aa r X i v : . [ h e p - t h ] N ov Lie Theory and Its Applications in Physics VIIed. V.K. Dobrev et al, Heron Press, Sofia, 2008
Lightlike Braneworlds
Eduardo Guendelman , Alexander Kaganovich , Emil Nissimov , Svetlana Pacheva Department of Physics, Ben-Gurion University of the Negev, P.O.Box 653, IL-84105 Beer-Sheva, Israel Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sci-ences, Boul. Tsarigradsko Chausee 72, BG-1784 Sofia, Bulgaria
Abstract
We propose a new class of p -brane models describing intrinsically light-like branes in any world-volume dimensions. Properties of the dynamicsof these lightlike p -branes in various gravitational backgrounds of interestin the context of braneworlds are briefly described. Codimenion two (andmore) lightlike braneworlds perform in their ground states non-trivial mo-tions in the extra dimensions in sharp contrast to standard (Nambu-Goto)braneworlds. Lightlike branes (LL-branes, for short) are of particular interest in general rel-ativity primarily due to their role: (i) in describing impulsive lightlike signalsarising in cataclysmic astrophysical events [1]; (ii) as basic ingredients in the socalled “membrane paradigm” theory [2] of black hole physics; (iii) in the contextof the thin-wall description of domain walls coupled to gravity [3, 4].More recently, LL-branes became significant also in the context of modernnon-perturbative string theory, in particular, as the so called H -branes describ-ing quantum horizons (black hole and cosmological) [5], as well appearing asPenrose limits of baryonic D (= Dirichlet) branes [6].In the original papers [3, 4] LL-branes in the context of gravity and cosmol-ogy have been extensively studied from a phenomenological point of view, i.e.,by introducing them without specifying the Lagrangian dynamics from whichthey may originate . On the other hand, we have proposed in a series of re-cent papers [8] a new class of concise Lagrangian actions, among them – Weyl-conformally invariant ones, providing a derivation from first principles of the In a recent paper [7] brane actions in terms of their pertinent extrinsic geometry have beenproposed which generically describe non-lightlike branes, whereas the lightlike branes are treated asa limiting case. ( p + 1) = odd world-volume dimensions.In Section 2 of the present paper we extend our previous construction to thecase of LL-brane actions for arbitrary world-volume dimensions. In Section 4we discuss the properties of LL-brane dynamics in generic static gravitationalbackgrounds, in particular, the case with two extra dimensions from the pointof view of “braneworld” scenarios [9] (for a review, see [10]). Unlike conven-tional braneworlds, where the underlying branes are of Nambu-Goto type (i.e.,describing massive brane modes) and in their ground state they position them-selves at some fixed point in the extra dimensions of the bulk space-time, ourlightlike braneworlds perform in the ground state non-trivial motions in the ex-tra dimensions – planar circular, spiral winding, etc. depending on the topologyof the extra dimensions. Finally, in the outlook section we briefly outline thetreatment of the special case of codimension one lightlike branes which play animportant role in the context of black hole physics. Also we comment on the roleof lightlike branes in Kaluza-Klein scenarios with singular bulk metrics [11]. The main ingredients of our construction of LL-brane actions for arbitrary ( p +1) world-volume dimensions are: • Alternative non-Riemannian integration measure density Φ( ϕ ) (volumeform) on the p -brane world-volume manifold: Φ( ϕ ) ≡ p + 1)! ε I ...I p +1 ε a ...a p +1 ∂ a ϕ I . . . ∂ a p +1 ϕ I p +1 (1)instead of the usual √− γ . Here (cid:8) ϕ I (cid:9) p +1 I =1 are auxiliary world-volumescalar fields; γ ab ( a, b = 0 , , . . ., p ) denotes the intrinsic Riemannian met-ric on the world-volume, and γ = det k γ ab k . • Auxiliary ( p − -rank antisymmetric tensor gauge field A a ...a p − on theworld-volume with p -rank field-strength and its dual: F a ...a p = p∂ [ a A a ...a p ] , F ∗ a = 1 p ! ε aa ...a p √− γ F a ...a p . (2)Note the simple identity: F a ...a p − b F ∗ b = 0 , (3)which will play a crucial role in what follows, and let us also introduce the short-hand notation: F ≡ F a ...a p F b ...b p γ a b . . . γ a p b p . (4)We now propose the following reparametrization invariant action describingintrinsically lightlike p -branes for any world-volume dimension ( p + 1) : S = − Z d p +1 σ Φ( ϕ ) h γ ab ∂ a X µ ∂ b X ν G µν ( X ) − L (cid:0) F (cid:1)i (5)uendelman,Kaganovich,Nissimov,Pacheva 3using the objects (1) and (2), where L (cid:0) F (cid:1) is arbitrary function of F (4) and G µν ( X ) denotes the Riemannian metric of the bulk space-time. Remark.
