Is there any symmetry left in gravity theories with explicit Lorentz violation?
aa r X i v : . [ g r- q c ] S e p Is there any symmetry left in gravity theories with explicit Lorentzviolation?
Yuri Bonder and Crist´obal CorralInstituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exicoApartado Postal 70-543, Ciudad de M´exico, 04510, M´[email protected]; [email protected]
Abstract
It is well known that a theory with explicit Lorentz violation is not invariant under diffeomorphisms. On the otherhand, for geometrical theories of gravity, there are alternative transformations, which can be best defined withinthe first-order formalism, and that can be regarded as a set of improved diffeomorphisms. These symmetries areknown as local translations and, among other features, they are Lorentz covariant off shell. It is thus interestingto study if theories with explicit Lorentz violation are invariant under local translations. In this work, an exampleof such a theory, known as the minimal gravity sector of the Standard Model Extension, is analyzed. Using arobust algorithm, it is shown that local translations are not a symmetry of the theory. It remains to be seen if localtranslations are spontaneously broken under spontaneous Lorentz violation, which are regarded as a more naturalalternative when spacetime is dynamic.
Conventional theories of gravity, when geometrical, are invariant under diffeomorphisms (Diff) and local Lorentztransformations (LLT). In addition, such theories are invariant under the so-called local translations (LT), whichcan be regarded as improved Diff in the sense that they are fully Lorentz covariant, among other propeties [1]. Asexpected, for theories that are invariant under LLT, on shell, invariance under Diff implies invariance under LTand vice versa [2, 3]. Thus, in light of this result, it is interesting to study the logical relation between Diff and LTwhen LLT is explicitly broken. This is the main goal of this work. Clearly, a symmetry is a very powerful tool thatsimplifies calculations and provides conceptual clarity, therefore, elucidating if there is a symmetry left in theorieswith explicit Lorentz violation can be extremely relevant.General relativity is thought to be a low energy limit of a more fundamental theory of gravity that incorporatesthe quantum principles [4]. Several interesting quantum gravity approaches have been developed. The leadingcandidates include string theory, loop quantum gravity, spin networks, noncommutative geometry, causal sets, andcausal dynamical triangulations (see the corresponding contributions in Ref. [5]). Notably, even though this theoryis still unknown, research fields like quantum cosmology are making steady progress by appealing to effects inspiredby quantum gravity [6, 7].On the other hand, there is an approach to quantum gravity where the priority is to make contact withexperiments and which is known as quantum gravity phenomenology. Within quantum gravity phenomenology, itis customary to look for traces of Lorentz violation. The motivation to do this stems from the fact that withinthe most prominent approaches to quantum gravity, it has been argued that LLT may not be a fundamentalsymmetry [8–12], and thus, it may be possible to look for empirical traces of quantum gravity by searching forviolations of LLT. Inspired by this possibility, a parametrization of Lorentz violation based on effective field theoryhas been developed [13, 14], which is known as the Standard Model Extension (SME).The SME action contains Lorentz-violating extensions to all sectors of conventional physics, including generalrelativity [15]; this latter sector is known as the gravity sector. Moreover, within the gravity sector, the part thatproduces second-order field equations for the metric is called minimal gravity sector, and, using a post-Newtonianexpansion [16], it has been tested with several interesting experiments including atom interferometry [17], frame1ragging [18], lunar ranging [19, 20], pulsar timing [21, 22], and planetary motion [23]. There are also tests in thecontext of cosmology [24]. Other experiments related to the SME are reported in Ref. [25].It should be stressed that the dominant position in the SME community is that, in the presence of gravity,Lorentz violation must occur spontaneously. This is assumed to deal with the severe restrictions arising fromthe contracted Bianchi identity [15], i.e., the fact that the Einstein tensor is divergence free. This, in turn, isclosely related to the invariance under Diff. In addition, some relations between Diff and LLT are known in theliterature. For example, it has been shown that spontaneous violation of Diff implies spontaneous violation ofLLT, and vice versa [26–28]. In these regards some surprising results have already been uncovered by using LT.In particular, in the unimodular theory of gravity [29], explicit breaking of Diff produces a breakdown of LT but,contrary to the expectations, LLT is unaffected [30]. Thus, it is interesting to analyze the fate of the LT whenLLT is spontaneously broken. Here, however, attention is restricted to explicit symmetry breaking in the minimalgravity sector of the SME. This assumption is adopted for simplicity, but also since the dynamics associated withspontaneous Lorentz violation may spoil the Cauchy initial value formulation [31]. Moreover, this setup allowsone to study if torsion, which modifies the Bianchi identities, can relax the restrictions that have driven the SMEcommunity to consider spontaneous Lorentz violation. Other alternatives to deal with these restrictions include aSt¨uckelberg-like mechanism [32] and the use of Finsler geometries [33].
