Gravity from Spontaneous Lorentz Violation

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provided by IUScholarWorks PHYSICAL REVIEW D 79, 065018 (2009) Gravity from spontaneous Lorentz violation

V. Alan Kostelecky´1 and Robertus Potting2 1Physics Department, Indiana University, Bloomington, Indiana 47405, USA 2CENTRA, Physics Department, FCT, Universidade do Algarve, 8000-139 Faro, Portugal (Received 13 January 2009; published 19 March 2009) We investigate a class of theories involving a symmetric two-tensor field in Minkowski spacetime with a potential triggering spontaneous violation of Lorentz symmetry. The resulting massless Nambu- 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 the Lorentz violation. The nonlinear theory generated from the self-consistent bootstrap is an alternative 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 the Riemann-spacetime metric constructed as a combination of the two-tensor field and the Minkowski metric. At high energies, the structure of the theory is qualitatively different from general relativity. Observable effects can arise in suitable gravitational experiments.

DOI: 10.1103/PhysRevD.79.065018 PACS numbers: 11.30.Cp, 04.50.h, 12.60.i

interpretations of the NG modes include a new spin- I. INTRODUCTION dependent interaction [11] and various new spin- The idea that physical Lorentz symmetry could be bro- independent forces [12]. For certain theories in Riemann- ken in a fundamental theory of nature has received much Cartan spacetimes, the NG modes can instead be absorbed attention in recent years. One attractive mechanism is into the spin connection via the Lorentz-Higgs effect [5]. spontaneous Lorentz violation, in which an interaction In the present work, we investigate the possibility that drives an instability that triggers the development of non- the full nonlinear structure of general relativity can be zero vacuum values for one or more tensor fields [1]. recovered from an alternative theory of gravity with spon- Unlike explicit breaking, spontaneous Lorentz violation taneous Lorentz violation in which the gravitons are fun- is compatible with conventional gravitational geometries damental excitations identified with the NG modes. [2], and it is therefore advantageous for model building. General relativity has the interesting feature that it can be However, spontaneous violation of a continuous global reconstructed uniquely from massless spin-2 fields by re- symmetry comes with massless excitations, the Nambu- quiring consistent self-coupling to the energy-momentum Goldstone (NG) modes [3]. Among the challenges facing tensor [13–17]. For example, the linearized theory describ- attempts to construct realistic models with spontaneous ing gravitational waves via a symmetric two-tensor h Lorentz violation is accounting for the role of the corre- propagating in a spacetime with Minkowski metric sponding NG modes and interpreting them phenomenolog- contains sufficient information to reconstruct the full non- ically. linearity of general relativity when self-consistency is im- Since the NG modes are intrinsically massless, they can posed. Here, we demonstrate that applying this bootstrap generate long-range forces. One intriguing possibility is method to a linearized theory with a symmetric two-tensor that they could reproduce one of the long-range forces field C and a potential VðC ;Þ inducing spontane- known to exist in nature. For electrodynamics, for ex- ous Lorentz violation yields an alternative theory of grav- ample, the Einstein-Maxwell equations in a fixed gauge ity, which we call cardinal gravity [18]. The coupling of the naturally emerge from the NG sector of certain gravita- cardinal field to the matter sector is derived, and constraints tionally coupled vector theories with spontaneous Lorentz from existing experiments are considered. We show that violation known as bumblebee models [4,5]. For gravity the action of cardinal gravity corresponds to the Einstein- itself, the gravitons can be interpreted as the NG modes Hilbert action at energies small compared to the Planck from spontaneous Lorentz violation in several ways. As scale. However, the structure of the theory at high energies fundamental field excitations, gravitons can be identified is qualitatively different from that of general relativity. Our with the NG modes of a symmetric two-tensor field C in results indicate that cardinal gravity is a viable alternative a theory with a potential inducing spontaneous Lorentz theory of gravity exhibiting some intriguing features in violation, which generates the linearized Einstein equa- extreme gravitational environments. tions in a fixed gauge [6]. Alternatively, gravitons as com- We begin this work in Sec. II by presenting the linear- posite objects can be understood as the NG modes of ized cardinal theory and a discussion of its correspondence spontaneous Lorentz violation arising from self- to linearized general relativity. Section III reviews the interactions of vectors [7], fermions [8], or scalars [9], bootstrap procedure for general relativity and obtains following related ideas for composite photons [10]. Other some generic results. For general relativity, the bootstrap

1550-7998=2009=79(6)=065018(21) 065018-1 Ó 2009 The American Physical Society V. ALAN KOSTELECKY´ AND ROBERTUS POTTING PHYSICAL REVIEW D 79, 065018 (2009) procedure yields a unique answer even if a potential for ð1; 1; 1; 1Þ as the only nonzero components. As usual, in h is allowed [16]. In the context of spontaneous Lorentz other coordinate systems the Minkowski metric takes dif- violation, the phase transition circumvents this uniqueness. ferent forms and covariant derivatives must be used. The However, the nontrivial integrability conditions required equations of motion obtained by varying Eq. (1) with for implementing the bootstrap constrain the form of the respect to C are potential V. In Sec. IV, we obtain differential equations V ¼ 0 expressing the integrability conditions and derive accept- KC : (3) able potentials V. This section also applies the bootstrap to C yield the full cardinal gravity. Certain aspects of the ex- The theory (1) has various symmetries. It is invariant trema of the potential are considered, and alternative boot- under translations and under global Lorentz transforma- strap procedures are discussed. The coupling of the tions. For infinitesimal parameters ¼, the latter cardinal field to the matter sector and some experimental take the form implications are studied in Sec. V. A summary of the ! þ þ ! results and a discussion of their broader implications is C C C C ; : provided in Sec. VI. Throughout this work, we use the (4) conventions of Ref. [2]. There are also local spacetime symmetries, including invariance under local Lorentz transformations on the tan- II. LINEARIZED ANALYSIS gent space at each point and invariance under In this section, the linear cardinal theory is defined and diffeomorphisms of the Minkowski spacetime. These local investigated. We show that its NG sector is equivalent to symmetries play a subsidiary role in the present context. conventional linearized gravity in a special gauge. In addition to the spacetime symmetries, the form of the kinetic operator (2) ensures that the kinetic term is by itself invariant under gauge transformations of C alone, A. Linear cardinal theory ! ! Consider first the action for the symmetric two-tensor C C @ @ ; : (5) cardinal field C defined in a background spacetime. For However, one or more of these four gauge symmetries may definiteness and simplicity, we take the background to be be explicitly broken by the potential V, so the Lagrange Minkowski spacetime with metric , although a more density (1) contains between zero and four gauge degrees general background could be countenanced and treated of freedom depending on the choice of V. Since C has with similar methods. We suppose the kinetic term in the ten independent components, it follows that there are be- action is quadratic in C, so the derivative operators in the tween six and ten physical or auxiliary fields. equation of motion are linear in C. The NG excitations The potential V for the theory (1) is a scalar function of of C subsequently play the role of the metric fluctuation the cardinal field C and the Minkowski metric . The in a linearized theory of gravity. The action is assumed to only scalars that can be formed from these two objects generate spontaneous violation of Lorentz symmetry involve traces of products of the combination C . The through a potential VðC; Þ. scalar Xm with m such products has the form ¼ tr½ð Þm 1. Basics Xm C : (6) The Lagrange density for the linear cardinal theory is Here, we have introduced a convenient matrix notation ð Þ 4 4 taken to be C C . Since C is a symmetric matrix, there are at most four independent scalars Xm,sowe L ¼ 1CK C VðC; Þ: C 2 (1) can restrict attention to the cases Xm ¼ 1, 2, 3, 4. It follows that the potential V can be written as Here, K is the usual quadratic kinetic operator for a massless spin-2 field. Allowing for an arbitrary scaling V ¼ VðX1;X2;X3;X4Þ (7) parameter to be chosen later, K can be written in without loss of generality. For definiteness, V is assumed to Cartesian coordinates as be positive everywhere except at its absolute minimum, 1 1 1 which is taken to be zero. K ¼ 2½ð þ 2 þ 2 @ @ Under the gauge transformation (5), each scalar Xm þ @ @ þ @ @ transforms nontrivially and therefore explicitly breaks one symmetry. For simplicity in what follows, we assume 1 1 2@@ 2@@ the potential V depends on all four independent scalars Xm, 1 @ @ 1 @ @ ; (2) so the gauge symmetry (5) is completely broken for ge- 2 2 neric field configurations. With this assumption, the theory where is the Minkowski metric with diagonal entries describes ten physical or auxiliary fields and zero gauge

065018-2 GRAVITY FROM SPONTANEOUS LORENTZ VIOLATION PHYSICAL REVIEW D 79, 065018 (2009) fields. This assumption could be relaxed, but the resulting contained in the fluctuations N defined by discussion would involve additional gauge-fixing ~ considerations. C N ¼ E c þ E c O E; (13) The potential V is taken to have a minimum in which C attains a nonzero vacuum value where hCic: (8) 1 O ¼ 2ð c þ c c c Þ: In this minimum, the scalars X have vacuum values m (14) m hXmixm ¼ tr½ðcÞ : (9) Since there are six independent fields in E, the ten These vacuum values spontaneously break particle Lorentz symmetric components of N must obey four identities. symmetry, but they leave unaffected the structure of ob- For c satisfying our assumed conditions, we find these server Lorentz and general coordinate transformations, identities can be expressed as which amount to coordinate choices. To avoid complica- tions with soliton-type solutions, we also suppose c is tr ½NðcÞj¼0; (15) constant, @c ¼ 0 (10) with j ¼ 0,1,2,3. In addition to the six NG modes in the field N, the in Cartesian coordinates. fluctuation C~ includes four massive modes. These are Given a vacuum value c, the freedom of coordinate contained in the field M given by choice can be used to adopt a canonical form. For definiteness and simplicity, we assume in what follows that the ~ M ¼ C N ; (16) matrix ðcÞ c has four inequivalent nonzero real eigenvalues. This implies, for example, invertibility subject to a suitable orthogonality condition. The symmet- and the existence of one timelike and three spacelike eigenvectors. It also implies that all six Lorentz transfor- ric field M has ten components but only four indepen- mations are spontaneously broken. The consequences of dent degrees of freedom, which we denote here by mj, ¼ 0 other possible choices may differ from the discussion j , 1, 2, 3. For some purposes, it is convenient to below and would be interesting to explore, but they lie expand M as beyond our present scope. M ¼ m0 þ m1c þ m2ðccÞ þ m3ðcccÞ : 2. Nambu-Goldstone and massive modes (17)

