Gravity from spontaneous Lorentz violation
aa r X i v : . [ g r- q c ] J a n Gravity from spontaneous Lorentz violation
V. Alan Kosteleck´y a and Robertus Potting ba Physics Department, Indiana University, Bloomington, IN 47405 b CENTRA, Physics Department, FCT,Universidade do Algarve, 8000-139 Faro, Portugal (Dated: IUHET 523, January 2009)
Abstract
We investigate a class of theories involving a symmetric two-tensor field in Minkowski spacetimewith a potential triggering spontaneous violation of Lorentz symmetry. The resulting masslessNambu-Goldstone modes are shown to obey the linearized Einstein equations in a fixed gauge.Imposing self-consistent coupling to the energy-momentum tensor constrains the potential for theLorentz violation. The nonlinear theory generated from the self-consistent bootstrap is an al-ternative theory of gravity, containing kinetic and potential terms along with a matter coupling.At energies small compared to the Planck scale, the theory contains general relativity, with theRiemann-spacetime metric constructed as a combination of the two-tensor field and the Minkowskimetric. At high energies, the structure of the theory is qualitatively different from general relativity.Observable effects can arise in suitable gravitational experiments. . INTRODUCTION The idea that physical Lorentz symmetry could be broken in a fundamental theory of na-ture has received much attention in recent years. One attractive mechanism is spontaneousLorentz violation, in which an interaction drives an instability that triggers the developmentof nonzero vacuum values for one or more tensor fields [1]. Unlike explicit breaking, sponta-neous Lorentz violation is compatible with conventional gravitational geometries [2], and itis therefore advantageous for model building. However, spontaneous violation of a contin-uous global symmetry comes with massless excitations, the Nambu-Goldstone (NG) modes[3]. Among the challenges facing attempts to construct realistic models with spontaneousLorentz violation is accounting for the role of the corresponding NG modes and interpretingthem phenomenologically.Since the NG modes are intrinsically massless, they can generate long-range forces. Oneintriguing possibility is that they could reproduce one of the long-range forces known toexist in nature. For electrodynamics, for example, the Einstein-Maxwell equations in a fixedgauge naturally emerge from the NG sector of certain gravitationally coupled vector theorieswith spontaneous Lorentz violation known as bumblebee models [4, 5]. For gravity itself,the gravitons can be interpreted as the NG modes from spontaneous Lorentz violation inseveral ways. As fundamental field excitations, gravitons can be identified with the NGmodes of a symmetric two-tensor field C µν in a theory with a potential inducing sponta-neous Lorentz violation, which generates the linearized Einstein equations in a fixed gauge[6]. Alternatively, gravitons as composite objects can be understood as the NG modes ofspontaneous Lorentz violation arising from self interactions of vectors [7], fermions [8], orscalars [9], following related ideas for composite photons [10]. Other interpretations of theNG modes include a new spin-dependent interaction [11] and various new spin-independentforces [12]. For certain theories in Riemann-Cartan spacetimes, the NG modes can insteadbe absorbed into the spin connection via the Lorentz-Higgs effect [5].In the present work, we investigate the possibility that the full nonlinear structure ofgeneral relativity can be recovered from an alternative theory of gravity with spontaneousLorentz violation in which the gravitons are fundamental excitations identified with the NGmodes. General relativity has the interesting feature that it can be reconstructed uniquelyfrom massless spin-2 fields by requiring consistent self-coupling to the energy-momentum2ensor [13, 14, 15, 16, 17]. For example, the linearized theory describing gravitational wavesvia a symmetric two-tensor h µν propagating in a spacetime with Minkowski metric η µν contains sufficient information to reconstruct the full nonlinearity of general relativity whenself consistency is imposed. Here, we demonstrate that applying this bootstrap methodto a linearized theory with a symmetric two-tensor field C µν and a potential V ( C µν , η µν )inducing spontaneous Lorentz violation yields an alternative theory of gravity, which we callcardinal gravity [18]. The coupling of the cardinal field to the matter sector is derived, andconstraints from existing experiments are considered. We show that the action of cardinalgravity corresponds to the Einstein-Hilbert action at energies small compared to the Planckscale. However, the structure of the theory at high energies is qualitatively different fromthat of general relativity. Our results indicate that cardinal gravity is a viable alternativetheory of gravity exhibiting some intriguing features in extreme gravitational environments.We begin this work in Sec. II by presenting the linearized cardinal theory and a discussionof its correspondence to linearized general relativity. Section III reviews the bootstrapprocedure for general relativity and obtains some generic results. For general relativity, thebootstrap procedure yields a unique answer even if a potential for h µν is allowed [16]. In thecontext of spontaneous Lorentz violation, the phase transition circumvents this uniqueness.However, the nontrivial integrability conditions required for implementing the bootstrapconstrain the form of the potential V . In Sec. IV, we obtain differential equations expressingthe integrability conditions and derive acceptable potentials V . This section also applies thebootstrap to yield the full cardinal gravity. Certain aspects of the extrema of the potentialare considered, and alternative bootstrap procedures are discussed. The coupling of thecardinal field to the matter sector and some experimental implications are studied in Sec.V. A summary of the results and a discussion of their broader implications is provided inSec. VI. Throughout this work, we use the conventions of Ref. [2]. II. LINEARIZED ANALYSIS
In this section, the linear cardinal theory is defined and investigated. We show that itsNG sector is equivalent to conventional linearized gravity in a special gauge.3 . Linear cardinal theory
Consider first the action for the symmetric two-tensor cardinal field C µν defined in a back-ground spacetime. For definiteness and simplicity, we take the background to be Minkowskispacetime with metric η µν , although a more general background could be countenanced andtreated with similar methods. We suppose the kinetic term in the action is quadratic in C µν ,so the derivative operators in the equation of motion are linear in C µν . The NG excitationsof C µν subsequently play the role of the metric fluctuation in a linearized theory of gravity.The action is assumed to generate spontaneous violation of Lorentz symmetry through apotential V ( C µν , η µν ).
1. Basics
The Lagrange density for the linear cardinal theory is taken to be L C = C µν K µναβ C αβ − V ( C µν , η µν ) . (1)Here, K µναβ is the usual quadratic kinetic operator for a massless spin-2 field. Allowingfor an arbitrary scaling parameter κ to be chosen later, K µναβ can be written in cartesiancoordinates as K µναβ = κ [( − η µν η αβ + η µα η νβ + η µβ η να ) ∂ λ ∂ λ + η µν ∂ α ∂ β + η αβ ∂ µ ∂ ν − η µα ∂ ν ∂ β − η να ∂ µ ∂ β − η µβ ∂ ν ∂ α − η νβ ∂ µ ∂ α ] , (2)where η µν is the Minkowski metric with diagonal entries ( − , , ,
1) as the only nonzerocomponents. As usual, in other coordinate systems the Minkowski metric takes differentforms and covariant derivatives must be used. The equations of motion obtained by varyingEq. (1) with respect to C µν are K µναβ C αβ − δVδC µν = 0 . (3)The theory (1) has various symmetries. It is invariant under translations and under global4orentz transformations. For infinitesimal parameters ǫ µν = − ǫ νµ , the latter take the form C µν → C µν + ǫ µα C αν + ǫ να C αµ ,η µν → η µν . (4)There are also local spacetime symmetries, including invariance under local Lorentz trans-formations on the tangent space at each point and invariance under diffeomorphisms of theMinkowski spacetime. These local symmetries play a subsidiary role in the present context.In addition to the spacetime symmetries, the form of the kinetic operator (2) ensuresthat the kinetic term is by itself invariant under gauge transformations of C µν alone, C µν → C µν − ∂ µ Λ ν − ∂ ν Λ µ ,η µν → η µν . (5)However, one or more of these four gauge symmetries may be explicitly broken by thepotential V , so the Lagrange density (1) contains between zero and four gauge degrees offreedom depending on the choice of V . Since C µν has ten independent components, it followsthat there are between six and ten physical or auxiliary fields.The potential V for the theory (1) is a scalar function of the cardinal field C µν and theMinkowski metric η µν . The only scalars that can be formed from these two objects involvetraces of products of the combination C µα η αν . The scalar X m with m such products has theform X m = tr [( Cη ) m ] . (6)Here, we have introduced a convenient matrix notation ( Cη ) µν ≡ C µα η αν . Since Cη is asymmetric 4 × X m , so we can restrictattention to the cases X m = 1 , , ,
4. It follows that the potential V can be written as V = V ( X , X , X , X ) (7)without loss of generality. For definiteness, V is assumed to be positive everywhere exceptat its absolute minimum, which is taken to be zero.Under the gauge transformation (5), each scalar X m transforms nontrivially and thereforeexplicitly breaks one symmetry. For simplicity in what follows, we assume the potential V depends on all four independent scalars X m , so the gauge symmetry (5) is completely broken5or generic field configurations. With this assumption, the theory describes ten physical orauxiliary fields and zero gauge fields. This assumption could be relaxed, but the resultingdiscussion would involve additional gauge-fixing considerations.The potential V is taken to have a minimum in which C µν attains a nonzero vacuumvalue h C µν i ≡ c µν . (8)In this minimum, the scalars X m have vacuum values h X m i ≡ x m = tr [( cη ) m ] . (9)These vacuum values spontaneously break particle Lorentz symmetry, but they leave un-affected the structure of observer Lorentz and general coordinate transformations, whichamount to coordinate choices. To avoid complications with soliton-type solutions, we alsosuppose c µν is constant, ∂ α c µν = 0 (10)in cartesian coordinates.Given a vacuum value c µν , the freedom of coordinate choice can be used to adopt acanonical form. For definiteness and simplicity, we assume in what follows that the matrix( cη ) µν ≡ c µα η αν has four inequivalent nonzero real eigenvalues. This implies, for example,invertibility and the existence of one timelike and three spacelike eigenvectors. It also impliesthat all six Lorentz transformations are spontaneously broken. The consequences of otherpossible choices may differ from the discussion below and would be interesting to explore,but they lie beyond our present scope.
