Tubelike Wormholes and Charge Confinement

Transcription

1 Tubelike Wormholes and Charge Confinement p.1/4 Tubelike Wormholes and Charge Confinement Talk at the 7th Mathematical Physics Meeting, Belgrade, Sept 09-19, 01 Eduardo Guendelman 1, Alexander Kaganovich 1, Emil Nissimov, Svetlana Pacheva 1 Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva, Israel Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria

2 7th Mathematical Physics Meeting, Belgrade, Sept 09-19, 01 Tubelike Wormholes and Charge Confinement p./4 Background material and further development of: E. Guendelman, A. Kaganovich, E.N. and S. Pacheva, (1) Asymptotically de Sitter and anti-de Sitter Black Holes with Confining Electric Potential, Phys. Lett. B704 (011) 30-33, erratum Phys. Lett. B705 (011) 545 ; () Hiding Charge in a Wormhole. The Open Nuclear and Particle Physics Journal 4 (011) 7-34 (arxiv: [hep-th]); (3) Hiding and Confining Charges via Tubelike Wormholes. Int.J. Mod. Phys. A6 (011) ; (4) Dynamical Couplings, Dynamical Vacuum Energy and Confinement/Deconfinement from R -Gravity. arxiv: [hep-th].

3 Introduction - Overview of Talk Tubelike Wormholes and Charge Confinement p.3/4 We consider gravity (including f(r)-gravity) coupled to non-standard nonlinear gauge field system containing f 0 F. The latter is known to produce in flat space-time a QCD-like confinement. Several interesting features: New mechanism for dynamical generation of cosmological constant due to nonlinear gauge field dynamics: Λ eff = Λ 0 + πf 0 (Λ 0 bare CC, may be absent at all) ; Non-standard black hole solutions of Reissner-Nordström-(anti-)de-Sitter type containing a constant radial vacuum electric field (in addition to the Coulomb one), in particular, in electrically neutral black holes of Schwarzschild-(anti-)de-Sitter type;

4 Introduction - Overview of Talk Tubelike Wormholes and Charge Confinement p.4/4 In case of vanishing effective cosmological constant Λ eff (i.e., Λ 0 < 0, Λ 0 = πf0 ) the resulting Reissner-Nordström-type black hole, apart from carrying an additional constant vacuum electric field, turns out to be non-asymptotically flat a feature resembling the gravitational effect of a hedgehog; Appearance of confining-type effective potential in charged test particle dynamics in the above black hole backgrounds; New tubelike solutions of Levi-Civita-Bertotti-Robinson type, i.e., with space-time geometry of the form M S, where M is a two-dimensional anti-de Sitter, Rindler or de Sitter space depending on the relative strength of the electric field w.r.t. the coupling f 0 of the square-root gauge field term.

5 Introduction - Overview of Talk Tubelike Wormholes and Charge Confinement p.5/4 When in addition one or more lightlike branes are self -consistently coupled to the above gravity/nonlinear-gaugefield system (as matter and charge sources) they produce ( thin-shell ) wormhole solutions dislaying two novel physically interesting effects: Charge-hiding effect - a genuinely charged matter source of gravity and electromagnetism may appear electrically neutral to an external observer a phenomenon opposite to the famous Misner-Wheeler charge without charge effect; Charge-confining tubelike wormhole with two throats occupied by two oppositely charged lightlike branes the whole electric flux is confined within the finite-extent middle universe of generalized Levi-Civita-Bertotti-Robinson type no flux is escaping into the outer non-compact universes.

6 Introduction - Overview of Talk Tubelike Wormholes and Charge Confinement p.6/4 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: Confinement-deconfinement transition due to appearance of flat region in the effective dilaton potential; The effective gauge couplings as well as the induced cosmological constant become dynamical depending on the dilaton v.e.v. In particular, a conventional Maxwell kinetic term for the gauge field is dynamically generated even if absent in the original theory;

7 Introduction - Overview of Talk Tubelike Wormholes and Charge Confinement p.7/4 Regular black hole solution (no singularity at r = 0) with confining vacuum electric field: the bulk space-time consist of two regions an interior de Sitter and an exterior Reissner-Nordström-type (with hedgehog asymptotics ) glued together along their common horizon occupied by a charged lightlike brane. The latter also dynamically determines the non-zero cosmological constant in the interior de-sitter region.