The physical degrees of freedom contained in the cardi- The fields N and M obey identities expressing a kind nal field C can be taken as fluctuations about the vacuum of orthogonality: value c. We write j C ¼ c þ C~: (11) tr ½NðMÞ ¼0; (18)

~ The fluctuation field C is symmetric and has ten inde- with j ¼ 0, 1, 2, 3. More generally, we find pendent components, which include both the NG modes and the massive modes in the theory. tr ½NFðc; MÞ ¼ 0; (19) To identify the NG modes, we can make virtual infinitesimal symmetry transformations using the broken gen- where Fðc; MÞ is an arbitrary matrix polynomial in c erators acting on field vacuum values, and then promote the and M. corresponding parameters to field excitations. An infini- With the expansion (17), the fluctuation C~ can be tesimal Lorentz transformation with parameters ¼ written yields h i! þ þ X3 C c c c : (12) ~ j C ¼ N þ mj½ðcÞ : (20) Since there are six Lorentz transformations (three rotations j¼0 and three boosts), there could in principle be up to six Lorentz NG modes, corresponding to the promotion of the Using this equation, the four massive modes mj can be E ¼E ~ k six parameters to fields [5,19]. For c expressed in terms of C . Multiplying by ½ðcÞ with satisfying our assumed conditions, the maximal set of six k ¼ 0, 1, 2, 3 and applying the identities (15) yields the NG modes appears. In general, the NG modes in C~ are four equations

065018-3 V. ALAN KOSTELECKY´ AND ROBERTUS POTTING PHYSICAL REVIEW D 79, 065018 (2009) 0 10 1 2 3 ~ 4tr½c tr½ðcÞ tr½ðcÞ m0 C ! N subject to the constraints (15). If desired, the B CB C B 2 3 4 CB C on-shell values of can be set to zero by a suitable choice B tr½c tr½ðcÞ tr½ðcÞ tr½ðcÞ CB m1 C j B CB C of initial conditions. Equivalent results could be obtained B tr½ð Þ2 tr½ð Þ3 tr½ð Þ4 tr½ð Þ5 CB C @ c c c c A@ m2 A via an alternative Lagrange density involving a potential V 3 4 5 6 tr½ðcÞ tr½ðcÞ tr½ðcÞ tr½ðcÞ m3 with quadratic Lagrange-multiplier terms instead [19]. In 0 1 tr½C~ any event, if the graviton is to be identified with the B ~ C Lorentz NG modes in the theory (1), it follows that the B tr½CðcÞ C field N must be the candidate graviton field. ¼ B C: (21) @ tr½C~ ðcÞ2 A 3 3. Equations of motion for NG modes tr½C~ ðcÞ The behavior of the candidate graviton field N is The traces tr½ðcÞp with p ¼ 5, 6 can be rewritten in determined by its equations of motion. In the pure NG terms of tr½ðcÞm with m ¼ 1, 2, 3, 4 using the sector with vanishing Lagrange multipliers, the theory (1) Hamilton-Cayley theorem. In terms of the eigenvalues c j with the potential (25) is equivalent to an effective of the matrix c, the determinant of the 4 4 matrix O on Lagrange density LNG for the independent degrees of the left-hand side takes the form freedom, which are the Lorentz NG modes E. We can Y3 therefore write 2 det½O¼ ðcj ckÞ : (22) 1 j;k¼0 L NG ¼ O E K O E : (26) j

065018-4 GRAVITY FROM SPONTANEOUS LORENTZ VIOLATION PHYSICAL REVIEW D 79, 065018 (2009) We thus see that only two combinations of the six h that yields directly a match to the NG effective massless Lorentz NG modes E propagate as physical Lagrange density (26). on-shell fields. The other four NG modes are auxiliary. The conditions fixing this alternative ‘‘cardinal’’ gauge With the full potential V replaced by the Lagrange- at linear order in h are multiplier limit V , the four Lagrange multipliers can be tr ½ ð Þj¼0 viewed as playing a role analogous to that of the four h c ; (40) frozen massive modes mj. where j ¼ 0, 1, 2, 3. In this expression, ðcÞ c is a constant matrix assumed to have four inequivalent B. Correspondence to linearized general relativity nonzero real eigenvalues, which we denote by cj, j ¼ 0, In this subsection, we show the correspondence between 1, 2, 3. This assumption ensures the four conditions (40) the restriction of the linear cardinal theory to the NG sector are independent. For the present purpose of matching to the and the usual weak-field limit of general relativity describ- linear cardinal theory (1), the quantity c is to be identi- ing a massless spin-2 graviton field h propagating in a fied with the vacuum value of C in Eq. (8), so we denote background Minkowski spacetime. it by the same symbol. Consider the Lagrange density for a free symmetric To show that the conditions (40) are indeed a choice of 0 massless spin-2 field h, which is of the form (1) with gauge, we can consider an arbitrary initial field h and C replaced by h and the potential V set to zero: seek quantities such that a gauge transformation of the form (37) generates the desired field h satisfying (40). In 1 L h ¼ 2h Kh : (33) momentum space, the gauge transformation (37) takes the form The definition of K in Eq. (2) implies h ¼ h0 ik ik: (41) L Kh G; (34) The requirements on become L where G is the Einstein tensor linearized in h . At this ¼ 1 tr½ 0 stage, the value of can be fixed by requiring a match to ik 2 h ; the conventional normalization of the linearized action for ik ðcÞ ¼ 1 tr½h0c; general relativity in the presence of a matter coupling given 2 ½ð Þ2 ¼ 1 tr½ 0ð Þ2 by ik c 2 h c ; ½ð Þ3 ¼ 1 tr½ 0ð Þ3 L L ¼ 1 ik c 2 h c : (42) T 2h TM; (35) This represents a set of four equations for the four un- TM where is the matter energy-momentum tensor. This match fixes to be knowns , which can be regarded as a matrix equation. The set has a unique solution if the 4 4 matrix generated 1 ¼ by the coefficients of is invertible. Then, the four 4- ; (36) ð Þ ½ð Þ2 ½ð Þ3 16GN vectors k, k c , k c , k c are line- arly independent, and so where GN is the Newton gravitational constant. A priori, h has ten degrees of freedom. However, the k kc kðccÞ kðcccÞ 0: (43) theory is invariant under the four gauge transformations Expanding the 4-vector k in terms of the eigenvectors eðaÞ ! h h @ @ ; (37) of the matrix c shows that this condition is indeed sat- ðaÞ ¼ so four gauge-fixing conditions can be imposed on h. isfied for generic k, for which all components k k ð Þ Numerous choices of gauge appear in the literature. For e a are nonzero. It follows that the cardinal gauge (40) can free wave propagation, a common choice is transverse- be attained everywhere in conventional linearized general traceless gauge, which imposes relativity, except for special k at which additional gauge fixing is required. This remnant gauge freedom is analo- ¼ 0 ¼ 0 nh ;hh ; (38) gous to that of axial gauge in electrodynamics [20]. Similarly, in the context of spontaneous Lorentz violation, for a unit timelike vector n . For suitable initial conditions, the linearized potential for the vector field in certain bum- the harmonic condition blebee models generates an NG-sector axial constraint with a related remnant gauge freedom [5,19]. For simplic- @h ¼ 0 (39) ity in what follows, we consider the case of generic k. then follows from the equations of motion. However, this Once the cardinal gauge (40) is imposed, the harmonic gauge is not the only possible choice. Here, we demon- condition (32) follows from the equations of motion. The strate the existence of an alternative gauge condition on latter are found from the Lagrange density (33)tobe

065018-5 V. ALAN KOSTELECKY´ AND ROBERTUS POTTING PHYSICAL REVIEW D 79, 065018 (2009)

L Kh G ¼ 0: (44) III. BOOTSTRAP PROCEDURE This section considers some generic features of the c Contracting these equations in turn with , , bootstrap procedure for self-consistent coupling to the ðccÞ ðcccÞ , and yields in momentum space the energy-momentum tensor. We summarize the Deser ver- four conditions sion [14] of the bootstrap for obtaining general relativity k hk ¼ 0; from the linear graviton theory (33), and we present some generic results that are useful for the subsequent analysis. kc h k ¼ 0; ¼ 0 A. Bootstrap for general relativity kc c h k ; k c c c hk ¼ 0: (45) 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 Collecting the coefficients of h k gives a 4 4 matrix that is invertible when Eq. (43) is satisfied, which is the Z ¼ 4 ðÞ case under the present assumptions. It follows that SGR d xg R : (48) hk ¼ 0, and hence that the gauge choice (40) obeys the harmonic condition (32). The equations of motion then Here, g is the tensor density of weight one defined in reduce to terms of the usual reciprocal metric g as qffiffiffiffiffiffi j j @ @h ¼ 0; (46) g g g : (49) and they describe the usual two graviton degrees of free- Its inverse is a tensor density of weight negative one, which dom propagating as massless spin-2 waves. we define as We now have all the ingredients in hand to verify the 1 g pffiffiffiffiffiffi g : (50) equivalence between the theory (33) for a propagating j j spin-2 field h and the theory (26) for the NG sector of g the cardinal model. Starting with the former, we can im- Also, pose the four cardinal gauge conditions (40) on the ten ðÞ¼ 1 1 independent graviton components h. The equations of R @ 2@ 2@ (51) motion (44) then imply the harmonic condition (32), which þ leaves two degrees of freedom that propagate as conventional massless modes. These results are paralleled in the is the curvature tensor for the connection . In this theory (26) for the NG sector of the cardinal model. The approach, both g and are viewed as independent E field N containing the Lorentz NG modes is subject fields at the level of the action, and the identification of to the constraints (15), so N matches the graviton h in with the Christoffel symbols arises on shell by solv- cardinal gauge, ing the equations of motion. In what follows, we define the fluctuation h of g $ h N : (47) about the Minkowski background as The harmonic condition holds for both N and h. The g ¼ þ h: (52) equations of motion (28) for the Lorentz NG modes E can be matched directly to the equations of motion (44) for Note the use of contravariant indices in this definition. the graviton h by multiplying the latter with O. Also, note that h can be identified at linear order with Evidently, the cardinal and graviton theories are in direct the usual trace-corrected field h : correspondence, even though their gauge structures differ. h h h þ 1h: (53) The presence of the potential in the linear cardinal theory 2 excludes the gauge symmetry of the graviton theory, but Given the linear graviton theory (33), the nonlinear the gauge freedom of the latter means that only six of the Einstein-Hilbert action can be derived by adding a cou- ten components of h are physical or auxiliary, thereby pling to the energy-momentum tensor T and requiring its matching the six Lorentz modes E in the NG sector of conservation be consistent order by order [13]. Deser has the cardinal theory. Note also that the gauge freedom of the shown that this bootstrap procedure can be performed in a graviton theory could be fixed to cardinal gauge in a single elegant step [14]. standard way, by adding suitable gauge-fixing terms to The starting point of the derivation is to note that the the Lagrange density. The parallel in the cardinal theory equations of motion (44) for h, obtained in the previous would be the presence of Lagrange multipliers for the section from the second-order Lagrange density (33), also constraints (15). follow from the linearized version of the first-order action

