡ | ~E | = | c F | > f √ (cid:16) s πf (cid:17) for Λ > − πf , or | F vη | ≡ | ~E | = | c F | > r π | Λ | for Λ < , | Λ | > πf . • Non-compact “right universe” for η > r when Λ > − πf (in particu- E. Guendelman, A. Kaganovich, E. Nissimov, S. Pacheva lar, when Λ = 0), or the exterior region of Reissner-Nordstr¨om- anti -de-Sitter-type black hole beyond the outer (Schwarzschild-type) hori-zon r in the case Λ < | Λ | > πf , or the exterior regionof Reissner-Nordstr¨om-“hedgehog” black hole for | Λ | = 2 πf (note: A ( η ) ≡ A RN − ((A)dS) ( r + η )): A ( η ) = 1 − √ π | Q | f − mr + η + Q ( r + η ) − Λ + 2 πf r + η ) , (25) C ( η ) = ( r + η ) , | F vη | ≡ | ~E | = f √ | Q |√ π ( r + η ) . The matching relations for the discontinuities of the metric and gaugefield components across the
LL-brane world-volume occupying the worm-hole “throat” (which are here derived self-consistently from a well-definedworld-volume Lagrangian action principle for the
LL-brane ) (14) determineall parameters of the wormhole solutions as functions of q (the LL-brane charge) and f (coupling constant of √− F ): Q = 0 , | c F | = | q | + f √ , (26)as well as the allowed range for the “bare” cosmological constant: − π (cid:16) | q | + f √ (cid:17) < Λ < π (cid:16) q − f (cid:17) , (27)The relations (26) (recall | F vη | ≡ | ~E | = | c F | in the “tubelike” “left uni-verse”) have profound consequences:(A) The non-compact “right universe” (25) becomes exterior region ofelectrically neutral Schwarzschild-( anti -)de-Sitter or purely Schwarzschildblack hole beyond the Schwarzschild horizon carrying a vacuum constantradial electric field | F vη | ≡ | ~E | = f √ .(B) Recalling that the dielectric displacement field is ~D = (cid:16) − f √ | ~E | (cid:17) ~E ,we find from the second relation (26) that the whole flux produced by thecharged LL-brane flows only into the “tubelike” “left universe” (24) (since ~D = 0 in the non-compact “right universe”). This is a novel propertyof hiding electric charge through a wormhole connecting non-compact to a“tubelike” universe from external observer in the non-compact “universe”.The charge-“hiding” wormhole geometry is visualized on Fig.1 below.Further, we find more interesting “two-throat” wormhole solution ex-hibiting QCD-like charge confinement effect – obtained from a self-consistent ravity, Nonlinear Gauge Fields and Charge Confinement/Deconfinement 9 - - - - Figure 1: Shape of t = const and θ = π slice of charge-“hiding” wormholegeometry: the whole electric flux produced by the charged LL-brane at the“throat” is expelled into the left infinitely long cylindric tube.coupling of the gravity/nonlinear-gauge-field system with two identical op-positely charged
LL-branes (Eq.(17) with N = 2). The total “two-throat”wormhole spacetime manifold is made of:(i) “Left-most” non-compact “universe” comprising the exterior regionof Reissner-Nordstr¨om-de-Sitter-type black hole beyond the middle Schwarzschild-type horizon r for the “radial-like” η -coordinate interval: − ∞ < η < − η ≡ − h π (cid:16) √ f | c F | − c F (cid:17) + Λ i − , (28)where: A ( η ) = A RNdS ( r − η − η ) =1 − √ π | Q | f − mr − η − η + Q ( r − η − η ) − Λ + 2 πf r − η − η ) , (29) C ( η ) = ( r − η − η ) , | F vη ( η ) | ≡ | ~E | = f √ | Q |√ π ( r − η − η ) . (ii) “Middle” “tube-like” “universe” of Levi-Civita-Bertotti-Robinsontype with geometry dS × S comprising the finite extent (w.r.t. η -coordinate) region between the two horizons of dS at η = ± η : − η < η < η ≡ h π (cid:16) √ f | c F | − c F (cid:17) + Λ i − , (30)where the metric coefficients and electric field are: A ( η ) = 1 − h π (cid:16) √ f | c F | − c F (cid:17) + Λ i η , A ( ± η ) = 0 , (31) C ( η ) = r = 14 πc F + Λ , | F vη | ≡ | ~E | = | c F | < f √ (cid:16) s πf (cid:17) , with Λ > − πf ;(iii) “Right-most” non-compact “universe” comprising the exterior re-gion of Reissner-Nordstr¨om-de-Sitter-type black hole beyond the middleSchwarzschild-type horizon r for the “radial-like” η -coordinate interval η < η < ∞ ( η as in (30)), where: A ( η ) = A RNdS ( r + η − η )= 1 − √ π | Q | f − mr + η − η + Q ( r + η − η ) − Λ + 2 πf r + η − η ) , (32) C ( η ) = ( r + η − η ) , | F vη ( η ) | ≡ | ~E | = f √ | Q |√ π ( r + η − η ) . As dictated by the