Local translations can be best defined in gauge theories of gravity [1,34,35] that consider two independent gravita-tional fields: the tetrad e aµ and the Lorentz connection ω abµ = − ω baµ . This setup includes the well-known Poincar´eGauge Theories [3, Chapter 3]. Here, spacetime indices are represented by Greek letters and tangent-space indiceswith Latin characters; the summation convention on repeated indices is understood. The four-dimensional space-time metric g µν is related to the tetrad by g µν = η ab e aµ e bν , where η ab = diag ( − , + , + , +). Note that η ab and itsinverse, η ab , can be used to lower and raise tangent-space indices, and that the tetrad and its inverse, E µa , whichis such that e aµ E µb = δ ab and e aµ E ν a = δ µν , can be used to map spacetime indices to tangent-space indices andvice versa. In addition, the Lorentz connection and the spacetime connection Γ λµν are related through the tetradpostulate ∂ µ e aν + ω abµ e bν = Γ λµν e aλ . (1)The left-hand side of this equation can be written as D µ e aν where D µ is the covariant derivative with respect tothe Lorentz connection .Curvature and torsion can be derived from the tetrad and the Lorentz connection through Cartan’s structureequations 12 R abµν = ∂ [ µ ω abν ] + ω ac [ µ ω cbν ] , (2)12 T aµν = D [ µ e aν ] , (3)where the squared brackets denote antisymmetrization of the n indices enclosed (with a 1 /n ! factor). Clearly, R abµν = R [ ab ][ µν ] and T aµν = T a [ µν ] and, from Eqs. (1) and (3), it can be verified that T aµν / λ [ µν ] e aλ .Furthermore, the Bianchi identities take the form D [ µ R abνλ ] − T c [ µν R abλ ] ρ E ρc = 0 , (4) D [ µ T aνλ ] − T b [ µν T aλ ] ρ E ρb = R a [ µνλ ] . (5)As usual, (active) infinitesimal Diff are implemented by the Lie derivative. Since the tetrad and the Lorentzconnection are 1-forms, it reads Diff = ( δ Diff ( ρ ) e aµ = ρ ν ∂ ν e aµ + ∂ µ ρ ν e aν ,δ Diff ( ρ ) ω abµ = ρ ν ∂ ν ω abµ + ∂ µ ρ ν ω abν . (6) The covariant derivative used here differs from the operator widely used in SME papers (e.g., Ref. [15]), which is represented byD µ , in that, when acting on a tensor, D µ does not add a connection term for each spacetime index. These two operators are discussedin Ref. [36] in a notation that does not coincide with the one used here. ( δ LLT ( λ ) e aµ = − λ ab e bµ ,δ LLT ( λ ) ω abµ = D µ λ ab . (7)The fact that Diff involves partial derivatives and not covariant derivatives under LLT already suggests that Diffare not Lorentz covariant. However, this observation also points to its cure: define a transformation replacing thepartial derivative by a covariant derivative. This is the most conventional way to introduce the LT [1] and theresult, for the case where the gauge group is LLT, isLT = ( δ LT ( ρ ) e aµ = D µ ρ a + ρ ν T aνµ ,δ LT ( ρ ) ω abµ = ρ ν R abνµ , (8)where ρ a = e aµ ρ µ . Remarkably, it is easy to verify that, acting on e aµ or ω abµ , δ Diff ( ρ ) = δ LT ( ρ ) + δ LLT (˜ λ ) , (9)where ˜ λ ab = ρ µ ω abµ . (10)From this relation it can be verified that Diff and LT are equivalent in theories that are invariant under LLT. Whatis more, if a theory is invariant under two of these symmetries, it has to be invariant under the third. Conversely,this suggests that, if a theory breaks a symmetry, as it is the case of the SME, it should break at least one of theremaining symmetries.Recall that general relativity is invariant under the two symmetry classes: LLT and Diff. The former acts locallyon the tangent space and it thus can be regarded as the gauge (or internal) symmetry of the theory, while the latterconnects different spacetime points. Note that, from Eq. (9), it seems that the LT act in both, the