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L 1 (48). The latter becomes ¼ 2½@h @h @h Z 1 L 4 L þ 2ð@ þ @ @Þ tr½h: (61) SGR ¼ d xLGR;

L The full nonlinear action of general relativity is obtained LGR ¼ ½h ð@ @ Þ by coupling the nonderivative part of h as a source for þ ð Þ ; (54) h , Z with h and viewed as independent fields. Variation ¼ L þ 4 ð Þ SGR SGR d xh : L of SGR with respect to these fields yields two sets of (62) equations of motion. These fix as the usual linearized Christoffel symbols, and they imply the linear equations of Variation of this action with respect to h yields the motion (44) for h obtained from the second-order Einstein equation R ¼ 0 in the form Lagrange density (33). L ðÞ¼1 þ The prescription for the bootstrap procedure is to require R 2h ; (63) that the energy-momentum tensor T obtained from the which implies action (54) is coupled as a source in a self-consistent L L manner. It turns out to be most convenient to work with Rð Þ¼8GNh: (64) the trace-reversed energy-momentum tensor , which in This verifies that coupling the nonderivative part of as the linear limit is related to T by h a source for h indeed produces the usual Einstein equa- 1 ¼ T 2T : (55) tions. Moreover, since the nonderivative part of h is independent of , it generates no additional contribution For a given Lagrange density L in Minkowski spacetime to the energy-momentum tensor and so no further iteration with metric , the tensor can be calculated via the steps are required. Rosenfeld method [22]. The procedure involves promoting the Minkowski metric to an auxiliary weight-one B. Generic bootstrap results metric density c and the partial derivative @ to the In this subsection, we outline some generic applications covariant derivative D formed using c , so that L of the bootstrap procedure, starting from an action given in becomes covariant in the auxiliary spacetime. The trace- Minkowski spacetime. The example relevant in our context reversed energy-momentum tensor is then found from ð0Þ ð1Þ is either an action S independent of h or an action S the expression linear in h. In each case, we seek to construct the

1 L corresponding action S that incorporates consistent self- ¼ : (56) coupling to h at all orders. 2 c c ! L ð0Þ For the linear theory with Lagrange density LGR, this 1. Case of S yields ð0Þ Consider first the case of an action S independent of 1 ¼ ð Þþ ð Þ h, such as a matter action. We write 2h h; ; Z (57) Sð0Þ ¼ d4xLð0Þ; (65) where is a total-derivative term given by where the Lagrange density 1 ðh; Þ¼2@½h þ h h L ð0Þ ¼ Lð0Þð ;f ;@ f Þ (66) a a þ h ð Þþh ð Þ is a function of the spacetime metric , a set of fields h ð Þ faðxÞ, and their derivatives @fa. For the purposes of this þ ð1 tr½ Þ 2 h h : (58) work, it suffices to suppose that the terms @fa are either derivatives of scalars or are gauge kinetic terms, so that On shell, can be expressed more elegantly as promotion of @ to the auxiliary covariant derivative has L L L ! ½c ¼ RðÞRð Þ; (59) no effect: @fa D fa @fa. This simplifying assumption avoids the need to consider terms of the L ðÞ where R is the linear part of the Ricci curvature type in the analysis. L 1 1 To obtain the energy-momentum tensor for the action RðÞ¼@ 2@ 2@ ; (60) (65), the Lagrange density Lð0Þ is promoted to a covariant and L is the linearized Christoffel symbol expression with respect to c ,

065018-7 V. ALAN KOSTELECKY´ AND ROBERTUS POTTING PHYSICAL REVIEW D 79, 065018 (2009) ð0Þ ð0Þ Lð0Þ L ! L ðc ;fa;@faÞ: (67) (69), the bootstrap procedure amounts to finding and then determining L via Eqs. (67) and (75). If instead a pure Lð0Þ ð0Þ To ensure premainsffiffiffiffiffiffiffiffi a density, multiplication by a factor matter action is specified by giving L , it suffices to of a power of jc j may be required as part of this promo- promote it according to Eq. (67) and obtain L via the tion, where c det½c . Using the definition (56) then identification (75). In this case, the bootstrap corresponds yields to the standard minimal-coupling procedure. For example, the usual Minkowski-spacetime energy-momentum tensor 1 Lð0Þ ð0Þ ¼ for Maxwell electrodynamics is : (68) 2 c c ! EM ¼ 1 ¼ ð0Þ ð0Þ T F F 4F F EM; (76) The bootstrap procedure requires that be consis- tently coupled as a source for h. The action Sð0Þ must with the latter equality following from conformal invari- therefore be supplemented by an additional term ance. The corresponding Lagrange density is Z Z ð0Þ 1 1 ð0Þ L ¼F F : (77) ð1Þ ¼ 4 Lð1Þ 4 h EM 4 S d x d x 2 ; (69) Promoting this according to Eq. (67) and making the up to a possible constant. However, in general the term Sð1Þ identification (75) directly yields the usual Lagrange den- ð1Þ L itself contributes a term to the energy-momentum sity EM for electrodynamics in curved spacetime, tensor, 1 L EM ¼ pffiffiffiffiffiffi g g F F ; ð0Þ (78) 1 Lð1Þ ð 1 Þ 4 jgj ð1Þ 2 ¼ ¼ h : (70) 2 c c ! c c ! where g det½g . Consistency of the coupling then requires that a further 2. Case of ð1Þ term Sð2Þ be added to the action, S Z Under some circumstances, the given starting point is Sð2Þ ¼ d4xLð2Þ; (71) instead an action Sð1Þ for a theory linear in h. To obtain the fully coupled action S, one can explicitly perform the where Lð2Þ is the solution to the differential equation iteration procedure above. However, a more efficient ‘‘inverse’’ method can be adopted instead. To implement this ð2Þ ð 1 ð0Þ Þ L 1 ð1Þ 2 method, we start by identifying the energy-momentum ¼ h : (72) ð0Þ ð1Þ h c ! 2 c c ! tensor from the specified action S written in the form (69), and we promote it to a covariant expression with We find respect to c : 1 ð0Þ ð Þ ð0Þ ð0Þ 1 2 1 1 ð1Þ L ð2Þ ¼ ¼ ðÞ!ðc Þ: (79) h h h ; 2 c c ! 2 2 pffiffiffiffiffiffiffiffi An appropriate multiplicative factor of jc j may be re- (73) quired to maintain the tensor transformation properties of ð0Þ up to a possible constant. . We then write the differential equation Iterating this procedure yields a series of terms summing ð0Þ L 1 ð0Þ L to the desired Lagrange density , ¼ ; (80) 2 c 1 1 ð0Þ X 1 nð Þ 2 nn L ¼ h11 hnn : which reduces to Eq. (68) in the limit c ! . The n! c 11 c n1n1 c ! n¼0 differential equation can be solved if the integrability (74) condition

The series can be constructed provided the integrability ð0Þ ð0Þ conditions are satisfied at each step, and it may terminate at ¼ (81) c some finite n. It represents a Taylor expansion of L, and c inspection reveals the identification is satisfied. Once the solution Lð0Þ is obtained, we can ð0Þ apply the identification (75) to obtain L and hence S. L ¼ L ðc ;fa;@faÞjc !g: (75) The above inverse trick is applied in some of the analysis The above derivation shows that knowledge of Lð0Þ in that follows. To illustrate it in a more familiar context, the form (66) suffices to determine L. If originally the consider the cosmological constant . In Minkowski matter-gravity coupling is specified in the linearized form spacetime, is associated with an effective energy-