For the special choice L (cid:0) F (cid:1) = (cid:0) F (cid:1) /p the action (5) becomesmanifestly invariant under Weyl (conformal) symmetry : γ ab −→ γ ′ ab = ρ γ ab , ϕ i −→ ϕ ′ i = ϕ ′ i ( ϕ ) with Jacobian det (cid:13)(cid:13)(cid:13) ∂ϕ ′ i ∂ϕ j (cid:13)(cid:13)(cid:13) = ρ .Rewriting the action (5) in the following equivalent form: S = − Z d p +1 σ χ √− γ h γ ab ∂ a X µ ∂ b X ν G µν ( X ) − L (cid:0) F (cid:1)i , χ ≡ Φ( ϕ ) √− γ (6)we see that the composite field χ plays the role of a dynamical (variable) branetension.The equations of motion obtained from (5) w.r.t. measure-building auxiliaryscalars ϕ I and γ ab read, respectively: γ cd ( ∂ c X∂ d X ) − L (cid:0) F (cid:1) = M (cid:16) = integration const (cid:17) , (7)
12 ( ∂ a X∂ b X ) − pL ′ (cid:0) F (cid:1) F aa ...a p − γ a b . . . γ a p − b p − F bb ...b p − = 0 , (8)where we have introduced short-hand notation for the induced metric: ( ∂ a X∂ b X ) ≡ ∂ a X µ ∂ b X ν G µν . (9)Let us note that Eqs.(8) can be viewed as p -brane analogues of the string Vira-soro constraints.Eqs.(7)–(8) have the following profound consequences. First, taking thetrace in (8) and comparing with (7) implies the following crucial relation forthe Lagrangian function L (cid:0) F (cid:1) : L (cid:0) F (cid:1) − pF L ′ (cid:0) F (cid:1) + M = 0 , (10)which determines F on-shell as certain function of the integration constant M ,i.e. F = F ( M ) = const . (11)The second and most important implication of Eqs.(8) is due to the identity(3) which implies that the induced metric (9) on the world-volume of the p -branemodel (5) is singular (as opposed to the ordinary Nambu-Goto brane): ( ∂ a X∂ b X ) F ∗ b = 0 , i . e . ( ∂ F X∂ F X ) = 0 , ( ∂ ⊥ X∂ F X ) = 0 , (12)where ∂ F ≡ F ∗ a ∂ a and ∂ ⊥ are derivatives along the tangent vectors in thecomplement of F ∗ a .Thus, we arrive at the following important conclusion: every point on thesurface of the p -brane (5) moves with the speed of light in a time-evolutionalong the vector-field F ∗ a . Therefore, we will name (5) by the acronym LL-brane (Lightlike-brane) model. LightlikeBraneworldsBefore proceeding let us point out that we can add to the LL-brane action(5) natural couplings to bulk Maxwell A µ and Kalb-Ramond A µ ...µ p +1 gaugefields: S = − Z d p +1 σ Φ( ϕ ) h γ ab ∂ a X µ ∂ b X ν G µν ( X ) − L (cid:0) F (cid:1)i − q Z d p +1 σε ab ...b p F b ...b p ∂ a X µ A µ ( X ) − β ( p + 1)! Z d p +1 σε a ...a p +1 ∂ a X µ . . . ∂ a p +1 X µ p +1 A µ ...µ p +1 . (13)The additional coupling terms to the bulk fields do not affect Eqs.(7) and (8),so that the conclusions about on-shell constancy of F (11) and the lightlikenature (12) of the p -branes under consideration remain unchanged. The secondChern-Simmons-like term in (13) is a special case of a class of Chern-Simmons-like couplings of extended objects to external electromagnetic fields proposed inref. [12].The Kalb-Ramond gauge field has special significance in D = p + 2 -dimensional bulk space-time. The single independent component F of its field-strength: F µ ...