tangent space,through LLT, and the manifold, via the Diff. However, the fact that the gravitational potentials transform underLT as Lorentz tensors, which does not occur under Diff, can be used to argue that they only act on the tangentspace.Now, it is possible to construct theories where the gauge group is different from LLT. Perhaps the most popularexamples of such theories are those where the gauge group is de Sitter or anti-de Sitter [36–39]. Fortunately, theLT definition can be generalized to theories invariant under arbitrary gauge transformations (GT). Thus, for thesake of generality, in the remaining of this section the role of the LLT is played by a general GT. To obtain thegeneralized LT one simply needs to replace the partial derivatives, in Eqs. (6), by covariant derivatives associated tothe corresponding gauge group. Equivalently, they can be defined [40] as the difference of δ Diff ( ρ ) and δ GT (˜ λ ), forthe corresponding ˜ λ ab , thus generalizing Eq. (9). It is important to remark that the action of LT on the geometricalfields depends on the gauge symmetry and, for theories invariant under generic GT, it does not need to coincidewith Eqs. (8).Interestingly, an algorithm has been developed [30] that, starting from the action, allows one to verify if thetheory is invariant under LT and GT and, if it does, it gives the corresponding transformations of the dynamicalfields (see also Ref. [41]). What is more, in certain cases, the algorithm leads to the corresponding contractedBianchi identities and the matter conservation laws. Since the algorithm is closely related to N¨other’s theorem, asit can be seen from Eqs. (16) and (17), it selects the fundamental symmetries of the theory. In fact, the algorithm’soutput contains the transformation laws of the dynamical fields under GT and LT; whether the theory is invariantunder Diff can be derived from such transformations. In this sense the GT and LT are more fundamental thanthe Diff. The covariant derivative associated with GT is denoted by ¯ D µ , and the situation in which GT is theLorentz group, which leads to Eqs. (8), arises as a particular case. Also, for simplicity, the algorithm is done in a4-dimensional spacetime that has no boundaries, and all dynamical fields besides the tetrad and Lorentz connection,which are denoted by Ψ, are assumed to be 0-forms in a nontrivial representation of the Lorentz group. Noticethat Ψ may be Dirac spinors. Also observe that the fact that the formalism is based on the action, and not theHamiltonian, allows one to disregard all issues related with spacetime foliations.The basic steps of the algorithm are (i) consider an action principle S [ e aµ , ω abµ , Ψ] = Z d x e L ( e aµ , ω abµ , Ψ) , (11)3here d x e is the covariant 4-volume element and L is an arbitrary Lagrangian. (ii) Perform an arbitrary variationof the action with respect to the dynamical fields δS = Z d x e (cid:0) δe aµ F µa + δω abµ F µab + δ Ψ F (cid:1) , (12)which implicitly defines F µa , F µab , and F . Note that F µa includes the variation of the volume element, namely, F µa = (1 /e ) δ ( e L ) /δe aµ , and that these objects vanish on shell. Then, (iii) apply a covariant derivative ¯ D µ to F µa , F µab , and F . Step (iv), verify if the resulting expressions can be written as linear combinations of F µa , F µab , and F , where the coefficients can be spacetime functions. If possible, it reads¯ D µ F µa = α µ F µa + β bµ F µab + γ a F, (13)¯ D µ F µab = π µ [ a F µb ] + θ µ F µab + µ [ ab ] F, (14)¯ D µ F = 0 , (15)where the fact that F is a 4-form is used. Note that the precise form of the coefficients in Eqs. (13) and (14)depends on the theory, and, in particular, on the