8 Tubelike Wormholes and Charge Confinement p.8/4 Introduction - Motivation for F Why F? t Hooft has shown that in any effective quantum gauge 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 (the latter appearing as an infrared counterterm ). The simplest way to realize these ideas in flat space-time: S = d 4 xl(f ), L(F ) = 1 4 F f 0 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 spontaneous breakdown of scale symmetry of the original scale-invariant Maxwell action with f 0 appearing as an integration

9 Tubelike Wormholes and Charge Confinement p.9/4 Introduction - Motivation for F For static field configurations the model (1) yields an electric displacement field D = E f 0 E E and the corresponding energy density turns out to be 1 E = 1 D + f 0 D f 0, so that it indeed contains a term linear w.r.t. D. The model (1) produces, when coupled to quantized fermions, a confining effective potential V (r) = β r + γr (Coulomb plus linear one with γ f 0) which is of the form of the well-known Cornell potential in the phenomenological description of quarkonium systems in QCD.

10 Gravity Coupled to Confining Nonlinear Gauge Field Tubelike Wormholes and Charge Confinement p.10/4 The action (R-scalar curvature; Λ 0 - bare CC, might be absent): S = d 4 x [ R Λ0 ] G + L(F ), L(F ) = 1 16π 4 F f 0 F, ( F F κλ F µν G κµ G λν, F µν = µ A ν ν A µ. The corresponding equations of motion read Einstein eqs.: R µν 1 G µνr + Λ 0 G µν = 8πT (F) µν, (3) T (F) µν = ( 1 f ) 0 F µκ F νλ G κλ 1 ( ) F + f 0 F G µν, (4) F 4 and nonlinear gauge field eqs.: ( G ( ν 1 f ) 0 )F κλ G µκ G νλ F = 0. (5)

11 Static Spherically Symmetric Solutions Non-standard Reissner-Nordström-(anti-)de-Sitter-type black holes depending on the sign of the dynamically generated CC Λ eff : ds = A(r)dt + dr A(r) + r( dθ + sin θdϕ ), (6) A(r) = 1 8π Q f 0 m r + Q r Λ eff 3 r, Λ eff = πf 0 + Λ 0, (7) with static spherically symmetric electric field containing apart from the Coulomb term an additional constant vacuum piece: F 0r = ε Ff 0 + Q 4π r, ε F sign(f 0r ) = sign(q), (8) corresp. to a confining Cornell potential A 0 = ε F f 0 r + Q 4π r. When Λ eff = 0, A(r) 1 8π Q f 0 for r ( hedgehog non-flat-spacetime asymptotics). Tubelike Wormholes and Charge Confinement p.11/4

12 Generalized Levi-Civita-Bertotti-Robinson Space-Times Three distinct types of static solutions of tubelike LCBR type with space-time geometry of the form M S, where M is some -dim manifold ((anti-)de Sitter, Rindler): ds = A(η)dt + dη A(η) + ( r 0 dθ + sin θdϕ ), < η <, (9) F 0η = c F = const, 1 r 0 = 4πc F + Λ 0 (= const). (10) (i) AdS S with constant vacuum electric field F 0η = c F : [ A(η) = 4π c F f 0 c F Λ ] 0 η (η Poincare patch coord), 4π ) provided either c F > f 0 ( Λ 0 for Λ πf0 0 πf0 or 1 c F > Λ 4π 0 for Λ 0 < 0, Λ 0 > πf0. (11) Tubelike Wormholes and Charge Confinement p.1/4

13 Generalized Levi-Civita-Bertotti-Robinson Space-Times Tubelike Wormholes and Charge Confinement p.13/4 (ii) Rind S with constant vacuum electric field F 0η = c F, where Rind is the flat -dim Rindler spacetime with: A(η) = η for 0 < η < or A(η) = η for < η < 0 (1) ) provided c F = f 0 ( Λ 0 for Λ πf0 0 > πf0. (iii) ds S with weak const vacuum electric field F 0η = c F, where ds is the -dim de Sitter space with: [ f0 A(η) = 1 4π c F c F + Λ ] 0 )η, (13) 4π ) when c F < f 0 ( Λ 0 πf0 [ two horizons at η = ±η 0 ± 4π ( f 0 c F c F for Λ 0 > πf 0. Note that ds has ) + Λ0 ] 1.