µ D = D∂ [ µ A µ ...µ D ] = F√− Gε µ ...µ D (14)when coupled to gravity produces a dynamical (positive) cosmological constant(cf. ref. [13] for D = 4 ; recall, here D = p + 2 ): K = 8 πG N p ( p + 1) F . (15)It remains to write down the equations of motion w.r.t. auxiliary world-volume gauge field A a ...a p − and X µ produced by the action (13): ∂ [ a (cid:0) F ∗ c γ b ] c χL ′ ( F ) (cid:1) + q ∂ a X µ ∂ b X ν F µν ( X ) = 0 ; (16) ∂ a (cid:0) χ √− γγ ab ∂ b X µ (cid:1) + χ √− γγ ab ∂ a X ν ∂ b X λ Γ µνλ ( X ) − qε ab ...b p F b ...b p ∂ a X ν F λν ( X ) G λµ ( X ) − β ( p + 1)! ε a ...a p +1 ∂ a X µ . . . ∂ a p +1 X µ p +1 F λµ ...µ p +1 ( X ) G λµ ( X ) = 0 . (17)Here χ is the dynamical brane tension as in (6), F µν = ∂ µ A ν − ∂ ν A µ , F λµ ...µ p +1 = ( p + 2) ∂ [ λ A µ ...µ p +1 ] (18)are the corresponding gauge field-strengths, Γ µνλ = 12 G µκ ( ∂ ν G κλ + ∂ λ G κν − ∂ κ G νλ ) (19)is the Christoffel connection for the external metric, and L ′ ( F ) denotes deriva-tive of L ( F ) w.r.t. the argument F .uendelman,Kaganovich,Nissimov,Pacheva 5 Invariance under world-volume reparametrizations allows to introduce the stan-dard synchronous gauge-fixing conditions: γ i = 0 ( i = 1 , . . . , p )) , γ = − . (20)In what follows we will also use a natural ansatz for the auxiliary world-volumegauge field-strength: F ∗ i = 0 ( i = 1 , . . ., p ) , i . e . F i ...i p − = 0 , (21)the only non-zero component of the dual strength being: F ∗ = 1 p ! ε i ...i p p γ ( p ) F i ...i p , (22) γ ( p ) ≡ det k γ ij k ( i, j = 1 , . . . , p ) , F = p ! (cid:0) F ∗ (cid:1) = const . According to (12) the meaning of the ansatz (21) is that the lightlike direction F ∗ a ∂ a ≃ ∂ ≡ ∂ τ , i.e., it coincides with the brane proper-time direction. Bianc-chi identity ∂ a F ∗ a = 0 together with (21)–(22) implies: ∂ F i ...i p = 0 −→ ∂ p γ ( p ) = 0 . (23)Using (20) and (21) the equations of motion (8), (16) and (17) acquire theform, respectively: ( ∂ X ∂ X ) = 0 , ( ∂ X ∂ i X ) = 0 , ( ∂ i X ∂ j X ) − a ( M ) γ ij = 0 (24)(Virasoro-like constraints), where the constant: a ( M ) ≡ F L ′ ( F ) (cid:12)(cid:12) F = F ( M ) (25)(it must be strictly positive); ∂ i χ + qa ( M ) ∂ X µ ∂ i X ν F µν = 0 , ∂ i X µ ∂ j X ν F µν = 0 , (26)with a ( M ) ≡ F ∗ L ′ ( F ) (cid:12)(cid:12) F = F ( M ) = const ; (27) − p γ ( p ) ∂ ( χ∂ X µ ) + ∂ i (cid:16) χ p γ ( p ) γ ij ∂ j X µ (cid:17) + χ p γ ( p ) (cid:0) − ∂ X ν ∂ X λ + γ kl ∂ k X ν ∂ l X λ (cid:1) Γ µνλ − q p ! F ∗ p γ ( p ) ∂ X ν F λν G λµ − β ( p + 1)! ε a ...a p +1 ∂ a X µ . . . ∂ a p +1 X µ p +1 F λµ ...µ p +1 G λµ = 0 . (28) LightlikeBraneworlds Let us split the bulk space-time coordinates as: ( X µ ) = ( x a , y α ) ≡ (cid:0) x , x i , y α (cid:1) (29) a = 0 , , . . . , p , i = 1 , . . . , p , α = 1 , . . . , D − ( p + 1) and consider static ( x -independent) background metrics G µν of the form: ds = − A ( y )( dx ) + C ( y ) g ij ( ~x ) dx i dx j + B αβ ( y ) dy α dy β . (30)Here we will discuss the simplest non-trivial ansatz for the LL-brane em-bedding coordinates: X a ≡ x a = σ a , X p + α ≡ y α = y α ( τ ) , τ ≡ σ . (31)With (30) and (31), the constraint Eqs.