PoS(QG-Ph)009 http://pos.sissa.it/ ∗ [email protected] Speaker. Spontaneous breaking of local Lorentzquires symmetry a occurs when nonzero a vacuum local expectationcontext vector value. of or tensor The theory. field effects These ac- of includesymmetry and such an generation associated breaking of massless spontaneous are Nambu-Goldstone breaking modes. examined ofanism The in diffeomorphism possibility is the of examined a as Higgs well, mech- andmassive it is gauge found fields) that does the not conventionalspacetime Higgs occur mechanism a in (giving Higgs rise a mechanism to Riemann involving .conventional the Higgs spin However, mechanism connection in in is a Riemann possible. Riemann-Cartan metric spacetime, can additional Despite appear massive the through modes lack unconventional involving processes oftheory. the that a The have effects no of analogue these in typesbumblebee nonabelian of model, processes gauge in are which a illustrated vector using field a acquires specific a model, vacuum known value. as a ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

From Quantum to Emergent Gravity: TheoryJune and 11-15 Phenomenology 2007 Trieste, Italy Robert Bluhm Effects of Spontaneous Lorentz Violation in Gravity Colby College, Waterville, ME 04901, USA E-mail: PoS(QG-Ph)009 ]. ], 7 5 , , 6 4 Robert Bluhm 2 ], as well as in the references cited within these works. 9 , 8 , 7 , 6 ]). At the same time, a comprehensive phenomenological investigation of 3 ]. Indeed, it was the idea that mechanisms in string field theory might lead to , 6 2 , 1 Many of the main results presented here, including background on bumblebee models, are One of the more elegant ideas for Lorentz violation is that this symmetry might be sponta- However, as soon as one begins to discuss the idea of spontaneous symmetry breaking, well It is these types of effects that are examined here for the case where it is local Lorentz sym- Interest in the possibility of Lorentz violation has increased significantly over the years, as described in greater detail in [ It is this type of modelis that then is followed used by here a to more illustrate general the discussion effects of of the phenomenology. NG and massive modes. This neously broken [ spontaneous Lorentz violation thatLorentz helped breaking. stimulate much Moreover, one of ofthat the the they current primary are interest interpretations vacuum in of expectationcouple the the values to coefficients conventional (combined topic matter in with of at the Yukawa low couplings) energy.gation SME Thus, of of is one Lorentz tensor of violation fields the using products that thebreaking of SME originating a is from comprehensive investi- the a Planck survey scale. for possible signals of spontaneous Lorentz known results from particle physicspearance immediately of come massless into Nambu-Goldstone play. (NG)the modes, These question the of include whether possibility the additional of possible massive a modes ap- (analogous Higgs to mechanism, the and Higgs boson)metry can that arise. is spontaneously broken.can Clearly, any have important processes implications generating for massless phenomenology. orbreaking massive This of modes is Lorentz particularly symmetry the in casemodes the for might context influence spontaneous gravitational of propagation gravity, or where alterThus, the the in form effects of addition of the to the static looking Newtonian NGHiggs potential. at and mechanism the can massive fate occur, it of is the importantin NG as the well modes context to of in look specific general at models the and thatexample role permit the is of an question for the examination the NG of of and case their whether massive effects of a modes on a gravity. vector The field, simplest where such models are known as bumblebee models [ which provides the most generalCPT effective violation. field-theoretical framework Using incorporating thethe Lorentz SME, context and detailed of high-energy investigations particle of physics, Lorentzastrophysics. gravitational breaking physics, can nuclear This and be on-going atomic conductedbreaking effort physics, in well and has beyond pushed levels associated experimentalsignals with bounds remain suppression untested. on by In the some addition Planck formsbreaking to mass. continues these of to phenomenological Nonetheless, Lorentz undergo investigations, many theoretical the scrutiny. ideaing In of in particular, Lorentz the possible context effects of of gravity Lorentzinteresting have break- prospects begun for to alternative be explanations explored, of and such in things certain as cases dark these matter ideas and can dark lead energy. to Lorentz violation has been initiated thatinvariance. has led This to was a number facilitated of by new high-precision development tests of of the Lorentz Standard-Model Extension (SME) [ it has been realized thatLorentz symmetry effects at arising low energy. at These thecommutative include Planck mechanisms geometry, in scale quantum string might gravity, theory, field give effectsreviews, theories rise due in see to non- to [ small modified breakings dispersion of relations, etc. (For Effects of Spontaneous Lorentz Violation in Gravity 1. Introduction PoS(QG-Ph)009 (2.1) (2.2) relates a µ e Robert Bluhm (where Latin indices denote abc T ]. Under an observer general coor- 4 . enters in covariant derivatives that act on . abc ab T ab µ c ν η ω e b ν b µ e e a µ 3 a λ e e transformations [ = = µν g λ µν T particle and observer (where Greek indices label the spacetime frame) . For