LL-brane dynamics [11] each of the two
LL-branes lo-cates itself on one of the two common horizons at η = ± η between “left”and “middle”, and between “middle” and “right” “universes”, respectively.The matching relations for the discontinuities of the metric and gaugefield components across the each of the two LL-brane world-volumes de-termine all parameters of the wormhole solutions as functions of ± q (theopposite LL-brane charges) and f (coupling constant of √− F ). Mostimportantly we obtain: Q = 0 , | c F | = | q | + f √ , (33)and the bare cosmological constant must be in the interval:Λ ≤ , | Λ | < π ( f − q ) → | q | < f √ , (34)in particular, Λ could be zero.Similarly to the charge-“hiding” case, relations (33) meaning: | ~E | middle universe = | q | + | ~E | left / right universe , have profound consequences: ravity, Nonlinear Gauge Fields and Charge Confinement/Deconfinement11 - - - - Figure 2: Shape of t = const and θ = π slice of charge-confining wormholegeometry: the whole electric flux produced by the two oppositely charged LL-branes is confined within the middle finite-extent cylindric tube betweenthe “throats”. • The “left-most” (29) and “right-most” (32) non-compact “universes”become two identical copies of the electrically neutral exterior region ofSchwarzschild-de-Sitter black hole beyond the Schwarzschild horizon.They both carry a constant vacuum radial electric field with magni-tude | ~E | = f √ pointing inbound towards the horizon in one of these“universes” and pointing outbound w.r.t. the horizon in the second“universe”. The corresponding electric displacement field ~D = 0, sothere is no electric flux there (recall ~D = (cid:16) − f √ | ~E | (cid:17) ~E ). • The whole electric flux produced by the two charged
LL-branes withopposite charges ± q at the boundaries of the above non-compact “uni-verses” is confined within the “tube-like” middle “universe” (31) ofLevi-Civita-Robinson-Bertotti type with geometry dS × S , wherethe constant electric field is | ~E | = f √ + | q | with associated non-zeroelectric displacement field | ~D | = | q | . This is QCD-like confinement .A simple visualization of the charge-confining wormhole geometry isgiven in Fig.2. R -Gravity Coupled to Confining Nonlinear Gauge Fieldand Dilaton Consider now coupling of f ( R ) = R + αR gravity (possibly with a barecosmological constant Λ ) to a “dilaton” φ and the nonlinear gauge fieldsystem containing √− F : S = Z d x √− g h π (cid:16) f ( R ( g, Γ)) − (cid:17) + L ( F ( g )) + L D ( φ, g ) i , (35) f ( R ( g, Γ)) = R ( g, Γ) + αR ( g, Γ) , R ( g, Γ) = R µν (Γ) g µν , (36) L ( F ( g )) = − e F ( g ) − f p − F ( g ) , (37) F ( g ) ≡ F κλ F µν g κµ g λν , F µν = ∂ µ A ν − ∂ ν A µ (38) L D ( φ, g ) = − g µν ∂ µ φ∂ ν φ − V ( φ ) . (39) R µν (Γ) is the Ricci curvature in the first order (Palatini) formalism, i.e. , thespacetime metric g µν and the affine connection Γ µνλ are a priori independentvariables.The equations of motion resulting from the action (35) read: R µν (Γ) = 1 f ′ R (cid:20) πT µν + 12 f ( R ( g, Γ)) g µν (cid:21) , f ′ R ≡ df ( R ) dR = 1 + 2 αR ( g, Γ) , (40) ∇ λ (cid:0) √− gf ′ R g µν (cid:1) = 0 , (41) ∂ ν (cid:16) √− g h /e − f p − F ( g ) i F κλ g µκ g νλ (cid:17) = 0 . (42)The total energy-momentum tensor is given by: T µν = h L ( F ( g )) + L D ( φ, g ) − π Λ i g µν + (cid:16) /e − f p − F ( g ) (cid:17) F µκ F νλ g κλ + ∂ µ φ∂ ν φ . (43)Eq.