Signals for Lorentz Violation in Post-Newtonian Gravity

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8-15-2006

Quentin G. Bailey Indiana University - Bloomington, [email protected]

V. Alan Kostelecký Indiana University - Bloomington

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Scholarly Commons Citation Bailey, Q. G., & Kostelecký, V. A. (2006). Signals for Lorentz Violation in Post-Newtonian Gravity. Physical Review D, 74(4). https://doi.org/10.1103/PhysRevD.74.045001

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Quentin G. Bailey and V. Alan Kostelecký Physics Department, Indiana University, Bloomington, IN 47405, U.S.A. (Dated: IUHET 489, March 2006; Physical Review D, in press) The pure-gravity sector of the minimal Standard-Model Extension is studied in the limit of Rie- mann spacetime. A method is developed to extract the modified Einstein field equations in the limit of small metric fluctuations about the Minkowski vacuum, while allowing for the dynamics of the 20 independent coefficients for Lorentz violation. The linearized effective equations are solved to obtain the post-newtonian metric. The corresponding post-newtonian behavior of a perfect fluid is studied and applied to the gravitating many-body system. Illustrative examples of the methodology are provided using bumblebee models. The implications of the general theoretical results are studied for a variety of existing and proposed gravitational experiments, including lunar and satellite laser ranging, laboratory experiments with gravimeters and torsion pendula, measurements of the spin precession of orbiting gyroscopes, timing studies of signals from binary pulsars, and the classic tests involving the perihelion precession and the time delay of light. For each type of experiment considered, estimates of the attainable sensitivities are provided. Numerous effects of local Lorentz violation can be studied in existing or near-future experiments at sensitivities ranging from parts in 104 down to parts in 1015.

I. INTRODUCTION dictions of realistic theories involving relativity modifications are therefore expressible in terms of the SME by At the classical level, gravitational phenomena are well specifying the SME coefficient values. In fact, the explicit described by general relativity, which has now survived form of all dominant Lorentz-violating terms in the SME nine decades of experimental and theoretical scrutiny. is known [4]. These terms consist of Lorentz-violating In the quantum domain, the Standard Model of parti- operators of mass dimension three or four, coupled to cle physics offers an accurate description of matter and coefficients with Lorentz indices controlling the degree nongravitational forces. These two field theories proof Lorentz violation. The subset of the theory contain- vide a comprehensive and successful description of na- ing these dominant Lorentz-violating terms is called the ture. However, it remains an elusive challenge to find a minimal SME. consistent quantum theory of gravity that would merge Since Lorentz symmetry underlies both general rela- them into a single underlying unified theory at the Planck tivity and the Standard Model, experimental searches for scale. violations can take advantage either of gravitational or of Since direct measurements at the Planck scale are in- nongravitational forces, or of both. In the present work, feasible, experimental clues about this underlying theory we initiate an SME-based study of gravitational experi- are scant. One practical approach is to search for prop- ments searching for violations of local Lorentz invariance. erties of the underlying theory that could be manifest To restrict the scope of the work to a reasonable size as suppressed new physics effects, detectable in sensi- while maintaining a good degree of generality, we limit tive experiments at attainable energy scales. Promising attention here to the pure-gravity sector of the minimal candidate signals of this type include ones arising from SME in Riemann spacetime. This neglects possible com- minuscule violations of Lorentz symmetry [1, 2]. plexities associated with matter-sector effects and with Effective field theory is a useful tool for describing ob- Riemann-Cartan [5] or other generalized spacetimes, but servable signals of Lorentz violation [3]. Any realistic it includes all dominant Lorentz-violating signals in ef- effective field theory must contain the Lagrange densi- fective action-based metric theories of gravity. ties for both general relativity and the Standard Model, The Minkowski-spacetime limit of the minimal SME possibly along with suppressed operators of higher mass [6] has been the focus of various experimental studies, dimension. Adding also all terms that involve operators including ones with photons [7–9], electrons [10–12], pro- for Lorentz violation and that are scalars under coordi- tons and neutrons [13–15], mesons [16], muons [17], neu- nate transformations results in an effective field theory trinos [18], and the Higgs [19]. To date, no compelling called the Standard-Model Extension (SME). The lead- evidence for nonzero coefficients for Lorentz violation has ing terms in this theory include those of general relativ- been found, but only about a third of the possible sig- ity and of the minimally coupled Standard Model, along nals involving light and ordinary matter (electrons, pro- with possible Lorentz-violating terms constructed from tons, and neutrons) have been experimentally investi- gravitational and Standard-Model fields. gated, while some of the other sectors remain virtually Since the SME is founded on well-established physics unexplored. Our goal here is to provide the theoretical and constructed from general operators, it offers an ap- basis required to extend the experimental studies into the proach to describing Lorentz violation that is largely in- post-newtonian regime, where asymptotically Minkowski dependent of the underlying theory. Experimental pre- spacetime replaces the special-relativistic limit. 2

Nonzero SME coefficients for Lorentz violation could periments, along with a discussion of existing bounds. In arise via several mechanisms. It is convenient to dis- Sec. V B, we focus on experiments involving laser rang- tinguish two possibilities, spontaneous Lorentz violation ing, including ranging to the Moon and to artificial satel- and explicit Lorentz violation. If the Lorentz violation lites. Section V C studies some promising laboratory ex- is spontaneous [20], the SME coefficients arise from un- periments on the Earth, including gravimeter measure- derlying dynamics, and so they must be regarded as ments of vertical acceleration and torsion-pendulum tests fields contributing to the dynamics through the varia- of horizontal accelerations. The subject of Sec. VD is the tion of the action. In contrast, if the Lorentz viola- precession of an orbiting gyroscope, while signals from tion is explicit, the SME coefficients for Lorentz vio- binary pulsars are investigated in Sec. V E. The role of lation originate as prescribed spacetime functions that the classic tests of general relativity is discussed in Sec. play no dynamical role. Remarkably, the geometry of V F. To conclude the paper, a summary of the main Riemann-Cartan spacetimes, including the usual Rie- results is provided in Sec. VI, along with a tabulation mann limit, is inconsistent with explicit Lorentz viola- of the estimated attainable experimental sensitivities for tion [4]. In principle, a more general non-Riemann ge- the SME coefficients for Lorentz violation. Some details ometry such as a Finsler geometry might allow for ex- of the orbital analysis required for our considerations of plicit violation [4, 21], but this possibility remains an laser ranging are relegated to Appendix A. Throughout open issue at present. We therefore limit attention to this work, we adopt the notation and conventions of Ref. spontaneous Lorentz violation in this work. Various sce- [4]. narios for the underlying theory are compatible with a spontaneous origin for Lorentz violation, including ones based on string theory [20], noncommutative field theo- II. THEORY ries [22], spacetime-varying fields [23], quantum gravity [24], random-dynamics models [25], multiverses [26], and A. Basics brane-world scenarios [27]. Within the assumption of spontaneous Lorentz break- The SME action with gravitational couplings is pre- ing, the primary challenge to extracting the post- sented in Ref. [4]. In the general case, the geomet- newtonian physics of the pure-gravity minimal SME lies ric framework assumed is a Riemann-Cartan spacetime, in accounting correctly for the fluctuations around the which allows for nonzero torsion. The pure gravitational vacuum values of the coefficients for Lorentz violation, in- part of the SME Lagrange density in Riemann-Cartan cluding the massless Nambu-Goldstone modes [28]. Ad- spacetime can be viewed as the sum of two pieces, one dressing this challenge is the subject of Sec. II of this Lorentz invariant and the other Lorentz violating: work. The theoretical essentials for the analysis are presented in Sec. II A, while Sec. II B describes our method- Lgravity = LLI + LLV. (1) ology for obtaining the effective linearized field equations for the metric fluctuations in a general scenario. The All terms in this Lagrange density are invariant un- post-newtonian metric of the pure-gravity minimal SME der observer transformations, in which all fields and is obtained in Sec. III A, and it is used to discuss modifi- backgrounds transform. These include observer local cations to perfect-fluid and many-body dynamics in Sec. Lorentz transformations and observer diffeomorphisms IIIB. In recent decades, a substantial effort has been in- or general coordinate transformations. The piece LLI vested in analysing weak-field tests of general relativity also remains invariant under particle transformations, in in the context of post-newtonian expansions of an arbi- which the localized fields and particles transform but the trary metric, following the pioneering theoretical works of backgrounds remain fixed. These include particle local Nordtvedt and Will [29, 30]. Some standard and widely Lorentz transformations and particle diffeomorphisms. used forms of these expansions are compared and con- However, for vanishing fluctuations of the coefficients for trasted to our results in Sec. III C. The theoretical part Lorentz violation, the piece LLV changes under particle of this work concludes in Sec. IV, where the key ideas for transformations and thereby breaks Lorentz invariance. our general methodology are illustrated in the context of The Lorentz-invariant piece LLI is a series in pow- a class of bumblebee models. ers of the curvature Rκλµν , the torsion Tλµν , the co- The bulk of the present paper concerns the implica- variant derivatives Dκ, and possibly other dynamical tions for gravitational experiments of the post-newtonian fields determining the pure-gravity properties of the the- metric for the pure-gravity sector of the minimal SME. ory. The leading terms in LLI are usually taken as This issue is addressed in Sec. V. To keep the scope the Einstein-Hilbert and cosmological terms in Riemann- reasonable, we limit attention to leading and subleading Cartan spacetime. The Lorentz-violating piece LLV is effects. We identify experiments that have the potential constructed by combining coefficients for Lorentz viola- to measure SME coefficients for Lorentz violation, and tion with gravitational field operators to produce indi- we provide estimates of the attainable sensitivities. The vidual terms that are observer invariant under both local analysis begins in Sec. V A with a description of some Lorentz and general coordinate transformations. The ex- general considerations that apply across a variety of explicit form of this second piece can also be written as a 3 series in the curvature, torsion, covariant derivative, and origin of the Lorentz violation. As mentioned in the possibly other fields: introduction, explicit Lorentz violation is incompatible with Riemann spacetime [4]. We therefore limit attention λµν κλµν LLV = e(kT ) Tλµν + e(kR) Rκλµν in this work to spontaneous Lorentz violation in Riemann αβγλµν µν κλµν +e(kT T ) TαβγTλµν spacetime, for which the coefficients u, s , t are dy- κλµν namical fields. Note that spontaneous local Lorentz vi- +e(kDT ) DκTλµν + ..., (2) olation is accompanied by spontaneous diffeomorphism a violation, so as many as 10 symmetry generators can be where e is the determinant of the vierbein eµ . The λµν κλµν broken through the dynamics, with a variety of interest- coefficients for Lorentz violation (kT ) , (kR) , µν αβγλµν κλµν ing attendant phenomena [28]. Note also that u, s , (kT T ) , (kDT ) can vary with spacetime posi- tκλµν may be composites of fields in the underlying the- tion. Since particle local Lorentz violation is always ac- ory. Examples for this situation are discussed in Sec. IV. companied by particle diffeomorphism violation [28], the The third term in Eq. (3) is the general matter action coefficients for Lorentz violation also control diffeomor- S′. In addition to determining the dynamics of ordinary phism violation in the theory. matter, it includes contributions from the coefficients u, In the present work, we focus on the Riemann- sµν , tκλµν , which for our purposes must be considered in spacetime limit of the SME, so the torsion is taken to some detail. The action S′ could also be taken to include vanish. We suppose that the Lorentz-invariant piece of the SME terms describing Lorentz violation in the matter the theory is the Einstein-Hilbert action, and we also sector. These terms, given in Ref. [4], include Lorentz- restrict attention to the leading-order Lorentz-violating violating matter-gravity couplings with potentially ob- terms. The gravitational terms that remain in this limit servable consequences, but addressing these effects lies form part of the minimal SME. The basic features of the beyond the scope of the present work. Here, we focus resulting theory are discussed in Ref. [4], and those rele- instead on effects from the gravitational and matter cou- vant for our purposes are summarized in this subsection. plings of the coefficients u, sµν , tκλµν in Eq. (5). The effective action of the minimal SME in this limit Variation with respect to the metric gµν while holding can be written as u, sµν , and tκλµν fixed yields the field equations ′ µν Rstu µν µν S = SEH + SLV + S . (3) G − (T ) = κ(Tg) . (6)

The first term in (3) is the Einstein-Hilbert action of In this expression, general relativity. It is given by Rstu µν 1 µ ν 1 ν µ µν 2 µν (T ) ≡ − 2 D D u − 2 D D u + g D u + uG 1 αβ µν 1 µ αν 1 ν αµ 1 4 + 2 s Rαβg + 2 DαD s + 2 DαD s SEH = d xe(R − 2Λ), (4) 2κ 1 2 µν 1 µν αβ 1 αβγµ ν Z − 2 D s − 2 g DαDβs + 2 t Rαβγ where R is the Ricci scalar, Λ is the cosmological con- 1 αβγν µ 1 αβγδ µν + 2 t Rαβγ + 2 t Rαβγδg stant, and κ =8πG. As usual, in the present context of a µανβ ναµβ Riemann spacetime, the independent degrees of freedom −DαDβt − DαDβt , (7) of the gravitational field are contained in the metric gµν . while the general matter energy-momentum tensor is de- Since we are ultimately focusing on the post-newtonian fined as usual by limit of (3), in which the effects of Λ are known to be 1 µν ′ negligible, we set Λ = 0 for the remainder of this work. 2 e(Tg) ≡ δL /δgµν , (8) The second term in Eq. (3) contains the leading where L′ is the Lagrange density of the general matter Lorentz-violating gravitational couplings. They can be action S′. written as 1 S = d4xe(−uR + sµν RT + tκλµν C ). (5) B. Linearization LV 2κ µν κλµν Z T In this equation, Rµν is the trace-free Ricci tensor and One of the central goals of this work is to use the SME Cκλµν is the Weyl conformal tensor. The coefficients for to obtain the newtonian and leading post-newtonian cor- Lorentz violation sµν and tκλµν inherit the symmetries rections to general relativity induced by Lorentz viola- of the Ricci tensor and the Riemann curvature tensor, tions. For this purpose, it suffices to work at linear order respectively. The structure of Eq. (5) implies that sµν in metric fluctuations about a Minkowski background. can be taken traceless and that the various traces of tκλµν We can therefore adopt the usual asymptotically inertial can all be taken to vanish. It follows that Eq. (5) contains coordinates and write 20 independent coefficients, of which one is in u, 9 are in g = η + h . (9) sµν , and 10 are in tκλµν . µν µν µν The coefficients u, sµν , and tκλµν typically depend on In this subsection, we derive the effective linearized field spacetime position. Their nature depends in part on the equations for hµν in the presence of Lorentz violation. 4