(24) yield: − A ( y ( τ )) + B αβ ( y ( τ )) . y α . y β = 0 , C ( y ( τ )) g ij − a ( M ) γ ij = 0 , (32)where . y α ≡ ddτ y α . Second Eq.(32) together with the last relation in (23) implies: ddτ C ( y ( τ )) = . y α ∂∂y α C (cid:12)(cid:12) y = y ( τ ) = 0 . (33)The second-order Eqs.(28) in the absence of couplings to bulk Maxwell andKalb-Ramond fields (which will be case we will consider in the present section)yield accordingly: ∂ τ χ + χA ( y ) . y β ∂∂y β A ( y ) (cid:12)(cid:12) y = y ( τ ) = 0 , (34) .. y α + . y β . y γ Γ αβγ + B αβ (cid:18) p a ( M ) C ( y ) ∂∂y β C ( y ) + 12 ∂∂y β A ( y ) (cid:19) (cid:12)(cid:12) y = y ( τ ) − . y α A ( y ) . y β ∂∂y β A ( y ) (cid:12)(cid:12) y = y ( τ ) = 0 . (35)where Γ αβγ is the Christoffel connection for the metric B αβ in the extra dimen-sions (cf. (30)).Here we will be interested in the case of constant brane tension: ∂ τ χ = 0 → . y α ∂∂y α A (cid:12)(cid:12) y = y ( τ ) = 0 from Eq . (34) . (36)Thus we arrive at the following compatible system of equations describing anontrivial motion of the LL-brane in the extra dimensions: . y α ∂∂y α A (cid:12)(cid:12) y = y ( τ ) = 0 , . y α ∂∂y α C (cid:12)(cid:12) y = y ( τ ) = 0 , (37) − A ( y ( τ )) + B αβ ( y ( τ )) . y α . y β = 0 , (38) .. y α + . y β . y γ Γ αβγ + B αβ (cid:18) p a ( M ) C ( y ) ∂∂y β C ( y ) + 12 ∂∂y β A ( y ) (cid:19) (cid:12)(cid:12) y = y ( τ ) = 0 (39)uendelman,Kaganovich,Nissimov,Pacheva 7 In this case: y α = ( ρ, φ ) , B αβ ( y ) dy α dy β = dρ + ρ dφ ; (40) A = A ( ρ ) , C = C ( ρ ) ; . ρ = 0 , i . e . ρ = ρ = const . (41)Eqs.(38) and (39) yield correspondingly: − A ( ρ ) + ρ . φ = 0 ; (42) − ρ . φ + (cid:18) p a ( M ) C ( ρ ) ∂ ρ C + 12 ∂ ρ A ) (cid:19) (cid:12)(cid:12) ρ = ρ = 0 , .. φ = 0 . (43)The last Eq.(43) implies: φ ( τ ) = ωτ , (44)which upon substituting into (42)–(43) gives: ω = A ( ρ ) ρ , A ( ρ ) = ρ (cid:18) p a ( M ) C ( ρ ) ∂ ρ C + 12 ∂ ρ A ) (cid:19) (cid:12)(cid:12) ρ = ρ . (45)Thus, we find that the LL-brane performs a planar circular motion in the flatextra dimensions whose radius ρ and angular velocity ω are determined from(45). This property of the LL-branes has to be contrasted with the usual caseof Nambu-Goto-type braneworlds which (in the ground state) occupy a fixedposition in the extra dimensions. In this case: y α = ( θ, φ ) , B αβ ( y ) dy α dy β = dθ + sin ( θ ) dφ ; (46) A = A ( θ ) , C = C ( θ ) ; . θ = 0 , i . e . θ = θ = const . (47)Eqs.(38) and (39) yield correspondingly: − A ( θ ) + sin ( θ ) . φ = 0 ; (48) − sin( θ ) cos( θ ) . φ + (cid:18) p a ( M ) C ( θ ) ∂ θ C + 12 ∂ θ A (cid:19) (cid:12)(cid:12) θ = θ = 0 , .. φ = 0 . (49)Therefore, once again we obtain: φ ( τ ) = ωτ , (50)which upon substituting into (48)–(49) gives: ω = A ( θ )sin ( θ ) , A ( θ ) = tan( θ ) (cid:18) p a ( M ) C ( θ ) ∂ θ C + 12 ∂ θ A (cid:19) (cid:12)(cid:12) θ = θ . (51)As in the case of flat extra dimensions, Eqs.