example, the spacetime λ µν T A fundamental premise is that a physical theory should always be observer independent. This In , Lorentz symmetry is a global symmetry consisting of rotations and boosts One way to reveal the transformation properties of vectors and tensors under both local Lorentz In considering theories with violation of Lorentz and diffeomorphism symmetry it is important includes even when Lorentz symmetry andbroken. diffeomorphisms are In either fact, explicitly the or SME spontaneously is based on this. It is formulated as a lagrangian-based field theory that dinate or local , vectors andbases tensors used remain to unchanged, define while their the components coordinate transform.transformations In change contrast, vectors particle diffeomorphisms and and tensors, Lorentz basis while vectors. leaving unchanged In the theories coordinateparticle systems with transformations and no are symmetry inversely breaking, relatedsymmetries the but are transformation otherwise spontaneously laws are broken for similar andthe observer transformation fields in laws and are form. for divided these However, into will if differ vacuum for the values the and observer and excitations, particle transformations. However, in curved spacetime, inIt a transforms gravitational local theory, vectors Lorentz andto symmetry tensors being is locally in a Lorentz the local invariant, tangentThese a symmetry. transformations plane gravitational act at theory on tensor each is and also spacetime vectorit fields invariant is point. defined the under diffeomorphism on diffeomorphisms. symmetry In the that spacetime addition is manifold. moreLorentz readily Typically, symmetry apparent in acts a gravitational only theory, since inare local important, local and frames. a complete However,examination discussion ultimately of of both its Lorentz effects types on breaking of diffeomorphisms in as transformations gravity well. theory should include an transformations and diffeomorphisms is by using a vierbein formalism. The vierbein Effects of Spontaneous Lorentz Violation in Gravity 2. Spontaneous Lorentz Breaking tensor components defined withcomponents respect with respect to to a a local frame), localsystem, to those e.g., basis, defined with e.g., respect to the spacetime coordinate Similarly, for an arbitrary tensor, A vierbein formalism also allowsallels spinors gauge to theory, be with incorporated Lorentzgroups. into symmetry In the and a theory, vierbein diffeomorphisms and formalism, both it the spin acting naturally connection as par- local symmetry metric and local Minkowski metric are related by to distinguish between local tensor components and plays the rolethe of metric the excitations act gauge as field the for gauge thea fields Lorentz for vierbein symmetry. the formalism, In diffeomorphism there contrast, symmetry. are When primarily workinggeometry with two (with geometries that no can torsion), bethat distinguished. the does In spin Riemannian not connection propagate. isconnection nondynamical. However, must in be Riemann-Cartan It treated geometry is as (with independent purely degrees nonzero an of torsion), freedom auxiliary the that field spin in principle can propagate. PoS(QG-Ph)009 (3.2) (2.6) (3.1) (2.5) (2.3) (2.4) Robert Bluhm , has a minimum when ) directly as a constraint, λ µν T 2.5 in the local frame is chosen, abc t , . 2 ) . 2 2 abc t . t t ± ∓ pqr can impose ( t , = . λ rc . αβγ a η T µ abc λ µν t t δ αβγ qb νγ T η = g = = νγ pa 4 > > µβ > g η g a µ µβ abc e λα λ µν abc T g t . Such a vev spontaneously breaks particle local Lorentz g T < < λα = abc < g t 2 λ µν t T = ( ∓ λ µν > ) span a degenerate space of possible vacuum solutions. Spon- T ∼ abc 2.5 V T < to the Lagrangian. For example, a potential of the form V To examine the fate of the NG modes in a theory with spontaneous Lorentz violation, a general In a theory with spontaneous breaking of a particle spacetime symmetry, the lagrangian still In a gravitational theory, local Lorentz symmetry is spontaneously broken when a local tensor Spontaneous Lorentz breaking can be introduced into a theory dynamically by adding a po- Alternatively, a potential with a Lagrange multipler field The spacetime tensor therefore has a vev as well, approach can first be followed.Lorentz frame, Consider for a example, theory with a tensor that has a nonzero vev in a local satisfying the condition 3. Nambu-Goldstone Modes symmetry. In addition, the vierbein alsoa has background a Minkowski constant spacetime, or fixed background value. For example, in taneous Lorentz breaking occurs when a particular vacuum value which also leads to spontaneous selection of a vacuum value Note that the sign oncomponents. the right-hand Solutions side of depends Eq. on ( the timelike or spacelike nature of the tensor remains invariant under the brokenHowever, symmetry, fixed and vacuum-valued the fields full appear equationsformations. that of cannot motion Interaction be remain terms transformed covariant. involving underequations these the of fixed particle motion, background trans- which vacuum by fieldsthese themselves also vacuum break appear fields the in that particle the cantested symmetry. lead in It to experiments. is physical the effects interaction of with the broken particle symmetryfield that acquires can a be vacuum expectationtensor, value (vev). For example, for the case of a three-component tential term Effects of Spontaneous Lorentz Violation in Gravity is fully invariant under observer generaltions. coordinate transformations and local Lorentz transforma- The vacuum of the theory thentaneously has breaks preferred the spacetime particle directions Lorentz in symmetry. the local frames, which spon- consisting of a quadratic function of products of the tensor components PoS(QG-Ph)009 (3.5) (3.3) (3.6) (3.4) . Next, . They V νµ χ − Robert Bluhm ]. In this form, = . 