1. Primary linearization With these first two assumptions, we can extract the linearized version of the field equations (7) in terms of hµν µν κλµν A key issue is the treatment of the dynamics of the and the fluctuationsu ˜,s ˜ , and t˜ . Some calculation coefficient fields u, sµν , and tκλµν . As described above, shows that the linearized trace-reversed equations can be these are assumed to induce spontaneous violation of lo- written in the form cal Lorentz invariance and thereby acquire vacuum ex- Rµν = κ(Sg)µν + Aµν + Bµν . (12) pectation values u, sµν , and tκλµν , respectively. Denot- ing the field fluctuations about these vacuum solutions In this expression, (Sg)µν is the trace-reversed energy- asu ˜,s ˜µν , and t˜κλµν , we can write momentum tensor defined by

u = u +˜u, 1 α (Sg)µν = (Tg)µν − 2 ηµν (Tg) α, (13) sµν = sµν +˜sµν , where (T ) is the linearized energy-momentum tensor κλµν κλµν ˜κλµν g µν t = t + t . (10) containing the energy-momentum density of both con- µν κλµν Note that the fluctuations include as a subset the Nambu- ventional matter and the fluctuationsu ˜,s ˜ , and t˜ . Goldstone modes for local Lorentz and diffeomorphism Also, the terms Aµν and Bµν in Eq. (12) are given by violation, which are described in Ref. [28]. For present A = −∂ ∂ u˜ − 1 η 2u˜ + ∂ ∂ s˜α − 1 2s˜ purposes it suffices to work at linear order in the fluc- µν µ ν 2 µν α (µ ν) 2 µν tuations, so in what follows nonlinear terms at O(h2), 1 α ˜ α β ˜αγβ + ηµν 2s˜ α − 2∂α∂βtµ ν + ηµν ∂α∂βt 2 4 γ O(hu˜), O(˜s ), etc., are disregarded, and we adopt the α β αβ α β standard practice of raising and lowering indices on lin- +s (µ∂αΓ ν)β + s ∂αΓ(µν)β − s (µ∂ Γν)βα ear quantities with the Minkowski metric ηµν . 1 α γ β α β γ + 2 ηµν s β∂ Γ γα − 4t(µ ν) ∂αΓ γβ, (14) Deriving the effective linearized field equations for hµν involves applying the expressions (9)-(10) in the asymp- and totically inertial frame. It also requires developing methods to account for effects on h due to the fluctuations 1 αβ α µν Bµν = uRµν − 2 ηµν s Rαβ + s (µRν)α u˜,˜sµν , t˜κλµν . In fact, as we show below, five key assump- αβγ 3 αβγδ tions about the properties of these fluctuations suffice for +2t (µRαβγν) − 2 ηµν t Rαβγδ this purpose. α β −2t R . (15) The first assumption concerns the vacuum expectation (µ ν) αβ µν κλµν values. We assume (i): the vacuum values u, s , t The connection coefficients appearing in Eq. (14) are lin- are constant in asymptotically inertial cartesian coordi- earized Christoffel symbols, where indices are lowered nates. Explicitly, we take with the Minkowski metric as needed. The terms R, Rαβ, and Rαβγδ appearing in Eqs. (12)-(15) and elsewhere be- ∂αu = 0, µν low are understood to be the linearized Ricci scalar, Ricci ∂αs = 0, tensor, and Riemann curvature tensor, respectively. κλµν ∂αt = 0. (11)

More general conditions could be adopted, but assump- 2. Treatment of the energy-momentum tensor tion (i) ensures that translation invariance and hence energy-momentum conservation are preserved in the To generate the effective equation of motion for hµν asymptotically Minkowski regime. The reader is cau- alone, the contributions from the fluctuationsu ˜,s ˜µν , tioned that this assumption is typically different from κλµν t˜ must be expressed in terms of hµν , its derivatives, the requirement of covariant constancy. µν κλµν The second assumption ensures small Lorentz- and the vacuum values u, s , t . In general, this is a violating effects. We suppose (ii): the dominant effects challenging task. We adopt here a third assumption that κλµν simplifies the treatment of the dynamics sufficiently to are linear in the vacuum values u, sµν , t . This as- permit a solution while keeping most interesting cases. sumption has been widely adopted in Lorentz-violation We assume (iii): the fluctuations u˜, s˜µν , t˜κλµν have no phenomenology. The basic reasoning is that any Lorentz relevant couplings to conventional matter. This assump- violation in nature must be small, and hence linearization is standard in alternate theories of gravity, where it tion typically suffices. Note, however, that in the present is desired to introduce new fields to modify gravity while context this assumption applies only to the vacuum val- maintaining suitable matter properties. It follows that ues of the coefficients for Lorentz violation in the SME (S ) can be split as action (3). Since these coefficients are related via un- g µν determined coupling constants to the vacuum values of (Sg)µν = (SM )µν + (Sstu)µν , (16) the dynamical fields in the underlying theory, assumption (ii) provides no direct information about the sizes of where (SM )µν is the trace-reversed energy-momentum the latter. tensor for conventional matter. 5

To gain insight about assumption (iii), consider the it vanishes in most theories, at least to linear order. How- µν κλµν contributions of the fluctuationsu ˜,s ˜ , t˜ to the ever, theories with nonzero Σµν do exist and may even be trace-reversed energy-momentum tensor (Sg)µν . A pri- generic. Some simple examples are discussed in Sec. IV. ori, the fields u, sµν , tκλµν can have couplings to mat- Nonetheless, it turns out that assumption (iv) suffices for ter currents in the action S′, which would affect the meaningful progress because in many models the nonzero µν κλµν energy-momentum contribution ofu ˜,s ˜ , t˜ . How- contributions from Σµν merely act to scale the effective ever, through the vacuum values u, sµν , tκλµν , these linearized Einstein equations relative to those obtained in couplings would also induce coefficients for Lorentz vi- the zero-Σµν limit. Models of this type are said to vio- olation in the SME matter sector, which are generically late assumption (iv) weakly, and their linearized behavior known to be small from nongravitational experiments. is closely related to that of the zero-Σµν limit. In con- The couplings of the fluctuationsu ˜,s ˜µν , t˜κλµν to the trast, a different behavior is exhibited by certain models matter currents are therefore also generically expected to with ghost kinetic terms for the basic fields. These have be small, and so it is reasonable to take the the matter nonzero contributions to Σµν that qualitatively change as decoupled from the fluctuations for present purposes. the behavior of the effective linearized Einstein equations Possible exceptions to the decoupling of the fluctua- relative to the zero-Σµν limit. Models with this feature tions and matter can arise. For example, in the case of are said to violate assumption (iv) strongly. It is reason- certain bumblebee models, including those described in able to conjecture that for propagating modes a strongly Sec. IV, the fluctuationss ˜µν can be written in terms of a nonzero Σµν is associated with ghost fields, but estab- vacuum value and a propagating vector field. This vec- lishing this remains an open issue lying outside the scope tor field can be identified with the photon in an axial of the present work. gauge [28]. A sufficiently large charge current jµ could then generate significant fluctuationss ˜µν , thereby com- peting with gravitational effects. However, for the various systems considered in this work, the gravitational 3. Decoupling of fluctuations fields dominate over the electrodynamic fields by a con- siderable amount, so it is again reasonable to adopt as- At this stage, the trace-reversed energy-momentum sumption (iii). tensor (S ) in Eq. (12) has been linearized and ex- Given the decomposition (16), the task of decoupling g µν pressed in terms of h , its derivatives, and the vacuum the fluctuationsu ˜,s ˜µν , t˜κλµν in (S ) reduces to ex- µν g µν µν κλµν pressing the partial energy-momentum tensor (Sstu)µν values u, s , t . The term Bµν in Eq. (12) already has the desired form, so it remains to determine A . in terms of hµν , its derivatives, and the vacuum values µν µν κλµν κλµν The latter explicitly contains the fluctuationss ˜ , t˜ , u, sµν , t . For this purpose, we can apply a set of andu ˜. By assumption (iii), the fluctuations couple only four identities, derived from the traced Bianchi identi- to gravity, so it is possible in principle to obtain them ties, that are always satisfied by the gravitational energy- as functions of h alone by solving their equations of momentum tensor [4]. The linearized versions of these µν motion. It follows that A can be expressed in terms conditions suffice here. They read µν of derivatives of these functions. Since the leading-order µ αβ αβγδ dynamics is controlled by second-order derivatives, the κ∂ (Tg)µν = −s ∂βRαν − 2t ∂δRαβγν . (17) leading-order result for Aµν is also expected to be second

Using the fact that (TM )µν is separately conserved, one order in derivatives. We therefore adopt assumption (v): can show that the four conditions (17) are satisfied by the undetermined terms in Aµν are constructed from linear combinations of two partial spacetime derivatives of α 1 αβ µν κλµν κ(Sstu)µν = −2s (µRν)α + 2 sµν R + ηµν s Rαβ hµν and the vacuum values ηµν , u, s , and t . More αβγ αβγδ explicitly, the undetermined terms take the generic form −4t (µRαβγν) +2ηµν t Rαβγδ αβγδ Mµν ∂α∂βhγδ. This assumption ensures a smooth α β +4t µ ν Rαβ +Σµν . (18) match to conventional general relativity in the limit of αβγδ vanishing coefficients Mµν . In this equation, the term Σµν obeys To constrain the form of Aµν , we combine assumption (v) with invariance properties of the action, notably ∂µ(Σ − 1 η Σα )=0. (19) µν 2 µν α those of diffeomorphism symmetry. Since we are consid- It represents an independently conserved piece of the lin- ering spontaneous symmetry breaking, which maintains earized energy-momentum tensor that is undetermined the symmetry of the full equations of motion, the origi- by the conditions (17). nal particle diffeomorphism invariance can be applied. In For calculational purposes, it is convenient in what fol- particular, it turns out that the linearized particle diffeo- lows to adopt assumption (iv): the independently con- morphism transformations leaves the linearized equations served piece of the trace-reversed energy momentum ten- of motion (12) invariant. To proceed, note first that the µν κλµν sor vanishes, Σµν = 0. Since Σµν is independent of other vacuum values u, s , t and ηµν are invariant under µ contributions to (Sstu)µν , one might perhaps suspect that a particle diffeomorphism parametrized by ξ [28], while 6 the metric fluctuation transforms as 4. Effective linearized Einstein equations

hµν → hµν − ∂µξν − ∂ν ξµ. (20) The final effective Einstein equations for the metric fluctuation hµν are obtained upon inserting Eqs. (15), (18), and (22) into Eq. (12). We can arrange the equa- The form of Eq. (18) then implies that (Sg)µν is invariant tions in the form at linear order, and from Eq. (15) we find Bµν is also invariant. It then follows from Eq. (12) that Aµν must R = κ(S ) +Φu +Φs +Φt . (23) be particle diffeomorphism invariant also. µν M µν µν µν µν

The invariance of Aµν can also be checked directly. The quantities on the right-hand side are given by Under a particle diffeomorphism, the induced transfor- u mations on the fluctuationss ˜µν , t˜κλµν ,u ˜ can be derived Φµν = uRµν , µν s 1 αβ α 1 from the original transformation of the fields u, s , and Φµν = ηµν s Rαβ − 2s Rαν) + sµν R κλµν 2 (µ 2 t in the same manner that the induced transforma- αβ +s Rαµνβ, tion of hµν is obtained from the transformation for gµν . We find t αβγ α β Φµν = 2t (µRν)γαβ +2tµ ν Rαβ 1 αβγδ κλµν κλµν αλµν κ καµν λ + ηµν t Rαβγδ t˜ → t˜ + t ∂αξ + t ∂αξ 2 κλαν µ κλµα ν = 0. (24) +t ∂αξ + t ∂αξ , µν µν µα ν αν µ s˜ → s˜ + s ∂αξ + s ∂αξ , Each of these quantities is independently conserved, as u˜ → u.˜ (21) required by the linearized Bianchi identities. Each is also invariant under particle diffeomorphisms, as can be ver- t ified by direct calculation. The vanishing of Φµν at the Using these transformations, one can verify explicitly linearized level is a consequence of these conditions and of that Aµν is invariant at leading order under particle dif- αβγδ feomorphisms. the index structure of the coefficient t , which implies the identity [31] t [γδδµ] = 0. The combination of assumption (v) and the imposition [αβ ν] u s t of particle diffeomorphism invariance suffices to extract In Eq. (23), the coefficients of Φµν ,Φµν ,Φµν are taken a covariant form for Aµν involving only the metric fluc- to be unity by convention. In fact, scalings can arise from tuation hµν . After some calculation, we find Aµν takes the terms in Eq. (22) or, in the case of models weakly the form violating assumption (iv), from a nonzero Σµν term in Eq. (18). However, these scalings can always be absorbed αβ α µν κλµν Aµν = −2auRµν + s Rα(µν)β − s (µRν)α into the definitions of the vacuum values u, s , t . In writing Eq. (23) we have implemented this rescaling of α β 1 αβγδ −bt µ ν Rαβ − 4 bηµν t Rαβγδ vacuum values, since it is convenient for the calculations αβγ to follow. However, the reader is warned that as a result −bt Rν)γαβ. (22) (µ the vacuum values of the fields u, sµν , tκλµν appearing in Eq. (5) differ from the vacuum values in Eqs. (24) by a Some freedom still remains in the structure of Aµν , as possible scaling that varies with the specific theory being evidenced in Eq. (22) through the presence of arbitrary considered. scaling factors a and b. These can take different values Since Eq. (23) is linearized both in h and in the in distinct explicit theories. µν vacuum values, the solution for hµν can be split into the We note in passing that, although the above derivation sum of two pieces, makes use of particle diffeomorphism invariance, an alter- E ˜ native possibility exists. One can instead apply observer hµν = hµν + hµν , (25) diffeomorphism invariance, which is equivalent to invari- one conventional and one depending on the vacuum val- ance under general coordinate transformations. This in- E variance is unaffected by any stage of the spontaneous ues. The first term hµν is defined by the requirement symmetry breaking, so it can be adapted to a version of that it satisfy the above reasoning. Some care is required in this proce- Rµν = κ(SM )µν , (26) dure. For example, under observer diffeomorphisms the vacuum values u, sµν , tκλµν transform nontrivially, and which are the standard linearized Einstein equations of the effects of this transformation on Eqs. (11) must be general relativity. The second piece controls the devia- µν taken into account. In contrast, the fluctuationss ˜ , tions due to Lorentz violation. Denoting by Rµν the Ricci ˜κλµν t ,u ˜ are unchanged at leading order under observer tensor constructed with h˜µν , it follows that this second diffeomorphisms. In any case, the result (22) provides piece is determined by the expression e the necessary structure of Aµν when the assumptions (i) u s to (v) are adopted. Rµν =Φµν +Φµν , (27)