14 Lagrangian Formulation of Lightlike Brane Dynamics In what follows we will consider bulk Einstein/non-linear gauge field system () self-consistently coupled to N 1 (distantly separated) charged codimension-one lightlike p-brane (LL-brane) sources (here p = ). World-volume LL-brane action in Polyakov-type formulation: S LL [q] = 1 d p+1 σ Tb p 1 [ 0 γ γ abḡ ab b 0 (p 1) ], (1 ḡ ab a X µ G µν b X ν 1 T ( au + qa a )( b u + qa b ), A a a X µ A µ. (1 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 0 is world-volume CC. Tubelike Wormholes and Charge Confinement p.14/4

15 Lagrangian Formulation of Lightlike Brane Dynamics Tubelike Wormholes and Charge Confinement p.15/4 X µ (σ) are the p-brane embedding coordinates in the bulk space-time; u is auxiliary world-volume scalar field defining the lightlike direction of the induced metric; T is dynamical (variable) brane tension; q the coupling to bulk spacetime gauge field A µ is LL-brane surface charge density. The on-shell singularity of the induced metric g ab, i.e., the lightlike property, directly follows from the eqs. of motion: g ab (ḡbc ( c u + qa c ) ) = 0. (16)

16 Gravity/Nonlinear Gauge Field Coupled to LL-Branes Tubelike Wormholes and Charge Confinement p.16/4 Full action of self-consistently coupled bulk Einstein/non-linear gauge field/ll-brane (L(F ) = 1 4 F f 0 F ): S = d 4 x [ R(G) Λ0 G 16π ] + L(F ) + N S LL [q (k) ], (17) k=1 where the superscript (k) indicates the k-th LL-brane. The corresponding equations of motion are as follows: R µν 1 [ G µνr + Λ 0 G µν = 8π T µν (F) + [ G ( ν 1 f ] 0 )F κλ G µκ G νλ F + N k=1 ] T µν (k), (18) N j µ (k) = 0. (19) k=1

17 Gravity/Nonlinear Gauge Field Coupled to LL-Branes Tubelike Wormholes and Charge Confinement p.17/4 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): ) δ (x (4) X (k) (σ) T µν (k) = d 3 σ T ḡ (k) (k) ḡ(k) ab a X µ G (k) bx(k) ν, (0) j µ (k) = q(k) ) d 3 σ δ (x X (4) (k) (σ) ḡ (k) ḡ(k) ab a X µ (k) b u (k) + q (k) A (k) b T (k). (1)

18 Gravity/Nonlinear Gauge Field Coupled to LL-Brane Thin-shell wormhole solutions of static spherically-symmetric type (in Eddington-Finkelstein coordinates dt = dv dη A(η) ): ds = A(η)dv + dvdη + C(η)h ij (θ)dθ i dθ j, F vη = F vη (η), () < η <, A(η (k) 0 ) = 0 for η (1) 0 <... < η (N) 0. (3) (i) Take vacuum solutions of (18) (19) (without delta-function LL-brane terms) in each space-time region ( <η<η (1) ) 0,..., ( (N) η 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. (iii) Match discontinuities of the derivatives of the metric and the gauge field strength across each horizon at η = η (k) 0 using the explicit expressions for the LL-brane stress-energy tensor and charge current density (0) (1). Tubelike Wormholes and Charge Confinement p.18/4

19 Charge- Hiding Wormhole First we will construct one-throat wormhole solutions to (17) with the charged LL-brane occupying the wormhole throat, which connects (i) a non-compact universe with Reissner-Nordström-(anti)-de-Sitter-type geometry (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. The whole electric 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-Sitter or purely Schwarzschild geometry. Therefore, an external observer in the non-compact universe detects a genuinely charged matter source (the charged LL-brane) as electrically neutral. Tubelike Wormholes and Charge Confinement p.19/4