8 α µν means that particle λ µ χ t ) λ µν να T χ + has 16 components. With no να h a µ 2 1 e + ( , ) . . α µν ) λ ν t χ αβγ ) t + λ µν t µα νγ e χ µν − h + µβ 2 1 e 5 λ µν µα T that contains the NG fields. Indeed, with the appro- + ( λα h , and antisymmetric components, e ) 1 2 . This fixed vacuum value for µν = ( νµ = µν η h abc + ( χ t = = λ µν + µν λ µν τ α T µν µν t µν ) h e h 2 1 ). A solution of this condition is given by the vierbein acting on acts on λα ( χ > 2.5 + a µ e λα h < 1 2 ( ≈ λ µν τ These results can be obtained as well using an expansion of the vierbein in terms of infinites- Spontaneous breaking of these symmetries implies that NG modes should appear (in the ab- A related question asks where the ten NG modes reside. In general, the answer depends on Inserting the expansion of the vierbein intoto this lowest equation order and gives solving for the tensor-field excitations priate gauge choices, an expansion of thisdiffeomorphisms combination shows as explicitly virtual that local the Lorentz NG transformations modes and reside in these components [ in terms of symmetric components, the local vev, and is equal to imal excitations about the vacuum.spacetime components For can such be small dropped excitations, (withvierbein the Greek (with letters lowered distinction indices) being between is used local then for and written both as from here on). The spontaneous Lorentz breaking, the sixbe Lorentz used and to reduce four the diffeomorphismgravitational vierbein degrees theory down of can to freedom have six can six independentin propagating degrees that metric of there modes; freedom. are however, general only (Noteten relativity two). that is spacetime a In special symmetries general contrast, are in broken,Therefore, a the theory a vierbein with second can spontaneous result have Lorentzmodes is 16 breaking, can that where propagating in all appear degrees a of andvierbein. freedom. theory all with of spontaneous them Lorentz can breaking, naturally up be to ten incorporated NG as degrees of freedom in the consider small excitations ofcomponent the tensor, these tensor have the field form about its vacuum value. For the case of a three- Evidently, it is the combination sence of a ).usual This rule raises is the that question therecase, of can maximal how symmetry-breaking be many would up NG yield to modes sixfeomorphisms. broken can as Therefore, Lorentz appear. many there generators NG The can and be four modes up broken as to dif- there ten NG are modes broken in symmetries.the general. choices In of this gauge.as However, a one simple natural counting choice argument is shows to is put possible. all ten The NG vierbein modes into the vierbein, Effects of Spontaneous Lorentz Violation in Gravity which is obtained when diffeomorphisms are spontaneously broken. Thus, a firstof general local result is Lorentz that symmetry spontaneous implies breaking spontaneous breaking of diffeomorphisms. The NG modes aretherefore the obey field the excitations condition that ( stay within the minimum of the potential PoS(QG-Ph)009 (4.1) 0 that 6= 0 while the V µν Robert Bluhm χ , as for example V ]. 9 only involves the scalar Higgs V . ··· + ]. This is because in the conventional 2 6 ) σ µν t σ ρλ ] that the form of the potential 6 Γ ( 6 ∼ 2 ) λ µν T ρ D , resulting in an alternative Higgs mechanism that has no direct ( V ]. However, the question of whether ghost modes are generated must also be 10 ), does permit quadratic terms involving excitations of the metric to appear. This results 2.4 . However, it was also pointed out in Ref. [ Summarizing for the case of diffeomorphisms, the general results are that there is no con- In the context of gravity, since Lorentz symmetry is a local symmetry, the possibility of a Higgs First, consider the case of diffeomorphisms. For this symmetry, the corresponding gauge µν h Since the connection consists ofmass terms derivatives generated of for the the metric. metric,metric. As and a not result, there the is metric no itself, conventional Higgs there mechanism are for the no ventional Higgs mechanism for thedue ; to however, mass the terms form involving of the the metric potential may arise fields (and not the gauge fields).is combinations However, in of the both case the of tensorterms. spontaneous field diffeomorphism and Since breaking, the the it metric field metric excitationspossible appears that in in acquire these quadratic principle mass terms, to butdiscontinuity generate not [ in mass the terms usual that Fierz-Pauli avoid form, the it becomes van Dam, Veltmann, and Zakharov parallel in nonabelian . in an alternative form oftheory. This the is Higgs because mechanism in that nonabelian gauge has theory, no the direct potential analogue in nonabelian gauge addressed. This typically becomes a model-dependentthe issue, tensor since fields the form can of influence the whethermodes kinetic the appear. terms massive-mode for In excitations some propagate models, andfields. the whether massive ghost modes However, do even not in propagate,interactions, these but including instead cases, possible remain modifications the auxiliary of massive the Newtonian modes potential can [ have an influence