(51) determine the position θ of thecircular orbit of the LL-brane and its angular velocity ω . LightlikeBraneworlds In this case: y α = ( θ, φ ) , ≤ θ, φ ≤ π , B αβ ( y ) dy α dy β = dθ + a dφ ; (52)Eqs.(37)–(39) assume the form: (cid:16) . θ ∂ θ A + . φ ∂ φ A (cid:17) (cid:12)(cid:12) θ = θ ( τ ) , φ = φ ( τ ) = 0 , (cid:16) . θ ∂ θ C + . φ ∂ φ C (cid:17) (cid:12)(cid:12) θ = θ ( τ ) , φ = φ ( τ ) = 0 ; (53) − A (cid:12)(cid:12) θ = θ ( τ ) , φ = φ ( τ ) + . θ + a . φ = 0 ; (54) .. θ + (cid:18) p a ( M ) C ∂ θ C + 12 ∂ θ A (cid:19) (cid:12)(cid:12) θ = θ ( τ ) , φ = φ ( τ ) = 0 , .. φ + (cid:18) p a ( M ) C ∂ φ C + 12 ∂ φ A (cid:19) (cid:12)(cid:12) θ = θ ( τ ) , φ = φ ( τ ) = 0 . (55)Eqs.(53) can be solved by taking A ( θ, φ ) and C ( θ, φ ) as functions of only onecombination ξ ( θ, φ ) such that: A = A (cid:0) ξ ( θ, φ ) (cid:1) , C = C (cid:0) ξ ( θ, φ ) (cid:1) (56) ddξ A (cid:12)(cid:12) θ = θ ( τ ) , φ = φ ( τ ) = 0 , ddξ C (cid:12)(cid:12) θ = θ ( τ ) , φ = φ ( τ ) = 0 . (57)Taking into account (57), Eqs.(55) imply: .. θ = 0 , .. φ = 0 , i . e . θ ( τ ) = ω τ , φ ( τ ) = ω τ . (58)Furthermore, taking into account the periodicity of A and C w.r.t. ( θ, φ ) we find: ξ ( θ, φ ) = θ − N φ , ω = N ω , (59)where N is abritrary positive integer. In other words, from (56)–(57) the admiss-able form of the background metric must be of the form: A = A ( θ − N φ ) , C = C ( θ − N φ ) , A ′ (0) = 0 , C ′ (0) = 0 , (60)whereas Eq.(54) determines the angular frequencies ω , in (58): ω = A (0)1 + a /N , ω = ω N . (61)A particular choice for A (and similarly for C ) respecting conditions (57) is: A = A sin (cid:0) θ − N φ (cid:1) + A , A , = positive const . (62)Thus, we conclude that the LL-brane performs a spiral motion in the toroidalextra dimensions with winding frequences as in (61).uendelman,Kaganovich,Nissimov,Pacheva 9 In the present paper we presented a systematic Lagrangian formulation oflightlike p -branes in arbitrary ( p + 1) world-volume dimensions allowing inaddition for natural (gauge-invariant) couplings to bulk electromagnetic andKalb-Ramond gauge fields. In the context of “brane-world scenarios” light-like braneworlds (of codimension two or more) in their ground state per-form non-trivial motions in the extra dimensions unlike ordinary Nambu-Gotobraneworlds which position themselves at certain fixed points in the extra di-mensions.The special case of codimension one LL-branes needs separate study whichis relegated to a subsequent paper. As already discussed in refs. [8] for light-like membranes ( p = 2 ) in D = 4 bulk space-time, the LL-brane dynamicsdictates that the bulk space-time must possess an event horizon which is auto-matically occupied by the LL-brane (an explicit dynamical realization of the“membrane paradigm” in black hole physics [2]). Extending our treatment inrefs. [8], we will study the important issue of