20 Charge- Hiding Wormhole ( Tubelike Left Universe ) Tubelike Wormholes and Charge Confinement p.0/4 Explicit form ds = A(η)dv + dvdη + C(η) ( dθ + sin θdϕ ) for the metric and the nonlinear gauge theory s electric field F vη (η): Left universe of Levi-Civita-Bertotti-Robinson ( tubelike ) type with geometry AdS S for η < 0: ( A(η) = 4π c F f 0 c F Λ ) 0 η, C(η) r 1 0 = 4π 4πc F + Λ 0 F vη E = c F > f ( Λ ) 0 for Λ πf0 0 > πf0,, (4 or F vη E = c F > 1 4π Λ 0 for Λ 0 < 0, Λ 0 > πf 0.

21 Tubelike Wormholes and Charge Confinement p.1/4 Charge- Hiding Wormhole (Non-Compact Right Universe ) Explicit form ds = A(η)dv + dvdη + C(η) ( dθ + sin θdϕ ) for the metric and the nonlinear gauge theory s electric field F vη (η): Non-compact right universe for η > 0 comprising the exterior region of RN-de-Sitter-type black hole beyond the middle (Schwarzschild-type) horizon r 0 when Λ 0 > πf 0 (in particular, when Λ 0 = 0), or the exterior region of RN-anti-de-Sitter-type black hole beyond the outer (Schwarzschild-type) horizon r 0 in the case Λ 0 < 0 and Λ 0 > πf 0, or the exterior region of RN- hedgehog black hole for Λ 0 = πf 0 (note: A(η) A RN ((A)dS) (r 0 + η)): A(η) = 1 8π Q f 0 m r 0 + η + Q (r 0 + η) Λ 0 + πf0 3 C(η) = (r 0 + η), F vη E = f 0 + (r 0 + η), ( Q 4π (r0 + η).

22 Charge Hiding Wormhole The matching relations for the discontinuities of the metric and gauge field components across the LL-brane world-volume occupying the wormhole throat (which are here derived self-consistently from a well-defined world-volume Lagrangian action principle for the LL-brane) determine all parameters of the wormhole solutions as functions of q (the LL-brane charge) and f 0 (coupling constant of F ): Q = 0, c F = q + f 0, (6) as well as the allowed range for the bare CC: ( 4π q + f ) ( 0 < Λ0 < 4π q f 0 ), (7) in particular, Λ 0 could be zero. Tubelike Wormholes and Charge Confinement p./4

23 Charge Hiding Wormhole Tubelike Wormholes and Charge Confinement p.3/4 The relations (6) (Q = 0, c F = q + f 0 ; recall F vη E = c F in the tubelike left universe ) have profound consequences: (A) The non-compact right universe becomes exterior region of electrically neutral Schwarzschild-(anti-)de-Sitter or purely Schwarzschild black hole beyond the Schwarzschild horizon carrying a vacuum constant radial electric field F vη E = f 0. (B) ( Recalling ) that the dielectric displacement field is D = 1 f 0 E, E we find from the second rel.(6) that the whole flux produced by the charged LL-brane flows only into the tubelike left universe (since D = 0 in the non-compact right universe ). This is a novel property of hiding electric charge through a wormhole connecting non-compact to a tubelike universe from external observer in the non-compact universe.

24 Visualizing Charge- Hiding Wormhole Tubelike Wormholes and Charge Confinement p.4/4 Shape of t = const and θ = π slice of charge- hiding wormhole geometry: the whole electric flux produced by the charged LL-brane at the throat is expelled into the left infinitely long cylindric tube

25 Charge Confining Wormhole (Non-Compact Left Universe ) There exist more interesting two-throat wormhole solution exhibiting QCD-like charge confinement effect obtained from a self-consistent coupling of the gravity/nonlinear-gauge-field system () with two identical oppositely charged LL-branes. The total two-throat wormhole space-time manifold is made of: (i) Left-most non-compact universe comprising the exterior region of RN-de-Sitter-type black hole beyond the middle Schwarzschild-type horizon r 0 for the radial-like η-coordinate [ interval < η < η 0 4π ( ) ] 1 f 0 c F c F + Λ0, where: 1 8π Q f 0 A(η) = A RNdS (r 0 η 0 η) = m r 0 η 0 η + Q (r 0 η 0 η) Λ 0 + πf0 3 C(η) = (r 0 η 0 η), F vη (η) E = f 0 + (r 0 η 0 η) Q 4π (r η η). Tubelike Wormholes and Charge Confinement p.5/4