2. Nambu-Goldstone and massive modes
The physical degrees of freedom contained in the cardinal field C µν can be taken asfluctuations about the vacuum value c µν . We write C µν = c µν + e C µν . (11)The fluctuation field e C µν is symmetric and has ten independent components, which includeboth the NG modes and the massive modes in the theory.6o identify the NG modes, we can make virtual infinitesimal symmetry transformationsusing the broken generators acting on field vacuum values, and then promote the correspond-ing parameters to field excitations. An infinitesimal Lorentz transformation with parameters ǫ µν = − ǫ νµ yields h C µν i → c µν + ǫ µα c αν + ǫ να c αµ . (12)Since there are six Lorentz transformations (three rotations and three boosts), there couldin principle be up to six Lorentz NG modes, corresponding to the promotion of the sixparameters ǫ µν to fields E µν = −E νµ [5, 19]. For c µν satisfying our assumed conditions, themaximal set of six NG modes appears. In general, the NG modes in e C µν are contained inthe fluctuations N µν defined by e C µν ⊃ N µν = E µα c αν + E ν α c αµ ≡ O µναβ E αβ , (13)where O µναβ = ( η µα c νβ + η να c µβ − η µβ c να − η νβ c µα ) . (14)Since there are six independent fields in E µν , the ten symmetric components of N µν mustobey four identities. For c µν satisfying our assumed conditions, we find these identities canbe expressed as tr (cid:2) N η ( cη ) j (cid:3) = 0 , (15)with j = 0 , , , N µν , the fluctuation e C µν includes fourmassive modes. These are contained in the field M µν given by M µν = e C µν − N µν , (16)subject to a suitable orthogonality condition. The symmetric field M µν has ten componentsbut only four independent degrees of freedom, which we denote here by m j , j = 0 , , , M µν as M µν = m η µν + m c µν + m ( cηc ) µν + m ( cηcηc ) µν . (17)The fields N µν and M µν obey identities expressing a kind of orthogonality:tr (cid:2) N η ( M η ) j (cid:3) = 0 , (18)7ith j = 0 , , ,
3. More generally, we findtr [
N η F ( cη, M η )] = 0 , (19)where F ( cη, M η ) is an arbitrary matrix polynomial in cη and M η .With the expansion (17), the fluctuation e C µν can be written e C µν = N µν + X j =0 m j [( cη ) j ] µα η αν . (20)Using this equation, the four massive modes m j can be expressed in terms of e C µν . Mul-tiplying by [ η ( cη ) k ] µν with k = 0 , , , cη ] tr[( cη ) ] tr[( cη ) ]tr[ cη ] tr[( cη ) ] tr[( cη ) ] tr[( cη ) ]tr[( cη ) ] tr[( cη ) ] tr[( cη ) ] tr[( cη ) ]tr[( cη ) ] tr[( cη ) ] tr[( cη ) ] tr[( cη ) ] m m m m = tr[ e Cη ]tr[ e Cη ( cη )]tr[ e Cη ( cη ) ]tr[ e Cη ( cη ) ] . (21)The traces tr[( cη ) p ] with p = 5 , cη ) m ] with m = 1 , , , c j of the matrix cη , thedeterminant of the 4 × O on the left-hand side takes the formdet [ O ] = Y j,k =0 j 4. Their explicit forms are unnecessary in thediscussion that follows, so we omit them here.8he above considerations reveal that the decomposition of the cardinal field C µν in termsof NG and massive modes is C µν = c µν + N µν + M µν . (23)The potential V can therefore be viewed as a function of N µν and M µν with constraintsadded to restrict these fields to their independent degrees of freedom, or equivalently as afunction of the Lorentz NG modes E µν and the massive modes m j : V ( C µν , η µν ) = V ( c µν , E µν , m , m , m , m , η µν ) . (24)To investigate the correspondence of the linear cardinal theory (1) to linearized generalrelativity, it is useful to restrict attention to the pure NG sector. This can be achieved byconsidering the limit of infinite mass for the fields m j . Alternatively, the potential V can bereplaced with the Lagrange-multiplier limit V λ given by V λ = X m =1 λ m ( X m − x m ) , (25)where the quantities λ m are four Lagrange-multiplier fields. This potential freezes all fluc-tuations of C µν away from the potential minimum. In this limit, the independent degrees offreedom in the field fluctuations e C µν are therefore restricted to the NG modes E µν or, equiva-lently, e C µν → N µν subject to the constraints (15). If desired, the on-shell values of λ j can beset to zero by a suitable choice of initial conditions. Equivalent results could be obtained viaan alternative Lagrange density involving a potential V with quadratic Lagrange-multiplierterms instead [19]. In any event, if the graviton is to be identified with the Lorentz NGmodes in the theory (1), it follows that the field N µν must be the candidate graviton field. 3. Equations of motion for NG modes The behavior of the candidate graviton field N µν is determined by its equations of mo-tion. In the pure NG sector with vanishing Lagrange multipliers, the theory (1) with thepotential (25) is equivalent to an effective Lagrange density L NG for the independent degreesof freedom, which are the Lorentz NG modes E µν . We can therefore write L NG = O µνρσ E ρσ K µναβ O αβγδ E γδ . (26)9arying L NG with respect to the independent degrees of freedom E µν yields the six equationsof motion O µνρσ K µναβ O αβγδ E γδ = 0 . (27)These can equivalently be written as O µνρσ K µναβ N αβ = 0 , (28)where the constraints (15) are understood.To solve these equations we can use Fourier decomposition, transforming to momentumspace with 4-momentum k µ . It is convenient to introduce the scalars K m,n and K m , definedby the matrix equations K m,n ≡ k ( cη ) m N η ( cη ) n k, K m ≡ k ( cη ) m k. (29)Note that K m,n = K n,m by virtue of the symmetry of N µν . Contraction of the equations ofmotion (28) with k ( cη ) m yields the following results, equivalent in content to the originalequations of motion: k K m +1 ,n + K m K ,n +1 + K n +1 K m, − k K m,n +1 − K n K m +1 , − K m +1 K ,n = 0 . (30)These expressions are solved by the on-shell condition k = 0 and the constraint k µ N µν = 0.We have verified that no physical off-shell solutions exist. The on-shell solutions are modesobeying the usual massless wave equation, ∂ λ ∂ λ N µν = 0 , (31)subject to the harmonic condition ∂ µ N µν = 0 . (32)The latter imposes four constraints on the six independent degrees of freedom in N µν .We thus see that only two combinations of the six massless Lorentz NG modes E µν propagate as physical on-shell fields. The other four NG modes are auxiliary. With the fullpotential V replaced by the Lagrange-multiplier limit V λ , the four Lagrange multipliers canbe viewed as playing a role analogous to that of the four frozen massive modes m j .10 . Correspondence to linearized general relativity In this subsection, we show the correspondence between the restriction of the linearcardinal theory to the NG sector and the usual weak-field limit of general relativity describinga massless spin-2 graviton field h µν propagating in a background Minkowski spacetime.Consider the Lagrange density for a free symmetric massless spin-2 field h µν , which is ofthe form (1) with C µν replaced by h µν and the potential V set to zero: L h = h µν K µναβ h αβ . (33)The definition of K µναβ in Eq. (2) implies K µναβ h αβ ≡ − κG Lµν , (34)where G Lµν is the Einstein tensor linearized in h µν . At this stage, the value of κ can be fixedby requiring a match to the conventional normalization of the linearized action for generalrelativity in the presence of a matter coupling given by L LT = h µν T M µν , (35)where T M µν is the matter energy-momentum tensor. This match fixes κ to be κ = 116 πG N , (36)where G N is the Newton gravitational constant. A priori, h µν has ten degrees of freedom. However, the theory is invariant under the fourgauge transformations h µν → h µν − ∂ µ ξ ν − ∂ ν ξ µ , (37)so four gauge-fixing conditions can be imposed on h µν . Numerous choices of gauge appearin the literature. For free wave propagation, a common choice is transverse-traceless gauge,which imposes n µ h µν = 0 , h ≡ h µµ = 0 , (38)for a unit timelike vector n µ . For suitable initial conditions, the harmonic condition ∂ µ h µν = 0 (39)11hen follows from the equations of motion. However, this gauge is not the only possiblechoice. Here, we demonstrate the existence of an alternative gauge condition on h µν thatyields directly a match to the NG effective Lagrange density (26).The conditions fixing this alternative ‘cardinal’ gauge at linear order in h µν aretr (cid:2) hη ( cη ) j (cid:3) = 0 , (40)where j = 0 , , , 3. In this expression, ( cη ) µν ≡ c µα η αν is a constant matrix assumed tohave four inequivalent nonzero real eigenvalues, which we denote by c j , j = 0 , , , 3. Thisassumption ensures the four conditions (40) are independent. For the present purpose ofmatching to the linear cardinal theory (1), the quantity c µν is to be identified with thevacuum value of C µν in Eq. (8), so we denote it by the same symbol.To show that the conditions (40) are indeed a choice of gauge, we can consider an arbitraryinitial field h ′ µν and seek quantities ξ µ such that a gauge transformation of the form (37)generates the desired field h µν satisfying (40). In momentum space, the gauge transformation(37) takes the form h µν = h ′ µν − ik µ ξ ν − ik ν ξ µ . (41)The requirements on ξ µ become ik µ ξ µ = tr [ ηh ′ ] ,ik α ( cη ) αµ ξ µ = tr [ ηh ′ ηc ] ,ik α [( cη ) ] αµ ξ µ = tr (cid:2) ηh ′ ( ηc ) (cid:3) ,ik α [( cη ) ] αµ ξ µ = tr (cid:2) ηh ′ ( ηc ) (cid:3) . (42)This represents a set of four equations for the four unknowns ξ µ , which can be regardedas a matrix equation. The set has a unique solution if the 4 × ξ µ is invertible. Then, the four 4-vectors k µ , k α ( cη ) αµ , k α [( cη ) ] αµ , k α [( cη ) ] αµ are linearly independent, and so ǫ µνρσ k µ k α c αν k β ( cηc ) βρ k γ ( cηcηc ) γσ = 0 . (43)Expanding the 4-vector k in terms of the eigenvectors e ( a ) of the matrix cη shows thatthis condition is indeed satisfied for generic k , for which all components k ( a ) = k · e ( a ) arenonzero. It follows that the cardinal gauge (40) can be attained everywhere in conventional12inearized general relativity, except for special k at which additional gauge fixing is required.This remnant gauge freedom is analogous to that of axial gauge in electrodynamics [20].Similarly, in the context of spontaneous Lorentz violation, the linearized potential for thevector field in certain bumblebee models generates an NG-sector axial constraint with arelated remnant gauge freedom [5, 19]. For simplicity in what follows, we consider the caseof generic k .Once the cardinal gauge (40) is imposed, the harmonic condition (32) follows from theequations of motion. The latter are found from the Lagrange density (33) to be K µναβ h αβ ≡ − κG Lµν = 0 . (44)Contracting these equations in turn with η µν , c µν , ( cηc ) µν , and ( cηcηc ) µν yields in momentumspace the four conditions k µ h µν k ν = 0 ,k α c αµ h µν k ν = 0 ,k α c αβ c βµ h µν k ν = 0 ,k α c αβ c βγ c γµ h µν k ν = 