on gravitational Higgs mechanism the quadratic termsfor (that the field give acquiring rise a to vev (the masscovariant tensor derivatives acting terms) field on in come the this tensor from case). field. the These However,to for kinetic kinetic gauge-covariant) diffeomorphism-covariant terms (as derivatives terms consist opposed it of is products the of connection that appears in quadratic form. For example, in Eq. ( 4. Gravitational Higgs Mechanism mechanism naturally arises. However, since therediffeomorphisms) are there two are sets of potentially broken twoalso symmetries the associated (Lorentz possibility Higgs and that mechanisms. additional massive modes Furthermore, can there arise is as field excitations obeying Effects of Spontaneous Lorentz Violation in Gravity the Lorentz NG modes candiffeomorphism NG be modes associated can be with most thein closely associated six with antisymmetric the components four gauge degrees of freedom fields are the metric (or vierbein) excitations,presumably and give therefore rise a to conventional Higgs mass mechanism terms would forHiggs the mechanism metric. involving However, it the was previously metric shown does that the not usual occur [ do not stay in the potential minimum. This section discusses these possibilities. PoS(QG-Ph)009 , µ B 25 is to , (5.1) (4.2) µ 24 B , 23 , Robert Bluhm 22 , 21 , . They can be defined 20 µ , b 19 ]. , 7 , 18 6 , , 17 M , gauge symmetry. This symmetry is ··· L ) 16 + 1 , 2 ( ) ) + U 15 2 , b αµν t ± 14 , µ α B ρ λ 13 7 µ ω , which in general has a functional form involving , ( B V ( 12 ∼ V , as the vector in a vector-tensor theory of gravity. In this 2 9 ) − µ , 0 8 B is the potential that induces spontaneous Lorentz breaking, that acquires a fixed vacuum value , L λ µν 7 µ V T , B = ρ 6 D ( L , µ ]. The relevant gauge field for the Lorentz symmetry is the . It appears ]. Incorporating Lorentz violation may lead to the appearance of additional mass terms, B . A model of this type is known as a bumblebee model [ 8 contains the kinetic terms, 11 V 0 contains additional interaction and matter terms. A particularly noteworthy feature that all L M The specific choice of kinetic terms is largely a reflection of how the bumblebee field The discussion in this section and the previous section shows that when Lorentz symmetry There are numerous examples of bumblebee models that have been explored in recent years. The next question is whether a Higgs mechanism can occur stemming from the broken local L ]). They all involve a vector field broken explicitly by the form ofproducts of the potential be interpreted. One approach is to view and bumblebee models share is that they do not have a local where 26 is spontaneously broken thereClearly, will any in theory general with be spontaneoustheir both Lorentz role massless breaking is NG must in modesalong account the and these for underlying massive lines these dynamics requires modes. modes described workingby and by in the what the the rank context of theory. the ofsimplest tensor a A acquiring example concrete more a is model. definite vev a and A investigation potential by theory given the with model forms is a of defined vector the kinetic field and that potential has terms. a The nonzero vacuum value induced by a (For some specific examples, see [ which could create new possibilitiespossibility for have been model carried building. out, but the Some search preliminary for viable investigations models of remains this largely an open problem. 5. Bumblebee Models generally in Riemann-Cartan spacetime, orbe restrictions considered. to A Riemann complete or definitionand Minkowski depends the spacetime on gravitational fields. the can choice A general of Lagrangian kinetic typically and has potential the terms form for Lorentz symmetry. For thecan Lorentz occur symmetry, [ it is found that a conventional Higgs mechanism Effects of Spontaneous Lorentz Violation in Gravity directly in expressions for covarianttensor derivatives that acting acquires a on vev, local quadratic mass tensorthe terms components. usual for the Higgs spin For mechanism. connection the For can example, local be kinetic generated, terms following of the form can generate quadratic terms forform the involving spin the connection. spin connection However, can a onlyThis viable requires occur Higgs that if mechanism there the is of spin nonzero this connection torsion and itselfa therefore is Higgs that a the mechanism dynamical geometry field. is for Riemann-Cartan. theConstructing Thus, spin a connection ghost-free model is with possible,problem a but [ propagating only spin in connection a is Riemann-Cartan known to geometry. be a challenging PoS(QG-Ph)009 µ ], V J µ 27 (5.5) (5.4) (5.3) (5.2) B is taken = µ imposes a M b L λ . One choice Robert Bluhm V ,  ν B ν D µ B µν µ and excludes massive-mode B D 3 µ µν τ B B 1 2 0). A second example is a linear 1 τ . + symmetry that gives rise to charge 6= 4 1 ν 0 µν ) B , 1 − B V µ 2 ( , ) R ) µν D 2 is that it is a generalized vector potential. U 2 µ ν b B b B µ B 1 4 µ ± B ± µ B µ µ − ). For simplicity, the vacuum value D 2 is chosen. To begin the analysis, a Riemannian 2 B B R σ τ 8 µ µ µ 5.4 2 1 G + J B B ( ( π µ 1 + κ µν λ B 1 2 16 R = ν = = B gauge symmetry in this class of models when a nonzero V µ V ) KS B 0 1 1 ( ). To allow for effects due to a massive mode, the potential L can be considered the more physically relevant quantity, and the σ U + 5.3 µν R B  are not directly relevant and can be dropped. . A generalization of this form adds