065018-8 GRAVITY FROM SPONTANEOUS LORENTZ VIOLATION PHYSICAL REVIEW D 79, 065018 (2009) momentum tensor given by independent of h, so the bootstrap method of the previous subsection applies. The net result of the bootstrap is ¼2 ¼ ð0Þ T : (82) the nonlinear constraint terms qffiffiffiffiffiffi qffiffiffiffiffiffiffi The challenge is to bootstrap this to the fully coupled L TT ¼ 2ð1Þð jgj jjÞþð2Þ n ðg Þ Lagrange density L. Following the inverse trick, we ð0Þ þ ð3Þ g ; (89) promote to qffiffiffiffiffiffiffiffi which correspond to a nonlinear form of the TT gauge ð0Þ ðc Þ¼2 jc jc ; (83) constraints, pffiffiffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffiffi jc j where the appropriate factor of has been introduced. jgj ¼ jj;ng ¼ n ; g ¼ 0: With the identities (90) c ¼c c c ; qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi (84) 1 jc j ¼ 2 jc jc c ; IV. BOOTSTRAP FOR CARDINAL GRAVITY the integrability condition (81) can be verified, so the At this stage, we are in a position to consider the non- differential equation (80) can be solved for Lð0Þðc Þ. linear extension of the cardinal theory (1). This section Making the identification (75) then yields begins by presenting a convenient first-order reformulation qffiffiffiffiffiffi of the linear cardinal theory. In this form, the bootstrap of L ¼2 jgj; (85) the kinetic terms is straightforward using the methods of the previous section. We investigate the bootstrap integra- in agreement with the usual result. Notice that the linear- bility conditions on an arbitrary potential term, which turn ized version of this is out to provide interesting constraints on the theory. Finally, the bootstrap of these terms is also presented. L 2 ¼2 þ ð1 ð0Þ Þ h h 2 ; (86) A. First-order action as expected from Eq. (82), and that the zeroth-order term To facilitate comparison with the bootstrap for general Lð0ÞðÞ is merely a constant in this example. Note also that relativity, a first-order form of the theory (1) is useful. To the first-order term Lð1ÞðÞ produces a linear instability in develop this, we introduce the trace-reversed cardinal field the action at this order. This could be avoided by initiating C as the bootstrap from a theory formulated in a suitable 1 C ¼C þ 2 C : (91) Riemann background spacetime [17]. As another example, consider the bootstrap procedure Note the signs, which are chosen to improve the corre- for the transverse-traceless (TT) gauge. A common form spondence to the conventions used in the analysis of gen- for this gauge involves the trace-corrected field h and a eral relativity. The field C plays a central role in what timelike unit vector n: follows. In terms of C , the second-order Lagrange density LC tr ½ ¼0 ¼ 0 ¼ 0 h ;nh ;@h : (87) yielding equivalent equations of motion to the theory (1) takes the form These standard linear gauge-fixing conditions can be ex- L 1 pressed in terms of h and using Eqs. (53) and (61). L C ¼ 2C KC VðC ;Þ: (92) The resulting expressions can then be implemented in the linearized action (54) via the addition of the linear Here, the quadratic operator K is given in Cartesian Lagrange density coordinates by 1 L L K ¼ 4½ð þ Þ@ @ L TT ¼ ð1Þ tr½hþð2Þnh þ ð3Þ ; (88) @@ @@ (93) þ þ where ð1Þ, ð2Þ, and ð3Þ are Lagrange multipliers. The @@ @@ bootstrap procedure can be applied to each of the three þ @ @ þ @ @ : terms independently. The first term is linear in h and of the same form as in Eq. (86), so the bootstrap is immediate. Note that acting with this operator on the fluctuation h L The second term is also linear in h , and the integrability produces the linearized Ricci curvature R: conditions are directly satisfied. The inverse trick de- L scribed above can therefore be applied. The third term is K h R: (94)

065018-9 V. ALAN KOSTELECKY´ AND ROBERTUS POTTING PHYSICAL REVIEW D 79, 065018 (2009) Note also that the quantities K C in Eq. (1) and ~ ~ 1 ~ C ¼C þ 2 C : (101) KC are related by trace reversal with a sign. In The analogue of Eq. (11) therefore becomes Eq. (92), the potential VðC ;Þ is determined by the ~ requirement that the equations of motion C ¼ c þ C : (102) V ¼ 0 An alternative expression for the linearized action (97)is K C (95) C therefore have the same content as the original equations of motion Z L 4 L S~ ¼ d xL~ ; (3). This requires that C C 1 ~ V V V LL ¼ ½C ð@ @ Þ ¼ þ : (96) ~ C C 2 C C þ ð Þ þ V To construct the first-order form of the linear cardinal L K~ þ V: (103) theory, we follow a similar path to that of the Palatini C formalism in general relativity discussed in Sec. III A. L L Note that the two linearized actions S and S~ are identi- Introducing an independent auxiliary field , the C C L Lagrange density (92) can be rewritten in terms of C cal, but by virtue of Eq. (100) the kinetic term K differs L from K~ by a total derivative. and in the equivalent form C Z The cardinal field C can be further decomposed into L ¼ 4 LL NG modes and massive modes, in parallel with Eq. (23). SC d x C; We write LL ¼ ½Cð@ @ Þ C C ¼ c þ N þ M; (104) þ ð Þ þ V where the trace-reversed NG field N is defined as KL þ V; (97) 1 N ¼N þ 2 N (105) where KL is the kinetic part of the Lagrange density. Variation of this action with respect to the independent and the trace-reversed massive-mode field is fields C and gives the equations of motion. With 1 M ¼M þ 2 M : (106) standard manipulations, the equations of motion determine the fields to be linearized Christoffel symbols of the The constraints in the NG sector corresponding to Eq. (15) conventional form but depending on C instead of h. can be written as They also imply linearized versions of the equations of tr ½NðcÞj¼0; (107) motion (95) for C obtained from the second-order Lagrange density (92). with j ¼ 0, 1, 2, 3, while the analogue of Eq. (19)is The linearized action (97) can be written in other equiva- tr ½ ð Þ ¼ 0 lent forms by decomposing the cardinal field C. In the NF c; M ; (108) minimum of the potential V, the field C acquires an where Fðc; MÞ is an arbitrary matrix polynomial in c expectation value c , and M. Another equivalent form for the action (97)is hCi¼c c þ 1c : (98) therefore 2 Z L ¼ 4 LL This satisfies the identities SN;M d x N;M; tr½ ¼tr½ c c ; LL ¼ ½ðN þ MÞð@ @ Þ N;M (109) tr½ð Þ2¼tr½ð Þ2 c c ; þ ð Þ þ V tr½ðcÞ3¼tr½ðcÞ3þ3 tr½c tr½ðcÞ21ðtr½cÞ3; 2 4 ¼ L þ KN;M V; tr½ðcÞ4¼tr½ðcÞ42tr½c tr½ðcÞ3 L 3 2 2 1 4 where KN;M denotes the kinetic term expressed in terms of þ 2ðtr½cÞ tr½ðcÞ 4ðtr½cÞ ; (99) N , M , and . and it also obeys B. Kinetic bootstrap @c ¼ 0 (100) With the linear cardinal theory massaged into a first- ~ by virtue of the assumption (10). The fluctuation C about order form paralleling that used for general relativity, we c is are in a position to investigate the bootstrap to nonlinear

065018-10 GRAVITY FROM SPONTANEOUS LORENTZ VIOLATION PHYSICAL REVIEW D 79, 065018 (2009) cardinal gravity. Since the bootstrap involves adding self- 2. Alternative bootstraps coupling order by order, it can be done independently for The derivation of the action SK;C in Eq. (111) is based on each part in the action. In particular, the bootstrap for the applying the bootstrap to the linearized cardinal action (97) kinetic part parallels the bootstrap for the linearized ver- for the cardinal field C. However, the spontaneous sion (54) of general relativity. Lorentz violation produces a phase transition that naturally separates the cardinal excitations into NG and massive 1. Primary bootstrap modes. One could therefore instead consider applying the It is perhaps most natural to apply the bootstrap proce- bootstrap to various choices of excitation in the effective dure to the linearized theory in the form (97), which holds theory describing the physics after the spontaneous sym- prior to the spontaneous Lorentz breaking. For the corre- metry breaking has occurred. In the remainder of this sponding kinetic term KL, the energy-momentum tensor subsection, we consider these alternative bootstrap proce- associated with C is of the same form as before, dures and their application to the kinetic term in the linear cardinal theory. 1ð Þ ¼ ð Þþ 2 C ; Suppose the bootstrap is instead applied to the alterna- (110) tive linearized cardinal action (103) for the fluctuation ~ C . This procedure has the possible disadvantage of and the nonlinear kinetic action SK;C is obtained by cou- requiring a preestablished value for the vacuum expecta- pling its nonderivative part as a source for C , ~ Z tion c . However, since C is a fluctuation, this proce- S ¼ SL þ d4xCð Þ dure does parallel more closely the usual bootstrap in K;C K;C general relativity, for which the relevant field h is also Z a fluctuation. The derivation of the nonlinear action S ~ ¼ d4xð þ CÞR ðÞ; (111) K;C from the linearized theory (103) proceeds as before. The result for this secondary theory is where RðÞ is the Ricci curvature defined via the auxil- Z iary field in the usual way, 4 ~ ~ ¼ L þ ð Þ SK;C S ~ d xC ðÞ¼ 1 1 K;C R @ 2@ 2@ Z 4 ~ þð Þ ¼ d xð þ C ÞR ðÞ: (116) : (112) Since the extra term in Eq. (111) is independent of ,no This is equivalent to the action S under a suitable further iteration steps are needed. K;C coordinate transformation. We thereby find that the sec- In the extremum of the potential V, the massive modes ondary bootstrap yields the same physics for the kinetic vanish and the result (111) for the kinetic bootstrap reduces term as did the primary bootstrap leading to Eq. (111). to A tertiary theory could also be countenanced, in which Z 4 the bootstrap is applied only to the NG modes N appear- S d xð þ c þ NÞR ðÞ: (113) K;C ing in the linearized action (109). While this procedure also requires a preestablished value for the vacuum expectation The combination ð þ cÞ can be viewed as playing the c, it has the possible advantage of matching more closely role of an effective background metric. Under a suitable the symmetry structure of the bootstrap for general rela- change of coordinates, this effective metric can be brought tivity. The key point is that the gauge transformation (5), to the Minkowski form, ð þ cÞ!. With the which fails to be a symmetry of the linearized theory due to identification the potential, nonetheless does define a symmetry for the h $ N; (114) pure NG sector because the potential vanishes for pure NG excitations. In linearized general relativity, the analogous which matches the linearized correspondence (47), it fol- gauge symmetry can be related to the conserved two-tensor lows that the kinetic action SK;C reduces to the Einstein- current, and it morphs into diffeomorphism symmetry Hilbert action in the limit of vanishing massive modes. The following the bootstrap procedure. In the present context, result (111) for the kinetic bootstrap thereby reveals that this symmetry structure is reproduced in the pure NG the nonlinear cardinal theory represents an alternative sector if the bootstrap is applied only to the NG excitation theory of gravity containing general relativity in a suitable N in the linearized action (109). low-energy limit. The correspondence For this tertiary bootstrap, the first step is to obtain the ~ energy-momentum tensor for the kinetic term KL in g $ þ C (115) N;M terms of the NG and massive modes. The calculations for provides the match between the metric density g of this step again parallel those for the bootstrap in general general relativity and fields in cardinal gravity. relativity. We find