0 . (45)Collecting the coefficients of h µν k ν gives a 4 × h µν k ν = 0, andhence that the gauge choice (40) obeys the harmonic condition (32). The equations ofmotion then reduce to ∂ λ ∂ λ h µν = 0 , (46)and they describe the usual two graviton degrees of freedom propagating as massless spin-2waves.We now have all the ingredients in hand to verify the equivalence between the theory(33) for a propagating spin-2 field h µν and the theory (26) for the NG sector of the cardinalmodel. Starting with the former, we can impose the four cardinal gauge conditions (40) onthe ten independent graviton components h µν . The equations of motion (44) then imply theharmonic condition (32), which leaves two degrees of freedom that propagate as conventionalmassless modes. These results are paralleled in the theory (26) for the NG sector of thecardinal model. The field N µν containing the Lorentz NG modes E µν is subject to the13onstraints (15), so N µν matches the graviton h µν in cardinal gauge, h µν ↔ N µν . (47)The harmonic condition holds for both N µν and h µν . The equations of motion (28) for theLorentz NG modes E µν can be matched directly to the equations of motion (44) for thegraviton h µν by multiplying the latter with O µνρσ .Evidently, the cardinal and graviton theories are in direct correspondence, even thoughtheir gauge structures differ. The presence of the potential in the linear cardinal theoryexcludes the gauge symmetry of the graviton theory, but the gauge freedom of the lattermeans that only six of the ten components of h µν are physical or auxiliary, thereby matchingthe six Lorentz modes E µν in the NG sector of the cardinal theory. Note also that the gaugefreedom of the graviton theory could be fixed to cardinal gauge in a standard way, byadding suitable gauge-fixing terms to the Lagrange density. The parallel in the cardinaltheory would be the presence of Lagrange multipliers for the constraints (15). III. BOOTSTRAP PROCEDURE This section considers some generic features of the bootstrap procedure for self-consistentcoupling to the energy-momentum tensor. We summarize the Deser version [14] of thebootstrap for obtaining general relavitity from the linear graviton theory (33), and we presentsome generic results that are useful for the subsequent analysis. A. Bootstrap for general relativity The analysis takes advantage of the first-order Palatini form [21] of the nonlinear Einstein-Hilbert action of general relativity, which can be written as S GR = Z d x κ g µν R µν (Γ) . (48)Here, g µν is the tensor density of weight one defined in terms of the usual reciprocal metric g µν as g µν ≡ p | g | g µν . (49)14ts inverse is a tensor density of weight negative one, which we define as g µν ≡ p | g | g µν . (50)Also, R µν (Γ) = ∂ α Γ αµν − ∂ µ Γ ανα − ∂ ν Γ αµα +Γ ββα Γ αµν − Γ αµβ Γ βνα (51)is the curvature tensor for the connection Γ αµν . In this approach, both g µν and Γ αµν areviewed as independent fields at the level of the action, and the identification of Γ αµν withthe Christoffel symbols arises on shell by solving the equations of motion.In what follows, we define the fluctuation h µν of g µν about the Minkowski background η µν as g µν = η µν + h µν . (52)Note the use of contravariant indices in this definition. Also, note that h µν can be identifiedat linear order with the usual trace-corrected field h µν : h µν ≈ − h µν ≡ − h µν + η µν h. (53)Given the linear graviton theory (33), the nonlinear Einstein-Hilbert action can be derivedby adding a coupling to the energy-momentum tensor T µν and requiring its conservationbe consistent order by order [13]. Deser has shown that this bootstrap procedure can beperformed in a single elegant step [14].The starting point of the derivation is to note that the equations of motion (44) for h µν ,obtained in the previous section from the second-order Lagrange density (33), also followfrom the linearized version of the first-order action (48). The latter becomes S L GR = Z d x L L GR , L L GR = κ [ h µν ( ∂ α Γ αµν − ∂ ν Γ αµα )+ η µν (Γ ββα Γ αµν − Γ αµβ Γ βνα )] , (54)with h µν and Γ αµν viewed as independent fields. Variation of S L GR with respect to these fieldsyields two sets of equations of motion. These fix Γ αµν as the usual linearized Christoffelsymbols, and they imply the linear equations of motion (44) for h µν obtained from thesecond-order Lagrange density (33). 15he prescription for the bootstrap procedure is to require that the energy-momentumtensor T µν obtained from the action (54) is coupled as a source in a self-consistent manner.It turns out to be most convenient to work with the trace-reversed energy-momentum tensor τ µν , which in the linear limit is related to T µν by τ µν = T µν − η µν T αα . (55)For a given Lagrange density L in Minkowski spacetime with metric η µν , the tensor τ µν can be calculated via the Rosenfeld method [22]. The procedure involves promoting theMinkowski metric η µν to an auxiliary weight-one metric density ψ µν and the partial derivative ∂ µ to the covariant derivative D µ formed using ψ µν , so that L becomes covariant in theauxiliary spacetime. The trace-reversed energy-momentum tensor τ µν is then found fromthe expression − τ µν = δ L L δψ µν (cid:12)(cid:12)(cid:12)(cid:12) ψ → η . (56)For the linear theory with Lagrange density L L GR , this yields − τ h µν = κ (Γ ββα Γ αµν − Γ αµβ Γ βνα ) + κσ µν ( h , Γ) , (57)where σ µν is a total-derivative term given by σ µν ( h , Γ) = − ∂ γ (cid:2) h µγ Γ ρ νρ + h νγ Γ ρ µρ − h µν Γ ρ γρ + h µρ (Γ ν γρ − Γ γ νρ ) + h νρ (Γ µ γρ − Γ γ µρ ) − h γρ (Γ µ νρ − Γ ν µρ )+ η µν ( tr [ h η ] Γ σ γσ − h ρσ Γ γρσ ) (cid:3) . (58)On shell, σ µν can be expressed more elegantly as σ µν = R Lµν (Γ) − R Lµν (Γ L ) , (59)where R Lµν (Γ) is the linear part of the Ricci curvature, R Lµν (Γ) = ∂ α Γ αµν − ∂ µ Γ ανα − ∂ ν Γ αµα , (60)and Γ L is the linearized Christoffel symbolΓ Lαµν = [ ∂ α h µν − ∂ µ h να − ∂ ν h µα + ( η µα ∂ ν + η να ∂ µ − η µν ∂ α )tr [ h η ]] . (61)16he full nonlinear action of general relativity is obtained by coupling the nonderivativepart of τ h µν as a source for h µν , S GR = S L GR + Z d x κ h µν (Γ ββα Γ αµν − Γ αµβ Γ βνα ) . (62)Variation of this action with respect to h µν yields the Einstein equation R µν = 0 in the form κR Lµν (Γ) = τ h µν + σ µν , (63)which implies R Lµν (Γ L ) = 8 πG N τ h µν . (64)This verifies that coupling the nonderivative part of τ h µν as a source for h µν indeed producesthe usual Einstein equations. Moreover, since the nonderivative part of τ h µν is independentof η µν , it generates no additional contribution to the energy-momentum tensor and so nofurther iteration steps are required. B. Generic bootstrap results In this subsection, we outline some generic applications of the bootstrap procedure, start-ing from an action given in Minkowski spacetime. The example relevant in our context iseither an action S (0) independent of h µν or an action S (1) linear in h µν . In each case, we seekto construct the corresponding action S that incorporates consistent self-coupling to h µν atall orders. 1. Case of S (0) Consider first the case of an action S (0) independent of h µν , such as a matter action. Wewrite S (0) = Z d x L (0) , (65)where the Lagrange density L (0) = L (0) ( η µν , f a , ∂ µ f a ) (66)is a function of the spacetime metric η µν , a set of fields f a ( x ), and their derivatives ∂ µ f a . Forthe purposes of this work, it suffices to suppose that the terms ∂ µ f a are either derivatives17f scalars or are gauge kinetic terms, so that promotion of ∂ µ to the auxiliary covariantderivative has no effect: ∂ µ f a → D µ [ ψ ] f a ≡ ∂ µ f a . This simplifying assumption avoids theneed to consider terms of the σ µν type in the analysis.To obtain the energy-momentum tensor for the action (65), the Lagrange density L (0) ispromoted to a covariant expression with respect to ψ µν , L (0) → L (0) ( ψ µν , f a , ∂ µ f a ) . (67)To ensure L (0) remains a density, multiplication by a factor of a power of p | ψ | may berequired as part of this promotion, where ψ ≡ det [ ψ µν ]. Using the definition (56) thenyields − τ (0) µν = δ L (0) δψ µν (cid:12)(cid:12)(cid:12)(cid:12) ψ → η . (68)The bootstrap procedure requires that τ (0) µν be consistently coupled as a source for h µν .The action S (0) must therefore be supplemented by an additional term S (1) = Z d x L (1) ≡ Z d x h µν ( − τ (0) µν ) , (69)up to a possible constant. However, in general the term S (1) itself contributes a term τ (1) µν tothe energy-momentum tensor, − τ (1) µν = δ L (1) δψ µν (cid:12)(cid:12)(cid:12)(cid:12) ψ → η = h αβ δ ( − τ (0) αβ ) δψ µν (cid:12)(cid:12)(cid:12)(cid:12) ψ → η . (70)Consistency of the coupling then requires that a further term S (2) be added to the action, S (2) = Z d x L (2) , (71)where L (2) is the solution to the differential equation δ L (2) δ h µν (cid:12)(cid:12)(cid:12)(cid:12) ψ → η = − τ (1) µν ≡ h αβ δ ( − τ (0) αβ ) δψ µν (cid:12)(cid:12)(cid:12)(cid:12) ψ → η . (72)We find L (2) = h αβ h γδ δ ( − τ (0) γδ ) δψ αβ (cid:12)(cid:12)(cid:12)(cid:12) ψ → η = h µν ( − τ (1) µν ) , (73)18p to a possible constant.Iterating this procedure yields a series of terms summing to the desired Lagrange density L , L = ∞ X n =0 n ! h α β · · · h α n β n δ n ( − τ (0) α n β n ) δψ α β · · · δψ α n − β n − (cid:12)(cid:12)(cid:12)(cid:12) ψ → η . (74)The series can be constructed provided the integrability conditions are satisfied at each step,and it may terminate at some finite n . It represents a Taylor expansion of L , and inspectionreveals the identification L = L (0) ( ψ µν , f a , ∂ µ f a ) (cid:12)(cid:12)(cid:12)(cid:12) ψ → g . (75)The above derivation shows that knowledge of L (0) in the form (66) suffices to determine L . If originally the matter-gravity coupling is specified in the linearized form (69), thebootstrap procedure amounts to finding L (0) and then determining L via Eqs. (67) and (75).If instead a pure matter action is specified by giving L (0) , it suffices to promote it accordingto Eq. (67) and obtain L via the identification (75). In this case, the bootstrap correspondsto the standard minimal-coupling procedure. For example, the usual Minkowski-spacetimeenergy-momentum tensor for Maxwell electrodynamics is T EM µν = F λµ F νλ − η µν F αβ F αβ = τ (0)EM µν , (76)with the latter equality following from conformal invariance. The corresponding Lagrangedensity is L (0)EM = − η αγ η βδ F αβ F γδ . (77)Promoting this according to Eq. (67) and making the identification (75) directly yields theusual Lagrange density L EM for electrodynamics in curved spacetime, L EM = − p | g | g αγ g βδ F αβ F γδ , (78)where g ≡ det [ g µν ]. 