e 7

u where it is understood that the terms on the right-hand As might be expected from the form of Φµν in Eq. (24), E side are those of Eqs. (24) in the limit hµν → hµν , so which involves a factor u multiplying the Ricci tensor, a that only terms at linear order in Lorentz violation are nonzero u acts merely to scale the post-newtonian metric kept. derived below. Moreover, since the vacuum value u is a Equation (27) is the desired end product of the lin- scalar under particle Lorentz transformations and is also earization process. It determines the leading corrections constant in asymptotically inertial coordinates, it plays to general relativity arising from Lorentz violation in a no direct role in considerations of Lorentz violation. For broad class of theories. This includes any modified the- convenience and simplicity, we therefore set u = 0 in ory of gravity that has an action with leading-order con- what follows. However, no assumptions are made about tributions expressible in the form (5) and satisfying as- the sizes of the coefficients for Lorentz violation, other sumptions (i)-(v). Note that the fields u, sµν , tκλµν can than assuming they are sufficiently small to validate the be composite, as occurs in the bumblebee examples dis- perturbation techniques adopted in Sec. II B. In terms cussed in Sec. IV. Understanding the implications of Eq. of the post-newtonian bookkeeping, we treat the coeffi- (27) for the post-newtonian metric and for gravitational cients for Lorentz violation as O(0). This ensures that we experiments is the focus of the remainder of this work. keep all possible Lorentz-violating corrections implied by the linearized field equations (23) at each post-newtonian order considered. III. POST-NEWTONIAN EXPANSION The choice of observer frame of reference affects the coefficients for Lorentz violation and must therefore be specified in discussions of physical effects. For immediate This section performs a post-newtonian analysis of the purposes, it suffices to assume that the reference frame linearized effective Einstein equations (27) for the pure- chosen for the analysis is approximately asymptotically gravity sector of the minimal SME. We first present the inertial on the time scales relevant for any experiments. post-newtonian metric that solves the equations. Next, In practice, this implies adopting a reference frame that the equations of motion for a perfect fluid in this met- is comoving with respect to the dynamical system un- ric are obtained. Applying them to a system of massive der consideration. The issue of specifying the observer self-gravitating bodies yields the leading-order accelera- frame of reference is revisited in more detail as needed in tion and the lagrangian in the point-mass limit. Finally, subsequent sections. a comparison of the post-newtonian metric for the SME As usual, the development of the post-newtonian se- with some other known post-newtonian metrics is pro- ries for the metric involves the introduction of certain vided. potentials for the perfect fluid [30]. For the pure-gravity sector of the minimal SME taken at O(3), we find that the following potentials are required: A. Metric ρ(~x′,t) U = G d3x′ , R Following standard techniques [32], we expand the lin- Z ρ(~x′,t)Rj Rk earized effective Einstein equations (26) and leading- U jk = G d3x′ , order corrections (27) in a post-newtonian series. The R3 Z relevant expansion parameter is the typical small velocity ρ(~x′,t)vj (~x′,t) V j = G d3x′ , v of a body within the dynamical system, which is taken R to be O(1). The dominant contribution to the metric Z ρ(~x′,t)vk(~x′,t)RkRj fluctuation hµν is the newtonian gravitational potential W j = G d3x′ , 2 R3 U. It is second order, O(2) = v ≈ GM/r, where M is Z the typical body mass and r is the typical system dis- ρ(~x′,t)vj (~x′,t)RkRl Xjkl = G d3x′ , tance. The source of the gravitational field is taken to R3 Z be a perfect fluid, and its energy-momentum tensor is ρ(~x′,t)vm(~x′,t)RmRj RkRl also expanded in a post-newtonian series. The dominant Y jkl = G d3x′ , (28) R5 term is the mass density ρ. The expansion for hµν begins Z at O(2) because the leading-order gravitational equation where Rj = xj − x′j and R = |~x − ~x′| with the euclidean is the Poisson equation ∇~ 2U = −4πGρ. norm. The focus of the present work is the dominant Lorentz- The potential U is the usual newtonian gravitational violating effects. We therefore restrict attention to the potential. In typical gauges, the potentials V j , W j oc- newtonian O(2) and sub-newtonian O(3) corrections in- cur in the post-newtonian expansion of general relativ- duced by Lorentz violation. For certain experimental ity, where they control various gravitomagnetic effects. jk jkl jkl applications, the O(4) metric fluctuation h00 would in In these gauges, the potentials U , X , and Y lie principle be of interest, but deriving it requires solving beyond general relativity. To our knowledge the latter the sub-linearized theory of Sec. II A and lies beyond the two, Xjkl and Y jkl, have not previously been considered scope of the present work. in the literature. However, they are needed to construct 8 the contributions to the O(3) metric arising from general Eqs. (26) and (27) can be written at this order as leading-order Lorentz violation. 00 jk jk 0j j A ‘superpotential’ χ defined by g00 = −1+2U +3s U + s U − 4s V + O(4), (35) 3 ′ ′ 0j 0k jk 7 1 00 j 3 jk k χ = −G d x ρ(~x ,t)R (29) g0j = −s U − s U − 2 (1 + 28 s )V + 4 s V Z 1 15 00 j 5 jk k − 2 (1 + 4 s )W + 4 s W is also used in the literature [30]. It obeys the identities 9 kl klj 15 kl jkl 3 kl klj + 4 s X − 8 s X − 8 s Y , (36) jk 00 jk jk jk gjk = δ + (2 − s )δ U ∂j ∂kχ = U − δ U, lm jk jl mk kl jm 00 jl km lm j j +(s δ − s δ − s δ +2s δ δ )U . ∂0∂j χ = V − W . (30) (37) In the present context, it is convenient to introduce two additional superpotentials. We define Although they are unnecessary for a consistent O(3) expansion, the O(3) terms for g0j and the O(2) terms for gjk are displayed explicitly because they are useful for part j 3 ′ ′ j ′ χ = G d x ρ(~x ,t)v (~x ,t)R (31) of the analysis to follow. The O(4) symbol in the ex- Z pression for g00 serves as a reminder of the terms missing and for a complete expansion at O(4). Note that the metric potentials for general relativity in the chosen gauge 3 ′ ′ j ′ j are recovered upon setting all coefficients for Lorentz vi- ψ = G d x ρ(~x ,t)v (~x ,t)R R. (32) olation to zero, as expected. Note also that a nonzero Z u¯ would merely act to scale the potentials in the above These obey several useful identities including, for exam- equations by an unobservable factor (1 +u ¯). ple, The properties of this metric under spacetime transformations are induced from those of the SME action. l jk l ljk ∂j ∂kχ = δ V − X , As described in Sec. II A, two different kinds of spacetime transformation can be considered: observer trans- ∂ ∂ ∂ ψ = 3δ(jkV l) − 3δ(jkW l) − 3X(jkl) +3Y jkl. j k l formations and particle transformations. The SME is (33) invariant under observer transformations, while the coefficients for Lorentz violation determine both the par- In the latter equation, the parentheses denote total sym- ticle local Lorentz violation and the particle diffeomor- metrization with a factor of 1/3. phism violation in the theory. Since the SME includes all In presenting the post-newtonian metric, it is neces- observer-invariant sources of Lorentz violation, the O(3) sary to fix the gauge. In our context, it turns out that post-newtonian metric of the minimal SME given in Eqs. calculations can be substantially simplified by imposing (35)-(37) must have the same observer symmetries as the the following gauge conditions: O(3) post-newtonian metric of general relativity. One relevant set of transformations under which the 1 metric of general relativity is covariant are the post- ∂j g0j = ∂0gjj , 2 galilean transformations [33]. These generalize the 1 galilean transformations under which newtonian gravity ∂ g = ∂ (g − g ). (34) j jk 2 k jj 00 is covariant. They correspond to Lorentz transformations in the asymptotically Minkowski regime. A post-galilean It is understood that these conditions apply to O(3). Al- transformation can be regarded as the post-newtonian though the conditions (34) appear superficially similar to product of a global Lorentz transformation and a possi- those of the standard harmonic gauge [32], the reader is ble gauge transformation applied to preserve the chosen warned that in fact they differ at O(3). post-newtonian gauge. Explicit calculation verifies that With these considerations in place, direct calculation the O(3) metric of the minimal SME is unchanged by an now yields the post-newtonian metric at O(3) in the cho- observer global Lorentz transformation, up to an over- sen gauge. The procedure is to break the effective lin- all gauge transformation and possible effects from O(4). earized equations (27) into temporal and spatial compo- This suggests that the post-newtonian metric of the min- nents, and then to use the usual Einstein equations to imal SME indeed takes the same form (35)-(37) in all ob- E eliminate the pieces hµν of the metric on the right-hand server frames related by post-galilean transformations, as side in favor of the potentials (28) in the chosen gauge, expected. keeping appropriate track of the post-newtonian orders. In contrast, covariance of the minimal-SME metric ˜ The resulting second-order differential equations for hµν (35)-(37) fails under a particle post-galilean transforma- can be solved in terms of the potentials (27). tion, despite the freedom to perform gauge transforma- After some work, we find that the metric satisfying tions. This behavior can be traced to the invariance (as 9 opposed to covariance) of the coefficients for Lorentz vio- For the temporal component ν = 0, we find lation appearing in Eqs. (35)-(37) under a particle post- ∗ 1 j j galilean transformation, which is a standard feature of 0 = ∂0[ρ (1+Π+ 2 v v + U)] ∗ 1 k k j j vacuum values arising from spontaneous Lorentz viola- +∂j[ρ (1+Π+ 2 v v + U)v + pv ] tion. The metric of the minimal SME therefore breaks −ρ∗∂ U − 2ρ∗vj ∂ U the particle post-galilean symmetry of ordinary general 0 j 1 ∗ 00 jk jk relativity. + 2 ∂0[ρ (3s U + s U )] 1 ∗ 00 kl kl j + 2 ∂j [ρ (3s U + s U )v ] 1 ∗ 00 jk jk − 2 ρ ∂0(3s U + s U ) B. Dynamics ∗ j 00 kl kl −ρ v ∂j (3s U + s U ). (42) Many analyses of experimental tests involve the equa- The first four terms of this equation reproduce the usual tions of motion of the gravitating sources. In particu- generalized Euler equations as expressed in the post- lar, the many-body equations of motion for a system of newtonian approximation. The terms involving sµν rep- massive bodies in the presence of Lorentz violation are resent the leading corrections due to Lorentz violation. necessary for the tests we consider in this work. Here, The key effects on the perfect fluid due to Lorentz vio- we outline the description of massive bodies as perfect lation arise in the spatial components ν = j of Eq. (39). fluids and obtain the equations of motion and action for These three equations can be rewritten in terms of ρ∗ the many-body dynamics in the presence of Lorentz vio- and then simplified by using the continuity equation (40). lation. The result is an expression describing the acceleration of the fluid elements. It can be written in the form dvj ρ∗ + Aj + Aj =0, (43) 1. The post-newtonian perfect fluid dt E LV j where AE is the usual set of terms arising in general Consider first the description of each massive body. j relativity and ALV contains terms that violate Lorentz Adopting standard assumptions [30], we suppose the ba- symmetry. sic properties of each body are adequately described by For completeness, we keep terms in Aj to O(4) and the usual energy-momentum tensor (T )µν for a perfect E M terms in Aj to O(3) in the post-newtonian expansion. fluid. Given the fluid element four-velocity uµ, the mass LV This choice preserves the dominant corrections to the density ρ, the internal energy Π, and the pressure p, the perfect fluid equations of motion due to Lorentz viola- energy-momentum tensor is j tion. Explicitly, we find that AE is given by µν µ ν µν j (TM ) = (ρ + ρΠ+ p)u u + pg . (38) ∗ AE = −ρ ∂j U + ∂j [p(1+3U)] −(∂ p)(Π + 1 vkvk + p/ρ∗) The four equations of motion for the perfect fluid are j 2 j ∗ −v (ρ ∂0U − ∂0p) µν ∗ j j Dµ(TM ) =0. (39) +4ρ d(Uv − V )/dt 1 ∗ j j + 2 ρ ∂0(V − W ) Note that the construction of the linearized effective Ein- ∗ k k ∗ stein equations (23) in Sec. II B ensures that this equation +4ρ v ∂j V − ρ ∂j Φ ∗ k k ∗ is satisfied in our context. −ρ (∂j U)(v v +3p/ρ ), (44) To proceed, we separate the temporal and spatial com- where Φ is a perfect-fluid potential at O(4) given by ponents of Eq. (39) and expand the results in a post- newtonian series using the metric of the minimal SME ρ(~x′,t) Φ = G d3x′ [2vkvk(~x′,t)+2U(~x′,t) given in Eqs. (35)-(37), together with the associated R Christoffel symbols. As usual, it is convenient to de- Z fine a special fluid density ρ∗ that satisfies the continuity + Π(~x′,t)+3p(~x′,t)/ρ(~x′,t)]. (45) equation Similarly, the Lorentz-violating piece Aj is found to be ∗ ∗ j LV ∂0ρ + ∂j (ρ v )=0. (40) j 1 ∗ 00 kl kl ALV = − 2 ρ ∂j (3s U + s U ) Explicitly, we have ∗ j0 0k jk −ρ ∂0(s U + s U ) ∗ k j0 0l jl ∗ 1/2 0 −ρ v ∂k(s U + s U ) ρ = ρ(−g) u , (41) ∗ k 0k 0l kl +ρ v ∂j (s U + s U ) ∗ 0k k where g is the determinant of the post-newtonian metric. +2ρ ∂j s V . (46) 10

We note in passing that Eqs. (40) and (43) can be used taking a point-particle limit. In this limit, some calcu- to demonstrate that the standard formula lation reveals that the coordinate acceleration of the ath body due to N other bodies is given by dΠ = −p∂ vk (47) dt k d2xj Gm rj a = −(1 + 3 s00) b ab dt2 2 r3 remains valid in the presence of Lorentz violation. b=6 a ab X k jk Gmbrab +s 3 rab 2. Many-body dynamics Xb=6 a j k l 3 kl Gmbrabrabrab − 2 s 5 As a specific relevant application of these results, we rab seek the equations of motion for a system of massive bod- Xb=6 a Gm (vkrk − 2vkrk ) ies described as a perfect fluid. Adopting the general +s0j b b ab a ab techniques of Ref. [30], the perfect fluid can be separated r3 b=6 a ab into distinct self-gravitating clumps of matter, and the X k j j k k j equations (43) can then be appropriately integrated to Gmb(2va r − v r + v r ) +s0k ab b ab b ab yield the coordinate acceleration of each body. r3 b=6 a ab The explicit calculation of the acceleration requires the X Gm vl rl rj rk introduction of various kinematical quantities for each +3s0k b b ab ab ab body. For our purposes, it suffices to define the conserved r5 b=6 a ab mass of the ath body as X + .... (52)

3 ′ ∗ ′ ma = d x ρ (t, ~x ), (48) The ellipses represent additional acceleration terms aris- Z ing at O(4) in the post-newtonian expansion. These where ρ∗ is specified by Eq. (41). Applying the continuity higher-order terms include both corrections from general ∗ j relativity arising via Eq. (44) and O(4) effects controlled properties of ρ , we can introduce the position xa, the velocity vj , and the acceleration aj of the ath body as by SME coefficients for Lorentz violation. a a Note that these point-mass equations can be generalized to include tidal forces and multipole moments using j −1 3 ∗ j xa = ma d xρ (t, ~x)x , (49) standard techniques. As an additional check, we verified Z the above result is also obtained by assuming that each dxj dxj vj = a = m−1 d3xρ∗(t, ~x) , (50) body travels along a geodesic determined by the pres- a dt a dt Z ence of all other bodies. The geodesic equation for the d2xj d2xj ath body then yields Eq. (52). aj = a = m−1 d3xρ∗(t, ~x) . (51) a dt2 a dt2 The coordinate acceleration (52) can also be derived as Z the equation of motion from an effective nonrelativistic The task is then to insert Eq. (43) into Eq. (51) and action principle for the N bodies, integrate. To perform the integration, the metric potentials are S = dtL. (53) split into separate contributions from each body using Z the definitions (49) and (50). We also expand each po- A calculation reveals that the associated lagrangian L tential in a multipole series. Some of the resulting terms can be written explicitly as involve tidal contributions from the finite size of each body, while others involve integrals over each individual Gmamb j L = 1 m ~v2 + 1 (1 + 3 s00 + 1 sjkrˆ rˆk ) body. In conventional general relativity, the latter van- 2 a a 2 r 2 2 ab ab a ab ish if equilibrium conditions are imposed for each body. X Xab Gm m In the context of general metric expansions it is known 1 a b 0j j 0j j k k − 2 (3s va + s râbva râb) that some parameters may introduce self accelerations rab Xab [30], which would represent a violation of the gravita- + ..., (54) tional weak equivalence principle. However, for the gravitational sector of the minimal SME, we find that no where it is understood that the summations omit b = a. j self contributions arise to the acceleration aa at post- Note that the O(2) potential has a form similar to that newtonian order O(3) once standard equilibrium condi- arising in Lorentz-violating electrostatics [34, 35]. tions for each body are imposed. Using standard techniques, one can show that there For many applications in subsequent sections, it suf- are conserved energy and momenta for this system of fices to disregard both the tidal forces across each body point masses. This follows from the temporal and spa- and any higher multipole moments. This corresponds to tial translational invariance of the lagrangian L, which 11

νλ in turn is a consequence of the choice ∂µs = 0 in Eq. C. Connection to other post-newtonian metrics (11). As an illustrative example, consider the simple case of This subsection examines the relationship between the two point masses m1 and m2 in the limit where only O(3) metric of the pure-gravity sector of the minimal SME, terms are considered. The conserved hamiltonian can be obtained under assumptions (i)-(v) of Sec. II B, and some written existing post-newtonian metrics. We focus here on two ~ 2 ~ 2 popular cases, the parametrized post-newtonian (PPN) P1 P2 Gm1m2 1 jk j k H = + − (1 + 2 s rˆ rˆ ) formalism [29] and the anisotropic-universe model [36]. 2m1 2m2 r The philosophies of the SME and these two cases are 0j j j 3Gs (m2P + m1P ) different. The SME begins with an observer-invariant + 1 2 2r action constructed to incorporate all known physics and Gs0j rj (m P k + m P k)rk fields through the Standard Model and general relativity. + 2 1 1 2 . (55) 2r3 It categorizes particle local Lorentz violations according to operator structure. Within this approach, the dom- In this expression, ~r = ~r1 − ~r2 is the relative separation inant Lorentz-violating effects in the pure-gravity sec- of the two masses, and the canonical momenta are given tor are controlled by 20 independent components in the ~ κλµν by Pn = ∂L/∂~rn for each mass. It turns out that the traceless coefficients u, sµν , t . No assumptions about total conserved momentum is the sum of the individual post-newtonian physics are required a priori. In con- canonical momenta. Explicitly, we find trast, the PPN formalism is based directly on the expan-