an additional fourth-order term in in ( µ G M B , which only allows NG excitations in π KS 1 0 ν 2 L b D 16 L ∓ − ], = 6 ν = , and its effects on dynamics must be understood along with those due to the NG 0 B µ µ λ 0) as well as massive excitations (obeying L 0. In this case, the theory has a global B D µ = is included in the full Lagrangian. However, it is common in this case to include = B 0 = µ V is a Lagrange-multiplier field. In this case, the Lagrange multiplier field V is a constant (of mass dimension zero). This type of potential allows both NG excitations J µν µ λ ]. In this type of approach, it is common to assume only gravitational couplings to matter, κ B D To illustrate the behavior of the NG and massive modes and for definiteness, consider the case In a similar way, there are different choices that can be made for the potential An alternative interpretation of the vector field 16 [ µ of a KS bumblebee model.with the In kinetic the term absence of a cosmological-constant term, it has a Lagrangian excitations. However, there is stillmultiplier field an additional degree ofmodes. freedom in the form of the Lagrange- 5.1 KS Bumblebee Model where constraint, where (obeying Lagrange-multiplier potential is chosen as the smooth quadratic potential in ( natural choice of kinetic termsand have Samuel an (KS) Einstein-Maxwell [ form, as first considered by Kostelecký conservation in the matter sector. is a smooth quadratic potential, as timelike, and an interaction of the form and therefore the terms In this case, the field strength potential spacetime geometry is assumed.later section. Generalization to a Riemann-Cartan geometry is deferred to a couplings to matter alongterms with involving some current basic couplings notion with charged of matter charge can in be the included matter by sector. defining, For example, Effects of Spontaneous Lorentz Violation in Gravity case, an appropriate kinetic term has a form similar to that investigated by Will and Nordvedt [ Note that there is still no local with where B PoS(QG-Ph)009 0, = (5.9) (5.6) (5.7) (5.8) (5.10) (5.11) V = 0 0, it acts as V Robert Bluhm 6= 0 V . ν B µ B 0 leads to a current-conservation V 2 . µν 0 B + . . µν ) = µ 0 . B , µν 0 Vg B µν B T V µν ) = − T 2 µ + + B αβ − GT 0 µν B µ π M V µν 9 M J T 8 2 ( T αβ µ = = − B = D ], the nonzero vacuum value is imposed as a nonlinear µ µν µν µν lead to conservation of the total energy-momentum ten- J is the total energy-momemtum tensor, which consists of g µν ( = 28 B G µ 4 1 ν T µν µν µν D D G − T T α µ ν D B ]. It is possible, however, to restrict the theory in such a way that µα 16 B , = 15 ), µν B T 5.7 0, these modes by themselves obey the same equations as in electrodynamics in a = 0 is the Einstein tensor and V µν G is the energy-momentum tensor for the matter sector, while the bumblebee energy-momentum A more complete determination of whether Einstein-Maxwell solutions can emerge from bum- The equations of motion for the KS bumblebee model are obtained by varying the Lagrangian The idea that might arise as NG modes due to spontaneous Lorentz breaking arose , which can provide physical signatures of Lorentz violation. µν µ M gauge choice in the context ofare a no theory physical with signatures local of U(1)do Lorentz gauge not invariance. have violation. local As U(1) The a gauge result bumblebeeb invariance. of models They this, also are there permit different matter in couplings that with the they vacuum value blebee models, requires understanding thedegree of role freedom of beyond those the of massive theof mode. NG the modes initial-value that It problem. must constitutes be However, accounted an ansince for. exact additional It they solution also of are alters the the highly equations form both nonlinear of the motion in NG is and form. not gravitational feasible, modes In bywith acting particular, this as a the extra source degree massive of of mode chargeallowed and freedom couples initial energy present, density. value nonlinearly Moreover, the (in to full a nonlinearunbounded Minkowski-spacetime limit) theory from for is below which [ known the to Hamiltonian have is at negative least and one The contracted Bianchi identities for sor, Similarly, the antisymmetry of the bumblebee field strength both a source of energy and charge density. However, in the absence of a massive mode, with respect to the metric and bumblebee fields. The results are gravitational background. This raises thetype interesting of possibility that model massless not photons asmodes arise a when in result Lorentz this of symmetry is gauge spontaneously invariance broken. but instead as ainitially result in of the the context appearance of special ofearly relativity, NG where models, Lorentz e.g., symmetry the is model a of global Nambu symmetry. [ In these Effects of Spontaneous Lorentz Violation in Gravity law following from ( Examination of these equations reveals that when a massive mode is present, with and the equations reduce tocondition the usual Einstein-Maxwell equations. Since the NG modes obey the two terms, Here, is given by T PoS(QG-Ph)009 ) ]. µ b 13 − µ will not B µν = ( B Robert Bluhm µ E is conserved and µ ]. Indeed, any La- J 8 ]. They obey a condition that can be 8 0, which resembles a type of axial-gauge ) = ]. Unlike the NG modes, it cannot be written in ν 9 b 10 µν h , symmetry under three Lorentz transformations and 2 1 µ b − µ E ( µ b as . µν h µν h and . It has, for example, been shown (in Minkowski spacetime) that nonpoly- V µ E A massive mode consisting of field