(41) leads to the relation ∇ λ ( f ′ R g µν ) = 0 and thus it implies transitionto the “physical” Einstein-frame metrics h µν via conformal rescaling of theoriginal metric g µν [12]: g µν = 1 f ′ R h µν , Γ µνλ = 12 h µκ ( ∂ ν h λκ + ∂ λ h νκ − ∂ κ h νλ ) . (44)Using (44) the R -gravity equations of motion (40) can be rewritten in theform of standard Einstein equations: R µν ( h ) = 8 π (cid:18) T eff µν ( h ) − δ µν T eff λλ ( h ) (cid:19) (45) ravity, Nonlinear Gauge Fields and Charge Confinement/Deconfinement13 with effective energy-momentum tensor of the following form: T eff µν ( h ) = h µν L eff ( h ) − ∂L eff ∂h µν . (46)The effective Einstein-frame matter Lagrangian reads (the dilaton kineticterm X ( φ, h ) ≡ − h µν ∂ µ φ∂ n φ will be ignored in the sequel): L eff ( h ) = − e ( φ ) F ( h ) − f eff ( φ ) p − F ( h )+ X ( φ, h )(1 + 16 παX ( φ, h )) − V ( φ ) − Λ / π α (8 πV ( φ ) + Λ ) (47)with the following dynamical φ -dependent couplings:1 e ( φ ) = 1 e + 16 παf α (8 πV ( φ ) + Λ ) , (48) f eff ( φ ) = f παX ( φ, h )1 + 8 α (8 πV ( φ ) + Λ ) . (49)Thus, all equations of motion of the original R -gravity system (35)–(39)can be equivalently derived from the following Einstein/nonlinear-gauge-field/dilaton action: S eff = Z d x √− h h R ( h )16 π + L eff ( h ) i , (50)where R ( h ) is the standard Ricci scalar of the metric h µν and L eff ( h ) is asin (47). Important observation . Even if ordinary kinetic Maxwell term − F is absent in the original system ( e → ∞ in (37)), such term is nevertheless dynamically generated in the Einstein-frame action (47)–(50), which is a combined effect of αR and − f √− F : S maxwell = − παf Z d x √− h F κλ F µν h κµ h λν α (8 πV ( φ ) + Λ ) . (51)In what follows we consider constant “dilaton” φ extremizing the effec-tive Lagrangian (47): L eff = − e ( φ ) F ( h ) − f eff ( φ ) p − F ( h ) − V eff ( φ ) , (52) V eff ( φ ) = V ( φ ) + Λ π α (8 πV ( φ ) + Λ ) , f eff ( φ ) = f α (8 πV ( φ ) + Λ ) , (53)1 e ( φ ) = 1 e + 16 παf α (8 πV ( φ ) + Λ ) . (54) Important observation . The dynamical couplings and effective poten-tial are extremized simultaneously – this is an explicit realization of “leastcoupling principle” of Damour-Polyakov [13]: ∂f eff ∂φ = − παf ∂V eff ∂φ , ∂∂φ e = − (32 παf ) ∂V eff ∂φ → ∂L eff ∂φ ∼ ∂V eff ∂φ . (55)Therefore at the extremum of L eff (52) φ must satisfy: ∂V eff ∂φ = V ′ ( φ )[1 + 8 α ( κ V ( φ ) + Λ )] = 0 . (56)There are two generic cases:(a) Confining phase : Eq.(56) is satisfied for some finite-value φ ex-tremizing the original potential V ( φ ): V ′ ( φ ) = 0.(b) Deconfinement phase : For polynomial or exponentially growingoriginal V ( φ ), so that V ( φ ) → ∞ when φ → ∞ , we have: ∂V eff ∂φ → , V eff ( φ ) → πα = const when φ → ∞ , (57) i.e. , for sufficiently large values of φ we find a “flat region” in V eff . This “flatregion” triggers a transition from confining to deconfinement dynamics .Namely, in the “flat-region” case ( V ( φ ) → ∞ ) we have from (53)–(54): f eff → , e → e (58)and the effective gauge field Lagrangian (52) reduces to the ordinary non-confining one (the “square-root” term √− F vanishes): L (0)eff = − e F ( h ) − πα (59)with an induced cosmological constant Λ eff = 1 / α , which is completelyindependent of the bare cosmological constant Λ .Within the physical “Einstein”-frame in the confining phase ( V ′ ( φ ) =0 , φ = finite) we find:(A) Reissner-Nordstr¨om-( anti -)de-Sitter type black holes, in particular,non-standard Reissner-Nordstr¨om type with non-flat “hedgehog” asymp-totics, generalizing solutions (6)–(8) in the ordinary Einstein-gravity case,where now the effective cosmological constant and the vacuum constantradial electric field read: Λ eff ( φ ) = Λ + 8 πV ( φ ) + 2 πe f α (cid:0) Λ + 8 πV ( φ ) + 2 πe f (cid:1) , (60) | ~E vac | = (cid:16) e + 16 παf α (8 πV ( φ ) + Λ ) (cid:17) − f / √