0j 0k k j sion of the metric in a post-newtonian series. It assumes j j 3Gm1m2s Gm1m2s r r isotropy in a special frame, and the primary terms are P = PN − − , (56) r r3 chosen based on simplicity of the source potentials. The where the first term is the usual newtonian center-of-mass corresponding effects involve 10 PPN parameters. The anisotropic-universe model is again different: it develops momentum P~N = m1~v1 + m2~v2. Defining M = m1 + m2, an effective N-body classical point-particle lagrangian, we can adopt V~ = P~N /M as the net velocity of the bound system. with leading-order effects controlled by 11 parameters. Further insight can be gained by considering the time These differences in philosophy and methodology mean average of the total conserved momentum for the case that a complete match cannot be expected between the where the two masses are executing periodic bound mo- corresponding post-newtonian metrics. However, partial tion. If we keep only results at leading order in coeffi- matches exist, as discussed below. cients for Lorentz violation, the trajectory ~r for an elliptical orbit can be used in Eq. (56). Averaging over one orbit then gives the mean condition 1. PPN formalism ~ ~ < P >= M < V > +~a, (57) Since the O(4) terms in g00 in the minimal-SME metric (35) remain unknown, while many of the 10 PPN param- where eters appear only at O(4) in g00, a detailed comparison Gm m of the pure-gravity sector of the minimal SME and the aj = − 1 2 [3e2s0j +ε(e)s P j +(1−e2)1/2ε(e)s Qj )] ae2 P Q PPN formalism is infeasible at present. However, the ba- (58) sic relationship can be extracted via careful matching of is a constant vector. In the latter equation, a is the the known metrics in different frames, as we demonstrate semimajor axis of the ellipse, ε(e) is a function of its ec- next. centricity e, P~ points towards the periastron, while Q~ is We first consider the situation in an observer frame a perpendicular vector determining the plane of the or- that is comoving with the expansion of the universe, often bit. The explicit forms of ε(e), P~, and Q~ are irrelevant called the universe rest frame. In this frame, the PPN for present purposes, but the interested reader can find metric has a special form. Adopting the gauge (34) and them in Eqs. (166) and (169) of Sec. VE. Here, we re- keeping terms at O(3), the PPN metric is found to be mark that Eq. (57) has a parallel in the fermion sector of the minimal SME. For a single fermion with only a g00 = −1+2U + O(4), µ 1 1 j 1 1 j nonzero coefficient for Lorentz violation of the a type, g0j = − 2 (6γ +1+ 2 α1)V − 2 (3 − 2γ + 2 α1)W , the nonrelativistic limit of the motion yields a momentum jk jk gjk = δ [1 + (3γ − 1)U] − (γ − 1)U . (59) having the same form as that in Eq. (57) (see Eq. (32) of the first of Refs. [6]). Note also that the conservation of Of the 10 parameters in the PPN formalism, only two P~ and the constancy of ~a imply that the measured mean appear at O(3) in this gauge and this frame. The reader net velocity < V~ > of the two-body system is constant, a is cautioned that the commonly used form of the PPN result consistent with the gravitational weak equivalence metric in the standard PPN gauge differs from that in principle. Eq. (59) by virtue of the gauge choice (34). 12

The PPN formalism assumes that physics is isotropic by requirements of simplicity at the level of the post- in the universe rest frame. In contrast, the SME allows newtonian metric (PPN). for anisotropies in this frame. To compare the corre- Further insight about the relationship between the sponding post-newtonian metrics in this frame therefore gravitational sector of the minimal SME and the PPN requires restricting the SME to an isotropic limit. In fact, formalism can be gained by transforming to a Sun- the combination of isotropy and assumption (ii) of Sec. centered frame comoving with the solar system. This II B, which restricts attention to effects that are linear in frame is of direct relevance for experimental tests. More the SME coefficients, imposes a severe restriction: of the important in the present context, however, is that the 19 independent SME coefficients contained in sµν and transformation between the universe rest frame and tκλµν , only one combination is observer invariant under the Sun-centered frame mixes terms at different post- spatial rotations and hence isotropic. Within this as- newtonian order, which yields additional matching infor- sumption, to have any hope of matching to the PPN for- mation. malism, the SME coefficients must therefore be restricted Suppose the Sun-centered frame is moving with a ve- to the isotropic limiting form locity ~w of magnitude | ~w| ≪ 1 relative to the universe rest frame. Conversion of a metric from one frame to the 00 s 0 0 0 other can be accomplished with an observer post-galilean 1 00 µν 0 3 s 0 0 transformation. A complete transformation would re- s =  1 00  , 0 0 3 s 0 quire using the transformation law of the metric, express-  0 0 0 1 s00   3  ing the potentials in the new coordinates, and transform- tκλµν = 0.  (60) ing also the SME coefficients for Lorentz violation or the PPN parameters, all to the appropriate post-newtonian As discussed in Sec. III A, the coefficient u is unobserv- order. For the minimal-SME metric (61) in the isotropic 00 able in the present context and can be set to zero. In the limit, this procedure would include transforming s , in- special limit (60), the minimal-SME metric in Eqs. (35), cluding also the change in the new gravitational coupling 00 (36), and (37) reduces to Gnew via its dependence on s . However, for some purposes it is convenient to perform only the first two of 10 00 g00 = −1+(2+ 3 s )U + O(4), these steps. Indeed, in the context of the PPN formalism, this two-step procedure represents the standard choice g = − 1 (7 + s00)V j − 1 (1 + 5 s00)W j , 0j 2 2 3 adopted in the literature [30]. In effect, this means the jk 2 00 4 00 jk gjk = δ [1 + (2 − 3 s )U]+ 3 s U . (61) PPN parameters appearing in the PPN metric for the Sun-centered frame remain expressed in the universe rest With these restrictions and in the universe rest frame, frame. For comparative purposes, we therefore adopt in a match between the minimal-SME and the PPN met- this subsection a similar procedure for the isotropic limit rics becomes possible. The gravitational constant in the of the minimal-SME metric (61). restricted minimal-SME metric (61) must be redefined as Explicitly, we find that the PPN metric in the Sun- 5 00 centered frame and in the gauge (34) is given by Gnew = G(1 + 3 s ). (62) 2 The match is then g00 = −1+2U − (α1 − α2 − α3)w U j k jk j j 16 00 −α2w w U + (2α3 − α1)w V α1 = − s , 3 +(2α − 1 α )wj (V j − W j )+ O(4), 4 00 2 2 1 γ = 1 − 3 s . (63) 1 j 1 k jk g0j = − 4 α1w U − 4 α1w U From this match, we can conclude that the pure-gravity 1 1 j 1 1 j − (6γ +1+ α1)V − (3 − 2γ + α1)W , sector of the minimal SME describes many effects that 2 2 2 2 jk jk lie outside the PPN formalism, since 18 SME coefficients gjk = δ [1 + (3γ − 1)U] − (γ − 1)U . (64) cannot be matched in this frame. Moreover, the con- verse is also true. In principle, a similar match in the This expression depends on four of the 10 PPN parame- universe rest frame could be made at O(4), where the ters. It includes all terms at O(3) and also all terms in minimal-SME metric in the isotropic limit can still be g00 dependent on ~w. Since ~w is O(1), some of the latter written in terms of just one coefficient s00, while the are at O(4), including those involving the two parameters PPN formalism requires 10 parameters. It follows that α2 and α3 that are absent from the PPN metric (59) in the PPN formalism in turn describes many effects that the universe rest frame. The dependence on ~w is a key lie outside the pure-gravity sector of the minimal SME, feature that permits a partial comparison of these O(4) since 9 PPN parameters cannot be matched in this frame. terms with the minimal SME and hence provides match- The mismatch between the two is a consequence of the ing information that is unavailable in the universe rest differing philosophies: effects that dominate at the level frame. of a pure-gravity realistic action (gravitational sector of For the minimal SME, the isotropic limit of the pure- the minimal SME) evidently differ from those selected gravity sector in the Sun-centered frame and the gauge 13

(34) is found to be

10 00 2 00 g00 = −1+(2+ 3 s )U +4w s U 4 j k 00 jk 16 00 j j + 3 w w s U + 3 s w V + O(4), 4 00 j 4 00 k jk g0j = 3 s w U + 3 s w U 1 00 j 1 5 00 j − 2 (7 + s )V − 2 (1 + 3 s )W , jk 2 00 4 00 jk gjk = δ [1 + (2 − 3 s )U]+ 3 s U , (65) where the coefficient s00 remains defined in the universe rest frame. Since the general results (35)-(37) for the minimal-SME metric take the same form in any post- galilean observer frame, Eq. (65) can be derived by transforming the coefficient s00 from the universe rest frame to the corresponding coefficients sµν in the Sun-centered frame and then substituting the results into Eqs. (35)- (37). Like the PPN metric (64), the expression (65) contains all terms at O(3) along with some explicit O(4) terms that depend on ~w. By rescaling the gravitational constant as in Eq. (62) and comparing the two post-newtonian metrics (64) and (65), we recover the previous matching results (63) and obtain two additional relationships: FIG. 1: Schematic relationship between the pure-gravity sec- 16 00 tor of the minimal SME and the PPN formalism. α1 = − 3 s , 4 00 α2 = − 3 s , α3 = 0, These equations can be used to relate the results in this 4 00 subsection to ones expressed in terms of the coefficients γ = 1 − 3 s . (66) sµν in the Sun-centered frame. The vanishing of α3 is unsurprising. This parameter is The relationship between the pure-gravity sector of the always zero in semiconservative theories [30, 37], while minimal SME and the PPN formalism can be represented assumption (i) of Sec. II B and Eq. (11) imply a constant as the Venn diagram in Fig. 1. The overlap region, which asymptotic value for s00, which in turn ensures global corresponds to the isotropic limit, is a one-parameter re- energy-momentum conservation. A more interesting is- gion in the universe rest frame. This overlap region en- sue is the generality of the condition compasses only a small portion of the effects governed by the pure-gravity sector of the minimal SME and by the α1 =4α2 (67) PPN formalism. Much experimental work has been done to explore the PPN parameters. However, the figure illus- implied by Eq. (66). It turns out that this condition trates that a large portion of coefficient space associated depends on assumption (iv) of Sec. II B, which imposes with dominant effects in a realistic action (SME) remains the vanishing of the independent energy-momentum con- open for experimental exploration. We initiate the theo- tribution Σµν . The relationship between the conditions µν retical investigation of the various possible experimental Σ 6= 0 and α1 6= 4α2 is considered further in Sec. IV searches for these effects in Sec. V. below. We note in passing that the above matching consid- We emphasize that all quantities in Eq. (66) are de- erations are derived for the pure-gravity sector of the fined in the universe rest frame. For experimental tests of minimal SME. However, even in the isotropic limit, the the SME, however, it is conventional to report measure- matter sector of the minimal SME contains numerous ad- ments of the coefficients for Lorentz violation in the Sun- ditional coefficients for Lorentz violation. Attempting a centered frame. The conversion between the two takes a match between the isotropic limit of the minimal SME simple form for the special isotropic limit involved here. µν with matter and the PPN formalism would also be of It can be shown that the minimal-SME coefficients sS in 00 interest but lies beyond our present scope. the Sun-centered frame and the isotropic coefficient sU in the universe rest frame are related by