gauge group. Step (v), multiply Eqs. (13) and (14), respectively,with gauge parameters ρ a ( x ) and ξ ab ( x ) = − ξ ba ( x ). Finally, (vi) integrate over spacetime using the appropriate4-volume element. At this stage one must use that ¯ D µ can be replaced by ∂ µ when acting on a scalar under thegauge group. Also, the Leibniz rule, Gauss’s theorem, and the assumption that there are no spacetime boundariesare utilized to take the equations to the form0 = Z d x ∂ µ ( e F µa ρ a )= Z d x e [ (cid:0) ρ a ∂ µ ln e + ¯ D µ ρ a + α µ ρ a (cid:1)| {z } δ LT ( ρ ) e aµ F µa + ρ [ a β b ] µ | {z } δ LT ( ρ ) ω abµ F µab + γ a ρ a | {z } δ LT ( ρ )Ψ F ] , (16)0 = Z d x ∂ µ (cid:0) e F µab ξ ab (cid:1) = Z d x e [ − π µb ξ ab | {z } δ GT ( ξ ) e aµ F µa + (cid:0) ξ ab ∂ µ ln e + ¯ D µ ξ ab + θ µ ξ ab (cid:1)| {z } δ GT ( ξ ) ω abµ F µab + µ ab ξ ab | {z } δ GT ( ξ )Ψ F ] . (17)Then, comparing with the action variation (12), it is possible to read off the field transformations under LT andGT. On the other hand, if it is impossible to write the covariant derivatives of F µa and F µab as in Eqs. (13)and (14), then one can identify which symmetries are broken and which terms are responsible for such breakdowns.In the next section, this algorithm is applied to the minimal gravity sector of the SME. A theory with explicit Lorentz violation is one in which identical experiments done in different inertial framescan produce different results. It is thus easy to imagine gedanken and realistic experiments designed to look forsuch violations. In fact, the Earth’s rotation (translation) gives a natural family of instantaneous inertial framesin which Lorentz violation can manifest themselves as signals with a daily (yearly) period. Moreover, there areconcrete models that incorporate Lorentz violation to account for astrophysical puzzles like the presence of cosmicrays above the GZK cutoff [42]. Here attention is restricted to a sector of the SME, which should be regarded as ageneric parametrization of Lorentz violation.The action of the minimal gravity sector of the SME, in the first-order formalism, takes the form S = S g + S m where the Lorentz-violating gravitational part is S g [ e aµ , ω abµ ] = 12 κ Z d x e E µa E ν b (cid:0) R abµν + k cdab R cdµν (cid:1) . (18)Here κ = 8 πG N is the gravitational coupling constant, and e = det e aµ . The first term of this action is theEinstein–Hilbert action, and k cdab is a nondynamical 0-form parametrizing possible Lorentz violations, and whosecomponents are known as the SME coefficients. The SME coefficients are typically expressed in terms of spacetimeindices, this can be achieved by using the tetrad and its inverse to translate the Latin indices in k cdab to spacetime4ndices. From the index symmetries of R abµν is it clear that k cdab = k [ cd ][ ab ] . However, from Eq. (5) it can be seenthat, in the presence of torsion, it is not necessary to assume that k a [ bcd ] = 0. Thus, there are 36 independentcomponents of k cdab ; this should be compared with the 20 independent components that are present when torsionvanishes. Note that the SME coefficients sensitive to torsion are studied in Ref. [15]. For simplicity, even thoughit has been shown that there is a York–Gibbons–Hawking term for the minimal gravity SME sector [43], spacetimeboundaries are not considered. Also, the SME coefficients are assumed to be extremely small in any relevantreference frame, thus, they should not damage the Cauchy initial value formulation of general relativity [44].The matter action is S m = Z d x e L m [ e aµ , ω abµ , Ψ] , (19)where Ψ denotes the matter fields, which are taken to be 0-forms in a nontrivial representation of the Lorentz