26 Charge Confining Wormhole ( Tubelike Middle Universe ) Tubelike Wormholes and Charge Confinement p.6/4 (ii) Middle tube-like universe of Levi-Civita-Bertotti-Robinson type with geometry ds S comprising the finite extent (w.r.t. η-coordinate) region between the two horizons of ds at η = ±η 0 : η 0 < η < η 0 [ 4π ( ) ] 1 f0 c F c F + Λ 0, (8) where the metric coefficients and electric field are: [ ( ) ] A(η) = 1 4π f0 c F c F + Λ 0 η, A(±η 0 ) = 0, C(η) = r0 1 = 4πc F + Λ, F vη E = c F < f ( Λ ) 0 πf0 with Λ 0 > πf 0 ;

27 Tubelike Wormholes and Charge Confinement p.7/4 Charge Confining Wormhole (Non-Compact Right Universe ) (iii) Right-most non-compact universe comprising the exterior region of RN-de-Sitter-type black hole beyond the middle Schwarzschild-type horizon r 0 for the radial-like η-coordinate interval η 0 < η < (η 0 as in (8)), where: = 1 8π Q f 0 m r 0 + η η 0 + A(η) = A RNdS (r 0 + η η 0 ) Q (r 0 + η η 0 ) Λ 0 + πf0 3 C(η) = (r 0 + η η 0 ), F vη (η) E = f 0 + (r 0 + η η 0 ) Q 4π (r0 + η η 0 As dictated by the LL-brane dynamics each of the two LL-branes locates itself on one of the two common horizons at η = ±η 0 between left and middle, and between middle and right universes, respectively.

28 Charge Confining Wormhole (Non-Compact Right Universe ) Tubelike Wormholes and Charge Confinement p.8/4 The matching relations for the discontinuities of the metric and gauge field components across the each of the two LL-brane world-volumes determine all parameters of the wormhole solutions as functions of ±q (the opposite LL-brane charges) and f 0 (coupling constant of F ). Most importantly we obtain: Q = 0, c F = q + f 0, (9) and bare cosmological constant must be in the interval: Λ 0 0, Λ 0 < π(f 0 q ) q < f 0, (30) in particular, Λ 0 could be zero.

29 Charge-Confining Wormhole Similarly to the charge- hiding case, rel.(9) Q = 0 and c F = q + f 0, which means: E middle universe = q + E left/right universe, have profound consequences: The left-most and right-most non-compact universes become two identical copies of the electrically neutral exterior region of Schwarzschild-de-Sitter black hole beyond the Schwarzschild horizon. They both carry a constant vacuum radial electric field with magnitude E = f 0 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, so there is no electric flux there (recall D ( ) = 1 f 0 E). Tubelike Wormholes and Charge Confinement p.9/4

30 Charge-Confining Wormhole Tubelike Wormholes and Charge Confinement p.30/4 The whole electric flux produced by the two charged LL-branes with opposite charges ±q at the boundaries of the above non-compact universes is confined within the tube-like middle universe of Levi-Civitta-Robinson-Bertotti type with geometry ds S, where the constant electric field is E = f 0 + q with associated non-zero electric displacement field D = q. This is QCD-like confinement.