00 4 j j 00 2. Anisotropic-universe model sS = (1+ 3 w w )sU , s0j = − 4 wj s00, S 3 U An approach adopting a somewhat different philosophy jk 1 jk j k 00 sS = 3 (δ +4w w )sU . (68) is the anisotropic-universe model [36]. This model is con- 14 servative and is based on the formulation of an effective post-galilean transformations. To our knowledge the ob- classical point-particle lagrangian. Possible anisotropies server transformation properties of the parameters in the in a preferred frame are parametrized via a set of spatial anisotropic-universe model remain unexplored in the lit- two-tensors and spatial vectors, and the post-newtonian erature, and adding this requirement may remove these metric is constructed. The model includes velocity- two extra degrees of freedom. At O(4), however, the dependent terms and three-body interactions up to O(4). full anisotropic-universe model contains additional two- The connection to the PPN formalism is discussed in Ref. tensors for which there are no matching additional coef- [36]. ficients in the pure-gravity sector of the minimal SME. A detailed match between the anisotropic-universe It therefore appears likely that the correspondence be- model and the classical point-particle limit of the pure- tween the two approaches is again one of partial overlap. gravity sector of the minimal SME is impractical at It is conceivable that effects from the SME matter sector present, since the O(4) terms for the latter are unde- or from nonminimal SME terms could produce a more termined. However, some suggestive features of the rela- complete correspondence. tionship can be obtained. For this purpose, it suffices to consider the restriction of the anisotropic-universe model to two-body terms at O(3). In this limit, the lagrangian IV. ILLUSTRATION: BUMBLEBEE MODELS is Gm m The analysis presented in Secs. II and III applies to L = 1 m ~v2 + 1 a b (1+2Ωjkrˆj rˆk ) AU 2 a a 2 r ab ab all theories with terms for Lorentz violation that can a ab X Xab be matched to the general form (5). An exploration of Gmamb j j j j k k its implications for experiments is undertaken in Sec. V. + (Φ va +Υ râbva · râb). (69) rab ab Here, we first take a short detour to provide a practical X illustration of the general methodology and to illuminate The anisotropic properties of the model at this order are the role of the five assumptions adopted in the lineariza- controlled by 11 parameters, collected into one symmetric tion procedure of Sec. II B. For this purpose, we consider spatial two-tensor Ωjk and two three-vectors Φj ,Υj . a specific class of theories, the bumblebee models. Note, To gain insight into the relationship, we can compare however, that the material in this section is inessential Eq. (69) with the O(3) point-particle lagrangian (54) ob- for the subsequent analysis of experiments, which is in- tained in Sec. III B. Consider the match in the preferred dependent of specific models. The reader can therefore frame of the anisotropic-universe model. Comparison of proceed directly to Sec. V at this stage if desired. Eqs. (69) and (54) gives the correspondence Bumblebee models are theories involving a vector field Bµ that have spontaneous Lorentz violation induced by Ωjk = 1 sjk − 1 s00δjk, 4 12 a potential V (Bµ). The action relevant for our present j 3 0j Φ = − 2 s , purposes can be written as j 1 0j Υ = − 2 s . (70) 4 1 µ ν SB = d x (eR + ξeB B Rµν ) jk 2κ The five parameters Ω are determined by the SME co- Z efficients sjk and s00, while the two three-vectors Φj ,Υj h − 1 αeBµν B + 1 βe(D B )(DµBν ) are determined by s0j . 4 µν 2 µ ν The SME is a complete effective field theory and so µ contains effects beyond any point-particle description, − eV (B )+ LM , (71) including that of the anisotropic-universe model. Even i in the point-particle limit, it is plausible that the pure- where α and β are real. In Minkowski spacetime and with gravity sector of the minimal SME describes effects out- vanishing potential V , a nonzero value of β introduces side the anisotropic-universe model because the latter is a Stückelberg ghost. However, the ghost term with β based on two-tensors and vectors while the SME contains potentially nonzero is kept in the action here to illustrate κλµν the four-tensor t . Nonetheless, in this limit the con- some features of the assumptions made in Sec. II B. In verse is also plausible: the anisotropic-universe model Eq. (71), the potential V is taken to have the functional is likely to contain effects outside the pure-gravity sec- form tor of the minimal SME. The point is that the match jk j j µ µ 2 (70) implies the 11 parameters Ω , Φ ,Υ are fixed V (B )= V (B Bµ ± b ), (72) by only 9 independent SME coefficients sµν . This suggests that two extra degrees of freedom appear in the where b2 is a real number. This potential induces a µ µ µ 2 anisotropic-universe model already at O(3). Some cau- nonzero vacuum value B = b obeying b bµ = ∓b . tion with this interpretation may be advisable, as the The theory (71) is understood to be taken in the limit of condition Φj = 3Υj implied by Eq. (70) in the context Riemann spacetime, where the field-strength tensor can of the minimal SME arises from the requirement that be written as Bµν = ∂µBν − ∂ν Bµ. Bumblebee mod- the form of the lagrangian be observer invariant under els involving nonzero torsion in the more general context 15 of Riemann-Cartan spacetime are investigated in Refs. underlying bumblebee field and the metric, given by [4, 28]. u = 1 ξBαBβg , Theories coupling gravity to a vector field with a vac- 4 αβ µν µ ν 1 α β µν uum value have a substantial history in the literature. s = ξB B − 4 ξB B gαβg . (73) A vacuum value as a gauge choice for the photon was discussed by Nambu [38]. Models of the form (71) with- This is a nonlinear relationship between the basic fields out a potential term but with a vacuum value for Bµ of a given model and the SME, a possibility noted in Sec. and nonzero values of α and β were considered by Will II A. and Nordtvedt [39] and by Hellings and Nordtvedt [40]. Following the notation of Sec. II B, the bumblebee field The idea of using a potential V to break Lorentz symme- Bµ can be expanded around its vacuum value bµ as try spontaneously and hence to enforce a nonzero vacuum value for Bµ was introduced by Kosteleckýand Bµ = bµ + B˜µ. (74) Samuel [41], who studied both the smooth ‘quadratic’ µ 2 2 case V = −λe(B Bµ ±b ) /2 and the limiting Lagrange- In an asympotically cartesian coordinate system, the vac- µ 2 multiplier case V = −λe(B Bµ ± b ) for β = 0 and uum value is taken to obey in N dimensions. The spontaneous Lorentz breaking is µ accompanied by qualitatively new features, including a ∂ν b =0. (75) Nambu-Goldstone sector of massless modes [28], the nec- This ensures that assumption (i) of Sec. II B holds. essary breaking of U(1) gauge invariance [4], and impli- µν κλµν cations for the behavior of the matter sector [42], the The expansion of the SME fields u, s , t about photon [28], and the graviton [4, 43]. More general po- their vacuum values is given in Eq. (10). At leading order, tentials have also been investigated, and some special the ξ-dependent term in the bumblebee action (71) is cases with hypergeometric V turn out to be renormal- reproduced by the Lorentz-violating piece (5) of the SME izable in Minkowski spacetime [44]. The situation with a action by making the identifications Lagrange-multiplier potential for a timelike b and both µ u = 1 ξbαb , α and β nonzero has been explored by Jacobson and col- 4 α 1 ˜α 1 α β laborators [45]. Related analyses have been performed in u˜ = 2 ξ(bαB + 2 b b hαβ), Refs. [46, 47]. µν µ ν 1 µν α s = ξ(b b − 4 η b bα), In what follows, we consider the linearized and post- s˜µν = ξ(bµB˜ν + bν B˜µ + 1 hµν bαb newtonian limits of the theory (71) for some special cases. 4 α 1 µν ˜α 1 µν α β The action for the theory contains a pure-gravity Einstein − 2 η bαB − 4 η b b hαβ), piece, a gravity-bumblebee coupling term controlled by κλµν t = 0, ξ, several terms determining the bumblebee dynamics, κλµν and a matter Lagrange density. Comparison of Eq. (71) t˜ = 0. (76) with the action (5) suggests a correspondence between the gravity-bumblebee coupling and the fields u and sµν , We see that assumption (ii) of Sec. II B holds if the com- µ ν with the latter being related to the traceless part of the bination ξb b is small. product BµBν . It is therefore reasonable to expect the The next step is to examine the field equations. The linearization analysis of the previous sections can be ap- nonlinear nature of the match (73) and its dependence plied, at least for actions with appropriate bumblebee on the metric imply that the direct derivation of the lin- dynamics. earized effective Einstein equations from the bumblebee theory (71) differs in detail from the derivation of the same equations in terms of sµν and u presented in Sec. II B. For illustrative purposes and to confirm the results obtained via the general linearization process, we pursue A. Cases with β = 0 the direct route here. Linearizing in both the metric fluctuation hµν and the bumblebee fluctuation B˜µ but keeping all powers of the Consider first the situation without a ghost term, vacuum value bµ, the linearized Einstein equations can β = 0. Suppose for definiteness that α = 1. The modi- be extracted from the results of Ref. [4]. Similarly, the fied Einstein and bumblebee field equations for this case linearized equations of motion for the bumblebee fluc- are given in Ref. [4]. To apply the general formalism de- tuations can be obtained. For simplicity, we disregard veloped in the previous sections, we first relate the bum- the possibility of direct couplings between the bumblebee blebee action (71) to the action (3) of the pure-gravity fields and the matter sector, which means that assump- sector of the minimal SME, and then we consider the tion (iii) of Sec. II B holds. The bumblebee equations of linearized version of the equations of motion. motion then take the form The match between the bumblebee and SME actions µν µ ′ ξ µ involves identifying u and s as composite fields of the ∂ Bµν =2V bν − κ b Rµν , (77) 16 where the prime denotes the derivative with respect to in the universe rest frame. Note that the condition (67) is the argument. Once the potential V is specified, Eq. valid both for the Lagrange-multiplier potential and for (77) for the bumblebee fluctuations can be inverted using the smooth potential. In fact, the β = 0 model with zero Fourier decomposition in momentum space. potential term V = 0 but a nonzero isotropic vacuum Consider, for example, the situation for a smooth po- value for Bµ also satisfies the condition (67) at leading µ 2 2 tential V = −λe(B Bµ ± b ) /2. For this case, Eq. (77) order [30, 40]. can be written as We note in passing that the limit of zero coupling ξ implies the vanishing of all the coefficients and fluctuations ˜µ (ηµν 2 − ∂µ∂ν − 4λbµbν )B = in Eq. (76). In the pure gravity-bumblebee sector, any α µ α β ξ α Lorentz-violating effects in the effective linearized the- −b ∂ (∂µhνα − ∂ν hµα)+2λbν b b hαβ − κ b Rαν . (78) ory must then ultimately be associated with the bumblebee potential V . Within the linearization assump- Converting to momentum space, the propagator for the tions we have made above, this result is compatible with B˜µ field for both timelike and spacelike bµ is found to be that obtained in Ref. [28] for the effective action of the bumblebee Nambu-Goldstone fluctuations, for which the µν µ ν ν µ µν η (b p + b p ) Einstein-Maxwell equations are recovered in the same G (p) = − 2 + 2 α p p bαp limit. Further insight can be obtained by examining α 2 ν µ the bumblebee trace-reversed energy-momentum tensor (4λb bα + p )p p − . (79) (S ) , obtained by varying the minimally coupled parts 4λp2(b pα)2 B µν α of the bumblebee action (71) with respect to the metric. In this equation, pµ is the four momentum and p2 ≡ This variation gives pµp . This propagator matches results from other anal- µ α α β α yses [44, 46]. (SB)µν = bµ∂ Bαν + bν ∂ Bαµ − ηµν b ∂ Bαβ 1 α ′ The propagator (79) can be used to solve for the har- −2(bµbν − 2 ηµν b bα)V . (81) monic bumblebee fluctuation B˜µ(p) in momentum space in terms of the metric. We find Note that the composite nature of u and sµν means that this expression cannot be readily identified with any of µ α β ˜µ µα p b b hαβ the various pieces of the energy-momentum tensor intro- B (p) = −h bα + α 2b pα duced in Sec. II B. Using the bumblebee field equations, µ µ µ α ′ ξb R ξp R ξp b bαR (SB)µν can be expressed entirely in terms of V and terms − 2 + α + 2 α 2κp 8κλb pα 2κp b pα proportional to ξ, whatever the chosen potential. For the smooth quadratic potential, insertion of the bumblebee ξb Rαµ ξpµbαbβR α αβ modes B˜µ obtained in Eq. (80) yields + 2 − 2 α . (80) κp κp b pα

′ α α β ξR This solution can be reconverted to position space and V = λ(2b B˜α + b b hαβ)= . (82) substituted into the linearized gravitational field equa- 4κ tions to generate the effective Einstein equations for h . µν This shows that all terms in (S ) are proportional to At leading order in the coupling ξ, these take the ex- B µν ξ, thereby confirming that these modes contribute no pected general form (23) with the identifications in Eq. Lorentz-violating effects to the behavior of h in the (76), except that the coefficient of Φu appears as −3 µν µν limit of vanishing ξ. Appropriate calculations for the rather than unity. To match the normalization conven- analogous modes of B˜µ in the case of the Lagrange- tions adopted in Eq. (23), a rescaling of the type dis- multiplier potential gives the same result. cussed in Sec. IIB must be performed, setting u →−u/3. The calculation shows that assumption (v) in Sec. II B holds. Moreover, the independently conserved piece Σµν B. Cases with β =6 0 of the energy-momentum tensor (Sstu)µν contains only a u trace term generating the rescaling of Φµν . This bumblebee theory therefore provides an explicit illustration of a As an example of a model that lies within the mini- model that weakly violates assumption (iv) of Sec. II B. mal SME but outside the mild assumptions of Sec. II B, A similar analysis can be performed for the Lagrange- consider the limit α = 0 and β = 1 of the theory (71). µ 2 multiplier potential V = −λe(B Bµ ±b ), for both time- The kinetic term of this theory is expected to include like and spacelike bµ. We find that the linearized limits of negative-energy contributions from the ghost term. these models are also correctly described by the general Following the path adopted in Sec. IV A, we expand formalism developed in the previous sections, including the bumblebee field about its vacuum value as in Eq. the five assumptions of Sec. II B. (74), impose the condition (75) of asymptotic cartesian It follows that the match (66) to the PPN formalism constancy, make the identifications (76), and disregard holds for all these bumblebee models in the context of the bumblebee couplings to the matter sector. It then follows isotropic limit, for which b0 is the only nonzero coefficient that the limit α = 0, β = 1 also satisfies assumptions (i), 17

(ii) and (iii) of Sec. II B. In this limit, the bumblebee In light of the results obtained above, we conjecture equations of motion become that quadratic ghost kinetic terms are associated with a nonzero value of Σµν that strongly violates assump- µ ′ ξ µ D DµBν =2V bν − κ b Rµν . (83) tion (iv) of Sec. II B and that violates the condition (67) at linear order in the isotropic limit. This conjecture For specific potentials, the linearized form of Eq. (83) is consistent with the results obtained above with the can be inverted by Fourier decomposition and the prop- Lagrange-multiplier potential and with the smooth po- agator obtained. Here, we consider for definiteness the tential, both for the case β = 0 and for the case β 6= 0. µ 2 2 potential V = −λe(B Bµ ±b ) /2. However, most of the Moreover, studies in the context of the PPN formalism of results that follow also hold for the Lagrange-multiplier various models with V = 0 but a nonzero vacuum value potential. for Bµ also suggest that ghost terms are associated with Inverting and substituting into the linearized modified violations of the condition (67). For example, this is true Einstein equations produces the effective equations for of the post-newtonian limit of the ghost model with van- hµν . For our purposes, it suffices to study the case ξ = 0. ishing V = 0 and ξ = 0 [39]. Similarly, inspection of In the harmonic gauge, we find the perturbation hµν is the general case with V = 0 but α 6= 0 and β 6= 0 [30] determined by the linearized effective equations shows that the condition (67) is satisfied at linear order in the PPN parameters when β vanishes. A proof of the R = κ[ 1 bαb 2h − 1 bαbβ∂ ∂ h µν 2 (µ ν)α 2 α (µ ν)β conjecture in the general context appears challenging to 1 α 2 1 α γ + 2 (b ∂α) hµν − 4 b b(µ∂ν)∂αh γ obtain but would be of definite interest. 1 α β − 4 ηµν b b 2hαβ + (SM )µν ]. (84) The right-hand side of this equation fails to match the V. EXPERIMENTAL APPLICATIONS generic form (23) of the linearized equations derived in Sec. IIB. In fact, the non-matter part can be regarded as The remainder of this paper investigates various grav- an effective contribution to the independently conserved itational experiments to determine signals for nonzero piece Σµν of the energy-momentum tensor (Sstu)µν . We coefficients for Lorentz violation and to estimate the at- can therefore conclude that the ghost model with α = 0, tainable sensitivities. The dominant effects of Lorentz β = 1 strongly violates assumption (iv) of Sec. II B. violation in these experiments are associated with new- The ghost nature of the bumblebee modes in this tonian O(2) and post-newtonian O(3) terms in g00 and model implies that an exploration of propagating solu- with post-newtonian O(2) terms in g0j. They are con- tions of Eq. (84) can be expected to encounter problems trolled by combinations of the 9 coefficients sµν . Effects with negative energies. Nonetheless, a post-newtonian from O(4) terms in g00 lie beyond the scope of the present expansion for the metric can be performed. For definite- analysis. However, some effects involving O(3) terms in ness, we take the effective coefficients for Lorentz viola- g0j or O(2) terms in gjk are accessible by focusing on tion κbµbν to be small, and we adopt the isotropic-limit specific measurable signals. For example, experiments in- assumption that in the universe rest frame only the coef- volving the classic time-delay effect on a photon passing ficient b0 is nonzero. A calculation then reveals that the near a massive body or the spin precession of a gyroscope post-newtonian metric at O(3) in the Sun-centered frame in curved spacetime can achieve sensitivity to terms at and in the gauge (34) is given by these orders.

0 2 2 0 2 j k jk We begin in Sec. VA with a discussion of our frame g00 = −1+2U − κ(b ) w U + κ(b ) w w U conventions and transformation properties, which are ap- −2κ(b0)2wj (V j − W j )+ O(4), plicable to many of the experimental scenarios considered 7 j 1 j below. The types of experimental constraints that might g0j = − 2 V − 2 W , jk be deduced from prior experiments are also summarized. gjk = δ (1+2U), (85) The remaining subsections treat distinct categories of experiments. Section V B examines measurements obtained where G has been appropriately rescaled. Comparison from lunar and satellite laser ranging. Section V C con- with the PPN metric (59) in the same gauge yields the siders terrestrial experiments involving gravimeter and following parameter values in this ghost model: laboratory tests. Orbiting gyroscopes provide another source of information, as described in Sec. V D. The im- α1 = 0, 0 2 plications for Lorentz violation of observations of binary- α2 = −κ(b ) , pulsar systems are discussed in Sec. V E. The sensitivies α3 = 0, of the classic tests, in particular the perihelion shift and γ = 1. (86) the time-delay eﬀect, are considered in Sec. V F. More speculative applications of the present theory, for exam- This model therefore fails to obey the condition (67). The ple to the properties of dark matter or to the Pioneer violation of assumption (iv) evidently aﬀects the general anomaly, are also of interest but their details lie beyond structure of the post-newtonian metric. the scope of this work and will be considered elsewhere. 18

typically suffices to include contributions to the metric fluctuations from the Sun and the Earth. Of particular interest for later applications are various sets of orthonormal basis vectors that can be defined in the context of the line element (89). One useful set is appropriate for an observer at rest, dXJ /dT = 0, at a given point (T, X~ ) in the Sun-centered frame. Denoting the four elements of this basis set by eµ with µ ≡ (t, j), we can write

T 1 ~ et = δ t[1 + 2 hT T (T, X)+ O(4)]eT , FIG. 2: Sun-centered celestial-equatorial frame. J 1 ~ J ~ ej = δ j [eJ − 2 hJK (T, X)eK ]+ δ j hTJ (T, X)eT . (90)

A. General Considerations Direct calculation shows that this basis satisﬁes

2 2 2 1. Frame conventions and transformations ds = −et + ej (91)