065018-11 V. ALAN KOSTELECKY´ AND ROBERTUS POTTING PHYSICAL REVIEW D 79, 065018 (2009) qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 1ð Þ ¼ ð Þ 2 N;M V ðC ;Þ! jc jVðC = jc j; jc jc Þ qffiffiffiffiffiffiffiffi þ ðN; ÞþðM; Þ; (117) ¼ jc jVðX1; X2; X3; X4Þ; (123) where is the total-derivative term given by Eq. (58)but with modified arguments as indicated. The prescription for where the four quantities X are now the tertiary bootstrap is then to couple the nonderivative m part of ð Þ as a source for N, N;M m X mðc Þ¼tr½ðCc Þ (124) ¼ L þ ð Þ K N;M KN;M N : (118) and are scalars with respect to c . In parallel with the bootstrap for the kinetic term, C is taken to be a tensor This prescription yields the tertiary kinetic action density with respect to c in constructing these Z 4 expressions. SK;N;M ¼ d xð þ N ÞRðÞ The energy-momentum tensor C is þ ð Þ M @ @ : (119) pffiffiffiffiffiffiffiffi 1 ð jc jVÞ Paralleling the case of general relativity, the extra term in ¼ : (125) 2 C c Eq. (118) is independent of , so no further iteration steps are needed. Note that the structure of this result implies the auxiliary field is no longer equivalent The bootstrap procedure requires this to be obtained from on shell to a Christoffel symbol. an action by varying with respect to C . We must therefore add to the Lagrange density a term V0 such that The tertiary kinetic action SK;N;M differs nontrivially from the primary one SK;C, and the physical content of pffiffiffiffiffiffiffiffi the two is also different. With the identification (114) and V0 1 ð jc jVÞ ¼ C ¼ : (126) in the pure NG sector, both actions match the Einstein- C 2 c Hilbert action of general relativity. Their linearized content is also the same as that of the linear cardinal theory (1). If V0 is smooth, then

C. Integrability conditions for the potential 2 0 2V0 V ¼ ; (127) Next, we investigate the integrability conditions re- CC CC quired to apply the bootstrap on the potential term. We V obtain constraints such that obeys the integrability con- which implies ditions, and we determine a general form of V satisfying these constraints. pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 2 2ð jc jVÞ ð jc jVÞ To proceed, start with the theory in the form (92)in ¼ : (128) terms of the cardinal field C. The potential is c C c C VðC ;Þ, and it is a scalar. The only scalars that can 0 be formed from C and involve traces of the matrix This is the integrability condition for the existence of V .It C. The scalar Xm with m such products has the form requires symmetry of the double partial derivative under ðÞ$ðÞ X ¼ tr½ðCÞm: (120) the interchange . m The double derivative appearing in the result (128) can Since C is a 4 4 matrix, only four of these are inde- be written as ð Þ pendent, so the potential V C ; can be written pffiffiffiffiffiffiffiffi 2 qffiffiffiffiffiffiffiffi ð Þ¼ ð Þ ð jc jVÞ V C ; V X1; X2; X3; X4 : (121) ¼ jc jðAmVm þ BmnVmnÞ; c C V C ¼ c X In the minimum of , and the scalars m have (129) expectation values m hXmi¼tr½ðcÞ xm: (122) where m and n are summed, with The next step is to determine the energy-momentum V 2V tensor C associated with the potential V and check V m ; Vmn ; (130) the integrability conditions. We therefore promote V to a Xm XmXn covariant expression with respect to the auxiliary metric density c , and with the coefficients Am and Bmn given by

065018-12 GRAVITY FROM SPONTANEOUS LORENTZ VIOLATION PHYSICAL REVIEW D 79, 065018 (2009) 2 1 X X Y0 ¼ 1; A ¼ c m þ m m 2 C c C 1 Y1 ¼ 2X1; 1 ¼ c ½c ðCc Þm1 Y ¼ 1ðX2 2X Þ (136) 2 m 2 8 1 2 ; 1 3 mX1 Y3 ¼ 48ðX1 6X1X2 þ 8X3Þ; ½c ð c Þk ½c ð c Þm1k m C C ; ¼ 1 ð 4 12 2 þ 12 2 þ 32 48 Þ ¼0 Y4 384 X1 X1X2 X2 X1X3 X4 : k 1 ¼ Xm Xn þ Xn Xm More generally, it follows that any polynomial obtained as Bmn ð qÞ 2 c C c C thepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi term at O C in the series expansion of 1 j det½1 þ Cc j is a solution. An expression for these ¼ ð½c ðCc Þm ½c ðCc Þn1 2 mn polynomials is n m1 1 q þ½c ðCc Þ ½c ðCc Þ Þ: (131) @ 2 3 4 1=2 Y q ¼ lim ðp1 þ p2 þ p3 þ p4Þ : !0 q! @q Inspection of these results reveals that the integrability (137) condition is satisfied if and only if the combined quantity For example, at q ¼ 5 a solution to Eq. (135) is the poly- 1 m1 Cmn ¼ 2mVm c ½c ðCc Þ nomial ½c ð c Þm ½c ð c Þn1 1 mnVmn C C Y ¼ ð3X5 þ 28X3X 36X X2 5 768 1 1 2 1 2 (132) 2 48X1X3 þ 32X2X3 þ 48X1X4Þ: (138) is symmetric under the interchange ðÞ$ðÞ. Using the Hamilton-Cayley theorem, we can write We conjecture that the polynomials obtained in this way are in fact unique solutions at each order q. 4 3 2 A general potential V that solves the differential equa- ½c ðCc Þ ¼ p1½c ðCc Þ p2½c ðCc Þ tions (135) can therefore be written as þ p3½c Cc p4 c ; (133) qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi X1 jc jV ¼ jc j qYq; (139) where q¼0

p1 ¼ X1; where the q are arbitrary real constants. For any fixed q, 1 2 1 a potential of this form satisfies the integrability conditions p2 ¼ X X2; 2 1 2 (128) required for the bootstrap procedure. Note that for ¼ 1 3 1 þ 1 p3 6X1 2X1X2 3X3; the special case q ¼ 0 for all q 0, the solution be- 1 4 1 2 1 2 1 1 comes p4 ¼ X X X2 þ X þ X1X3 X4: (134) 24 1 4 1 8 2 3 4 qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jc j ¼ j det½c þ j Adopting this result and requiring symmetry of the combi- V 0 C : (140) nation (132) reveals that the integrability condition imposes the following six partial differential equations on the potential V: D. Bootstrap for integrable potential In this subsection, we first apply the bootstrap procedure V2 þ 8p4V24 ¼V11 4p3V14; to the integrable potential (139). We then consider some 3 aspects of extrema of the resulting theory, provide a con- V3 þ 12p4V34 ¼2V12 þ 4p2V14; 2 struction for a local minimum, and offer some remarks 2V4 þ 16p4V44 ¼3V13 4p1V14; about alternative bootstrap procedures for the potential. (135) 3V13 12p3V34 ¼4V22 þ 8p2V24; 1. Potential bootstrap 4V14 16p3V44 ¼6V23 8p1V24; The bootstrap procedure using the cardinal field C can 8V24 þ 16p2V44 ¼9V33 12p1V34: be explicitly performedpffiffiffiffiffiffiffiffi term by term on the potential (139). For each q, jc jY is a coefficient in the expansion Solutions of these equations that are polynomials in X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q m j det½c þ j can be found by construction, and they are conveniently of C . In Sec. III B 2, a bootstrap procedure has been performed that leads to the potential (85) propor- classified according to the power q of X1 appearing in the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j det½c þ j polynomial. With some calculation, we have established tional to C . It followsp fromffiffiffiffiffiffiffiffi this analysis that that the unique polynomial solutions for q 4 are the bootstrap applied to the term jc jYq generates for

065018-13 V. ALAN KOSTELECKY´ AND ROBERTUS POTTING PHYSICAL REVIEW D 79, 065018 (2009) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi each q the full result j det½c þ Cj minus the sum of all A vacuum of VC can also be identified by the values xm terms of orders less than q: taken by the four scalars Xm, as in Eq. (122). The restric- qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tion of the potential VC to the NG sector can then be jc jY0 ! j det½c þ Cj; achieved by replacing VC with the Lagrange-multiplier qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi potential jc jY1 ! j det½c þ Cj jc jY0; (141) X4 qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi V ¼ mðXm xmÞ; (146) jc jY2 ! j det½c þ Cj jc jðY0 þ Y1Þ; m¼1 and so on, with the general term being which excludes fluctuations away from the extremum. If desired, the on-shell values of the Lagrange multipliers m qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 1 qX can be set to zero by suitable boundary conditions. This jc j ! j det½c þ j jc j Yq C Yk: (142) potential facilitates the identification of the NG and mas- ¼0 k sive modes. The NG modes N are the nonzero compo- Applying the bootstrap to the general potential (139) nents of C that preserve the constraints obtained from yields the bootstrap potential VC, the Lagrange-multiplier equations of motion, while the massive modes are the components of C that are con- qffiffiffiffiffiffiffiffi X1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi qX1 strained to zero. Note that the potential V is dynamically jc jVC ¼ q j det½c þ Cj jc j Yk 0 q¼0 k¼0 equivalent to a potential V expressed using the integrable polynomials (137), given by qffiffiffiffiffiffiffiffi X1 X1 qffiffiffiffiffiffiffiffi X1 ¼ jc j q Yk ¼ jc j kYk; X4 ¼0 ¼ ¼0 0 q k q k V 0 ¼ mðYm ymÞ; (147) (143) m¼1 where ym are the values of Ym for C ¼ c . The where the real coefficients k are given as Lagrange-multiplier constraints are equivalent by direct Xk comparison, while the dynamical properties under varia- ¼ k q: (144) tion with respect to C are equivalent when the Lagrange ¼0 q multipliers are identified by the nonsingular set of linear