31 Visualizing Charge-Confining Wormhole Tubelike Wormholes and Charge Confinement p.31/4 Shape of t = const and θ = π slice of charge-confining wormhole geometry: the whole electric flux produced by the two oppositely charged LL-branes is confined within the middle finite-extent cylindric tube between the throats

32 Dynamical Couplings & Confinement/Deconfinement from R -Gravity Tubelike Wormholes and Charge Confinement p.3/4 Consider now coupling of f(r) = R + αr gravity (possibly with a bare cosmological constant Λ 0 ) to a dilaton φ and the nonlinear gauge field system containing F : S = d 4 x [ 1 g (f ( R(g, Γ) ) ) Λ 0 16π ] + L(F (g)) + L D (φ,g), (31) f ( R(g, Γ) ) = R(g, Γ) + αr (g, Γ), R(g, Γ) = R µν (Γ)g µν, (3) L(F (g)) = 1 (g) f 0 F (g), 4e F (33) F (g) F κλ F µν g κµ g λν, F µν = µ A ν ν A µ (34) L D (φ,g) = 1 gµν µ φ ν φ V (φ). (35) R µν (Γ) is the Ricci curvature in the first order (Palatini) formalism, i.e., the space-time metric g µν and the affine connection Γ µ νλ are a priori independent variables.

33 Dynamical Couplings & Confinement/Deconfinement from R -Gravity Tubelike Wormholes and Charge Confinement p.33/4 The equations of motion resulting from the action (31) read: R µν (Γ) = 1 [κ T f R µν + 1 f( R(g, Γ) ) ] g µν, f R df(r) = 1+αR(g, Γ), dr (36) ν ( g [1/e ( λ gf R g µν) = 0, (37) f ) 0 ]F κλ g µκ g νλ = 0. (38) F (g) The total energy-momentum tensor is given by: [ T µν = L(F (g)) + L D (φ,g) 1 κ Λ 0 ( + 1/e f 0 F (g) ] g µν ) F µκ F νλ g κλ + µ φ ν φ. (39)

34 Dynamical Couplings & Confinement/Deconfinement from R -Gravity Tubelike Wormholes and Charge Confinement p.34/4 Eq.(37) leads to the relation λ (f R g µν) = 0 and thus it implies transition to the physical Einstein-frame metrics h µν via conformal rescaling of the original metric g µν : g µν = 1 h f R µν, Γ µ νλ = 1 hµκ ( ν h λκ + λ h νκ κ h νλ ). (40) Using (40) the R -gravity eqs. of motion (36) can be rewritten in the form of standard Einstein equations: ( R ν(h) µ µ = 8π T eff ν(h) 1 ) δµ λ νt eff λ(h) with effective energy-momentum tensor of the following form: (41) T eff µν(h) = h µν L eff (h) L eff. (4) h µν

35 Tubelike Wormholes and Charge Confinement p.35/4 Dynamical Couplings & Confinement/Deconfinement from R -Gravity The effective Einstein-frame matter lagrangian reads (here X(φ,h) 1 hµν µ φ n φ, to be ignored in the sequel): L eff (h) = 1 4e eff (φ)f (h) 1 f eff(φ) F (h) + X(φ,h)( παX(φ,h) ) V (φ) Λ 0 /8π 1 + 8α (8πV (φ) + Λ 0 ) (43) with the following dynamical φ-dependent couplings: 1 e eff (φ) = 1 e + 16παf α (8πV (φ) + Λ 0 ), (44) f eff (φ) = f παX(φ,h) 1 + 8α (8πV (φ) + Λ 0 ). (45)

36 Dynamical Couplings & Confinement/Deconfinement from R -Gravity Tubelike Wormholes and Charge Confinement p.36/4 Thus, all eqs. of motion of the original R -gravity system (31) (35) can be equivalently derived from the following Einstein/nonlinear-gauge-field/dilaton action: S eff = d 4 x [ R(h) ] h 16π + L eff(h), (46) where R(h) is the standard Ricci scalar of the metric h µν and L eff (h) is as in (43). Important observation. Even if ordinary kinetic Maxwell term 1 4 F is absent in the original system (e in (33)), such term is nevertheless dynamically generated in the Einstein-frame action (43) (46) combined effect of αr and f 0 F : S maxwell = 4παf0 d 4 x F κλ F µν h κµ h λν h 1 + 8α (8πV (φ) + Λ 0 ). (47)