19 . Case of S (1) Under some circumstances, the given starting point is instead an action S (1) for a theorylinear in h µν . To obtain the fully coupled action S , one can explicitly perform the iterationprocedure above. However, a more efficient ‘inverse’ method can be adopted instead. Toimplement this method, we start by identifying the energy-momentum tensor τ (0) µν from thespecified action S (1) written in the form (69), and we promote it to a covariant expressionwith respect to ψ µν : τ (0) µν ( η ) → τ (0) µν ( ψ ) . (79)An appropriate multiplicative factor of p | ψ | may be required to maintain the tensor trans-formation properties of τ (0) µν . We then write the differential equation − τ (0) µν = δ L (0) δψ µν , (80)which reduces to Eq. (68) in the limit ψ µν → η µν . The differential equation can be solved ifthe integrability condition δτ (0) µν δψ αβ = δτ (0) αβ δψ µν (81)is satisfied. Once the solution L (0) is obtained, we can apply the identification (75) to obtain L and hence S .The above inverse trick is applied in some of the analysis that follows. To illustrate it ina more familiar context, consider the cosmological constant Λ. In Minkowski spacetime, Λis associated with an effective energy-momentum tensor given by T Λ µν = − κ Λ η µν = − τ (0)Λ µν . (82)The challenge is to bootstrap this to the fully coupled Lagrange density L Λ . Following theinverse trick, we promote τ (0)Λ µν to τ (0)Λ µν ( ψ ) = 2 κ Λ p | ψ | ψ µν , (83)where the appropriate factor of p | ψ | has been introduced. With the identities δψ µν = − ψ µα ψ νβ δψ αβ ,δ p | ψ | = p | ψ | ψ αβ δψ αβ , (84)20he integrability condition (81) can be verified, so the differential equation (80) can be solvedfor L (0) ( ψ ). Making the identification (75) then yields L Λ = − κ Λ p | g | , (85)in agreement with the usual result. Notice that the linearized version of this is L Λ ≈ − κ Λ − κ Λ h µν η µν = − κ Λ + h µν ( − τ (0)Λ µν ) , (86)as expected from Eq. (82), and that the zeroth-order term L (0) ( η ) is merely a constant inthis example. Note also that the first-order term L (1) ( η ) produces a linear instability inthe action at this order. This could be avoided by initiating the bootstrap from a theoryformulated in a suitable Riemann background spacetime [17].As another example, consider the bootstrap procedure for the transverse-traceless (TT)gauge. A common form for this gauge involves the trace-corrected field h µν and a timelikeunit vector n µ : tr (cid:2) hη (cid:3) = 0 , n µ h µν = 0 , ∂ µ h µν = 0 . (87)These standard linear gauge-fixing conditions can be expressed in terms of h µν and Γ Lαµν using Eqs. (53) and (61). The resulting expressions can then be implemented in the linearizedaction (54) via the addition of the linear Lagrange density L L TT = λ (1) tr [ h η ] + λ (2) ν n µ h µν + λ (3) α η µν Γ Lαµν , (88)where λ (1) , λ (2) ν , and λ (3) α are Lagrange multipliers. The bootstrap procedure can be appliedto each of the three terms independently. The first term is linear in h µν and of the sameform as in Eq. (86), so the bootstrap is immediate. The second term is also linear in h µν ,and the integrability conditions are directly satisified. The inverse trick described abovecan therefore be applied. The third term is independent of h µν , so the bootstrap method ofthe previous subsection applies. The net result of the bootstrap is the nonlinear constraintterms L TT = 2 λ (1) ( p | g | − p | η | ) + λ (2) ν n µ ( g µν − η µν )+ λ (3) α g µν Γ αµν , (89)which correspond to a nonlinear form of the TT gauge constraints, p | g | = p | η | , n µ g µν = n µ η µν , g µν Γ αµν = 0 . (90)21 V. BOOTSTRAP FOR CARDINAL GRAVITY At this stage, we are in a position to consider the nonlinear extension of the cardinaltheory (1). This section begins by presenting a convenient first-order reformulation of thelinear cardinal theory. In this form, the bootstrap of the kinetic terms is straightforwardusing the methods of the previous section. We investigate the bootstrap integrability con-ditions on an arbitrary potential term, which turn out to provide interesting constraints onthe theory. Finally, the bootstrap of these terms is also presented. A. First-order action To facilitate comparison with the bootstrap for general relativity, a first-order form ofthe theory (1) is useful. To develop this, we introduce the trace-reversed cardinal field C µν as C µν = − C µν + η µν C αα . (91)Note the signs, which are chosen to improve the correspondence to the conventions used inthe analysis of general relativity. The field C µν plays a central role in what follows.In terms of C µν , the second-order Lagrange density L C yielding equivalent equations ofmotion to the theory (1) takes the form L C = C µν K µναβ C αβ − V ( C µν , η µν ) . (92)Here, the quadratic operator K µναβ is given in cartesian coordinates by K µναβ = κ [ − ( η µα η νβ + η µβ η να ) ∂ λ ∂ λ − η µν ∂ α ∂ β − η αβ ∂ µ ∂ ν + η µα ∂ ν ∂ β + η να ∂ µ ∂ β + η µβ ∂ ν ∂ α + η νβ ∂ µ ∂ α ] . (93)Note that acting with this operator on the fluctuation h αβ produces the linearized Riccicurvature R Lµν : K µναβ h αβ ≡ κR Lµν . (94)22ote also that the quantities K µναβ C αβ in Eq. (1) and K µναβ C αβ are related by trace reversalwith a sign. In Eq. (92), the potential V ( C µν , η µν ) is determined by the requirement thatthe equations of motion K µναβ C αβ − δ V δ C µν = 0 (95)have the same content as the original equations of motion (3). This requires that δ V δ C µν = − δVδC µν + η µν η αβ δVδC αβ . (96)To construct the first-order form of the linear cardinal theory, we follow a similar pathto that of the Palatini formalism in general relativity discussed in Sec. III A. Introducingan independent auxiliary field Γ αµν , the Lagrange density (92) can be rewritten in terms of C µν and Γ αµν in the equivalent form S L C = Z d x L L C , L L C = κ [ C µν ( ∂ α Γ αµν − ∂ ν Γ αµα )+ η µν (Γ ββα Γ αµν − Γ αµβ Γ βνα )] + V ≡ K L + V , (97)where K L is the kinetic part of the Lagrange density. Variation of this action with respectto the independent fields C µν and Γ αµν gives the equations of motion. With standard ma-nipulations, the equations of motion determine the fields Γ αµν to be linearized Christoffelsymbols of the conventional form but depending on C µν instead of h µν . They also implylinearized versions of the equations of motion (95) for C µν obtained from the second-orderLagrange density (92).The linearized action (97) can be written in other equivalent forms by decomposing thecardinal field C µν . In the minimum of the potential V , the field C µν acquires an expectationvalue c µν , h C µν i = c µν ≡ − c µν + η µν c αα . (98)23his satisfies the identitiestr [ c η ] = tr [ cη ] , tr (cid:2) ( c η ) (cid:3) = tr (cid:2) ( cη ) (cid:3) , tr (cid:2) ( c η ) (cid:3) = − tr (cid:2) ( cη ) (cid:3) + tr [ cη ] tr (cid:2) ( cη ) (cid:3) − (tr [ cη ]) , tr (cid:2) ( c η ) (cid:3) = tr (cid:2) ( cη ) (cid:3) − cη ] tr (cid:2) ( cη ) (cid:3) + (tr [ cη ]) tr (cid:2) ( cη ) (cid:3) − (tr [ cη ]) , (99)and it also obeys ∂ α c µν = 0 (100)by virtue of the assumption (10). The fluctuation e C µν about c µν is e C µν = − e C µν + η µν e C αα . (101)The analogue of Eq. (11) therefore becomes C µν = c µν + e C µν . (102)An alternative expression for the linearized action (97) is therefore S L e C = Z d x L L e C , L L e C = κ [ e C µν ( ∂ α Γ αµν − ∂ ν Γ αµα )+ η µν (Γ ββα Γ αµν − Γ αµβ Γ βνα )] + V ≡ K L e C + V . (103)Note that the two linearized actions S L C and S L e C are identical, but by virtue of Eq. (100) thekinetic term K L differs from K L e C by a total derivative.The cardinal field C µν can be further decomposed into NG modes and massive modes, inparallel with Eq. (23). We write C µν = c µν + N µν + M µν , (104)where the trace-reversed NG field N µν is defined as N µν = − N µν + η µν N αα (105)24nd the trace-reversed massive-mode field is M µν = − M µν + η µν M αα . (106)The constraints in the NG sector corresponding to Eq. (15) can be written astr[ N η ( cη ) j ] = 0 , (107)with j = 0 , , , 3, while the analogue of Eq. (19) istr [ N η F ( c η, M η )] = 0 , (108)where F ( c η, M η ) is an arbitrary matrix polynomial in c η and M η . Another equivalent formfor the action (97) is therefore S L N , M = Z d x L L N , M , L L N , M = κ [( N µν + M µν )( ∂ α Γ αµν − ∂ ν Γ αµα )+ η µν (Γ ββα Γ αµν − Γ αµβ Γ βνα )] + V = K L N , M + V , (109)where K L N , M denotes the kinetic term expressed in terms of N µν , M µν , and Γ αµν . B. Kinetic bootstrap With the linear cardinal theory massaged into a first-order form paralleling that usedfor general relativity, we are in a position to investigate the bootstrap to nonlinear cardinalgravity. Since the bootstrap involves adding self-coupling order by order, it can be doneindependently for each part in the action. In particular, the bootstrap for the kinetic partparallels the bootstrap for the linearized version (54) of general relativity. 1. Primary bootstrap It is perhaps most natural to apply the bootstrap procedure to the linearized theory inthe form (97), which holds prior to the spontaneous Lorentz breaking. For the correspondingkinetic term K L , the energy-momentum tensor associated with C µν is of the same form asbefore, − ( τ C ) µν = κ (Γ ββα Γ αµν − Γ αµβ Γ βνα ) + κσ µν , (110)25nd the nonlinear kinetic action S K , C is obtained by coupling its nonderivative part as asource for C µν , S K , C = S L K , C + Z d x κ C µν (Γ ββα Γ αµν − Γ αµβ Γ βνα )= Z d x κ ( η µν + C µν ) R µν (Γ) , (111)where R µν (Γ) is the Ricci curvature defined via the auxiliary field Γ αµν in the usual way, R µν (Γ) = ∂ α Γ αµν − ∂ µ Γ ανα − ∂ ν Γ αµα +(Γ ββα Γ αµν − Γ αµβ Γ βνα ) . (112)Since the extra term in Eq. (111) is independent of η µν , no further iteration steps are needed.In the extremum of the potential V , the massive modes vanish and the result (111) forthe kinetic bootstrap reduces to S K , C ⊃ Z d x κ ( η µν + c µν + N µν ) R µν (Γ) . (113)The combination ( η µν + c µν ) can be viewed as playing the role of an effective backgroundmetric. Under a suitable change of coordinates, this effective metric can be brought to theMinkowski form, ( η µν + c µν ) → η µν . With the identification h µν ↔ N µν , (114)which matches the linearized correspondence (47), it follows that the kinetic action S K , C reduces to the Einstein-Hilbert action in the limit of vanishing massive modes. The result(111) for the kinetic bootstrap thereby reveals that the nonlinear cardinal theory representsan alternative theory of gravity containing general relativity in a suitable low-energy limit.The correspondence g µν ↔ η µν + e C µν (115)provides the match between the metric density g µν of general relativity and fields in cardinalgravity. 