The comparative analysis of signals for Lorentz viola- to post-newtonian order. tion from different experiments is facilitated by adopt- Another useful set of basis vectors, appropriate for an ing a standard inertial frame. The canonical reference observer in arbitrary motion, can be obtained from the frame for SME experimental studies in Minkowski space- basis set (90) by applying a local Lorentz transformation. time is a Sun-centered celestial-equatorial frame [34]. In Denoting this new set of vectors by eµˆ, we have the present context of post-newtonian gravity, the stan- e =Λν (τ)e . (92) dard inertial frame is chosen as an asymptotically iner- µˆ µˆ ν tial frame that is comoving with the rest frame of the It is understood that all quantities on the right-hand side solar system and that coincides with the canonical Sun- of this equation are to be evaluated along the observer’s centered frame. The cartesian coordinates in the Sun- worldline, which is parametrized by proper time τ. centered frame are denoted by For the experimental applications in the present work, it suffices to expand the local Lorentz transformation in xΞ = (T,XJ ) = (T,X,Y,Z) (87) Eq. (92) in a post-newtonian series. This gives and are labeled with capital Greek letters. By defini- e = δt (1 + 1 v2)e + vj e , tion, the Z axis is aligned with the rotation axis of the tˆ tˆ 2 t j Earth, while the X axis points along the direction from e = δj vkRkj e + δj (δkl + 1 vkvl)Rlj e . (93) ˆj ˆj t ˆj 2 k the center of the Earth to the Sun at the vernal equinox. The inclination of the Earth’s orbit is denoted η. The Ξ The components (etˆ) coincide with the observer’s four- origin of the coordinate time T is understood to be the velocity uΞ in the Sun-centered frame. In this expression, time when the Earth crosses the Sun-centered X axis at vj is the coordinate velocity of the observer as measured the vernal equinox. This standard coordinate system is in the frame (90), and Rjk is an arbitrary rotation. The depicted in Fig. 2. The corresponding coordinate basis reader is cautioned that the coordinate velocities vj and vectors are denoted vJ typically differ at the post-newtonian level. The explicit relationship can be derived from (90) and is found eΞ = (eT , eJ ) = (∂T , ∂J ). (88) to be

In the Sun-centered frame, as in any other inertial vj = δ j vJ (1 + 1 h )+ 1 δ j h vK + O(4). (94) frame, the post-newtonian spacetime metric takes the J 2 T T 2 J JK form given in Eqs. (35)-(37). This metric, along with the point-mass equations of motion (52), forms the basis 2. Current bounds of our experimental studies to follow. The corresponding line element can be written in the form In most modern tests of local Lorentz symmetry with 2 2 gravitational experiments, the data have been analyzed ds = −[1 − hT T (T, X~ )+ O(4)]dT in the context of the PPN formalism. The discussion in ~ J +2hTJ (T, X)dT dX Sec. IIIC shows there is a correspondence between the J K +[δJK + hJK (T, X~ )]dX dX , (89) PPN formalism and the pure-gravity sector of the minimal SME in a special limit, so it is conceivable a priori where hT T is taken to O(3), hTJ is taken to O(3) and that existing data analyses could directly yield partial hJK is taken to O(2). For the purposes of this work, it information about SME coefficients. Moreover, a given 19 experiment might in fact have sensitivity to one or more A more interesting constraint can be obtained by con- SME coefficients even if the existing data analysis fails sidering implications of the observed close alignment of to identify it, so the possibility arises that new informa- the Sun’s spin axis with the angular-momentum axis of tion can be extracted from available data by adopting the planetary orbits. The key point is that the metric the SME context. The specific SME coefficients that term sjkU jk violates angular-momentum conservation. could be measured by the reanalysis of existing exper- As a result, a spinning massive body experiences a self- iments depend on details of the experimental procedure. torque controlled by the coefficients sjk. For the Sun, this Nonetheless, a general argument can be given that pro- effect induces a precession of the spin axis, which over the vides a crude estimate of the potential sensitivities and lifetime of the solar system would produce a misalign- the SME coefficients that might be constrained. ment of the spin axis relative to the ecliptic plane. This The standard experimental analysis for Lorentz viola- idea was introduced by Nordtvedt and used to bound tion in gravitational experiments is based on the assumed the PPN parameter α2 [50]. In the context of the pure- existence of a preferred-frame vector ~w. Usually, this is gravity sector of the minimal SME, the analysis is similar taken to have a definite magnitude and orientation for and so is not presented here. It turns out that the final the solar system and is identified with the velocity of the result involves a particular combination of SME coeffi- solar system with respect to the rest frame of the cosmic cients in the Sun-centered frame given by microwave background radiation [30]. The strength of the coupling of the vector to gravity is then determined JK JL ˆK ˆL JK ˆJ ˆK 2 sSSP = st st S S − (st S S ) . (97) by the two PPN parameters α and α . 1 2 q Performing a data analysis of this type under the as- In this expression, SˆJ are the Sun-frame components of sumption that the vector ~w is the source of isotropy vi- the unit vector pointing in the direction of the present olations is equivalent in the SME context to supposing JK µν solar spin. The coefficients st are the traceless compo- that only parallel projections of the coefficients s along nents of sJK , given by the unit vectorw ˆ contribute to any signals. This holds regardless of the choice of ~w. For the coefficients sµν , JK JK 1 JK T T st = s − δ s . (98) there are two possible parallel projections, given by 3 JK 0j j k 0k Under the assumption that the coefficients st are small sk =w ˆ wˆ s , enough for perturbation theory to be valid, we find an −13 jk j k l m lm approximate bound of sSSP < 10 . sk =w ˆ wˆ wˆ wˆ s . (95) ∼

κλµν Also, the coefficients t play no role in the analysis B. Lunar and Satellite Ranging at this order, in accordance with the discussion in Sec. II B 4. It therefore follows that sensitivity to at most 2 of Lunar laser ranging is among the most sensitive tests µν κλµν the 19 SME coefficients s and t could be extracted of gravitational physics within the solar system to date. from these types of analyses of experimental data. To- Dominant orbital perturbations in the context of the κλµν gether with the 10 coefficients t , the 7 perpendicular PPN metric were obtained in Ref. [51], while oscilla- projections given by tions arising from a subset of the parameters for the anisotropic-universe model were calculated in Ref. [52]. 0j 0k j k 0k s⊥ = s − wˆ wˆ s , Here, we obtain the dominant effects on the motion of a jk satellite orbiting the Earth that arise from nonzero coef- s = sjk − wˆj wˆkwˆlwˆmslm (96) ⊥ ficients sµν for Lorentz violation. Our results are appli- remain unexplored in experimental analyses to date. cable to the Earth-Moon system as well as to artificial satellites. The two independent measurements of parallel projec- The analysis is performed in the Sun-centered frame. tions of SME coefficients could be obtained, for example, The satellite motion is affected by the Earth, the Sun, from a reanalysis of existing data from lunar laser rang- and other perturbing bodies. For the analysis, it is useful ing [48, 49]. A crude estimate of the sensitivity to be to introduce quantities to characterize the masses, posi- expected can be obtained by assuming that the bounds tions, sidereal frequencies, etc., relevant for the motion. attained on a particular term of the PPN metric in the These are listed in Table 1. The various position vectors Sun-centered frame roughly parallel those that might be are depicted in Fig. 3. achieved for the corresponding term in the pure-gravity This subsection begins by presenting the coordinate minimal-SME metric. This procedure gives s0j < 10−8 k ∼ acceleration of the Earth-satellite separation, which is jk < −11 and sk ∼ 10 . However, these estimates are untrust- the primary observable in laser-ranging experiments. To worthy because the substantial differences between the gain insight into the content of the resulting expressions, two metrics imply unknown effects on the data analysis, we next perform a perturbative analysis that extracts the so these constraints are no more than a guide to what dominant frequencies and corresponding amplitudes for might be achieved. oscillations driven by Lorentz violation. This analysis is 20

somewhat lengthy, and it is largely relegated to Appendix A, with only the primary results presented in the main text. Finally, we estimate the likely sensitivities attainable in experimental analyses of ranging to the Moon and to various artiﬁcial satellites.

FIG. 3: The Earth-satellite system in the Sun-centered frame.

Quantity Deﬁnition

m1 satellite mass m2 Earth mass M = m1 + m2 total Earth-satellite mass δm = m2 − m1 Earth-satellite mass diﬀerence mn mass of nth perturbing body M⊙ Sun mass J r1 satellite position r0 mean satellite orbital distance to Earth J r2 Earth position R mean Earth orbital distance to Sun

R⊕ Earth radius J rn position of nth perturbing body J J J r = r1 − r2 = (x,y,z) Earth-satellite separation, with magnitude r = |~r1 − ~r2| δr deviation of Earth-satellite distance r from the mean r0 J J J R = (m1r1 + m2r2 )/M position of Earth-satellite newtonian center of mass J J J Rn = R − rn separation of newtonian center of mass and nth perturbing body ω mean satellite frequency

ω0 anomalistic satellite frequency ωn nth harmonic of satellite frequency 3 Ω⊕ = GM⊙/R mean Earth orbital frequency v0 p mean satellite orbital velocity V⊕ =Ω⊕R mean Earth orbital velocity J J J J v = v1 − v2 = dr /dT relative Earth-satellite velocity J J J V = (m1v1 + m2v2 )/M velocity of Earth-satellite newtonian center of mass 2 Q = J2GMR⊕ Earth quadrupole moment, with J2 ≈ 0.001 Table 1. Deﬁnitions for analysis of laser ranging to the Moon and to artiﬁcial satellites.

1. Earth-satellite dynamics bodies, we can write

2 J J d r αES ≡ 2 The coordinate acceleration αJ of the relative Earth- dT ES J J J J satellite separation can be obtained from Eq. (52). Al- = αN + αT + αQ + αLV + . . . lowing for perturbative eﬀects from the Sun and other (99) 21

The first three terms in this expression are newtonian In contrast, for artificial satellite orbits the corresponding effects that are independent of Lorentz violation. The Lorentz-violating tidal effects from the Moon and Sun first is the newtonian acceleration for a point mass in are suppressed by about six orders of magnitude and can the newtonian gravitational field of the Earth-satellite safely be ignored. system. It is given by Finally, the ellipses in Eq. (99) represent higher-order GM tidal corrections from perturbing bodies, higher multi- αJ = − rJ . (100) N r3 pole corrections from the Earth’s potential, and general relativistic corrections entering at the O(4) post- In this expression and what follows, G has been rescaled T T newtonian level. These smaller effects are ignored in the by a factor of (1 − 3s /2), which is unobservable in analysis that follows. this context. The second term is the newtonian tidal quadrupole term, which takes the form

J K K J Gmn(3Rˆ Rˆ r − r ) J n n 2. Perturbative expansion αT = 3 . (101) Rn nX=16 ,2 This expression basically represents the leading effects To fit laser-ranging data, Eq. (99) could be entered from external bodies at the newtonian level of approxi- into an appropriate computer code, along with model- mation. The third term is ing data for other systematics of the satellite orbit [48]. However, to get a sense of the types of sensitivities at- 3QzδZJ 3Q(5z2 + r2)rJ J tainable, it is useful to study analytically the acceleration αQ = − 5 − 7 . (102) r 2r (99) in a perturbation scheme. Here, we focus on oscilla- It represents the acceleration due to the Earth’s tory corrections to the observable relative Earth-satellite quadrupole moment. separation r ≡ |~r1 − ~r2| in the Sun-centered frame. Note J The term αLV in Eq. (99) contains the leading Lorentz- that the boost between the Sun-centered frame and the violating effects. It can be split into two pieces, Earth, where observations are in fact performed, introduces corrections to r only at O(2) and hence corrections αJ = αJ + αJ . (103) LV LV,ES LV,tidal to Eq. (99) at O(4), as can be verified by determining the The first of these is given by proper distance measured by an Earth observer. Sim- ilarly, modifications to the deviation δr of r from the GMsJK rK 3GMsKLrK rLrJ αJ = − mean Earth-satellite distance r0 also enter only at O(2) LV,ES r3 2r5 and can therefore be neglected for our analysis. See, for (3GMV K sTK +2GδmvK sTK )rJ example, Ref. [51]. + r3 Due to its length, the explicit calculation of the (GMV K rK sTJ +2GδmvK rK sTJ ) Lorentz-violating corrections to r is relegated to Ap- − r3 pendix A. We focus here on the main results. In pre- GMV J sTK rK 3GMV K sTLrK rLrJ senting them, some definitions associated with the satel- − + . lite orbit are needed. These are depicted in Fig. 4. The r3 r5 orientation of the orbit is described by two angles, the (104) longitude of the node α and the inclination of the orbit It contains the Lorentz-violating accelerations arising β. It is also useful to introduce the circular orbit phase J θ, which represents the angle at time T = 0 subtended from the Earth-satellite system alone. The term αLV,tidal represents the Lorentz-violating tidal accelerations due between the position vector ~r0 and the line of ascending to the presence of other bodies. For the Earth-Moon nodes in the plane of the unperturbed orbit. Note that case, the dominant Lorentz-violating tidal accelerations corrections to all these angles arising from the boost fac- are due to the Sun, and we find the leading contributions tor between the Sun-centered frame and the Earth are are negligible in the present context. The oscillatory radial corrections δr arising from the αJ = Ω2 sJK (rK − 3RˆKRˆLrL) LV,tidal ⊕ Lorentz-violating terms in Eq. (99) take the generic form 3 2 KL J ˆK ˆL K ˆL ˆJ − 2 Ω⊕s (r R R +2r R R M M K L J − 5Rˆ r Rˆ Rˆ Rˆ ) δr = [An cos(ωnT + φn)+ Bn sin(ωnT + φn)]. (106) 2 TK J K K J δm J K n +2Ω⊕s (r V + v R + M r v X ˆL L ˆJ K δm ˆL L ˆJ K − 3R r R V − 3 M R r R v ) The dominant amplitudes An and Bn and the values of 2 TJ K K K K δm K K the corresponding phases φn are listed in Table 2. The −2Ω⊕s (r V + v R + r v M amplitudes are taken directly from Eqs. (A38). The − 3RˆLrLRˆK V K − 3 δm RˆLrLRˆK vK ). M phases φn are related as indicated to the circular-orbit (105) phase θ shown in Fig. 4. 22

cients for Lorentz violation as

s11 − s22 = (cos2 α − sin2 α cos2 β) sXX +(sin2 α − cos2 α cos2 β) sY Y − sin2 β sZZ +2 sin α cos α(1 + cos2 β) sXY +2 sin β cos β sin α sXZ −2 sin β cos β cos α sY Z , s12 = − sin α cos α cos β sXX +cos α sin α cos β sY Y +(cos2 α − sin2 α)cos β sXY +cos α sin β sXZ + sin α sin β sY Z , s01 = cos α sT X + sin α sT Y , s02 = − sin α cos β sT X + cos α cos β sT Y T Z FIG. 4: Satellite orbital parameters in the Sun-centered + sin β s . (107) frame. For simplicity, the Earth is shown as if it were translated to the origin of the Sun-centered coordinates. Different combinations of coefficients for Lorentz violation also appear in Table 2, associated with the amplitudes of the oscillations at frequency Ω⊕. The relevant combinations of these coefficients in the Sun-centered Amplitude Phase frame are explicitly given by 1 11 22 A2ω = − (s − s )r0 2θ 1 12 sΩ c = (sin α sin η sin β cos β 1 12 ⊕ 2 B2ω = − 6 s r0 2θ 2 11 22 + sin α cos α cos η cos β A2ω−ω0 = −ωer0(s − s )/16(ω − ω0) 2θ − sin α cos η cos α)sT X B = −ωer s12/8(ω − ω ) 2θ 2ω−ω0 0 0 1 2 02 −(3 cos η + cos η sin α Aω = −ω(δm)v0r0s /M(ω − ω0) θ 2 01 1 Bω = ω(δm)v0r0s /M(ω − ω0) θ + 2 cos α sin η sin β cos β 1 2 2 T Y AΩ⊕ = V⊕r0(b1/b2)sΩ⊕c 0 + 2 cos η cos α cos β)s 1 BΩ⊕ = V⊕r0(b1/b2)sΩ⊕s 0 −(3 sin η + 2 cos α cos η sin β cos β 1 2 T Z + 2 sin β sin η)s , 1 2 2 1 2 T X sΩ⊕s = −(3 + 2 sin α cos β + 2 cos α)s 1 2 T Y Table 2. Dominant Earth-satellite range oscillations. − 2 cos α sin α sin βs 1 T Z + 2 sin α sin β cos βs . (108)