group(recall that Ψ includes spinors). As it is customary, the energy-momentum and spin densities are respectively definedby τ µa = 1 e δ ( e L m ) δe aµ , (20) σ µab = 2 δ L m δω abµ . (21)Also, since the gravity action does not depend on Ψ, it is possible to define, using the notation of the previoussection, F = δ L m /δ Ψ. The main assumption on S m is that it is invariant under Diff and LLT. This leads to twooff-shell conservation laws ( D µ + T µ ) τ µa = T bµν E µa τ ν b + 12 R bcµν E µa σ ν bc − F E µa ∂ µ Ψ − E µa ω bcµ [( D ν + T ν ) σ ν bc − g νρ E νb τ ρc ] , (22)( D µ + T µ ) σ µab = 2 g µν E µ [ a τ ν b ] + F J ab Ψ , (23)where T µ = T aµν E νa and J ab are the generators of LLT associated with Ψ, i.e., δ LLT ( λ )Ψ = − λ ab J ab Ψ /
2. Noticethat the presence of a T µ term next to every D µ is closely related to the presence of ∂ µ ln e in the transformationsthat are read off from Eqs. (16) and (17).Equation (22) is the generalization of the energy-momentum conservation law. In this equation, the issuesassociated with Diff invariance, which are mentioned in Sec. 2, become evident: there is a partial derivative of Ψand the Lorentz connection appears explicitly, and these two terms are not covariant under LLT. Of course, this isnot an issue when Eq. (23) is valid because, when these two equations are put together, the term with the Lorentzconnection gets replaced by a term with J ab that combines with the partial derivative to transform covariantly. Inaddition, since these terms are multiplied by F , they vanish on shell. Of course, the LT are precisely built in sucha way that the problematic terms do not even arise.An arbitrary variation of the total action is given by Eq. (12) with F µa = − τ µa − κ (cid:20) G µa − k bcde R bcρσ (2 E µe E ρa E σd + E µa E ρd E σe ) (cid:21) , (24) F µab = − σ µab + 12 κ h ( T cρσ + 2 e cρ T σ ) E ρ [ a E σb ] E µc + k abcd T eρσ E ρc E σd E µe − E νc E µd ( D ν + T ν ) k abcd i , (25)where G aµ = R bcρσ ( δ ab δ ρµ E σc − e aµ E ρb E σc /
2) is the Einstein tensor in the presence of torsion. Thus, on shell,Eq. (24) is the generalization of the Einstein equation, which is not necessarily symmetric, and Eq. (25) plays therole of the so-called Cartan equation.The key step in the algorithm presented above is to take the covariant derivative of Eqs. (24) and (25). Theresult are the contracted Bianchi identities, which, using Eqs. (22) and (23), can be casted into the form( D µ + T µ ) F µa = T bµν E µa F νb + R bcµν E µa F νbc − E µa (cid:18) ∂ µ Ψ + 12 ω abµ J ab Ψ (cid:19) F − κ R bcρσ E µa E ρd E σe D µ k bcde , (26)( D µ + T µ ) F µab = 2 g µν E µ [ a F νb ] + 1 κ (cid:16) R cdµν k cd [ ae E µb ] E νe − R c [ a | µν | k b ] cde E µd E νe (cid:17) . (27)5learly, the term with D µ k abcd in Eq. (26) breaks LT invariance since it cannot be written as a linear combinationof F µa , F µab or F . Analogously, in Eq. (27), the terms in the parenthesis break LLT. This answers the centralquestion of this paper: the theory is generically not invariant under any of the symmetries considered. Still, bymaking the last term in Eq. (26) equal to zero, while letting the parenthesis in Eq. (27) be nonzero, one can breakLLT while the theory is invariant under LT. Clearly, it is also possible to have invariance under LLT while LTis broken. However, tuning the values of the SME coefficients goes against the SME philosophy of keeping themarbitrary until they are constrained by experiments.Remarkably, from Eq. (25) it can be realized that k abcd acts as a torsion source, as it occurs in theories withnonminimal scalar couplings (see Ref. [45] and references therein). Thus, even in vacuum, the presence of k abcd generates