self-consistent solutions for bulkgravity-matter systems (e.g., Einstein-Maxwell-type) coupled to lightlike braneswhere the latter serves as a source for gravity, electromagnetism, dynamicallyproduces space-varying cosmological constant and triggers non-trivial matchingof two different space-time geometries across common event horizon spannedby the lightlike brane itself.Let us mention the observation in ref. [14], that large extra dimensions couldbe rendered undetectable (due to the zero eigenvalue of the induced metric) ifour world is considered as a lightlike brane moving in D > bulk space – pre-cisely the brane-world scenario obtained in the present paper from the consistentunified dynamical (Lagrangian) description of lightlike branesTo stress again, in our formalism we consider the intrinsic metric γ ab onthe world-volume of the lightlike brane to be the metric that defines the geom-etry experienced by an observer confined to the brane. This is in contrast tothe induced metric (9), which as a result of lightlike nature of the brane is nec-essarily singular , having spacelike components and a zero eigenvalue, i.e. alightlike instead of timelike one. Nevertheless, it is possible to ascribe a phys-ical role to singular induced metrics provided they possess an additional time-like (diagonal) component. The latter can be achieved by considering LL-branewith ( p + 2) -dimensional world-volume (cf. (5)) embedded in a D -dimensional( D > p + 2 ) bulk space with two timelike dimensions ( G µν having signature ( − , − , + , . . . , +) ). Repeating the steps in Section 2 we will get an induced ( p + 2) × ( p + 2) metric (9) with signature (0 , − , + , . . . , +) , i.e., with one light-like, one timelike and p spacelike dimensions. Then, applying the formalismfor degenerate metrics proposed in ref. [11], one can employ the resulting in-duced metric as a starting point for construction of a Kaluza-Klein model withthe pertinent lightlike brane (with ( p + 2) -dimensional world-volume) as a to-tal Kaluza-Klein space with naturally unobservable extra dimension (the firstlightlike one) from the point of view of the “normal” ( p + 1) world-volume di-mensions. Also, let us note that the use of an embedding space-time with two0 LightlikeBraneworldstimelike coordinates has an advantage if we want to obtain a Lorentz-invariantground state, since there is the possibility of having one additional time, notinvolved in the motion of the lightlike brane. Acknowledgments
E.N. is sincerely grateful to the organizers of the 7-th International Workshop“Lie Theory and Its Applications in Physics”, Varna (2007). E.N. and S.P. aresupported by European RTN network “Forces-Universe” (contract No.MRTN-CT-2004-005104). They also received partial support from Bulgarian NSF grantF-1412/04. Finally, all of us acknowledge support of our collaboration throughthe exchange agreement between the Ben-Gurion University of the Negev (Beer-Sheva, Israel) and the Bulgarian Academy of Sciences.