Note that the coefficient k for fixed k acquires nonvanish- equations ing contributions from any nonvanishing coefficients q ð1Þmþ1 X4 with q k. ¼ 0 y m 2 p pm (148) For nonlinear cardinal gravity, the above discussion m p¼m reveals that the potential term appearing in the bootstrap 1 4 action takes the form with m . Using the potential (146), the NG modes N are seen Z X1 Z directly to be the solutions of the equations X ¼ x , ¼ 4 ¼ 4 m m SV;C d xVC k d xYk: (145) which can be written as nonlinear generalizations of Eq. ¼0 k (107), This potential term combines with the kinetic term S in K;C 0 ¼ tr½N; Eq. (111) to form the primary cardinal action. 0 ¼ 2tr½Ncþtr½ðNÞ2; 2. Extrema of the potential 2 2 3 0 ¼ 3tr½NðcÞ þ3tr½ðNÞ cþtr½ðNÞ ; Vacuum solutions of nonlinear cardinal gravity are ex- 0 ¼ 4tr½NðcÞ3þ3tr½ðNÞ2ðcÞ2þ3tr½ðNcÞ2 tremal solutions of the potential VC. In an extremum, the 3 4 cardinal field C acquires a vacuum value that may differ þ 4tr½ðNÞ cþtr½ðNÞ : (149) from any extrema generated by the potential V in the linearized theory and defined in Eq. (98). By mild abuse The ten independent components of N are constrained of notation, in what follows we adopt the same notation by these four equations, leaving the expected six NG modes. The four massive modes can be denoted by Mm C ¼ c for a vacuum value in an extremum of VC. Similarly, we adopt the same notation as in Eq. (104) for and specified as ~ the decomposition of the cardinal field C and its fluctu- M ¼ X x ¼ tr½ðc þ CÞmtr½ðcÞm: (150) ~ m m m ations C into the NG excitations N of Eq. (105) and the massive excitations M of Eq. (106). However, line- They are contained in the symmetric tensor M, which is arized results for N and M such as Eqs. (107) and obtained by subtraction of the NG modes N from the ~ (108) no longer hold. cardinal fluctuation field C ¼ C c.

065018-14 GRAVITY FROM SPONTANEOUS LORENTZ VIOLATION PHYSICAL REVIEW D 79, 065018 (2009) In the absence of coupling to matter, the equations of 0 ¼ tr½h; motion for cardinal gravity are obtained by varying the 0 ¼ 2tr½ þtr½ð Þ2 sum of the kinetic and potential actions (111) and (145) hc h ; with respect to the independent fields. Eliminating the 0 ¼ 3tr½hðcÞ2þ3tr½ðhÞ2cþtr½ðhÞ3; auxiliary field yields the field equations in the ab- 0 ¼ 4tr½hðcÞ3þ3tr½ðhÞ2ðcÞ2þ3tr½ðhcÞ2 sence of matter as þ 4tr½ðhÞ3cþtr½ðhÞ4 (155)

vac R ¼ 2; Xm ¼ xm; (151) obtained by the replacement N ! h in Eq. (149). The bootstrap for general relativity transforms the gauge vac symmetry (37) of the linearized theory into diffeomor- where is given by phism invariance of the Einstein-Hilbert action, involving particle transformations of the metric density g. In the

1 X4 linear cardinal theory, the analogue of the gauge symmetry vac @VC @Xm ¼ ¼ V j ! (37) is the symmetry (5) of the kinetic term alone. The 2 C;m C c @C C!c m¼1 @C prebootstrap potential V explicitly breaks this symmetry, X4 so the potential term (145) can be expected to exhibit m1 ¼ m½ðcÞ VC;mjC!c: (152) diffeomorphism breaking under particle transformations ¼1 ð þ Þ m of the analogue metric density ( C ).p Thisffiffiffiffiffiffiffiffi is jc j ! preflected,ffiffiffiffiffiffiffi for example, in the presence of a factor j j ¼ 1 Note that VC;m ¼ m in the Lagrange-multiplier limit. The in the measure of Eq. (145). However, as ex- vac quantity represents a kind of vacuum energy- pected from the match to general relativity, the pure NG momentum tensor density. Trace-reversing yields the field sector of cardinal gravity with zero on-shell Lagrange equations for cardinal gravity in the absence of matter, multipliers does exhibit the usual diffeomorphism invari- which can be written in the form ance because the potential vanishes in this sector. Note also that cardinal gravity remains invariant under diffeomorphisms of the Minkowski spacetime. G ¼ 2Tvac: (153) Both general relativity and cardinal gravity are invariant under (observer) general coordinate transformations. The match between the two theories in the pure NG Here, G is the Einstein tensor for the metric obtained limit involves a coordinate transformation taking from the metric density ( þ C ), while the vacuum ð þ c Þ! in the kinetic term (113). There energy-momentum tensor Tvac is obtained by the corre- vac is therefore a corresponding transformation taking sponding trace reversal of . The conservation law 1 !½ð1 þ cÞ in the potential term. For example,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the general coordinate invariance ensures a factor 1 DTvac ¼ 0 (154) jð1 þ cÞ j appears in the measure of Eq. (145). However, the vanishing of the potential in the pure NG sector makes this factor irrelevant for the match to general follows by virtue of the Bianchi identities. This conserva- relativity. tion remains true in the presence of matter couplings, provided the matter-sector energy-momentum tensor is 3. Stability of the extrema independently conserved. If the Lagrange multipliers m Given a bootstrap potential V , an interesting issue is vanish, or more generally if Vm vanishes, then the vacuum C energy-momentum tensor is zero and the usual form of whether it admits an extremum that is stable. The question general relativity is recovered. Otherwise, there is a posi- of overall stability for any given theory with Lorentz tive or negative contribution to the vacuum energy- violation is involved [23]. Even for the comparatively momentum tensor. This may play a role in cosmology simple bumblebee theories the issue remains open, and the interpretation of dark energy. although considerable recent progress has been made In the pure NG sector with zero on-shell Lagrange- [24]. A full analysis for cardinal gravity lies outside the multiplier fields, the effective potential vanishes and non- scope of this work. Instead, this subsection provides a few linear cardinal gravity reduces to the kinetic term (113). As remarks on stability in the specific context of the potential already noted, this limit reproduces general relativity, with term. the identification N $ h in Eq. (114). The Einstein- In the vacuum, the extremal solutions obey 4 Hilbert action is recovered in a fixed gauge, the nonlinear @V X 0 ¼ C ¼ ½ ð Þm1 j cardinal gauge, which is defined by the four nonlinear m c VC;m C!c; (156) @C ! gauge conditions C c m¼1

065018-15 V. ALAN KOSTELECKY´ AND ROBERTUS POTTING PHYSICAL REVIEW D 79, 065018 (2009) where VC;m @VC=@Xm. By assumption, the matrix c reconstructed in terms of suitable combinations of the has four inequivalent nonzero eigenvalues. Working in the polynomial basis (137). basis in which c is diagonal, this implies the generic Given the potential VC in the form (162), the issue of conditions for a vacuum are finding a solution with positive definite hessian can be resolved numerically. Investigation shows that there is a V C;mjC!c ¼ 0: (157) subspace of coefficients amn for which the integrability conditions are satisfied and the hessian is positive definite. A vacuum of VC is stable if it is a Morse critical point with positive definite hessian. For simplicity, we introduce An explicit example is the potential the explicit diagonal basis X8 V ¼ Y ; (163) ¼ C k k C C (158) k¼1

(no sum on ), where the four quantities C are the with the coefficients given by eigenvalues of C. Then 1 ’2:81;2 ’5:46;3 ’ 13:1;4 ’ 19:3; X3 m 5 ’24:7;6 ’29:6;7 ’ 16:0;8 ’ 17:1: X m ¼ ðCjÞ ; (159) j¼0 (164) and in the vacuum Cj ¼ cj, with all four values cj inequi- The local minimum is found to lie at valent and nonzero. In the diagonal basis, stability depends X 1 ’ 0:250; X2 ’ 2:06; X3 ’ 0:578; X4 ’ 1:44: on the hessian (165) 2 X4 @ VC m1 n1 Hjk ¼ ¼ mnðcjÞ ðckÞ VC;mnjC!c: The eigenvalues of the corresponding hessian are found to @C @C ! j k C c m;n¼1 be (160) H1 ’ 2:80;H2 ’ 0:927;H3 ’ 0:104;H4 ’ 0:0579; If the discriminant is nonzero and the four eigenvalues Hm (166) of the hessian are positive, the extremum is a local minimum. demonstrating positivity. This example therefore repre- An analytical derivation of a potential with a positive sents a potential VC having a local minimum. definite hessian in terms of the polynomial basis (137)is challenging. Instead, we proceed by ansatz using the 4. Alternative potential bootstraps shifted variables The bootstrap procedure discussed above holds for the ~ X ¼ X x : (161) potential prior to the development of a vacuum value for m m m the cardinal field C. Alternative options for the potential For the ansatz, we adopt the form of a Taylor expansion term, applicable following spontaneous Lorentz violation instead, include a secondary bootstrap using the cardinal 1 1 ¼ ~ ~ þ ~ ~ ~ þ ... ~ V C 2amnXmXn 6amnpXmXnXp ; (162) fluctuation C and a tertiary one using only the NG modes N where the coefficients a ;a ; ...are real constants. The . The explicit construction of these potentials lies mn mnp outside the scope of this work. Instead, this subsection potential V in Eq. (145) is a combination of integrable C contains a few brief comments about some aspects of these partial potentials, so the expression (162) must be inte- alternative bootstrap procedures, following from the analy- grable too. We can therefore constrain the coefficients by sis of the primary case. imposing the integrability conditions (135)onV itself at C To perform an alternative bootstrap procedure, the cor- ~ ¼ 0 ~ Xm . At second order in Xm, this imposes six condi- responding integrable potential must first be constructed. tions on the ten degrees of freedom amn. The four degrees For the secondary bootstrap involving the cardinal fluctua- of freedom a 4 can be taken as unconstrained at this order. ~ m tion C introduced in Eq. (102), the promotion of the To impose the integrability conditions at third order, it is potential V to a covariant expression with respect to the convenient to take partial derivatives of Eqs. (135) with auxiliary metric density c involves the four scalars X respect to each X . This produces 24 equations, which m m given by combine with the second-order equations to yield 16 independent constraints on the 20 third-order coefficients a . ~ m mnp X mðc Þ¼tr½ðcc þ Cc Þ : (167) The four degrees of freedom am44 can be taken as unconstrained at this order. Proceeding in this way, we find a The energy-momentum tensor must now be obtained from ~ 4ðn 1Þ-dimensional solution space for the potential VC an action by varying with respect to C . The basic inte- up to order n. As a check, the resulting solutions can be grability condition is found to be

