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February5,2008 15:20 WSPC-ProceedingsTrimSize:9.75inx6.5in BraneMG11 1 7 Lorentz symmetry from Lorentz violation in the bulk 0 0 2 OrfeuBertolami1,2 andCarlaCarvalho1,2 n 1 Departamento de F´ısica, Instituto Superior Técnico, Avenida Rovisco Pais 1, 1049-001 a Lisboa, Portugal J 8 2 Centro de F´ısica dos Plasmas, Instituto Superior Técnico, Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal 1 v E-mail addresses: [email protected], [email protected] 1 9 Weconsiderthemechanismofspontaneous symmetrybreakingofabulkvectorfieldto 1 studysignaturesofbulkdimensionsinvisibletothestandardmodelconfinedtothebrane. 1 Byassigninganon-vanishingvacuumexpectationvaluetothevectorfield,adirectionis 0 singledoutinthebulkvacuum,thusbreakingthebulkLorentzsymmetry.Wepresentthe 7 condition for induced Lorentz symmetryonthe brane, as phenomenologically required, 0 notingthatitisrelatedtothevalueoftheobservedcosmologicalconstant. / h p - 1. Introduction o r Braneworld scenarios have changed our view of the extra dimensions. The various t s modelspredictthatgravityinourbraneworldcanexhibitsignificantdeviationsfrom a thatdescribedbyEinstein’sgeneralrelativity.Inparticular,instringtheoryinspired : v scenarios which assume that the background bulk spacetime is anti-de Sitter, it is i X possible to cancel out any 4-dimensional brane contribution to the cosmological r constant(seee.g.[1]andreferencestherein).Althoughnotonitsownasolutionfor a thecosmologicalconstantproblem,itissuggestivethatbraneworldscenariosmight be an important feature of a consistent description of the world. It is therefore relevant to investigate the implications of the braneworldscenar- ios to the formulation of fundamental symmetries, another fundamental ingredient of the physical description. Lorentz symmetry, being from the phenomenological point of view one of the most well and stringently tested symmetries of physics, is particularly suitable to test the relation between bulk and brane symmetries as a possible signature for the existence of extra dimensions. The possibility ofviolationofLorentz invariancehas been extensivelydiscussed in the recent literature (see e.g. [4]) and in particular its astrophysicalimplications have been studied.5 Furthermore, a connection between the cosmological constant and the violation of Lorentz invariance has been conjectured in the context of the string field theory.6 In this contribution we report on a recent study whose motivation was to un- February5,2008 15:20 WSPC-ProceedingsTrimSize:9.75inx6.5in BraneMG11 2 derstand the way spontaneous Lorentz violation in the bulk is related to Lorentz symmetry on the brane.2 We consider a bulk vector field coupled non-minimally to the graviton which, upon acquiring a non-vanishing expectation value in the vacuum, introducesspacetime anisotropiesinthe gravitationalfield equationsthrough the couplingwiththegraviton.3 Afterderivingtheequationsofmotioninthebulk, we project them parallel and orthogonal to the surface of the brane. The brane is assumed to be a distribution of Z –symmetric stress-energy about a shell of thick- 2 ness 2δ in the limit δ →0. Derivatives of quantities discontinuous across the brane will generate singular distributions on the brane which relate to the localization of the stress-energy. This relation is encompassed by the matching conditions across the brane obtained by the integration of the corresponding equation of motion in the direction normal to the brane. The matching conditions provide the boundary conditions on the brane for the bulk fields, thus constraining the parallel projected equations to produce the induced equations on the brane. Spontaneous symmetry breaking is then treated by assuming that the bulk vector field acquires a non- vanishing expectationvalue whichreflectsonthe branethe breakingofthe Lorentz symmetry in the bulk. 2. Bulk Vector Field Coupled to Gravity Aiming to examine the gravitational effects of the breaking of Lorentz symmetry in a braneworld scenario, we consider a bulk vector field B with a non-minimal coupling to the gravitonin a five-dimensionalanti-de Sitter space.The Lagrangian densityconsistsoftheHilbertterm,thecosmologicalconstantterm,thekineticand potential terms for B and the B–gravitoninteraction term, as follows 1 1 L= R−2Λ+ξBµBνR − B Bµν −V(BµB ±b2), (1) κ2 µν 4 µν µ (5) where B = ∇ B −∇ B is the tensor field associated with B and V is the µν µ ν ν µ µ potential which induces the spontaneous global symmetry breaking when the B field is drivento the minimum at BµB ±b2 =0, b2 being a realpositive constant. µ Here, κ2 = 8πG = M3 , M is the five-dimensional Planck mass and ξ is a (5) N Pl Pl dimensionless coupling constant that we have inserted to track the effect of the interaction. In the cosmological constant term Λ = Λ +Λ we have included (5) (4) both the bulk vacuum value Λ and that of the brane