2. Alternative bootstraps The derivation of the action S K , C in Eq. (111) is based on applying the bootstrap to thelinearized cardinal action (97) for the cardinal field C µν . However, the spontaneous Lorentz26iolation produces a phase transition that naturally separates the cardinal excitations intoNG and massive modes. One could therefore instead consider applying the bootstrap tovarious choices of excitation in the effective theory describing the physics after the sponta-neous symmetry breaking has occurred. In the remainder of this subsection, we considerthese alternative bootstrap procedures and their application to the kinetic term in the linearcardinal theory.Suppose the bootstrap is instead applied to the alternative linearized cardinal action(103) for the fluctuation e C µν . This procedure has the possible disadvantage of requiring apre-established value for the vacuum expectation c µν . However, since e C µν is a fluctuation,this procedure does parallel more closely the usual bootstrap in general relativity, for whichthe relevant field h µν is also a fluctuation. The derivation of the nonlinear action S K , e C fromthe linearized theory (103) proceeds as before. The result for this secondary theory is S K , e C = S L K , e C + Z d x κ e C µν (Γ ββα Γ αµν − Γ αµβ Γ βνα )= Z d x κ ( η µν + e C µν ) R µν (Γ) . (116)This is equivalent to the action S K , C under a suitable coordinate transformation. We therebyfind that the secondary bootstrap yields the same physics for the kinetic term as did theprimary bootstrap leading to Eq. (111).A tertiary theory could also be countenanced, in which the bootstrap is applied onlyto the NG modes N µν appearing in the linearized action (109). While this procedure alsorequires a pre-established value for the vacuum expectation c µν , it has the possible advantageof matching more closely the symmetry structure of the bootstrap for general relativity. Thekey point is that the gauge transformation (5), which fails to be a symmetry of the linearizedtheory due to the potential, nonetheless does define a symmetry for the pure NG sectorbecause the potential vanishes for pure NG excitations. In linearized general relativity, theanalogous gauge symmetry can be related to the conserved two-tensor current, and it morphsinto diffeomorphism symmetry following the bootstrap procedure. In the present context,this symmetry structure is reproduced in the pure NG sector if the bootstrap is applied onlyto the NG excitation N µν in the linearized action (109).For this tertiary bootstrap, the first step is to obtain the energy-momentum tensor forthe kinetic term K L N , M in terms of the NG and massive modes. The calculations for this step27gain parallel those for the bootstrap in general relativity. We find − ( τ N , M ) µν = κ (Γ ββα Γ αµν − Γ αµβ Γ βνα )+ κσ µν ( N , Γ) + κσ µν ( M , Γ) , (117)where σ µν is the total-derivative term given by Eq. (58) but with modified arguments asindicated. The prescription for the tertiary bootstrap is then to couple the nonderivativepart of ( τ N , M ) µν as a source for N µν , K N , M = K L N , M + κ N µν (Γ ββα Γ αµν − Γ αµβ Γ βνα ) . (118)This prescription yields the tertiary kinetic action S K , N , M = Z d x κ ( η µν + N µν ) R µν (Γ)+ κ M µν ( ∂ α Γ αµν − ∂ ν Γ αµα ) . (119)Paralleling the case of general relativity, the extra term in Eq. (118) is independent of η µν ,so no further iteration steps are needed. Note that the structure of this result implies theauxiliary field Γ αµν is no longer equivalent on shell to a Christoffel symbol.The tertiary kinetic action S K , N , M differs nontrivially from the primary one S K , C , andthe physical content of the two is also different. With the identification (114) and in thepure NG sector, both actions match the Einstein-Hilbert action of general relativity. Theirlinearized content is also the same as that of the linear cardinal theory (1). C. Integrability conditions for potential Next, we investigate the integrability conditions required to apply the bootstrap on thepotential term. We obtain constraints such that V obeys the integrability conditions, andwe determine a general form of V satisfying these constraints.To proceed, start with the theory in the form (92) in terms of the cardinal field C µν . Thepotential is V ( C µν , η µν ), and it is a scalar. The only scalars that can be formed from C µν and η µν involve traces of the matrix C η . The scalar X m with m such products has the form X m = tr [( C η ) m ] . (120)28ince C η is a 4 × V ( C µν , η µν )can be written V ( C µν , η µν ) = V ( X , X , X , X ) . (121)In the minimum of V , C µν = c µν and the scalars X m have expectation values h X m i = tr [( c η ) m ] ≡ x m . (122)The next step is to determine the energy-momentum tensor τ C µν associated with thepotential V and check the integrability conditions. We therefore promote V to a covariantexpression with respect to the auxiliary metric density ψ αβ , V ( C µν , η µν ) → p | ψ | V ( C µν / p | ψ | , p | ψ | ψ µν )= p | ψ | V ( X , X , X , X ) , (123)where the four quantities X m are now X m ( ψ ) = tr [( C ψ ) m ] (124)and are scalars with respect to ψ µν . In parallel with the bootstrap for the kinetic term, C µν is taken to be a tensor density with respect to ψ µν in constructing these expressions.The energy-momentum tensor τ C µν is − τ C µν = δ ( p | ψ | V ) δψ µν . (125)The bootstrap procedure requires this to be obtained from an action by varying with respectto C µν . We must therefore add to the Lagrange density a term V ′ such that δ V ′ δ C µν = − τ C µν = δ ( p | ψ | V ) δψ µν . (126)If V ′ is smooth, then δ V ′ δ C µν δ C αβ = δ V ′ δ C αβ δ C µν , (127)which implies δ ( p | ψ | V ) δψ µν δ C αβ = δ ( p | ψ | V ) δψ αβ δ C µν . (128)This is the integrability condition for the existence of V ′ . It requires symmetry of the doublepartial derivative under the interchange ( µν ) ↔ ( αβ ).29he double derivative appearing in the result (128) can be written as δ ( p | ψ | V ) δψ µν δ C αβ = p | ψ | ( A mµναβ V m + B mnµναβ V mn ) , (129)where m and n are summed, with V m ≡ δ V δ X m , V mn ≡ δ V δ X m δ X n , (130)and with the coefficients A mµναβ and B mnµναβ given by A mµναβ = ψ µν δ X m δ C αβ + δ X m δψ µν δ C αβ = mψ µν [ ψ ( C ψ ) m − ] αβ − m m − X k =0 [ ψ ( C ψ ) k ] µα [ ψ ( C ψ ) m − − k ] νβ ,B mnµναβ = (cid:16) δ X m δψ µν δ X n δ C αβ + δ X n δψ µν δ X m δ C αβ (cid:17) = − mn (cid:16) [ ψ ( C ψ ) m ] µν [ ψ ( C ψ ) n − ] αβ + [ ψ ( C ψ ) n ] µν [ ψ ( C ψ ) m − ] αβ (cid:17) . (131)Inspection of these results reveals that the integrability condition is satisfied if and only ifthe combined quantity C mnµναβ = m V m ψ µν [ ψ ( C ψ ) m − ] αβ − mn V mn [ ψ ( C ψ ) m ] µν [ ψ ( C ψ ) n − ] αβ (132)is symmetric under the interchange ( µν ) ↔ ( αβ ).Using the Hamilton-Cayley theorem, we can write[ ψ ( C ψ ) ] µν = p [ ψ ( C ψ ) ] µν − p [ ψ ( C ψ ) ] µν + p [ ψ C ψ ] µν − p ψ µν , (133)where p = X ,p = X − X ,p = X − X X + X ,p = X − X X + X + X X − X . (134)30dopting this result and requiring symmetry of the combination (132) reveals that theintegrability condition imposes the following six partial differential equations on the potential V : V + 8 p V = − V − p V , V + 12 p V = − V + 4 p V , V + 16 p V = − V − p V , − V − p V = − V + 8 p V , − V − p V = − V − p V , − V + 16 p V = − V − p V . (135)Solutions of these equations that are polynomials in X m can be found by construction, andthey are conveniently classified according to the power q of X appearing in the polynomial.With some calculation, we have established that the unique polynomial solutions for q ≤ Y = 1 , Y = X , Y = ( X − X ) , Y = ( X − X X + 8 X ) , Y = ( X − X X + 12 X + 32 X X − X ) . (136)More generally, it follows that any polynomial obtained as the term at O ( C q ) in the seriesexpansion of p | det [1 + C ψ ] | is a solution. An expression for these polynomials is Y q = lim ǫ → q ! ∂ q ∂ǫ q ( ǫp + ǫ p + ǫ p + ǫ p ) / . (137)For example, at q = 5 a solution to Eq. (135) is the polynomial Y = ( − X + 28 X X − X X − X X + 32 X X + 48 X X ) . (138)We conjecture that the polynomials obtained in this way are in fact unique solutions at eachorder q . 31 general potential V that solves the differential equations (135) can therefore be writtenas p | ψ | V = p | ψ | ∞ X q =0 α q Y q , (139)where the α q are arbitrary real constants. For any fixed α q , a potential of this form satisfiesthe integrability conditions (128) required for the bootstrap procedure. Note that for thespecial case α q = α for all q ≥ 0, the solution becomes p | ψ | V = α p | det [ ψ + C ] | . (140) D. Bootstrap for integrable potential In this subsection, we first apply the bootstrap procedure to the integrable potential (139).We then consider some aspects of extrema of the resulting theory, provide a constructionfor a local minimum, and offer some remarks about alternative bootstrap procedures for thepotential. 1. Potential bootstrap The bootstrap procedure using the cardinal field C µν can be explicitly performed termby term on the potential (139). For each q , p | ψ | Y q is a coefficient in the expansion of p | det [ ψ + C ] | . In Sec. III B 2, a bootstrap procedure has been performed that leads tothe potential (85) proportional to p | det [ ψ + C ] | . It follows from this analysis that thebootstrap applied to the term p | ψ | Y q generates for each q the full result p | det [ ψ + C ] | minus the sum of all terms of orders less than q : p | ψ | Y → p | det [ ψ + C ] | , p | ψ | Y → p | det [ ψ + C ] | − p | ψ | Y , p | ψ | Y → p | det [ ψ + C ] | − p | ψ | ( Y + Y ) , (141)and so on, with the general term being p | ψ | Y q → p | det [ ψ + C ] | − p | ψ | q − X k =0 Y k . (142)32pplying the bootstrap to the general potential (139) yields the bootstrap potential V C , p | ψ | V C = ∞ X q =0 α q (cid:16)p | det [ ψ + C ] | − p | ψ | q − X k =0 Y k (cid:17) = p | ψ | ∞ X q =0 α q ∞ X k = q Y k = p | ψ | ∞ X k =0 δ k Y k , (143)where the real coefficients δ k are given as δ k = k X q =0 α q . (144)Note that the coefficient δ k for fixed k acquires nonvanishing contributions from any nonva-nishing coefficients α q with q ≤ k .For nonlinear cardinal gravity, the above discussion reveals that the potential term ap-pearing in the bootstrap action takes the form S V , C = Z d x V C = ∞ X k =0 δ k Z d x Y k . (145)This potential term combines with the kinetic term S K , C in Eq. (111) to form the primarycardinal action. 