065018-16 GRAVITY FROM SPONTANEOUS LORENTZ VIOLATION PHYSICAL REVIEW D 79, 065018 (2009) pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 2ð jc jVÞ 2ð jc jVÞ V. COUPLING TO MATTER ¼ : ~ ~ (168) c C c C At the linear level, the cardinal field C must couple to other fields in the Minkowski spacetime via a symmetric ~ However, since the cardinal fluctuation C is merely a two-tensor current. Given our gravitational interpretation constant shift of the cardinal field C, we have of the cardinal field, the other fields in the theory can be regarded as the matter. They provide one natural two- @X @X tensor current, the energy-momentum tensor TM in the m ¼ m : ~ (169) @C @C Minkowski spacetime. We can therefore expect the linearized theory (1) to incorporate the matter interaction This in turn means that the integrability condition is sat- L 1 L ¼ C TM : (173) isfied for the same symmetry requirement on the same M;C 2 expression (132) as before. The integrable potential for No coupling constant is necessary for this interaction, since the secondary bootstrap therefore takes the same form it can be absorbed in the scaling factor already present in (139) as for the primary case. the original theory (1). A similar situation holds for the tertiary bootstrap involving the NG modes N in the decomposition (104). In A. Primary bootstrap this case, the four relevant scalars are The bootstrap procedure involving the cardinal field C m can be applied to the matter interaction (173) to determine X m ¼ tr½ðcc þ Nc þ Mc Þ : (170) the form of the matter coupling for cardinal gravity. For The energy-momentum tensor is required to arise by vary- this purpose, the interaction (173) is conveniently ex- ing an action with respect to N. This generates the pressed in terms of the trace-reversed energy-momentum integrability condition tensor M for the matter. This tensor arises by variation of the Lagrange density LM for the matter fields via pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 2 2ð jc j Þ ð jc jVÞ V 1 LMð ! c Þ ¼ : (171) ¼ c M (174) c N N 2 c c !

However, the form of Eq. (104) implies in the usual way. We can therefore write L 1 L M ¼C M (175) @X @X ;C 2 m ¼ m : (172) @N @C for the matter interaction with the cardinal field C. To perform the bootstrap, the techniques of Sec. III B It follows that the integrability condition is again satisfied can be applied. The Lagrange density (175) is linear in C for the same symmetry requirement on the same expression and so has the form (69), for which the bootstrap yields (132), and the integrable potential for the tertiary bootstrap Eq. (75). The bootstrap therefore generates the Lagrange takes the same form (139) as before. density Although the integrable potentials (139) are the same, L ¼ L j the alternative bootstrap procedures differ from each other M;C M !þC: (176) and from the primary one presented above. Moreover, Some insight into the physical content of this result can performing these bootstrap procedures involves additional be obtained by expanding about an extremum of the boot- choices because integration with respect to the linear car- strap potential. Writing C ¼ c in the extremum and dinal fluctuation or the linear NG modes can either be ~ denoting the corresponding fluctuations by C ¼ N þ continued at all orders or can be adjusted at each order to M as before, we obtain incorporate the induced nonlinearities. Any of these boot- L ¼ L j ~ strap procedures could in principle be performed using the M;C M !þcþC: (177) methods presented in Sec. III. An extremum of an alternative bootstrap potential is A comparison of this result to the matter coupling of achieved for vanishing massive modes. It can therefore general relativity can be performed by adopting the be represented by a suitable Lagrange-multiplier potential. Lagrange-multiplier bootstrap potential (146). The mas- In particular, in the pure NG limit the potential vanishes for sive modes vanish, M ! 0, and as before a suitable on-shell multipliers, and so the resulting effective theory is change of coordinates must be performed to implement controlled by the corresponding kinetic term. This means the transformation ð þ cÞ! and thereby ensure that general relativity is also recovered in the low-energy the kinetic term (113) contains the conventional LNG limits of the nonlinear theories arising in these alternative Minkowski metric. The resulting Lagrange density M;C GR bootstrap procedures. then matches the usual matter term LM in general rela-

065018-17 V. ALAN KOSTELECKY´ AND ROBERTUS POTTING PHYSICAL REVIEW D 79, 065018 (2009) tivity, couplings between matter and the massive modes. Each NG GR also contains a term involving the cardinal vacuum value L ¼ LMj ! þ $ L ¼ LMj ! ; (178) M;C N M g c and the energy-momentum tensor. This last term re- when the correspondence g $ þ N of Eq. (115) mains as an unconventional expression in the Lagrange ! 0 is adopted. density in the pure NG limit M , and for the match We can therefore conclude that the pure NG sector of to general relativity it therefore represents an unconven- cardinal gravity with zero on-shell Lagrange multipliers tional contribution to the matter sector. exactly reproduces general relativity, including the matter Couplings involving tensor vacuum values appear natu- coupling. When the massive modes are included, the mat- rally in the standard-model extension (SME), which pro- ter coupling deviates from that in general relativity by vides a general framework for the description of Lorentz terms that are suppressed by the scale of the massive violation using effective field theory [2,25]. The matter modes. sector of the SME includes Lorentz-violating operators controlled by coefficients that are symmetric observer two-tensors and that can be related to c. Numerous B. Alternative bootstraps experimental measurements have been performed on the Alternative bootstrap procedures for the matter coupling coefficients for Lorentz violation [26]. This offers an in- can be countenanced instead. We consider here the sec- teresting opportunity to identify constrains on the alterna- ondary and tertiary procedures discussed above for the tive bootstrap theories. kinetic and potential terms. We also examine some experi- Consider first an example illustrating the connection mental implications of the results for the pure NG sector between the cardinal matter coupling and the SME frame- and the match to general relativity. work, involving a matter Lagrange density for a complex The secondary bootstrap involving the cardinal fluctua- scalar field in Minkowski spacetime given by ~ C tion starts from the matter coupling (173) in the form L 0 ¼ y ð y Þ @ @ U : (183) L 1 ~ 1 L ~ ¼ c ð M ÞþC ð M Þ: (179) M;C 2 2 Here, UðyÞ is an effective Lorentz-invariant potential that can include mass and self-interaction terms. The cor- The bootstrap can be performed using the methods of 0 Sec. III B. The first term in Eq. (179) involves c but is responding energy-momentum tensor T is ~ independent of C , while the only dependence on the 0 ¼ y þ y þ L0 T @ @ @ @ : (184) Minkowski metric appears in M. The effect of the bootstrap on this term is therefore to replace Mð Þ with Introducing the cardinal coupling (173) and restricting ð þ ~ Þ attention to the vacuum value c adds the term qMffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C , introducing a suitable power of ~ ~ 1 0 1 0 j þ Cj as needed. The second term is linear in C L c ¼ 2c T ¼ c ð2Þ; (185) and hence is of the form (69), for which the bootstrap gives 0 0 where is the trace-reversed form of T . Performing Eq. (75). We therefore obtain either of the alternative bootstraps in the NG limit yields 1 L ~ ¼ ð j ~ ÞþL j ~ M;C c 2M !þC M !þC (180) the contribution of the cardinal-scalar coupling to the full theory, as the secondary bootstrap matter coupling. qffiffiffiffiffiffi qffiffiffiffiffiffi For the tertiary bootstrap, the starting point is the matter L ¼ j j½ ð1 0 Þj ¼ 1 j j 0 j c;N g c 2 !g 2 g c T !g coupling in the form qffiffiffiffiffiffi qffiffiffiffiffiffi 1 1 1 1 ¼ j j T 0 j þ j j tr½ tr½ 0j L L ¼ð þ Þð Þþ ð Þ 2 g c T !g 8 g cg T !gg ; M;N;M c M 2M N 2M : (181) (186) þ Here, we bootstrap only the field N containing the where we denote the bootstrap metric density N linearized NG modes, without correcting at each order. by g and the corresponding metric by g. For the last Using the techniques in Sec. III B, we find the Lagrange expression in this equation, the coefficient c has been density separated into traceless and trace pieces for convenience in what follows, via the definitions 1 L M;N;M ¼ðc þ M Þð2Mj!þNÞþLMj!þN c ¼ cT þ 1 tr½cgg; tr½cTg¼0: (187) (182) 4 L as the result of the tertiary bootstrap. We can compare the result for c;N to that obtained in The alternative results (180) and (182) for the matter the SME framework for the Lorentz-violating theory of a coupling contain terms corresponding to the usual mini- complex scalar field in Riemann spacetime with Lagrange mally coupled Lagrange density for matter and additional density [2]