37 Dynamical Couplings & Confinement/Deconfinement from R -Gravity In what follows we consider constant dilaton φ extremizing the effective Lagrangian (43): L eff = 1 4e eff (φ)f (h) 1 f eff(φ) F (h) V eff (φ), (48) V eff (φ) = V (φ) + Λ 0 8π 1 + 8α (8πV (φ) + Λ 0 ), f eff(φ) = f α (8πV (φ) + Λ 0 ), (49) 1 e eff (φ) = 1 e + 16παf α (8πV (φ) + Λ 0 ). (50) The dynamical couplings and effective potential are extremized simultaneously explicit realization of least coupling principle of Damour-Polyakov: f eff φ = 64παf V eff 0 φ, φ 1 e eff = (3παf 0 ) V eff φ L eff φ V eff φ. (51) Tubelike Wormholes and Charge Confinement p.37/4

38 Dynamical Couplings & Confinement/Deconfinement from R -Gravity Therefore at the extremum of L eff (48) φ must satisfy: V eff φ = V (φ) [1 + 8α (κ V (φ) + Λ 0 )] = 0. (5) There are two generic cases: (a) Confining phase: Eq.(5) is satisfied for some finite-value φ 0 extremizing the original potential V (φ): V (φ 0 ) = 0. (b) Deconfiment phase: For polynomial or exponentially growing original V (φ), so that V (φ) when φ, we have: V eff φ 0, V eff(φ) 1 64πα = const when φ, (53) i.e., for sufficiently large values of φ we find a flat region in V eff. This flat region triggers a transition from confining to deconfinement dynamics. Tubelike Wormholes and Charge Confinement p.38/4

39 Tubelike Wormholes and Charge Confinement p.39/4 Dynamical Couplings & Confinement/Deconfinement from R -Gravity Namely, in the flat-region case we have: f eff 0, e eff e (54) and the effective gauge field Lagrangian (48) reduces to the ordinary non-confining one (the square-root term F vanishes): L (0) eff = 1 4e F (h) 1 64πα with an induced cosmological constant Λ eff = 1/8α, which is completely independent of the bare cosmological constant Λ 0. (55)

40 Static spherically symmetric solutions of R -gravity Tubelike Wormholes and Charge Confinement p.40/4 Within the physical Einstein -frame in the confining phase: (A) Reissner-Nordström-(anti-)de-Sitter type black holes, in particular, non-standard Reissner-Nordström type with non-flat hedgehog asymptotics, where now: E vac = Λ 0 + 8πV (φ 0 ) + πe f0 Λ eff (φ 0 ) = 1 + 8α (Λ 0 + 8πV (φ 0 ) + πe f0), (56) ( 1 e + 16παf ) 1 0 f 0 / 1 + 8α (8πV (φ 0 ) + Λ 0 ) 1 + 8α (8πV (φ 0 ) + Λ 0 ). (57) (B) Levi-Civitta-Bertotti-Robinson type tubelike space-times with geometries AdS S, Rind S and ds S where now (using short-hand Λ(φ 0 ) 8πV (φ 0 ) + Λ 0 ): 1 r 0 = 4π [( 1 + 8α ( ) ) Λ(φ 0 ) + πf0 E + 1 ] 1 + 8αΛ(φ 0 ) 4π Λ(φ 0). (58)

41 Conclusions Inclusion of the non-standard nonlinear square-root gauge field term explicit realization of the old classic idea of t Hooft about the nature of low-energy confinement dynamics. Coupling of nonlinear gauge theory 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 cosmological constant; New non-standard black hole solutions of RN-(anti-)de-Sitter type carrying an additional constant vacuum electric field, in particular, non-standard RN type black holes with asymptotically non-flat hedgehog behaviour; Cornell -type confining potential in charged test particle dynamics; Tubelike Wormholes and Charge Confinement p.41/4

42 Conclusions Tubelike Wormholes and Charge Confinement p.4/4 Coupling to a charged lightlike brane produces a charge- hiding wormhole, where a genuinely charged matter source is detected as electrically neutral by an external observer; Coupling to two oppositely charged lightlike brane sources produces a two- throat wormhole displaying a genuine QCD-like charge confinement. When coupled to f(r) = R + αr gravity plus scalar dilaton, the F term triggers a transition from confining to deconfinement phase. Standard Maxwell kinetic term for the gauge field is dynamically generated even when absent in the original bare theory. The above are cumulative effects produced by the simultaneous presence of αr and F terms.