excitations that do not stay in the potential minimum can First, considering the diffeomorphism NG mode, it is found that it drops out completely from For the purposes considered here, with the aim of illustrating the effects of the NG and massive Solutions for the diffeomorphism and Lorentz NG modes can be obtained directly in the lin- In contrast, the Lorentz NG modes are found to consist of two propagating transverse massless It may be possible as well to alter the stability of the theory by adding nonrenormalizable also occur as a solution of the equations of motion [ and the metric excitations the linearized theory and does not propagate as a physical massless mode [ condition in inthe the Lorentz presence NG of modes gravity.again behave like that Hence, photons the as in bumblebeephysical expected, curved models signatures spacetime. of it in Lorentz is violation, general However, so found itequivalent the have to NG that must additional Einstein-Maxwell sector be coupled theory. matter to stressed matter couplings is not that strictly can speaking provide one diffeomorphism are spontaneously broken.and Thus, one there diffeomorphism can NG be mode. upsmall to Using virtual three a transformations Lorentz vierbein away NG formalism, from modes the thebe vacuum NG made solution. modes that can Or, leave alternatively, be gauge the written choices NG as can modes as combinations of the bumblebee excitations modes in a gravitational theory, it sufficesand to conserved consider the matter bumblebee currents. model withthe a Hamiltonian It KS is kinetic also positive term suffices (infeature to a of Minkowski-spacetime work behaving limit), in as while a the the sourceexamining linearized of massive the both mode effects limit. of charge retains the and its In massive energy mode density. such on Hence, a the this gravitational limit, 5.2 limit interactions. is NG suitable and for Massive Modes earized approximation. With a vector vev modes and one auxiliary mode that does not propagate [ Ultimately, however, the potential instability isbumblebee as not an likely to effective theory begravity, arising relevant the for from apparent instabilities physics. a would more merely Viewing fundamental the tering reflect at an and energy incomplete stable scales knowledge quantum above of that the theory ofquantum physics of the theory, en- it effective theory. is However, not in possible the to absence pursue of a these fundamental questions further. grangian formed out ofcontain contractions an of NG mode the for curvature the broken tensor diffeomorphisms. and the field strength written in terms of terms to the potential Effects of Spontaneous Lorentz Violation in Gravity the Hamiltonian remains positive. Onedoes assumption not is mix with that the theAs effective matter stated bumblebee current above, charge this stemming requires from that the thematter), matter nonlinear and sector field therefore has interactions. the a conserved theory charge hascan (as a then expected global in be ordinary U(1) chosen symmetry. thatregion With separate of this the phase assumption, full space initial phase that values space maintains into a regions positive Hamiltonian that can do be not selected. mix. In particular, a nomial potentials can lead to spontaneous Lorentz breaking and that such potentials are stable [ PoS(QG-Ph)009 0. and = r t (5.14) (5.15) (5.12) (5.13) (5.16) π 4 . In the / q q Robert Bluhm = q . It is this extra Φ does not hold for β 2 2 b b κ ∓ and charge 4 obey Poisson equations . However, the specific = − m ) µ m x ~ = ( B Φ B β µ 0. ρ B = and . β q 0 , the massive mode can be identified Φ . = ) µν , this condition shows that the massive . ν ) h B ~ B b 0 , , Φ B 0 µν can be determined from the field equations ρ . and , h . Both g 0 Gb 0 2 1 − r µ , 2 , π Φ / B b b ' E 4 − ) = β µ , and solve for its components using the linearized − 11 Gm ∇Φ x ~ = ( µ ~ E µ ( m ( ∂ − E µ B − µ µ Φ ν b q b Φ b = ∂ 2 = ∓ m − ∇ ~ g ∇Φ ~ 0), a nonzero massive mode can generate a nonzero electric ν Φ = Φ − E = β µ that acts as a source of density = q ∂ ) E ~ x ~ = ( β µν fields can then be determined for the case of a static point particle. It F B ~ and E ~ for the bumblebee massive mode. It is also defined by a Poisson equation, B Φ that has no time dependence. As a result, its value is fixed by the initial conditions at Electric and magnetic fields can be defined with the usual form, but as functions of the bumble- Similarly, the modified gravitational potential To lowest order, the equations of motion reveal a condition the massive mode must obey, As an example of this, consider a static point particle with mass ) x . However, the static electric field is modified by the presence of the massive-mode potential. ~ ( µ is found that the fields are modified by the presence of the massive mode and are given as field equations. The bee excitations. First define the Newtonian gravitational potential where it is the massivedegree mode of freedom that enters inand the gravitational equations static of potentials. motion and alters the form of the electromagnetic Evidently, there is no static magnetic fieldb generated for the case of aEven purely for timelike a vacuum neutral value point mass (with field. of motion. For the case of a point mass, it is found to have the form It is clearly independent of the Lorentz NG modes, which obey For the case of a timelikemode does vector not field, propagate with as a freeβ field in the linearized limit.Although Instead, it it is does purely not an propagate, auxiliary