065018-18 GRAVITY FROM SPONTANEOUS LORENTZ VIOLATION PHYSICAL REVIEW D 79, 065018 (2009) qffiffiffiffiffiffi qffiffiffiffiffiffi y y The similarity of the matches (189) and (193) between L g ¼ jgjg @ @ jgjUð Þ T qffiffiffiffiffiffi c and certain traceless SME coefficients for Lorentz 1 y y violation is no accident. Consider a theory in which the þ 2 jgjk ð@ @ þ @ @Þ: (188) spacetime metric in the gravity sector is g. If the theory In this model, k is a symmetric coefficient for Lorentz has Lorentz violation, the matter-sector metric could differ þ violation, which is normally taken to satisfy tr½kg¼0 from g. Denote the matter-sector metric by g k, because a nonzero trace is Lorentz invariant. Inspection where the coefficient k for Lorentz violation is symmet- reveals the identification ric and traceless. For small k, the matter-sector Lagrange density LMðg þ kÞ can be expanded as cT k (189) LMðgÞ L Mðg þ kÞ¼LMðgÞþk þ ... between the cardinal vacuum value and the SME coeffi- g cient for Lorentz violation. Note that the conformally 1 invariant case satisfies tr½T0g¼0, in which case the two ¼ LMðgÞþ kTM þ ...; (194) models (186) and (188) match exactly. 2 As another example with direct physical application, where TM is the energy-momentum tensor for the Lð0Þ consider the Maxwell Lagrange density EM for photons Lagrange density LMðgÞ. We see that the piece of the in Minkowski spacetime, given in Eq. (77). The corre- cardinal coupling (173) involving cT can always be EM sponding energy-momentum tensor T is presented in matched at leading order to a term involving a traceless Eq. (76). The cardinal-photon coupling is shift k in the matter-sector metric of a theory with Lorentz violation. L ¼ 1 EM ¼ ð1 EMÞ c 2c T c 2 ; (190) The same line of reasoning also yields a path to experimental constraints on cT. The key point is that a suitable and the bootstrap generates the result þ ! 0 qffiffiffiffiffiffi qffiffiffiffiffiffi choice of coordinates can convert g k g, L EM ¼ j j½ ð1 EMÞj ¼ 1 j j T thereby making the matter sector Lorentz invariant at c;N g c 2 !g 2 g c F F: leading order in k. The price for this transformation is (191) 0 the conversion of the gravity-sector metric g ! g T In this example, only the traceless part c appears in the k, which means that signals from Lorentz violation final answer because the photon action is conformally could be detectable in suitable gravitational experiments. invariant. This result can be compared to the CPT-even In particular, at leading order we find part of the photon sector in the minimal SME [27]. The 0 L cardinal g R ðÞ!g R ðÞþk R ðÞ: corresponding coefficients for Lorentz violation form an observer four-tensor ðkFÞ , which has the symmetries (195) of the Riemann tensor. This four-tensor can be decom- The last term matches the standard form for one type of posed in parallel with the decomposition of the Riemann Lorentz violation in the gravity sector of the minimal SME, tensor into the Weyl tensor, the traceless Ricci tensor, and controlled by the coefficient s for Lorentz violation [2]. the scalar curvature. The scalar part is Lorentz invariant. This coefficient can be studied experimentally in various The Weyl part involves an observer four-tensor that con- ways [28,29]. Most components of related coefficients trols birefringence of light induced by Lorentz violation. have been constrained to parts in 105 to 1010 via reanalysis The traceless Ricci part determines the anisotropies in the of several decades of data from lunar laser ranging [30] and propagation of light due to Lorentz violation, and it is by laboratory tests with atom interferometry [31]. We can specified by the traceless observer two-tensor kF therefore conclude that the traceless part of the vacuum ð Þ kF . Only the latter effects are relevant for present value of the cardinal field is constrained at the same level in purposes. Restricting attention to these coefficients pro- both the secondary and the tertiary cardinal theories. duces in Riemann spacetime the Lagrange density [2] qffiffiffiffiffiffi qffiffiffiffiffiffi L ¼1 j j þ 1 j j VI. SUMMARY AND DISCUSSION EM 4 g F F 2 g kF F F qffiffiffiffiffiffi qffiffiffiffiffiffi This work constructs an alternative theory of gravity, ¼1 j j þ 1 j j EM 4 g F F 2 g kF T ; (192) which we call cardinal gravity, based on the idea that gravitons are massless NG modes originating in sponta- where the tracelessness of kF has been used. Comparison neous Lorentz violation. The starting point is the simple of this result with Eq. (191) shows the match theory (1) of a symmetric two-tensor cardinal field C in cT k; (193) Minkowski spacetime with a potential triggering sponta- F neous Lorentz violation [6]. Requiring consistent self- in analogy with that of Eq. (189). coupling to the energy-momentum tensor constrains the

065018-19 V. ALAN KOSTELECKY´ AND ROBERTUS POTTING PHYSICAL REVIEW D 79, 065018 (2009) form of the potential to the form (139). It also defines a tions appear to standard solutions and also because the bootstrap procedure that permits the construction of a self- vacuum energy-momentum tensor (152) can appear. consistent nonlinear theory. These various investigations may be most effectively When the bootstrap is applied to the original theory prior undertaken in the nonlinear cardinal gauge (155), for to the spontaneous Lorentz violation, cardinal gravity which the form of conventional general-relativistic solu- emerges. This theory has kinetic term SK;C given by tions remains to be obtained. Eq. (111), potential term SV;C given by Eq. (145), and In more extreme situations, such as near the singularities matter coupling LM;C given by Eq. (176). At low energies of black holes or in the very early Universe, the contribu- compared to the scale of the massive modes, the potential tions from the massive modes could be sufficient to change can be approximated by its extremal Lagrange-multiplier qualitatively the usual general-relativistic behavior. The form (146) that allows only NG excitations about the additional propagating modes can be expected to affect vacuum. In this limit, the nonlinear cardinal action reduces features such as inflation and to change the cosmic gravi- to the Einstein-Hilbert action of general relativity with tational background. At sufficiently high temperatures the conventional matter coupling and possibly a vacuum potential changes shape [32] to restore exact Lorentz sym- energy-momentum term (152), all expressed in the non- metry, with an extremum having a zero value for C. This linear cardinal gauge given by Eq. (155). reverse phase transition converts the NG modes into mas- If instead the bootstrap is applied to the effective action sive modes, so the graviton excitations acquire Planck for the spontaneously broken theory, alternative cardinal masses and the nature of gravity at the big bang is radically theories are generated. Using the fluctuation field about the changed. cardinal vacuum value as the basis for the bootstrap yields Cardinal gravity has general coordinate invariance and a secondary cardinal gravity. This has kinetic term given by diffeomorphism symmetry of the background spacetime at Eq. (116) and matter coupling given by Eq. (180). Using all scales, as discussed in the context of the gauge-fixing instead only the NG excitations to perform the bootstrap conditions (155). Diffeomorphism invariance involving the þ N produces a tertiary cardinal gravity, with kinetic term given analogue metric density ( ) emerges in the low- energy limit, where the match to general relativity occurs. by Eq. (119) and matter coupling given by Eq. (182). The This feature of cardinal gravity has some appeal. The actions of these alternative cardinal theories also reduce to aesthetic and mathematical advantages of the diffeomor- the Einstein-Hilbert action in the pure NG limit and in the phism invariance of general relativity are maintained in the nonlinear cardinal gauge (155). However, unconventional low-energy limit of cardinal gravity, while at high energies matter coupling terms remain in this limit. These can be the presence of the original background spacetime may constrained by suitable gravitational experiments, and ex- offer conceptual and calculational advantages for under- isting results limit the magnitude of components of the 105 1010 standing the physics. One example might involve the vac- cardinal vacuum value to parts in to . uum value of the metric, which is presumably set by All forms of cardinal gravity differ from general rela- processes at the Planck scale. In general relativity one tivity in certain respects. One is the presence of the massive can ask why the vacuum value of the metric is nonzero. modes M . The scale of these modes is set by the curva- Since the metric is the fundamental field and the Einstein- ture of the potential about the Lorentz-violating extremum. Hilbert action has diffeomorphism invariance, it might The natural scale in the theory is the Planck mass, which seem natural for the metric field to vanish in the vacuum. enters via the Newton gravitational constant in the usual In contrast, in cardinal gravity at high energies the back- way, so it is plausible that the fluctuations of the modes ground spacetime is nondynamical, and the gravitational M are also of Planck mass. At low energies, their properties at high energies are controlled instead by the propagation can therefore be neglected, and they can be cardinal field. The vacuum value of the cardinal field integrated out of the action to yield their effective contri- affects the physics but not the existence of spacetime bution. The form of the kinetic term (111) suggests the properties. Another example might be improved prospects corrections to the Einstein-Hilbert action appear in part as for quantum calculations at high energies, although this the square of the Ricci tensor suppressed by the square of would require revisiting the analysis in the present work the mass of the modes M. A suppressed effective matter with quantum physics in mind. For instance, our derivation self-interaction that is quadratic in the energy-momentum of the integrable potential is based on purely classical tensor also appears. Investigation of the resulting sublead- considerations, and the effect of radiative corrections is ing corrections to the Einstein equations, some of which an open issue. In the context of bumblebee theories, re- are proportional to the Ricci tensor and hence vanish in the quiring one-loop stability under the renormalization group vacuum, is an open topic. A post-Newtonian study of the restricts the form of the potential and shows that those experimental consequences for laboratory and solar- producing spontaneous Lorentz breaking are generic [33]. system situations, including gravitational-wave searches, The analogue of this for cardinal gravity represents an would be of definite interest. A study of the implications independent condition on the potential that is likely to for cosmology would also be worthwhile because correc- constrain further its form.

065018-20 GRAVITY FROM SPONTANEOUS LORENTZ VIOLATION PHYSICAL REVIEW D 79, 065018 (2009) We conclude this discussion by noting an interesting cardinal-bumblebee theory in which the graviton and the possibility implied by the present work. We have demon- photon simultaneously emerge as NG modes from sponta- strated here that nonlinear gravitons in general relativity neous Lorentz violation. This would represent an alterna- can be interpreted as NG modes from spontaneous Lorentz tive unified framework for understanding the long-range violation. It is also known that photons can be interpreted forces in nature. as NG modes from spontaneous Lorentz violation, even in the presence of gravity: the Einstein-Maxwell equations ACKNOWLEDGMENTS are reproduced at low energies by a suitable bumblebee This work was supported in part by the United States theory [5]. Both the graviton and the photon have two Department of Energy under Grant No. DE-FG02- physical propagating modes. However, spontaneous 91ER40661 and by the Fundaçaõ para a Cieˆncia e a Lorentz violation and the accompanying diffeomorphism Tecnologia in Portugal. violation can generate up to ten NG modes [5], so the possibility exists in principle of developing a combined

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