References [1] C. Barrab´es and P. Hogan, “Singular Null-Hypersurfaces in General Relativity” ,World Scientific, Singapore (2004).[2] K. Thorne, R. Price and D. Macdonald (eds.), “Black Holes: The MembraneParadigm” , Yale Univ. Press, New Haven, CT (1986).[3] W. Israel, NuovoCim.
B44 , (1966) 1; erratum, ibid
B48 , 463 (1967).[4] C. Barrab´es and W. Israel, Phys. Rev.
D43 , 1129 (1991); T. Dray and G. ‘t Hooft,Class.QuantumGrav. , 825 (1986).[5] I. Kogan and N. Reis, Int.J.Mod.Phys. A16 , 4567 (2001) (hep-th/0107163).[6] D. Mateos and S. Ng, JHEP , 005 (2002) (hep-th/0205291).[7] C. Barrab´es and W. Israel, Phys.Rev.
D71 , 064008 (2005) (gr-qc/0502108).[8] E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva, hep-th/0409078; in“SecondWorkshoponGravity,AstrophysicsandStrings”, edited by P. Fiziev et.al.(Sofia Univ. Press, Sofia, Bulgaria, 2005) (hep-th/0409208); in “Third Internat.SchoolonModernMath.Physics”, Zlatibor (Serbia and Montenegro), edited by B.Dragovich and B. Sazdovich (Belgrade Inst. Phys. Press, 2005) (hep-th/0501220);Phys. Rev.
D72 , 0806011 (2005) (hep-th/0507193);
Self-Consistent Solutionsfor Bulk Gravity-Matter Systems Coupled to Lightlike Branes , hep-th/0611022;Fortschritte der Physik , 579 (2007) (hep-th/0612091); in “Fourth Internat.SchoolonModernMath.Physics”, Belgrade (Serbia and Montenegro), edited by B.Dragovich and B. Sazdovich (Belgrade Inst. Phys. Press, 2007) (hep-th/0703114).[9] V. Rubakov and M. Shaposhnikov, Phys. Lett. , 139 (1983); M. Visser, Phys.Lett. , 22 (1985); P. Horava and E. Witten, Nucl. Phys. B460 , 506 (1996)(hep-th/9510209); Nucl. Phys.
B475 , 94 (1996) (hep-th/9603142); I. Antoniadis,N. Arkani-Hamed, S. Dimopoulos and G. Dvali,Phys. Lett. , 257 (1998); L.Randall and R. Sundrum, Phys.Rev.Lett. , 3370, 4690 (1999).[10] V. Rubakov, Phys. Usp. , 871 (2001) (hep-th/0104152); P. Brax and C. van deBruck, Class.QuantumGrav. , R201 (2003) (hep-th/0303095).[11] T. Searight, Gen.Rel.Grav. , 791 (2003) (hep-th/0405204).[12] A. Davidson and E. Guendelman, Phys.Lett. , 250 (1990).[13] A. Aurilia, H. Nicolai and P. Townsend, Nucl.Phys. B176 , 509 (1980); A. Aurilia,Y. Takahashi and P. Townsend, Phys.Lett. , 265 (1980)[14] M. Beciu, Europhys.Lett.12