it field can nevertheless alter the form ofabsence the of static Lorentz potentials. violation the static potentials are the usual Coulombthat potential determine the form ofcharge these density. potentials. Each In has the athird presence source potential of given by spontaneous the Lorentz point-particle violation, mass it or is convenient to introduce a form that the potential takes dependsopens on up the the choice possibility of of exploring the modified initial forms value of for the the gravitational massive potential mode. in This search of, for Effects of Spontaneous Lorentz Violation in Gravity terms of the vierbein and vacuum values alone, since the condition Clearly, the gravitational potential is altered by the massive mode the massive mode. At linear order,as and the in combination terms of PoS(QG-Ph)009 r / (5.18) (5.17) Robert Bluhm . σ is a scalar depending on b ρ χ b ) νσ µ . ω , where α b − χ ) µ can be written in terms of the vierbein . However, in fact, this behavior is to ∂ α µ µσν µ β E µν = ω B ω merely reflect the fact that the field has 1 µ β )( remain nonzero through the dependence on E ν µ e E µ νρµ E . Indeed, the same behavior appears in the usual − ω m 12 α appear in the lagrangian, which perturbatively have Φ − ν β ω α β µρν 0, and the static potentials reduce to the usual Coulomb µ β µ ω ω e = ( vanishes because the bumblebee density cancels the charge 1 4 B , the field strength = ( Φ µ µν b , and hence has a modified Newtonian potential. The solution of F ≈ − µν m 0 can be considered as well. In these cases, both the static grav- B Φ µν 6= B 6= B g µν Φ Φ eB 4 1 0. In this case, − = . It has t q Φ has a vacuum value ], a number of models with generalized kinetic terms for the spin connection were ]. The field strength becomes extremely large, approaching the Planck scale, for example. Here again, the 8 − µ 9 [ 2 B is squared, quadratic terms in = . However, the bumblebee excitations b q q B κ Φ 0 at time µν ρ 4 ] due to the formation of shock discontinuities in Φ B = = 16 In Ref. [ Other examples with In Riemann-Cartan spacetime, the possibility of a Higgs mechanism involving the Lorentz An illustration is provided by the KS bumblebee model in Riemann-Cartan spacetime. In this Given the lack of specific experimental guidance, a natural choice of initial value would be to and | β . This type of solution has an unusual behavior that has been identified as potentially flawed , 2 m m 15 M Φ the bumblebee field has the form of a total derivative itational and Coulomb potentials arechoice modified by the massive mode. One simple example is the It is these quadraticLorentz terms NG modes that by allow the spin a connection. Higgs mechanism toconsidered. occur involving Finding absorption a physical ofdifficulty model is the with in no finding ghosts, kinetic however, terms remains describing an propagating open modes problem. that The are compatible with Eq. electromagnetic and gravitational potentials approachparticular their reveal that conventional the values. usual Einstein-Maxwell Thesethe solutions results correct (describing static massless in potentials) photons can as emerge wellphotons from as a in theory this that case has are noand local the thus U(1) has Lorentz gauge little NG symmetry. effect The modes, on the and static the potentials. massive mode remains extremely heavy and Newtonian expressions.| This holds true as well with a nonzero massive mode if the scale NG modes becomes a possibility. Inas this mechanism a it result is the of spinmechanism connection spontaneous that can Lorentz gains in mass breaking. terms principlefields. As be In long viable, practice, as however, leading it the to is spin difficult physical to connection massive construct is a propagating model dynamical, spin-connection thatcase, this is when ghost- and -free. density Φ [ be expected. For a point charge,dependence the stemming singularities from in its dependence on solutions of Einstein-Maxwell theory in an appropriately chosen gauge. 5.3 Higgs Mechanism for the Spin Connection and spin connection as When set Effects of Spontaneous Lorentz Violation in Gravity example, an alternative explanation ofapproach, dark since matter. the only In experimental fact,Newtonian constraints there potential are over is that probed considerable distance the freedom scales. potential in must this agree with the usual the form PoS(QG-Ph)009 (5.19) Robert Bluhm ]. However, in the 29 U(1) gauge-invariant × S(2) × . according to their spin-parity projections. λκµν R λ µν ω λκµν R 13 4 1 = grav , 0 L 0 propagate as massless modes. When this is combined with the mass 6= λ with , couples with conventional Standard-Model or gravitational fields. As a result, i ]. In many cases, it is sufficient to restrict the full SME to minimal extensions λ µν ω 12 λ µν T ), some of the propagating modes are converted to massive modes. Other examples can h 5.18 ) as a mass term. If ghosts are permitted, then the mechanism is straightforward. For example, Massive modes can arise in two ways, either through a Higgs mechanism in Riemann-Cartan Any remaining signals of spontaneous Lorentz violation would involve couplings with matter A group of theorists centered at Indiana University initiated a comprehensive phenomenolog- The effects described in this paper originating from spontaneous Lorentz breaking can provide The NG modes either lead to additional gravitational modes in the vierbein that would differ 5.18 any possible signal originating in thisis way because would the be SME allows