2. Extrema of the potential Vacuum solutions of nonlinear cardinal gravity are extremal solutions of the potential V C .In an extremum, the cardinal field C µν acquires a vacuum value that may differ from anyextrema generated by the potential V in the linearized theory and defined in Eq. (98). Bymild abuse of notation, in what follows we adopt the same notation C µν = c µν for a vacuumvalue in an extremum of V C . Similarly, we adopt the same notation as in Eq. (104) for thedecomposition of the cardinal field C µν and its fluctuations e C µν into the NG excitations N µν of Eq. (105) and the massive excitations M µν of Eq. (106). However, linearized results for N µν and M µν such as Eqs. (107) and (108) no longer hold.A vacuum of V C can also be identified by the values x m taken by the four scalars X m , asin Eq. (122). The restriction of the potential V C to the NG sector can then be achieved by33eplacing V C with the Lagrange-multiplier potential V λ = X m =1 λ m ( X m − x m ) , (146)which excludes fluctuations away from the extremum. If desired, the on-shell values ofthe Lagrange multipliers λ m can be set to zero by suitable boundary conditions. Thispotential facilitates the identification of the NG and massive modes. The NG modes N µν are the nonzero components of C µν that preserve the constraints obtained from the Lagrange-multiplier equations of motion, while the massive modes are the components of C µν that areconstrained to zero. Note that the potential V λ is dynamically equivalent to a potential V λ ′ expressed using the integrable polynomials (137), given by V λ ′ = X m =1 λ ′ m ( Y m − y m ) , (147)where y m are the values of Y m for C µν = c µν . The Lagrange-multiplier constraints areequivalent by direct comparison, while the dynamical properties under variation with respectto C µν are equivalent when the Lagrange multipliers are identified by the nonsingular set oflinear equations λ m = ( − m +1 m X p = m λ ′ p y p − m (148)with 1 ≤ m ≤ N µν are seen directly to be the solutions of theequations X m = x m , which can be written as nonlinear generalizations of Eq. (107),0 = tr [ N η ] , N ηcη ] + tr (cid:2) ( N η ) (cid:3) , (cid:2) N η ( cη ) (cid:3) + 3tr (cid:2) ( N η ) cη (cid:3) + tr (cid:2) ( N η ) (cid:3) , (cid:2) N η ( cη ) (cid:3) + 3tr (cid:2) ( N η ) ( cη ) (cid:3) + 3tr (cid:2) ( N ηcη ) (cid:3) +4tr (cid:2) ( N η ) cη (cid:3) + tr (cid:2) ( N η ) (cid:3) . (149)The ten independent components of N µν are constrained by these four equations, leavingthe expected six NG modes. The four massive modes can be denoted by M m and specifiedas M m = X m − x m = tr h ( c η + e C η ) m i − tr [( c η ) m ] . (150)34hey are contained in the symmetric tensor M µν , which is obtained by subtraction of theNG modes N µν from the cardinal fluctuation field e C = C µν − c µν .In the absence of coupling to matter, the equations of motion for cardinal gravity areobtained by varying the sum of the kinetic and potential actions (111) and (145) with respectto the independent fields. Eliminating the auxiliary field Γ αµν yields the field equations inthe absence of matter as R µν = 2 κτ vac µν , X m = x m , (151)where τ vac µν is given by − τ vac µν = ∂ V C ∂ C µν (cid:12)(cid:12)(cid:12) C → c = X m =1 ∂ X m ∂ C µν V C ,m (cid:12)(cid:12)(cid:12) C → c = X m =1 m [ η ( c η ) m − ] µν V C ,m | C → c . (152)Note that V C ,m = λ m in the Lagrange-multiplier limit. The quantity τ vac µν represents a kindof vacuum energy-momentum tensor density. Trace-reversing yields the field equations forcardinal gravity in the absence of matter, which can be written in the form G µν = 2 κT µν vac . (153)Here, G µν is the Einstein tensor for the metric obtained from the metric density ( η µν + C µν ),while the vacuum energy-momentum tensor T µν vac is obtained by the corresponding tracereversal of τ vac µν . The conservation law D µ T µν vac = 0 (154)follows by virtue of the Bianchi identities. This conservation remains true in the presenceof matter couplings, provided the matter-sector energy-momentum tensor is independentlyconserved. If the Lagrange multipliers λ m vanish, or more generally if V m vanishes, thenthe vacuum energy-momentum tensor is zero and the usual form of general relativity isrecovered. Otherwise, there is a positive or negative contribution to the vacuum energy-momentum tensor. This may play a role in cosmology and the interpretation of dark energy.In the pure NG sector with zero on-shell Lagrange multiplier fields, the effective potentialvanishes and nonlinear cardinal gravity reduces to the kinetic term (113). As already noted,35his limit reproduces general relativity, with the identification N µν ↔ h µν in Eq. (114). TheEinstein-Hilbert action is recovered in a fixed gauge, the nonlinear cardinal gauge, which isdefined by the four nonlinear gauge conditions0 = tr [ h η ] , h ηcη ] + tr (cid:2) ( h η ) (cid:3) , (cid:2) h η ( cη ) (cid:3) + 3tr (cid:2) ( h η ) cη (cid:3) + tr (cid:2) ( h η ) (cid:3) , (cid:2) h η ( cη ) (cid:3) + 3tr (cid:2) ( h η ) ( cη ) (cid:3) + 3tr (cid:2) ( h ηcη ) (cid:3) +4tr (cid:2) ( h η ) cη (cid:3) + tr (cid:2) ( h η ) (cid:3) (155)obtained by the replacement N µν → h µν in Eq. (149).The bootstrap for general relativity transforms the gauge symmetry (37) of the linearizedtheory into diffeomorphism invariance of the Einstein-Hilbert action, involving particle trans-formations of the metric density g µν . In the linear cardinal theory, the analogue of thegauge symmetry (37) is the symmetry (5) of the kinetic term alone. The pre-bootstrappotential V explicitly breaks this symmetry, so the potential term (145) can be expected toexhibit diffeomorphism breaking under particle transformations of the analogue metric den-sity ( η µν + C µν ). This is reflected, for example, in the presence of a factor p | ψ | → p | η | = 1in the measure of Eq. (145). However, as expected from the match to general relativity,the pure NG sector of cardinal gravity with zero on-shell Lagrange multipliers does exhibitthe usual diffeomorphism invariance because the potential vanishes in this sector. Note alsothat cardinal gravity remains invariant under diffeomorphisms of the Minkowski spacetime.Both general relativity and cardinal gravity are invariant under (observer) general coor-dinate transformations. The match between the two theories in the pure NG limit involvesa coordinate transformation taking ( η µν + c µν ) → η µν in the kinetic term (113). There istherefore a corresponding transformation taking η µν → [(1 + c η ) − η ] µν in the potential term.For example, the general coordinate invariance ensures a factor p | (1 + c η ) − η | appears inthe measure of Eq. (145). However, the vanishing of the potential in the pure NG sectormakes this factor irrelevant for the match to general relativity.36 . Stability of the extrema Given a bootstrap potential V C , an interesting issue is whether it admits an extremumthat is stable. The question of overall stability for any given theory with Lorentz violationis involved [23]. Even for the comparatively simple bumblebee theories the issue remainsopen, although considerable recent progress has been made [24]. A full analysis for cardinalgravity lies outside the scope of this work. Instead, this subsection provides a few remarkson stability in the specific context of the potential term.In the vacuum, the extremal solutions obey0 = ∂ V C ∂ C µν (cid:12)(cid:12)(cid:12) C → c = X m =1 m [ η ( c η ) m − ] µν V C ,m (cid:12)(cid:12)(cid:12) C → c , (156)where V C ,m ≡ ∂ V C /∂ X m . By assumption, the matrix c η has four inequivalent nonzeroeigenvalues. Working in the basis in which c η is diagonal, this implies the generic conditionsfor a vacuum are V C ,m (cid:12)(cid:12)(cid:12) C → c = 0 . (157)A vacuum of V C is stable if it is a Morse critical point with positive definite hessian. Forsimplicity, we introduce the explicit diagonal basis C µλ η λν = C µ δ µν (158)(no sum on µ ), where the four quantities C µ are the eigenvalues of C η . Then X m = X j =0 ( C j ) m , (159)and in the vacuum C j = c j , with all four values c j inequivalent and nonzero. In the diagonalbasis, stability depends on the hessian H jk = ∂ V C ∂ C j ∂ C k (cid:12)(cid:12)(cid:12) C → c = X m,n =1 mn ( c j ) m − ( c k ) n − V C ,mn (cid:12)(cid:12)(cid:12) C → c . (160)37f the discriminant is nonzero and the four eigenvalues H m of the hessian are positive, theextremum is a local minimum.An analytical derivation of a potential with a positive definite hessian in terms of thepolynomial basis (137) is challenging. Instead, we proceed by ansatz using the shiftedvariables e X m = X m − x m . (161)For the ansatz, we adopt the form of a Taylor expansion V C = a mn e X m e X n + a mnp e X m e X n e X p + . . . , (162)where the coefficients a mn , a mnp , . . . are real constants. The potential V C in Eq. (145) is acombination of integrable partial potentials, so the expression (162) must be integrable too.We can therefore constrain the coefficients by imposing the integrability conditions (135)on V C itself at e X m = 0. At second order in e X m , this imposes six conditions on the tendegrees of freedom a mn . The four degrees of freedom a m can be taken as unconstrained atthis order. To impose the integrability conditions at third order, it is convenient to takepartial derivatives of Eqs. (135) with respect to each X m . This produces 24 equations, whichcombine with the second-order equations to yield 16 independent constraints on the 20 third-order coefficients a mnp . The four degrees of freedom a m can be taken as unconstrained atthis order. Proceeding in this way, we find a 4( n − V C up to order n . As a check, the resulting solutions can be reconstructed in termsof suitable combinations of the polynomial basis (137).Given the potential V C in the form (162), the issue of finding a solution with positivedefinite hessian can be resolved numerically. Investigation shows that there is a subspace ofcoefficients a mn for which the integrability conditions are satisfied and the hessian is positivedefinite. An explicit example is the potential V C = X k =1 δ k Y k , (163)with the coefficients given by δ ≃ − . , δ ≃ − . , δ ≃ . , δ ≃ . ,δ ≃ − . , δ ≃ − . , δ ≃ . , δ ≃ . . (164)38he local minimum is found to lie at X ≃ . , X ≃ . , X ≃ . , X ≃ . . (165)The eigenvalues of the corresponding hessian are found to be H ≃ . , H ≃ . , H ≃ . , H ≃ . , (166)demonstrating positivity. This example therefore represents a potential V C having a localminimum. 