The mass, position, frequency, and velocity variables 3. Experiment appearing in Table 2 are defined in Table 1. The eccentricity of the orbit is denoted e, and for the analysis it is assumed to be much less than unity. The quantities A feature of particular potential interest for experi- b1 and b2 are defined in Eq. (A23) of Appendix A. The ments is the dependence of the coefficients in Table 2 on anomalistic frequency ω0 is the frequency of the natural the orientation of the orbital plane. To illustrate this, eccentric oscillations. Note that ω0 differs from ω in the consider first an equatorial satellite, for which β = 0. presence of perturbing bodies and quadrupole moments. The observable combinations of coefficients for Lorentz In the lunar case, for example, 2π/(ω0 − ω) ≈ 8.9 years violation then reduce to [53]. s11 − s22 = cos2α(sXX − sY Y )+2sin2α sXY , Some of the combinations of SME coefficients appear- s12 = − 1 sin2α(sXX − sY Y ) ing in Table 2 are labeled with indices 1 and 2, which 2 refer to projections on the orbital plane. The label 1 rep- +cos2α sXY . (109) resents projection onto the line of ascending nodes, while the label 2 represents projection onto the perpendicular This implies that the 2ω and 2ω − ω0 frequency bands of direction in the orbital plane. Explicitly, the combina- this orbit have sensitivity to the coefficients sXX , sY Y , tions are given in terms of the basic Sun-centered coeffi- and sXY . If instead a polar satellite is considered, for 23

11 22 which β = π/2, the same frequency bands have sensitiv- parts in 1010 on the combination (s − s )$, and parts JK 7 01 02 ity to all six coefficients in s : in 10 on the coefficients s , s , sΩ⊕s, sΩ⊕c. These sensitivities may be substantially improved with the new 11 22 1 XX Y Y 1 XX Y Y s − s = 2 (s + s )+ 2 cos2α(s − s ) Apache Point Observatory Lunar Laser-ranging Opera- +sin2α sXY − sZZ , tion (APOLLO), which is currently under development in New Mexico with the goal of millimeter-ranging pre- s12 = cos α sXZ + sin α sY Z . (110) cision [55]. These examples show that satellites with different orbits Next, we consider ranging to artificial satellites. In can place independent bounds on coefficients for Lorentz practice, the realistic modeling of satellite orbits is more violation. A similar line of reasoning shows sensitivity to challenging than the Moon due to substantial perturba- the sTJ coefficients can be acquired as well. tive effects. However, we show here that ranging to artificial satellites has the potential to be particularly use- For lunar laser ranging, the even-parity coefficients JK sJK can be extracted from Eqs. (107) using the appro- ful for measuring independent combinations of the s priate values of the longitude of the node and the incli- coefficients that are less readily accessible to lunar laser nation. For definiteness and to acquire insight, we adopt ranging due to the approximately fixed orientation of the the values α ≈ 125◦ and β ≈ 23.5◦. However, these an- plane of the lunar orbit. gles vary for the Moon due to comparatively large new- The relevant coefficients for a generic satellite orbit are tonian perturbations, so some caution is needed in using given in Eqs. (107) in terms of the right ascension α and the equations that follow. In any case, with these values inclination β. For the odd-parity coefficients, a similar we find the even-parity coefficients are given by argument applies as for the lunar case, so any one satellite can in principle measure all 3 coefficients sTJ . Using 11 22 XX Y Y ZZ (s − s )$ = 0.08(s + s − 2s ) reasoning similar to that for lunar laser ranging, we find XX Y Y XY that any one satellite can make only two independent −0.31(s − s ) − 1.7s measurements of combinations of the even-parity coef- +0.60sXZ +0.42sY Z , ficients sJK . It follows that any 2 satellites at different (s12) = 0.43(sXX − sY Y ) − 0.31sXY orientations can in principle measure 3 of the 4 combina- $ JK XZ Y Z tions of components of s to which lunar laser ranging −0.23s − 0.33s . (111) is insensitive. The reason that only 3 of the 4 can be measured is Evidently, lunar laser ranging measures two independent that the combination sXX + sY Y + sZZ = sT T is ab- combinations of the sJK coefficients. sent from Eqs. (107) and hence only 5 combinations of For the odd-parity coefficients sTJ , adopting the same the 6 coefficients in sJK are measurable via the am- values for α and β shows that lunar laser ranging offers plitudes in Table 2 for any set of orbit orientations. sensitivity to the combinations However, higher harmonics in ω, Ω, and other frequen- 01 T X T Y cies are likely to arise from the eccentric motion of the (s )$ = −0.60s +0.82s , Moon or satellite. Although the corresponding ampli- 02 T Y T X T Z (s )$ = −0.53s − 0.75s +0.40s , tudes would be suppressed compared to those in Table T Y T Z T X 2, they could include terms from which the combination (sΩ⊕c)$ = −3.1s − 1.1s +0.094s , sXX +sY Y +sZZ = sT T might be measured. The numer- (s ) = −3.4sT X +0.037sT Y +0.15sT Z . (112) Ω⊕s $ ical code used to fit the data would include these amplitudes. Note that the strength of some signals might be This reveals that measuring the amplitudes at frequency enhanced by a suitable orientation of the satellite orbit, ω and phase θ provides sensitivity to 2 combinations of since the amplitudes in Table 2 depend on the orientation the 3 independent coefficients sTJ . Provided one can angles α and β through the quantities b and b defined also independently measure the amplitudes at frequency 1 2 in Eq. (A23). Ω and zero phase, the third independent coefficient in TJ s can also be accessed. Satellite No. r (m) β Ref. Together the above considerations imply that lunar 0 laser ranging can in principle measure at least 5 inde- ETALON 2 2.5 × 107 65◦ [56] pendent combinations of coefficients for Lorentz viola- GALILEO 30 3.0 × 107 56◦ [57] tion. In particular, the odd-parity coefficients sTJ can GLONASS 3 2.5 × 107 65◦ [56] be completely determined. GPS 2 2.6 × 107 55◦ [56] To estimate the experimental sensitivity attainable LAGEOSI 1 1.2 × 107 110◦ [56] in lunar laser ranging, we assume ranging precision at 7 ◦ the centimeter level, which has already been achieved LAGEOSII 1 1.2 × 10 53 [56] 8 [54]. With the standard lunar values r0 = 3.8 × 10 m, −4 Table 3. Some high-orbit satellites. V⊕ = 1.0 × 10 , ω/(ω − ω0) ≈ 120, and e = 0.055, we find that lunar laser ranging can attain the follow- Some promising high-orbit satellite missions, both cur- 12 ing sensitivities: parts in 1011 on the coefficient (s )$, rent and future, are listed in Table 3. Although beyond 24 our present scope, it would be interesting to determine written in the form whether data from ranging to any of these satellites could 2 0ˆ 2 ˆj ˆj ˆj ˆj 2 ˆj ˆj 2 be adapted to search for the signals in Table 2. As an ds = −(dx ) 1+2a x + (a x ) + (ω x ) estimate of attainable sensitivities, suppose centimeter- 2 ˆj ˆj ˆj kˆ −ω x x+ R ˆx x 7 0ˆˆj0ˆk ranging sensitivity is feasible and take r0 ≈ 10 m, 3 0ˆ ˆj ˆjkˆˆl kˆ ˆl 2 kˆ ˆl ω/(ω − ω0) ≈ 2 × 10 , and e = 0.010. These values +2dx dx (ǫ ω x − 3 R0ˆkˆˆjˆlx x ) produce the following estimated sensitivities from laser ˆj kˆ 1 ˆl mˆ ranging to artificial satellites: parts in 109 on s11 − s22, +dx dx (δˆjkˆ − 3 Rˆjlˆkˆmˆ x x ). (113) 10 12 8 01 02 parts in 10 on s , parts in 10 on s , s , and parts 0ˆ 5 Here, we have introduced x , which coincides along the in 10 on sΩ⊕s, sΩ⊕c. Interesting possibilities may also exist for observing worldline with the observer’s proper time. secular changes to the orbits of near-Earth satellites. From this metric, an effective Lagrange density can Sensitivities comparable to some of those mentioned be established for test particle dynamics in the proper above may be attainable for certain coefficients [58]. A reference frame of an accelerated and rotated observer. detailed analysis of this situation would require incorpo- The action for a test particle of mass m takes the form rating also contributions to the Earth-satellite accelera- 2 J 0ˆ −ds tion αES proportional to the spherical moment of iner- S = −m dx (dx0ˆ)2 tia of the Earth. These contributions, which can arise Z s from Lorentz-violating effects proportional to the poten- 0ˆ tial U JK in Eq. (28), are small for high-orbit satellites = dx L. (114) or the Moon and so have been neglected in the analysis Z above, but they can be substantial for near-Earth orbits. For present purposes, it suffices to express the lagrangian as a post-newtonian series. Denote the coordinate position of the test particle by xˆj and its coordinate velocity ˆj C. Laboratory experiments byx ˙ , where the dot signifies derivative with respect to x0ˆ. This velocity and the rotation velocity of the Earth’s surface are taken to be O(1). The lagrangian then be- This section considers some sensitive laboratory ex- comes periments on the Earth. Among ones already performed 1 are gravimeter tests, analyzed in Refs. [59, 60]. For this m−1L = 1 (~x˙)2 − ~a · ~x + (~ω × ~x)2 case, the basic idea is to measure the force on a test 2 2 mass held fixed above the Earth’s surface, seeking ap- ˙ 1 ˆj kˆ +~x · (~ω × ~x) − Rˆˆˆˆx x + .... (115) parent variations in the locally measured value of New- 2 0j0k ton’s gravitational constant G. The coefficient space of In this expression, ~a is the observer acceleration, ~ω is the pure-gravity sector of the minimal SME to which this the observer angular velocity, and the various curvature experiment is sensitive is explored below. Other sensitive components are taken to contain only the dominant con- 2 existing laboratory experiments use various torsion pen- tributions proportional to ∇~ g00. The ellipses represent dula, with some already being applied to probe Lorentz terms that contribute at O(4) via post-newtonian quan- violation in the fermion sector [11]. This section also dis- tities and sub-dominant curvature terms. cusses some torsion-pendulum tests that could perform The dominant Lorentz-violating effects arise through sensitive measurements of Lorentz symmetry in the grav- ~a. To establish the local acceleration in the proper ref- itational sector. erence frame in the presence of Lorentz violation, consider first the observer’s four acceleration in an arbitrary post-newtonian coordinate frame. This is given by the 1. Theory standard formula duµ aµ = +Γµ uν uκ. (116) To explore these ideas, we must first determine the dy- dτ νκ namics of a test particle and establish the local accelera- We then choose the post-newtonian frame to be the Sun- tion at a point on the Earth’s surface in the presence of centered frame described in Sec. V A. In this frame, the Lorentz violation controlled by the coefficients sµν . The spacetime metric is given by Eq. (89). results are required in a suitable observer coordinate sys- To express the local acceleration in the Sun-centered tem for experiments. frame, the observer’s trajectory xΞ is required. This can An observer on the surface of the Earth is accelerat- be adequately described to post-newtonian order by ing and rotating with respect to the asymptotic inertial space. The appropriate coordinate system is therefore Ξ Ξ k˜ Ξ x = e ˜ξ + x⊕. (117) the proper reference frame of an accelerated and rotated k observer [61, 62]. At second order in the coordinate dis- Ξ k˜ In this equation, x⊕ is the worldline of the Earth and ξ is tance xˆj , the metric for this coordinate system can be the spatial coordinate location of the observer in a frame 25 centered on the Earth, for which indices are denoted with In these equations, the reference gravitational accelera- a tilde. The components of the comoving spatial basis tion g is in that frame are denoted eΞ . They can be obtained k˜ 2 jk j g = GM⊕/R⊕, (122) from Eq. (93) with R = 0 and the Earth velocity v⊕ J J given by Eq. (94) with V = V⊕ . To sufficient post- and the quantities i1, i2, i3 are defined by newtonian accuracy, the observer’s coordinate location ˜ I ξk in the Earth frame can be taken as a vector pointing ⊕ i⊕ = 2 , to the observer’s location and rotating with the Earth. M⊕R⊕ 1 Thus, modeling the Earth as a sphere of radius R⊕, we i1 = 1+ 3 i⊕, can write 5 i2 = 1 − 3 i⊕, ξ~ = R⊕(sin χ cos(ω⊕T + φ), sin χ sin(ω⊕T + φ), cos χ), i3 = 1 − i⊕. (123) (118) where χ is the colatitude of the observer and φ is a stan- Note that the analysis here disregards possible tidal dard phase fixing the time origin (see Ref. [34], Appendix changes in the Earth’s shape and mass distribution [30, 63], which under some circumstances may enhance C): φ = ω⊕(T⊕ − T ), where T⊕ is measured from one of the times when they ˆ and Y axis coincide. the observability of signals for Lorentz violation. To obtain explicitly the local acceleration, the poten- To obtain the corresponding expressions in the ob- tials (28) must be evaluated for the Earth. They can server’s frame, it suffices to project Eq. (121) along the comoving basis vectors e = eΞ e of the accelerated be written as functions of the spatial position X~ in the µˆ µˆ Ξ and rotated observer on the Earth’s surface. Explicit ex- Sun-centered frame, and the ones relevant here are given pressions for these basis vectors are given by Eq. (93), by j jk with the observer velocity vo and the rotation R taken GM as U = ⊕ , ~ j j J j ˜j |X − ~x⊕| v = δ V + δ dξ /dT + O(3), o J ⊕ ˜j J K JK GM⊕(X − x⊕) (X − x⊕) jk jk U = R = R (ω⊕T ). (124) 3 Z |X~ − ~x⊕| To sufficient approximation, we may use GI⊕ J K − [3(X − x⊕) (X − x⊕) 3|X~ − ~x |5 ⊕ V~⊕ = V⊕(sin Ω⊕T, − cos η cosΩ⊕T, − sin η cosΩ⊕T ), JK 2 (125) − δ |X~ − ~x⊕| ], where V⊕ is the mean Earth orbital speed and η is the J inclination of the Earth’s orbit, shown in Fig. 2. GM⊕V V J = ⊕ + .... (119) We are free to use the observer rotational invariance ~ |X − ~x⊕| of the metric (113) to choose the observer’s spatial co- ˆj In these equations, M is the mass of the Earth. The ordinates x to coincide with the standard SME conven- ⊕ tions for an Earth laboratory (see Ref. [34], Appendix C). quantity I⊕ is the spherical moment of inertia of the Earth, given by the integral Thus, at a given point on the Earth’s surface,z ˆ points towards the zenith,y ˆ points east, andx ˆ points south.