torsion. This is very unusual since, in most theories, vacuum torsion vanishes. Moreover, this torsionfield could be probed with spinors. In fact, for a Dirac spinor ψ , the equation of motion for ψ takes the form [46] iE µa γ a (cid:18) ∂ µ ψ + i ω bcµ σ bc ψ (cid:19) + E µa A µ γ γ a ψ = 0 , (28)where γ a are the Dirac matrices satisfying Clifford’s algebra { γ a , γ b } = − η ab , σ ab = iγ [ a γ b ] , and γ = iγ γ γ γ .In addition, ˜ ω abµ is the torsion-free spin connection (i.e., it satisfies the torsion-free tetrad postulate) and A µ = ǫ ρσνµ T ρσν /
6, with ǫ µνρσ the Levi-Civita tensor. Notably, bounds on vacuum torsion like those discussed in Ref. [46],could be motivated by the presence of torsion in the minimal gravity sector of the SME. On the other hand, iftorsion is assumed to be nonzero, one can use the very stringent bounds on the matter sector of the SME (seeRef. [25]) to put limits on k abcd ; this is similar to what can be done using field redefinitions in the matter-gravitySME sector [47]. Furthermore, the conventional SME phenomenology is recovered when torsion and spin densityare both set to zero. Notice that the former can be rigorously turned off with a Lagrange multiplier [48]. However,even in this case, it is expected that the LT will not be a symmetry of the theory since the structure of Eqs. (26)and (27) does not change by the presence of this Lagrange multiplier.Finally, as it is mentioned in the introduction, in the SME community, Eq. (26) is seen as a strong restrictionlinking matter, geometry, and k abcd . However, when torsion and spin density are considered, this condition isrelaxed in the sense that it involves more degrees of freedom of both, the matter and the geometry. This may be avaluable alternative in addition to spontaneous Lorentz violation. In this work, the so-called local translations are studied in a gravity theory with explicit Lorentz violation, whichis introduced by a nondynamical 0-form k abcd . It was known that the theory is not diffeomorphism invariant, and,in this work, it is shown that the theory is also not invariant under translational invariance. This is interestingsince the local translations can be regarded as improved diffeomorphisms in the sense that they are fully covariantunder local Lorentz transformations, thus having the potential to be unaffected by Lorentz violation. Anotherinteresting aspect that becomes evident is that the minimal gravity sector of the SME, in the presence of torsion,has additional coefficients which generate vacuum torsion.There are interesting issues that could be addressed in future contributions. For example, what happens withthe local translations in theories with spontaneous Lorentz violation? In this case k abcd would be dynamical and itthus transforms under all symmetries. Furthermore, even if the local translations are indeed spontaneously broken,there could still be advantages of using them when there is spontaneous Lorentz violation. The ultimate role ofthese local translations needs to be carefully analyzed but, if some traces of them remain, they could play veryimportant roles. Acknowledgements
The authors acknowledge getting valuable input from D. Gonz´alez, A. Kosteleck´y, and M. Montesinos. This researchwas funded by UNAM-DGAPA-PAPIIT Grant No. RA101818 and UNAM-DGAPA postdoctoral fellowship.6 eferences [1] F. W. Hehl, J. D. McCrea, E. W. Mielke, and Y. Ne’eman. Metric affine gauge theory of gravity: Fieldequations, Noether identities, world spinors, and breaking of dilation invariance.
Phys. Rept. , 258:1, 1995.[2] F. W. Hehl, P. Von Der Heyde, G. D. Kerlick, and J. M. Nester. General relativity with spin and torsion:Foundations and prospects.