Λ , described by a brane (5) (4) tension σ localized on the locus of the brane, Λ =σδ(N). (4) The Einstein equation is given by: 1 1 G +Λg −ξL −ξΣ = T , (2) κ2 µν µν µν µν 2 µν (5) where 1 L = g BρBσR −(B BρR +R BρB ), (3) µν µν ρσ µ ρν µρ ν 2 1 Σ = ∇ ∇ (B Bρ)+∇ ∇ (B Bρ) −∇2(B B )−g ∇ ∇ (BρBσ) (4) µν µ ρ ν ν ρ µ µ ν µν ρ σ 2 (cid:2) (cid:3) February5,2008 15:20 WSPC-ProceedingsTrimSize:9.75inx6.5in BraneMG11 3 are the contributions from the interaction term and 1 T =B B ρ+4V′B B +g − B Bρσ−V (5) µν µρ ν µ ν µν ρσ (cid:20) 4 (cid:21) isthecontributionfromthevectorfieldforthestress-energytensor.Fortheequation of motion for the vector field B, we find that ∇ν(∇ B −∇ B )−2V′B +2ξBνR =0. (6) ν µ µ ν µ µν whereV′ =dV/dB2.Projectingtheequationsparallel(A)andorthogonal(N)tothe surfaceof the brane,we proceededto integratethem in the normaldirectionto the extract the matching conditions. These conditions constrain the parallel projected equations to yield the induced equations on the brane. The generalfeatures of this procedure have been previously discussed.7 When the bulk vector field B acquires a non-vanishing, covariantly conserved3 vacuum expectation value by spontaneous symmetry breaking, the bulk vacuum acquires an intrinsic direction determined by hB i, thus inducing the breaking A of the Lorentz symmetry in the bulk. In order to obtain a vanishing cosmological constantandensurethatLorentzinvarianceholdsonthebrane,wetaketheEinstein equation induced on the brane and impose respectively that 1 Λ = (1−2(ξ−1))Kσ (7) (5) 2 and that 1 1 2K K − +ξ−1 K K κ2 (cid:20) AC BC (cid:18)2 (cid:19) AB (5) 1 + g R(ind)−2K K −(1−2(ξ−1))K2 AB CD CD 2 (cid:16) (cid:17)(cid:21) ξ 5 2 = −2+ hB ihB iR(ind)+hB ihB iR(ind) −(4ξ+2)K K hB ihB i 2(cid:20)(cid:18)2 ξ(cid:19)(cid:16) A C CB B C AC (cid:17) AC BD C D (cid:21) ξ + g hB ihB iR(ind)+2(ξ−1)K K hB ihB i , (8) 2 AB(cid:20) C D CD CE ED C D (cid:21) which for ξ =1 reduce to the results presented in [2]. 3. Discussion and Conclusions In this contributionwe examine the spontaneous symmetry breakingof Lorentzin- variance in the bulk and its effects on the brane. For this purpose, we considered a bulk vectorfieldsubjectto apotentialwhichendowsthe fieldwitha non-vanishing vacuumexpectationvalue,thusallowingforthespontaneousbreakingoftheLorentz symmetry in the bulk. This bulk vector field is directly coupled to the Ricci tensor so that, after the breaking of Lorentz invariance, the breaking of this symmetry is transmitted to the gravitational sector. We assign a non-vanishing vacuum expectation value to the component of B parallel to the brane (the generality of this procedurehasbeendiscussedin[8]).Weobservethatthereisaconnectionbetween February5,2008 15:20 WSPC-ProceedingsTrimSize:9.75inx6.5in BraneMG11 4 the vanishing of the cosmological constant and the reproduction of the Lorentz invariance on the brane. The conditions above were enforced so that the higher dimensional signatures encapsulated in the induced geometry of the brane cancel the Lorentz symmetry breaking inevitably induced on the brane, thus reproduc- ing the observed geometry. Naturally, the first condition, Eq. (7), can be modified to account for any non-vanishing value for the cosmological constant induced on the brane.A much more elaborate fine-tuning, however,is requiredfor the Lorentz symmetry to be observed on the brane, as expressed by the condition Eq. (8). We believe that this is a new feature in braneworld models, as in most such models Lorentz invariance is a symmetry shared by both the bulk and the brane. Notice that a connection between the cosmological constant and Lorentz symmetry had been conjectured long ago.6 We shall examine further implications of this mecha- nismina forthcomingpublicationwherewe willalsodiscuss the inclusionofabulk scalar field.9 References 1. O. Bertolami, “The Adventuresof Spacetime”, gr-qc/0607006. 2. O. Bertolami and C. Carvalho, Phys.Rev. D74(2006) 084020. 3. V.A. Kostelecky´, Phys. Rev. D69 (2004) 105009; R. Bluhm and V.A. Kostelecky´, Phys.Rev.D71(2005)065008;O.BertolamiandJ.Páramos,Phys.Rev.D72(2005) 044001. 4. CPT and Lorentz Symmetry III, Alan Kostelecky´, ed. (World Scientific, Singapore, 2005); O. Bertolami, Gen. Rel. Gravitation 34 (2002) 707; O. Bertolami, Lect. Notes Phys. 633 (2003) 96, hep-ph/0301191; D. Mattingly, Liv. Rev. Rel. 8 (2005) 5, gr-qc/0502097; R. Lehnert, “CPT- and Lorentz-symmetry breaking: a review”, hep-ph/0611177. 5. H. Sato, T. Tati, Prog. Theor. Phys. 47 (1972) 1788; S. Coleman and S.L. Glashow, Phys. Lett. B405 (1997) 249; Phys. Rev. D59 (1999) 116008; O. Bertolami and C. Carvalho, Phys. Rev. D61 (2000) 103002. 6. O. Bertolami, Class. Quantum Gravity 14 (1997) 2785. 7. M. Bucher and C. Carvalho, Phys.Rev. D71 (2005) 083511. 8. O. Bertolami and C. Carvalho, “Brane Lorentz Symmetry from Lorentz Breaking in theBulk”, gr-qc/0612129. 9. O. Bertolami and C. Carvalho, in preparation.