4. Alternative potential bootstraps The bootstrap procedure discussed above holds for the potential prior to the developmentof a vacuum value for the cardinal field C µν . Alternative options for the potential term,applicable following spontaneous Lorentz violation instead, include a secondary bootstrapusing the cardinal fluctuation e C µν and a tertiary one using only the NG modes N µν . Theexplicit construction of these potentials lies outside the scope of this work. Instead, thissubsection contains a few brief comments about some aspects of these alternative bootstrapprocedures, following from the analysis of the primary case.To perform an alternative bootstrap procedure, the corresponding integrable potentialmust first be constructed. For the secondary bootstrap involving the cardinal fluctuation e C µν introduced in Eq. (102), the promotion of the potential V to a covariant expression withrespect to the auxiliary metric density ψ αβ involves the four scalars X m given by X m ( ψ ) = tr h ( c ψ + e C ψ ) m i . (167)The energy-momentum tensor must now be obtained from an action by varying with respectto e C µν . The basic integrability condition is found to be δ ( p | ψ | V ) δψ µν δ e C αβ = δ ( p | ψ | V ) δψ αβ δ e C µν . (168)However, since the cardinal fluctuation e C µν is merely a constant shift of the cardinal field C µν , we have ∂ X m ∂ e C µν = ∂ X m ∂ C µν . (169)39his in turn means that the integrability condition is satisfied for the same symmetry re-quirement on the same expression (132) as before. The integrable potential for the secondarybootstrap therefore takes the same form (139) as for the primary case.A similar situation holds for the tertiary bootstrap involving the NG modes N µν in thedecomposition (104). In this case, the four relevant scalars are X m = tr [( c ψ + N ψ + M ψ ) m ] . (170)The energy-momentum tensor is required to arise by varying an action with respect to N µν .This generates the integrability condition δ ( p | ψ | V ) δψ µν δ N αβ = δ ( p | ψ | V ) δψ αβ δ N µν . (171)However, the form of Eq. (104) implies ∂ X m ∂ N µν = ∂ X m ∂ C µν . (172)It follows that the integrability condition is again satisfied for the same symmetry require-ment on the same expression (132), and the integrable potential for the tertiary bootstraptakes the same form (139) as before.Although the integrable potentials (139) are the same, the alternative bootstrap proce-dures differ from each other and from the primary one presented above. Moreover, perform-ing these bootstrap procedures involves additional choices because integration with respectto the linear cardinal fluctuation or the linear NG modes can either be continued at allorders or can be adjusted at each order to incorporate the induced nonlinearities. Any ofthese bootstrap procedures could in principle be performed using the methods presented inSec. III.An extremum of an alternative bootstrap potential is achieved for vanishing massivemodes. It can therefore be represented by a suitable Lagrange-multiplier potential. Inparticular, in the pure NG limit the potential vanishes for on-shell multipliers, and so theresulting effective theory is controlled by the corresponding kinetic term. This means thatgeneral relativity is also recovered in the low-energy limits of the nonlinear theories arisingin these alternative bootstrap procedures. 40 . COUPLING TO MATTER At the linear level, the cardinal field C µν must couple to other fields in the Minkowskispacetime via a symmetric two-tensor current. Given our gravitational interpretation of thecardinal field, the other fields in the theory can be regarded as the matter. They provide onenatural two-tensor current, the energy-momentum tensor T M µν in the Minkowski spacetime.We can therefore expect the linearized theory (1) to incorporate the matter interaction L L M ,C = C µν T M µν . (173)No coupling constant is necessary for this interaction, since it can be absorbed in the scalingfactor κ already present in the original theory (1). A. Primary bootstrap The bootstrap procedure involving the cardinal field C µν can be applied to the matterinteraction (173) to determine the form of the matter coupling for cardinal gravity. Forthis purpose, the interaction (173) is conveniently expressed in terms of the trace-reversedenergy-momentum tensor τ M µν for the matter. This tensor arises by variation of the Lagrangedensity L M for the matter fields via − τ M µν = δ L M ( η → ψ ) δψ µν (cid:12)(cid:12)(cid:12)(cid:12) ψ → η (174)in the usual way. We can therefore write L L M , C = − C µν τ M µν (175)for the matter interaction with the cardinal field C µν .To perform the bootstrap, the techniques of Sec. III B can be applied. The Lagrangedensity (175) is linear in C µν and so has the form (69), for which the bootstrap yields Eq.(75). The bootstrap therefore generates the Lagrange density L M , C = p | η + C | L L M , C (cid:12)(cid:12) η → η + C . (176)Some insight into the physical content of this result can be obtained by expanding aboutan extremum of the bootstrap potential. Writing C µν = c µν in the extremum and denoting41he corresponding fluctuations by e C µν = N µν + M µν as before, we obtain L M , C = q | η + c + e C | L L M , C (cid:12)(cid:12) η → η + c + e C . (177)A comparison of this result to the matter coupling of general relativity can be performedby adopting Lagrange-multiplier bootstrap potential (146). The massive modes vanish, M µν → 0, and as before a suitable change of coordinates must be performed to implementthe transformation ( η µν + c µν ) → η µν and thereby ensure the kinetic term (113) containsthe conventional Minkowski metric. The resulting Lagrange density L NGM , C then matches theusual matter term L GRM in general relativity, L NGM , C = p | η + N | L L M , C (cid:12)(cid:12) η → η + N ↔ L GRM = p | g | L L M (cid:12)(cid:12) η → g , (178)when the correspondence g µν ↔ η µν + N µν of Eq. (115) is adopted.We can therefore conclude that the pure NG sector of cardinal gravity with zero on-shellLagrange multipliers exactly reproduces general relativity, including the matter coupling.When the massive modes are included, the matter coupling deviates from that in generalrelativity by terms that are suppressed by the scale of the massive modes. B. Alternative bootstraps Alternative bootstrap procedures for the matter coupling can be countenanced instead.We consider here the secondary and tertiary procedures discussed above for the kinetic andpotential terms. We also examine some experimental implications of the results for the pureNG sector and the match to general relativity.The secondary bootstrap involving the cardinal fluctuation e C µν starts from the mattercoupling (173) in the form L L M , e C = c µν ( − τ M µν ) + e C µν ( − τ M µν ) . (179)The bootstrap can be performed using the methods of Sec. III B. The first term in Eq. (179)involves c µν but is independent of e C µν , while the only dependence on the Minkowski metricappears in τ M µν . The effect of the bootstrap on this term is therefore to introduce a factor42f q | η + e C | and to replace τ M µν ( η µν ) with τ M µν ( η µν + e C µν ). The second term is linear in e C µν and hence is of the form (69), for which the bootstrap gives Eq. (75). We therefore obtain L M , e C = q | η + e C | c µν ( − τ M µν (cid:12)(cid:12) η → η + e C )+ q | η + e C | L L M , C (cid:12)(cid:12) η → η + e C (180)as the secondary bootstrap matter coupling.For the tertiary bootstrap, the starting point is the matter coupling in the form L L M , N , M = ( c µν + M µν )( − τ M µν ) + N µν ( − τ M µν ) . (181)Here, we bootstrap only the field N µν containing the linearized NG modes, without correctingat each order. Using the techniques in Sec. III B, we find the Lagrange density L M , N , M = p | η + N | ( c µν + M µν )( − τ M µν (cid:12)(cid:12) η → η + N )+ p | η + N | L L M , N , M (cid:12)(cid:12) η → η + N (182)as the result of the tertiary booststrap.The alternative results (180) and (182) for the matter coupling contain terms correspond-ing to the usual minimally coupled Lagrange density for matter and additional couplingsbetween between matter and the massive modes. Each also contains a term involving thecardinal vacuum value c µν and the energy-momentum tensor. This last term remains as anunconventional expression in the Lagrange density in the pure NG limit M µν → 0, and forthe match to general relativity it therefore represents an unconventional contribution to thematter sector.Couplings involving tensor vacuum values appear naturally in the Standard-Model Ex-tension (SME), which provides a general framework for the description of Lorentz violationusing effective field theory [2, 25]. The matter sector of the SME includes Lorentz-violatingoperators controlled by coefficients that are symmetric observer two-tensors and that canbe related to c µν . Numerous experimental measurements have been performed on the coeffi-cients for Lorentz violation [26]. This offers an interesting opportunity to identify constrainson the alternative bootstrap theories.Consider first an example illustrating the connection between the cardinal matter couplingand the SME framework, involving a matter Lagrange density for a complex scalar field φ 43n Minkowski spacetime given by L φ = − η µν ∂ µ φ † ∂ ν φ − U ( φ † φ ) . (183)Here, U ( φ † φ ) is an effective Lorentz-invariant potential that can include mass and self-interaction terms. The corresponding energy-momentum tensor T µν is T µν = ∂ µ φ † ∂ ν φ + ∂ ν φ † ∂ µ φ + η µν L φ . (184)Introducing the cardinal coupling (173) and restricting attention to the vacuum value c µν adds the term L φc = c µν T µν = c µν ( − τ µν ) , (185)where τ µν is the trace-reversed form of T µν . Performing either of the alternative bootstrapsin the NG limit yields the contribution of the cardinal-scalar coupling to the full theory, L φ c , N = p | g | [ c µν ( − τ µν )] (cid:12)(cid:12) η → g = p | g | c µν T µν (cid:12)(cid:12) η → g = p | g | c T µν T µν (cid:12)(cid:12) η → g + p | g | tr [ cg ] tr h T (cid:12)(cid:12) η → g g i , (186)where we denote the bootstrap metric density η µν + N µν by g µν and the corresponding metricby g µν . For the last expression in this equation, the coefficient c µν has been separated intotraceless and trace pieces for convenience in what follows, via the definitions c µν = c T µν + tr [ cg ] g µν , tr (cid:2) c T g (cid:3) = 0 . (187)We can compare the result for L φ c , N to that obtained in the SME framework for theLorentz-violating theory of a complex scalar field in Riemann spacetime with Lagrangedensity [2] L φg = − p | g | g µν ∂ µ φ † ∂ ν φ − p | g | U ( φ † φ )+ p | g | k µν ( ∂ µ φ † ∂ ν φ + ∂ ν φ † ∂ µ φ ) . (188)In this model, k µν is a symmetric coefficient for Lorentz violation, which is normally takento satisfy tr [ kg ] = 0 because a nonzero trace is Lorentz invariant. Inspection reveals theidentification c T µν ≡ k µν (189)44etween the cardinal vacuum value and the SME coefficient for Lorentz violation. Note thatthe conformally invariant case satisfies tr [ T g ] = 0, in which case the two models (186) and(188) match exactly.As another example with direct physical application, consider the Maxwell Lagrangedensity L (0)EM for photons in Minkowski spacetime, given in Eq. (77). The correspondingenergy-momentum tensor T EM µν is presented in Eq. (76). The cardinal-photon coupling is L φc = c µν T EM µν = c µν ( − τ EM µν ) , (190)and the bootstrap generates the result L EM c , N = p | g | [ c µν ( − τ EM µν )] (cid:12)(cid:12) η → g = p | g | c T µν F αµ F να . (191)In this example, only the trace part c T µν appears in the final answer because the photonaction is conformally invariant. This result can be compared to the CPT-even part of thephoton sector in the minimal SME [27]. The corresponding coefficients for Lorentz violationform an observer four-tensor ( k F ) αλµν , which has the symmetries of the Riemann tensor.This four-tensor can be decomposed in parallel with the decomposition of the Riemanntensor into the Weyl tensor, the tracless Ricci tensor, and the scalar curvature. The scalarpart is Lorentz invariant. The Weyl part involves an observer four-tensor that controlsbirefringence of light induced by Lorentz violation. The traceless Ricci part determines theanisotropies in the propagation of light due to Lorentz violation, and it is specified by thetraceless observer two-tensor k µνF ≡ ( k F ) αµαν . Only the latter effects are relevant for presentpurposes. Restricting attention to these coefficients produces in Riemann spacetime theLagrange density [2] L EM = − p | g | F µν F µν + p | g | k µνF F αµ F αν = − p | g | F µν F µν + p | g | k µνF T EM µν , (192)where the tracelessness of k µνF has been used. Comparison of this result with Eq. (191) showsthe match c T µν ≡ k µνF , (193)in analogy with that of Eq. (189). 