Rev. Mod. Phys. , 48:393, 1976.[3] M. Blagojevic.
Gravitation and gauge symmetries . Institute of Physics, 2002.[4] C. P. Burgess. Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory.
Liv. Rev.Rel. , 7:5, 2004.[5] D. Oriti, editor.
Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter .Cambridge University Press, 2009.[6] C. Kiefer and M. Kr¨amer. Quantum gravitational contributions to the cosmic microwave background anisotropyspectrum.
Phys. Rev. Lett. , 108:021301, 2012.[7] S. Capozziello and O. Luongo. Entanglement inside the cosmological apparent horizon.
Phys. Lett. A , 378:2058,2014.[8] V. A. Kosteleck´y and S. Samuel. Spontaneous breaking of Lorentz symmetry in string theory.
Phys. Rev. D ,39:683, 1989.[9] V. A. Kosteleck´y and R. Potting. CPT and strings.
Nucl. Phys. B , 359:545, 1991.[10] R. Gambini and J. Pullin. Nonstandard optics from quantum space-time.
Phys. Rev. D , 59:124021, 1999.[11] J. Alfaro, H. A. Morales-T´ecotl, and L. F. Urrutia. Quantum gravity corrections to neutrino propagation.
Phys. Rev. Lett. , 84:2318, 2000.[12] S. M. Carroll, J. A. Harvey, V. A. Kosteleck´y, C. D. Lane, and T. Okamoto. Noncommutative field theoryand Lorentz violation.
Phys. Rev. Lett. , 87:141601, 2001.[13] D. Colladay and V. A. Kosteleck´y. CPT violation and the standard model.
Phys. Rev. D , 55:6760, 1997.[14] D. Colladay and V. A. Kosteleck´y. Lorentz-violating extension of the standard model.
Phys. Rev. D , 58:116002,1998.[15] V. A. Kosteleck´y. Gravity, Lorentz violation, and the standard model.
Phys. Rev. D , 69:105009, 2004.[16] Q. G. Bailey and V. A. Kosteleck´y. Signals for Lorentz violation in post-Newtonian gravity.
Phys. Rev. D ,74:045001, 2006.[17] H. M¨uller, S. W. Chiow, S. Herrmann, S. Chu, and K. Y. Chung. Atom-interferometry tests of the isotropy ofpost-Newtonian gravity.
Phys. Rev. Lett. , 100:031101, 2008.[18] Q. G. Bailey, R. D. Everett, and J. M. Overduin. Limits on violations of Lorentz symmetry from GravityProbe B.
Phys. Rev. D , 88:102001, 2013.[19] J. B. R. Battat, J. F. Chandler, and C. W. Stubbs. Testing for Lorentz Violation: Constraints on Standard-Model-Extension Parameters via Lunar Laser Ranging.
Phys. Rev. Lett. , 99:241103, 2007.[20] A. Bourgoin, A. Hees, S. Bouquillon, C. Le Poncin-Lafitte, G. Francou, and M. C. Angonin. Testing LorentzSymmetry with Lunar Laser Ranging.
Phys. Rev. Lett. , 117:241301, 2016.[21] L. Shao. Tests of Local Lorentz Invariance Violation of Gravity in the Standard Model Extension with Pulsars.
Phys. Rev. Lett. , 112:111103, 2014.[22] L. Shao. New pulsar limit on local Lorentz invariance violation of gravity in the standard-model extension.
Phys. Rev. D , 90:122009, 2014.[23] A. Hees, Q. G. Bailey, C. Le Poncin-Lafitte, A. Bourgoin, A. Rivoldini, B. Lamine, F. Meynadier, C. Guerlin,and P. Wolf. Testing Lorentz symmetry with planetary orbital dynamics.
Phys. Rev. D , 92:064049, 2015.724] Y. Bonder and G. Le´on. Inflation as an amplifier: The case of Lorentz violation.
Phys. Rev. D , 96:044036,2017.[25] V. A. Kosteleck´y and N. Russell. Data tables for Lorentz and
CP T violation.