21 + 8 α (8 πV ( φ ) + Λ ) . (61) ravity, Nonlinear Gauge Fields and Charge Confinement/Deconfinement15 (B) Levi-Civita-Bertotti-Robinson type “tubelike” spacetimes with ge-ometries AdS × S , Rind × S and dS × S generalizing (9)–(13), wherenow (using short-hand notation Λ( φ ) ≡ πV ( φ ) + Λ ):1 r = 4 π α Λ( φ ) h(cid:16) α (cid:0) Λ( φ ) + 2 πf (cid:1)(cid:17) ~E + 14 π Λ( φ ) i . (62)
6. Discussion
Inclusion of the non-standard nonlinear “square-root” gauge field term pro-vides explicit realization of the old “classic” idea of ‘t Hooft [9] about thenature of low-energy confinement dynamics. Coupling of nonlinear gaugetheory containing √− F to gravity (Einstein or f ( R ) = R + αR plus scalar“dilaton”) leads to a variety of remarkable effects: • Dynamical effective gauge couplings and dynamical induced cosmo-logical constant; • New non-standard black hole solutions of Reissner-Nordstr¨om-( anti -)de-Sitter type carrying an additional constant vacuum electric field,in particular, non-standard Reissner-Nordstr¨om type black holes withasymptotically non-flat “hedgehog” [6] behavior; • “Cornell”-type [10] confining potential in charged test particle dynam-ics; • Coupling to a charged lightlike brane produces a charge-“hiding” worm-hole, where a genuinely charged matter source is detected as electri-cally neutral by an external observer; • Coupling to two oppositely charged lightlike brane sources producesa two-“throat” wormhole displaying a genuine QCD-like charge con-finement. • When coupled to f ( R ) = R + αR gravity plus scalar “dilaton”,the √− F term triggers a transition from confining to deconfinementphase. Standard Maxwell kinetic term for the gauge field is dynami-cally generated even when absent in the original “bare” theory. Theabove are cumulative effects produced by the simultaneous presenceof αR and √− F terms.Let us conclude with a brief remark concerning the thermodynamicproperties of the non-standard black hole solutions described above. Tothis end, let us recall that for any static spherically symmetric metric of theform (6) with Schwarzschild-type horizon r , i.e. , A ( r ) = 0 , ∂ r A | r > surface gravity κ proportional to Hawking temperature T h ( e.g. [14], Ch. 12.5) is given by κ = 2 πT h = ∂ r A | r . With A ( r ) ofthe general form A ( r ) = 1 − c ( Q i ) − m/r + A ( r ; Q i ), where Q i are therest of the black hole parameters apart from the mass m , and c ( Q i ) isgenerically a non-zero constant as in (7) (responsible for the “hedgehog”non-flat spacetime asymptotics), one can straightforwardly derive the first law of black hole thermodynamics for the above class of solutions: δm = 18 π κδA H + e Φ i δQ i , A H = 4 πr , e Φ i = r ∂∂Q i (cid:16) A ( r ; Q i ) − c ( Q i ) (cid:17) . (63)In the special case of non-standard Reissner-Nordstr¨om-(anti-)de-Sitter typeblack holes (6)–(7) with parameters ( m, Q ) the conjugate potential in (63): e Φ = √ π (cid:16) Q √ πr − f √ r (cid:17) = √ πA | r = r (64)is (up to a constant factor) the electric field potential of the nonlinear gaugesystem on the horizon. Acknowledgments
E.N. is sincerely grateful to Prof. Branko Dragovich and the organizers of theSeventh Meeting in Modern Mathematical Physics (Belgrade, Sept 2012) for cor-dial hospitality. E.N. and S.P. are supported in part by Bulgarian NSF grant
DO 02-257 . Also, all of us acknowledge support of our collaboration through theexchange agreement between the Ben-Gurion University of the Negev and theBulgarian Academy of Sciences.
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