identical for toStandard-Model all a and observer-independent signal violations gravitational arising of in fields. Lorentz theenergy, symmetry but SME. it involving It This also provides is a connection defined todimension the as Planck [ a scale through general operators of effective nonrenormalizable field theory at low spacetime, or as field excitations thatcould do in not principle remain be in detectable the ashowever, potential new the previously minumum. massive unobserved In modes propagating either might particles. case, remain Alternatively, they auxiliaryof fields the that KS bumblebee. do not Their propagate, influencestatic then as potentials. appears in to the In be example either limited to ofmodes altering these is the scenarios, extremely form it high, of the is and relevant therefore likely their that observable the consequences scale are associated likely with tofields. the be For massive quite example, small. at low energy, signalsvev, of e.g., physical Lorentz violation would occur when a tensor involving only, for example, power-counting renormalizable or SU(3) ical investigation of Lorentz violation more than a decade ago. These investigations span virtually latter case, there would beleading order) no other observable than consequences the of existence of the the NG previously modes known massless themselves gauge (at particles. least at terms. To consider experiments insector. atomic Similarly, limits physics of it the often SME suffices thatwith include to gravity (or restrict either exclude) a the gravity Riemann can SME be or to defined, Riemann-Cartan its and geometry in QED can the be case assumed. term ( all the fields be studied as well, aided by decomposing the fields clear signals of physical Lorentzthose violation. arising In from the NG end, modes, there fromlogical are massive implications basically modes, of three and each of classes from these matter of can couplings. signals: be The considered. phenomeno- from the usual forms ofthey can gravitational be radiation interpreted predicted as in known gauge general fields relativity, such or as the in certain or cases graviton [ Evidently, the incorporation of spontaneousnew Lorentz arena in violation the in search theories forfind models with viable with torsion propagating models provides massive that modes. a do The not challenge, allow however, ghosts. is to 6. Phenomenology of Lorentz Violation Effects of Spontaneous Lorentz Violation in Gravity ( with a kinetic term in the gravitational sector of the form PoS(QG-Ph)009 ], neutri- eds., 34 , Robert Bluhm , World CPT and 33 ], hadrons [ 32 ], muons [ CPT and Lorentz Symmetry III 31 , 30 , World Scientific, Singapore, 1999; 14 Special Relativity: Will It Survive the Next 101 Years? , 5 (2005). 8 2 experiments in Penning traps, Hughes-Drever experiments, modern- − ]. Many of these efforts are on-going, with plans for attaining significantly CPT and Lorentz Symmetry g , World Scientific, Singapore, 2002; 36 ], and photons [ (Springer, Berlin, 2006). Scientific, Singapore, 2005. Lorentz Symmetry II 35 This work has examined possible consequences of spontaneous Lorentz violation in the con- This work has been supported by NSF grant PHY-0554663 and by the European Science Foun- [3] J. Ehlers and C. Lämmerzahl, eds., [2] D. Mattingly, Living Rev. Rel. [1] V.A. Kostelecký, ed., improved sensitivities in the coming years. 7. Summary and Conclusions text of gravity. Much of thepossibility focus of has a been Higgs on mechanism, questionsis and concerning found the the that fate appearance in of of theories the additional with NGcan spontaneous massive modes, Lorentz modes. all the violation, be In up incorporated general, to naturally ten it Lorentz NG in NG modes the modes can vierbein. appear. can They propagate Fornisms like the can photons example in occur, of one an a axial associated KSHowever, gauge. with it bumblebee has broken In model, been diffeomorphisms, principle, shown the the that two a other Higgsthe conventional with Higgs mecha- metric mechanism Lorentz does (for symmetry. not diffeomorphisms) involving occur.nism If (for the the Lorentz geometry symmetry) can is occur, Riemann-Cartan, inin which then a the a Riemann spin connection geometry, conventional acquires this Higgs a type mass. mecha- type of of However, Higgs Higgs mechanism mechanism can is occur, not leadinging possible. to the the metric Nonetheless, appearance field. an of These additional alternative can massivenumerous modes lead involv- phenomenological to altered questions forms that of arise the inof gravitational potential. these Lorentz Clearly, processes. breaking there However, are at all lowpursued relevant energies signals comprehensively using invloving the couplings SME. to known Standard-Model fields can be Acknowledgments dation network programme “Quantum Geometry and Quantum Gravity.” References Effects of Spontaneous Lorentz Violation in Gravity all areas of physics.neous The Lorentz violation scope (as of well as thesestimulated other, a investigations e.g., number includes explicit, of forms searching new of for and LorentzCPT improved signals breaking). symmetry, experiments. such of This These as work sponta- include has classicday tests Michelson-Morley of experiments, Lorentz and as wellastrophysical new sources. types of Together they tests, cover such aSome as wide specific in range examples space of include particle tests satellites sectors with ornos electrons in with [ [ the Standard Model. 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