45he similarity of the matches (189) and (193) between c T µν and certain traceless SMEcoefficients for Lorentz violation is no accident. Consider a theory in which the spacetimemetric in the gravity sector is g µν . If the theory has Lorentz violation, the matter-sectormetric could differ from g µν . Denote the matter-sector metric by g µν + k µν , where thecoefficient k µν for Lorentz violation is symmetric and traceless. For small k µν , the matter-sector Lagrange density L M ( g + k ) can be expanded as L M ( g + k ) = L M ( g ) + k µν δ L M ( g ) δg µν + . . . = L M ( g ) + k µν T µν M + . . . , (194)where T µν M is the energy-momentum tensor for the Lagrange density L M ( g ). We see that thepiece of the cardinal coupling (173) involving c T µν can always be matched at leading orderto a term involving a traceless shift k µν in the matter-sector metric of a theory with Lorentzviolation.The same line of reasoning also yields a path to experimental constraints on c T µν . The keypoint is that a suitable choice of coordinates can convert g µν + k µν → g ′ µν , thereby makingthe matter sector Lorentz invariant at leading order in k µν . The price for this transformationis the conversion of the gravity-sector metric g µν → g ′ µν − k µν , which means that signals fromLorentz violation could be detectable in suitable gravitational experiments. In particular,at leading order we find L cardinal ⊃ κ g µν R µν (Γ) → κ g ′ µν R µν (Γ) + κk µν R µν (Γ) . (195)The last term matches the standard form for one type of Lorentz violation in the gravitysector of the minimal SME, controlled by the coefficient s µν for Lorentz violation [2]. This co-efficient can be studied experimentally in various ways [28, 29]. Most components of relatedcoefficients have been constrained to parts in 10 to 10 via reanalysis of several decadesof data from lunar laser ranging [30] and by laboratory tests with atom interferometry [31].We can therefore conclude that the traceless part of the vacuum value of the cardinal fieldis constrained at the same level in both the secondary and the tertiary cardinal theories.46 I. SUMMARY AND DISCUSSION This work constructs an alternative theory of gravity, which we call cardinal gravity,based on the idea that gravitons are massless NG modes originating in spontaneous Lorentzviolation. The starting point is the simple theory (1) of a symmetric two-tensor cardinalfield C µν in Minkowski spacetime with a potential triggering spontaneous Lorentz violation[6]. Requiring consistent self-coupling to the energy-momentum tensor constrains the formof the potential to the form (139). It also defines a bootstrap procedure that permits theconstruction of a self-consistent nonlinear theory.When the bootstrap is applied to the original theory prior to the spontaneous Lorentzviolation, cardinal gravity emerges. This theory has kinetic term S K , C given by Eq. (111),potential term S V , C given by Eq. (145), and matter coupling L M , C given by Eq. (176). Atlow energies compared to the scale of the massive modes, the potential can be approximatedby its extremal Lagrange-multiplier form (146) that allows only NG excitations about thevacuum. In this limit, the nonlinear cardinal action reduces to the Einstein-Hilbert ac-tion of general relativity with conventional matter coupling and possibly a vacuum energy-momentum term (152), all expressed in the nonlinear cardinal gauge given by Eq. (155).If instead the bootstrap is applied to the effective action for the spontaneously brokentheory, alternative cardinal theories are generated. Using the fluctuation field about thecardinal vacuum value as the basis for the bootstrap yields a secondary cardinal gravity. Thishas kinetic term given by Eq. (116) and matter coupling given by Eq. (180). Using insteadonly the NG excitations to perform the bootstrap produces a tertiary cardinal gravity, withkinetic term given by Eq. (119) and matter coupling given by Eq. (182). The actions of thesealternative cardinal theories also reduce to the Einstein-Hilbert action in the pure NG limitand in the nonlinear cardinal gauge (155). However, unconventional matter coupling termsremain in this limit. These can be constrained by suitable gravitational experiments, andexisting results limit the magnitude of components of the cardinal vacuum value to parts in10 to 10 .All forms of cardinal gravity differ from general relativity in certain respects. One is thepresence of the massive modes M µν . The scale of these modes is set by the curvature ofthe potential about the Lorentz-violating extremum. The natural scale in the theory is thePlanck mass, which enters via the Newton gravitational constant in the usual way, so it is47lausible that the fluctuations of the modes M µν are also of Planck mass. At low energies,their propagation can therefore be neglected, and they can be integrated out of the action toyield their effective contribution. The form of the kinetic term (111) suggests the correctionsto the Einstein-Hilbert action appear in part as the square of the Ricci tensor suppressedby the square of the mass of the modes M µν . A suppressed effective matter self-interactionthat is quadratic in the energy-momentum tensor also appears. Investigation of the re-sulting subleading corrections to the Einstein equations, some of which are proportionalto the Ricci tensor and hence vanish in the vacuum, is an open topic. A post-newtonianstudy of the experimental consequences for laboratory and solar-system situations, includ-ing gravitational-wave searches, would be of definite interest. A study of the implicationsfor cosmology would also be worthwhile because corrections appear to standard solutionsand also because the vacuum energy-momentum tensor (152) can appear. These variousinvestigations may be most effectively undertaken in the nonlinear cardinal gauge (155), forwhich the form of conventional general-relativistic solutions remains to be obtained.In more extreme situations, such as near the singularities of black holes or in the veryearly Universe, the contributions from the massive modes could be sufficient to change qual-itatively the usual general-relativistic behavior. The additional propagating modes can beexpected to affect features such as inflation and to change the cosmic gravitational back-ground. At sufficiently high temperatures the potential changes shape [32] to restore exactLorentz symmetry, with an extremum having a zero value for C µν . This reverse phase transi-tion converts the NG modes into massive modes, so the graviton excitations acquire Planckmasses and the nature of gravity at the big bang is radically changed.Cardinal gravity has general coordinate invariance and diffeomorphism symmetry of thebackground spacetime at all scales, as discussed in the context of the gauge-fixing conditions(155). Diffeomorphism invariance involving the analogue metric density ( η µν + N µν ) emergesin the low-energy limit, where the match to general relativity occurs. This feature of cardinalgravity has some appeal. The aesthetic and mathematical advantages of the diffeomorphisminvariance of general relativity are maintained in the low-energy limit of cardinal gravity,while at high energies the presence of the original background spacetime may offer conceptualand calculational advantages for understanding the physics. One example might involve thevacuum value of the metric, which is presumably set by processes at the Planck scale. Ingeneral relativity one can ask why the vacuum value of the metric is nonzero. Since the metric48s the fundamental field and the Einstein-Hilbert action has diffeomorphism invariance, itmight seem natural for the metric field to vanish in the vacuum. In contrast, in cardinalgravity at high energies the background spacetime is nondynamical, and the gravitationalproperties at high energies are controlled instead by the cardinal field. The vacuum value ofthe cardinal field affects the physics but not the existence of spacetime properties. Anotherexample might be improved prospects for quantum calculations at high energies, althoughthis would require revisiting the analysis in the present work with quantum physics inmind. For instance, our derivation of the integrable potential is based on purely classicalconsiderations, and the effect of radiative corrections is an open issue. In the context ofbumblebee theories, requiring one-loop stability under the renormalization group restrictsthe form of the potential and shows that those producing spontaneous Lorentz breaking aregeneric [33]. The analogue of this for cardinal gravity represents an independent conditionon the potential that is likely to constrain further its form.We conclude this discussion by noting an interesting possibility implied by the presentwork. We have demonstrated here that nonlinear gravitons in general relativity can be in-terpreted as NG modes from spontaneous Lorentz violation. It is also known that photonscan be interpreted as NG modes from spontaneous Lorentz violation, even in the presence ofgravity: the Einstein-Maxwell equations are reproduced at low energies by a suitable bum-blebee theory [5]. 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