Rev. Mod. Phys. , 83:11, 2011.The 2018 version can be found at arxiv:0801.0287v11.[26] R. Bluhm and V. A. Kosteleck´y. Spontaneous Lorentz violation, Nambu–Goldstone modes, and gravity.
Phys.Rev. D , 71:065008, 2005.[27] R. Bluhm, S. H. Fung, and V. A. Kosteleck´y. Spontaneous Lorentz and diffeomorphism violation, massivemodes, and gravity.
Phys. Rev. D , 77:065020, 2008.[28] V. A. Kosteleck´y and M. Mewes. Lorentz and diffeomorphism violations in linearized gravity.
Phys. Lett. B ,779:136, 2018.[29] Y. Bonder and C. Corral. Unimodular Einstein–Cartan gravity: Dynamics and conservation laws.
Phys. Rev.D , 97:084001, 2018.[30] C. Corral and Y. Bonder. Local translations in modified gravity theories. Preprint at arXiv:1808.01497.[31] Y. Bonder and C. A. Escobar. Dynamical ambiguities in models with spontaneous Lorentz violation.
Phys.Rev. D , 93:025020, 2016.[32] R. Bluhm. Gravity theories with background fields and spacetime symmetry breaking.
Symmetry , 9:230, 2017.[33] V. A. Kosteleck´y. Riemann–finsler geometry and lorentz-violating kinematics.
Phys. Lett. B , 701:137, 2011.[34] Yu. N. Obukhov. Poincare gauge gravity: Selected topics.
Int. J. Geom. Meth. Mod. Phys. , 3:95, 2006.[35] M. Blagojevi´c and F. W. Hehl, editors.
Gauge Theories of Gravitation . World Scientific Publishing, 2013.[36] M. Hassaine and J. Zanelli.
Chern–Simons (Super)Gravity . World Scientific Publishing, 2016.[37] S. W. MacDowell and F. Mansouri. Unified geometric theory of gravity and supergravity.
Phys. Rev. Lett. ,38:739, 1977. Erratum:
Phys. Rev. Lett. , , 1977.[38] K. S. Stelle and P. C. West. De Sitter gauge invariance and the geometry of the Einstein–Cartan theory. J.Phys. A , 12:L205, 1979.[39] R. Troncoso and J. Zanelli. Higher dimensional gravity, propagating torsion and AdS gauge invariance.
Class.Quantum Grav. , 17:4451, 2000.[40] Yu. N. Obukhov and G. F. Rubilar. Invariant conserved currents in gravity theories: Diffeomorphisms andlocal gauge symmetries.
Phys. Rev. D , 76:124030, 2007.[41] M. Montesinos, D. Gonz´alez, M. Celada, and B. D´ıaz. Reformulation of the symmetries of first-order generalrelativity.
Class. Quantum Grav. , 34:205002, 2017.[42] S. Coleman and S. L. Glashow. High-energy tests of Lorentz invariance.
Phys. Rev. D , 59:116008, 1999.[43] Y. Bonder. Lorentz violation in the gravity sector: The t puzzle. Phys. Rev. D , 91:125002, 2015.[44] H. Ringstr¨om.
On the topology and future stability of the universe . Oxford Science Publications, 2013.[45] J. Barrientos, F. Cordonier-Tello, F. Izaurieta, P. Medina, D. Narbona, E. Rodr´ıguez, and O. Valdivia. Non-minimal couplings, gravitational waves, and torsion in Horndeski’s theory.
Phys. Rev. D , 96:084023, 2017.[46] V. A. Kosteleck´y, N. Russell, and J. D. Tasson. Constraints on Torsion from Bounds on Lorentz Violation.
Phys. Rev. Lett. , 100:111102, 2008.[47] Y. Bonder. Lorentz violation in a uniform Newtonian gravitational field.
Phys. Rev. D , 88:105011, 2013.[48] S. del Pino, G. Giribet, A. Toloza, and J. Zanelli. From Lorentz-Chern–Simons to Massive Gravity in 2 + 1dimensions.