3 2 With these conventions, we find the desired expression I⊕ = d yρ(~y)~y (120) for the local acceleration to be Z ˆj 3 T T 3 zˆzˆ ˆjzˆ ˆjzˆ over the volume of the Earth, where ~y is the vector dis- a = g(1 + 2 i1s + 2 i2s )δ − gi3s tance from the center of the Earth and ρ(~y) is the density 2 2 ˆjzˆ ˆjxˆ −ω R⊕(sin χδ + sin χ cos χδ ) of the Earth. The ellipses in Eq. (119) represent neglected T ˆj zˆ T zˆ ˆj gravitomagnetic effects. +gi3(s V⊕ + s V⊕) Inserting Eq. (117) into Eq. (116) yields the follow- ˆ −3g(i sTJ V J + i sT zˆV zˆ)δjzˆ + .... (126) ing components of the observer’s acceleration in the Sun- 1 ⊕ 2 ⊕ centered frame: The ellipses represent newtonian tidal corrections from T the Sun and Moon, along with O(4) terms. The various a = O(3), projections of quantities along the local axes appearing J 3 T T 3 KL K L ˆk˜ ˆ˜l J ˆ˜j in Eq. (126) are obtained using the comoving basis vec- a = g(1 + 2 i1s + 2 i2s δ k˜δ l˜ξ ξ )δ ˜j ξ Ξ T ˆj ˆj T Ξ ˜ ˜ ˜ tors e = e e . For example, s = e s . Typically, −gi sJK δK ξˆk + gi (sTJ V K δK ξˆk + sTK V J δK ξˆk) µˆ µˆ Ξ Ξ 3 k˜ 3 ⊕ k˜ ⊕ k˜ these projections introduce time dependence on T . Note, TK K J ˆ˜j TK L K L J ˆk˜ ˆ˜l ˆ˜j however, that the coefficient sT T carries no time depen- −3g(i1s V⊕ δ ˜ξ + i2s V⊕ δ ˜δ ˜δ ˜ξ ξ ξ ) j k l j dence up to O(3) and hence represents an unobservable J 2 ˜j 2 +δ ˜j d ξ /dT + O(4). (121) scaling. 26

2. Gravimeter tests The time dependence of the apparent variations in G arises from the terms involving the indicesz ˆ and the TJ By restricting attention to motion along thez ˆ axis, the components s . It can be decomposed in frequency ac- expression (126) for the local acceleration can be used to cording to obtain the dominant Lorentz-violating contributions to δG the apparent local variation δG of Newton’s gravitational = [C cos(ω T + φ )+ D sin(ω T + φ )]. G n n n n n n constant, as would be measured with a gravimeter. This n X gives (129)

zˆ 3 T T 1 zˆzˆ 2 2 a = g(1 + 2 i1s + 2 i4s ) − ω R⊕ sin χ Explicit calculation of the time dependence using the co- T zˆ zˆ TJ J moving basis vectors yields the expressions for the am- −gi4s V⊕ − 3gi1s V⊕ , (127) plitudes Cn, Dn and phases φn listed in Table 3. In this where table, the angle χ is the colatitude of the laboratory, and the other quantities are deﬁned in the previous subsec- i4 = 1 − 3i⊕. (128) tion.

Gravimeter Amplitudes Amplitude Phase 1 XX Y Y 2 C2ω = 4 i4(s − s ) sin χ 2φ 1 XY 2 D2ω = 2 i4s sin χ 2φ 1 XZ Cω = 2 i4s sin2χ φ 1 Y Z Dω = 2 i4s sin2χ φ 1 T Y 2 C2ω+Ω = − 4 i4V⊕s (cos η − 1) sin χ 2φ 1 T X 2 D2ω+Ω = 4 i4V⊕s (cos η − 1) sin χ 2φ 1 T Y 2 C2ω−Ω = − 4 i4V⊕s (1 + cos η) sin χ 2φ 1 T X 2 D2ω−Ω = 4 i4V⊕s (1 + cos η) sin χ 2φ T Y 1 2 T Z 2 CΩ = V⊕[s cos η( 2 i4 sin χ +3i1)+ s sin η(3i1 + i4 cos χ)] 0 T X 1 2 DΩ = −V⊕s ( 2 i4 sin χ +3i1) 0 1 T X Cω+Ω = 4 i4V⊕s sin2χ sin η φ 1 T Z T Y Dω+Ω = − 4 i4V⊕[s (1 − cos η) − s sin η]sin2χ φ 1 T X Cω−Ω = 4 i4V⊕s sin η sin2χ φ 1 T Z T Y Dω−Ω = 4 i4V⊕[s (1 + cos η)+ s sin η]sin2χ φ Table 4. Gravimeter amplitudes.

Assuming each amplitude in Table 4 can be separately 3. Torsion-pendulum tests extracted from real data, it follows that gravimeter experiments can measure up to 4 independent coefficients A complementary class of laboratory tests could ex- JK TJ in s and all 3 coefficients in s . Moreover, gravime- ploit the anisotropy of the locally measured acceleration ter experiments have attained impressive accuracies [60] in the horizontal directions instead of the vertical one. that would translate into stringent sensitivities on the Explicitly, the accelerations in thex ˆ andy ˆ directions are pure-gravity coefficients in the minimal SME. A crude es- given by timate suggests that gravimeter experiments fitting data JK xˆ xˆzˆ 2 to Eq. (129) could achieve sensitivity to s at parts in a = −gi3s − ω R⊕ sin χ cos χ 1011 and to sTJ at parts in 107. Possible systematics that T zˆ xˆ T xˆ zˆ +gi3s V⊕ + gi3s V⊕, would require attention include tidal influences from the yˆ yˆzˆ T zˆ yˆ T yˆ zˆ Sun and Moon. These can, for example, generate oscil- a = −gi3s + gi3s V⊕ + gi3s V⊕ (130) lations contributing to Eq. (129) at frequencies 2ω⊕ and to order O(3). In effect, the Lorentz-violating contribu- ω⊕. tions to these expressions represent slowly varying accelerations in the horizontal directions as the Earth rotates and orbits the Sun. In contrast, the conventional cen- trifugal term in Eq. (130) is constant in time. 27

These quantities depend on the number N of masses used. Note that the eﬀective potential (131) depends only on the distribution of the N masses and is independent of the disk mass M.

The first part of Eq. (131) is proportional to θ2. This term contains contributions both from Lorentz violation and from the Earth’s rotation. It acts as an effective restoring force for the disk, producing a time-dependent shift to the effective spring constant of the system that primarily affects free oscillations. The second part of Eq. (131) is proportional to θ, and it contains a time- independent contribution from the Earth’s rotation along with a Lorentz-violating piece. The latter produces a slowly varying Lorentz-violating torque τ = −dV/dθ on the disk.

FIG. 5: Idealized apparatus with N = 3 masses. The oscillations of the system are determined by the second-order diﬀerential equation

To gain insight into the experimental possibilities for testing the effects contained in Eqs. (130), consider an d2θ dθ idealized scenario with a disk of radius R and mass M I 2 +2γI + κθ = τ, (133) suspended at its center and positioned to rotate freely dT dT about its center of symmetry in thex ˆ-ˆy plane. We suppose that N spherical masses, each of mass mj , j = 1, 2,...,N, are attached rigidly to the disk at ra- where I is the total moment of inertia of the disk and dius r0 < R and initially located at angular position θj masses, and γ is the damping parameter of the torsion relative to thex ˆ axis. For convenience, we write the fiber. In writing this equation, we have made use of the 0ˆ 0ˆ masses as mj = µj m, where m is a basic mass and µj relation d/dx = (dT/dx )d/dT =(1+ O(2))d/dT . are dimensionless numbers. In what follows, we assume a torsion suspension and focus attention on small angular The presence of the damping term in Eq. (133) ensures deviations θ of the disk relative to its initial angular ori- that free oscillations vanish in the steady-state solution. entation. The apparatus with N = 3 is depicted in Fig. In the absence of additional torques, the steady-state so- 5. Note that the locations of the center of mass and the lution is given by center of suspension are typically different. In practice, the technical challenge of keeping the disk level might be overcome in several ways, perhaps involving rigid suspen- i mr g θ(T ) = 3 0 sion or magnetic support, but this is a secondary issue in 2 2 2 2 2 n I (ω0 − ωn) +4γ ωn the present context and is disregarded here. X Using Eq. (115), the effective potential energy for the × [Epn sin αn − Fn cos αn] sin(ωnT + βn) apparatus can be constructed. Up to an overall constant, we find −[En cos αn + Fn sin αn] cos(ωnT + βn) , 1 xˆ yˆ 2 2 2 (134) V = − 2 mr0[a CN + a SN + r0ω⊕ sin χ(2CN − 1)]θ xˆ yˆ 1 2 2 −mr0(a SN − a CN + 2 aω⊕ sin χSN2)θ. (131) In this expression, we have introduced the following where quantities:

N 2 2 CN = µj cos θj , βn =2γωn/(ωn − ω0) (135) j=1 X N SN = µj sin θj , and, as before, T is the time coordinate in the Sun- j=1 X centered frame deﬁned in Sec. V A 1. Table 5 provides N the amplitudes En, Fn and the phases αn in terms of SN2 = µj sin2θj. (132) quantities deﬁned above. j=1 X 28

Torsion-Pendulum Amplitudes Amplitude Phase XY 1 XX Y Y E2ω = s CN sin χ − 4 (s − s )SN sin2χ 2φ 1 XX Y Y 1 XY F2ω = − 2 (s − s )CN sin χ − 2 s SN sin2χ 2φ Y Z XZ Eω = s CN cos χ − s SN cos2χ φ XZ ZY Fω = −s CN cos χ − s SN cos2χ φ 1 T X 1 T Y E2ω+Ω = − 2 V⊕s CN (1 − cos η) sin χ − 4 V⊕s SN (1 − cos η)sin2χ 2φ 1 T Y 1 T X F2ω+Ω = − 2 V⊕s CN (1 − cos η) sin χ + 4 V⊕s SN (1 − cos η)sin2χ 2φ 1 T X 1 T Y E2ω−Ω = 2 V⊕s CN (1 + cos η) sin χ + 4 V⊕s SN (1 + cos η)sin2χ 2φ 1 T Y 1 T X F2ω−Ω = 2 V⊕s CN (1 + cos η) sin χ − 4 V⊕s SN (1 + cos η)sin2χ 2φ 1 T Y T X 1 2 Eω+Ω = 2 V⊕s CN sin η cos χ + V⊕s SN sin η( 2 − cos χ) φ 1 T Z + 2 V⊕s CN (cos η − 1)cos χ 1 T X T Z 1 2 Fω+Ω = − 2 V⊕s CN sin η cos χ − V⊕s SN (1 − cos η)( 2 − cos χ) φ T Y 1 2 + V⊕s SN sin η( 2 − cos χ) 1 T Z 1 T Y Eω−Ω = 2 V⊕s CN (1 + cos η)cos χ + 2 V⊕s CN sin η cos χ φ T X 1 2 + V⊕s SN sin η( 2 − cos χ) 1 T X T Y 1 2 Fω−Ω = − 2 V⊕s CN sin η cos χ + V⊕s SN sin η( 2 − cos χ) φ T Z 1 2 + V⊕s SN (1 + cos η)( 2 − cos χ) 1 T Y T Z EΩ = − 2 V⊕s SN cos η sin2χ + V⊕s SN sin η sin2χ 0 1 T X FΩ = 2 V⊕s SN sin2χ 0 Table 5. Torsion-pendulum amplitudes.

To estimate the attainable sensitivities to Lorentz vio- disk could be replaced with a ring of holes instead. A lation, we suppose the angular deflection of the apparatus more radical possibility is to change the geometry of the can be measured at the nrad level, which in a different apparatus. For example, one could replace the disk and context has already been achieved with a torsion pendu- masses with a thin rod lying in the horizontal plane and lum [64]. For definiteness, we suppose that the disk ra- hanging from a torsion fiber. If the rod is suspended off dius is R = 10 cm and that the N = 3 masses have equal center by some means while able to rotate freely about magnitude mj = M and are located at radius r0 ≈ R the vertical axis of the fiber, the steady-state oscillations at angular locations θ1 = π/2, θ2 = 4π/3, θ3 = 5π/3. again depend on the coefficients for Lorentz violation. In −2 Assuming a natural oscillation frequency ω0 ≈ 10 Hz fact, the explicit results for this scenario can be obtained and a small damping parameter γ ≈ 10−3, and taking from Eq. (134) and Table 5 by making the following re- i3 = 2/5 and m ≈ M, we find the apparatus could placements: attain sensitivity to the coefficients sJK at the level of 15 TJ 11 1 parts in 10 and to the s coefficients at parts in 10 . r0 → 2 (L − 2d), These are idealized sensitivities, which are unlikely to be CN → cos θ0, achieved in practice. Nonetheless, the result is of interest S → sin θ . (136) because it is about four orders of magnitude better than N 0 the (realistic) sensitivities that could be attained using Here, L is the total length of the rod, and d is the distance current data from lunar laser ranging. from the end of the rod to the point of suspension. Other Provided the amplitudes in Table 5 can be separately quantities still appear but are reinterpreted: m is the extracted, and assuming measurements can be made with mass of the rod, I is its moment of inertia, and θ0 is two different initial orientations θ1, it follows that these the initial angle the rod subtends to thex ˆ axis. Again, types of torsion-pendulum experiments can measure 4 co- the practical issue of suspending a rod off center could efficients in sJK and all 3 independent coefficients in conceivably be addressed with a rigid support system or sTJ . As before, major systematics may include solar and magnetic levitation. The former method might allow an lunar tidal effects. increase in sensitivity by reducing the effective torsion Modifications of the apparatus proposed above could fiber frequency ω0 of the system. In any case, it would lead to improved sensitivities. One theoretically simple be of definite interest to investigate various experiments possibility is to decrease the effective spring constant of of this type. the system. Different kinds of pendulum could also be Another potentially important issue is whether any of used. For example, the ring of point masses around the the existing torsion-pendulum experiments have sensitiv- 29 ity to Lorentz-violating effects. Consider, for example, an gravity [65, 66] may also be sensitive to these types of experiment designed to search for deviations from New- effects. ton’s law of gravity at sub-millimeter distances [64]. The basic component of this experiment is a pair of disks positioned one above the other and with symmetric rings of D. Gyroscope Experiment holes in each. The upper disk is suspended as a torsion pendulum, in a manner analogous to that depicted in Fig. 5, while the lower one rotates uniformly. A key effect of In curved spacetime, the spin of a freely falling test the holes is to produce a time-varying torque on the pen- body typically precesses relative to asymptotically flat dulum that varies with the vertical separation between spacetime [67, 68]. The primary relativistic effects for the two disks. Measurements of the torques at the rota- a gyroscope orbiting the Earth are the geodetic or de tion frequency and its harmonics offers high sensitivity to Sitter precession about an axis perpendicular to the short-distance deviations from Newton’s law of gravity. orbit and the gravitomagnetic frame-dragging or Lens- The question of interest here is whether the presence of Thirring precession about the spin axis of the Earth. In Lorentz violation would change the predicted newtonian the present context of the post-newtonian metric for the torque in an observable way. A detailed analysis of this pure-gravity sector of the minimal SME given in Eqs. experiment is involved and lies beyond our present scope. (35)-(37), we find that precession also occurs due to However, some insight can be gained by using a simpli- Lorentz violation. In particular, there can be precession fied model that treats the rings of holes as two rings of of the spin axis about a direction perpendicular to both point masses and comparable radii positioned one above the spin axis of the Earth and the angular-momentum the other, with the lower ring rotating. axis of the orbit. Consider first the case of two point masses m1 and m2 at coordinate locations ~x1 and ~x2. The modified newtonian potential V predicted by the pure-gravity sector of 1. Theory the minimal SME is To describe the motion of an orbiting gyroscope, Gm1m2 1 ˆj kˆ ˆjkˆ V = − (1 + 2 xˆ xˆ s ), (137) we adopt standard assumptions [62] and use the |~x1 − ~x2| Fermi-Walker transport equation given in coordinate- independent form by wherex ˆ = (~x1 − ~x2)/|~x1 − ~x2|. The non-newtonian effects are controlled by the SME coefficients for Lorentz ∇uS = u(a · S), (138) violation sˆjkˆ. Although Eq. (137) maintains the inverse- distance behavior of the usual newtonian potential, the where u is the four-velocity of the gyroscope, S is the associated force is misaligned relative to the vectorx ˆ be- spin four-vector and a is the four-acceleration. The dot tween the two point masses. It is therefore conceivable denotes the inner product with the metric two-tensor g. that an unconventional vertical dependence of the torque To obtain the equation of motion for the spin vector S, might arise in the experiment discussed above. To inves- we first construct a local frame eµˆ that is comoving with tigate this rigorously, the result (137) would be inserted S but is nonrotating with respect to the fixed stars in the into the computer code that models the experiment and asymptotically flat spacetime. The construction of this determines the predicted torque [64]. frame proceeds as in Sec. V A, using the post-newtonian In the context of the simplified model involving two metric in Eqs. (89). The basis can be taken from Eq. (93) rings of point masses, the difference between the newto- by setting the rotation terms to zero. Using Eq. (138), nian and Lorentz-violating torques can be explored as- we can then find the proper-time derivative of S · e . It ˆˆ ˆj suming a single nonzero coefficient sjk in Eq. (137). We is given by examined the Fourier components of the time-varying torque on the upper disk at harmonics of the rotational dS · eˆj = S · (∇ue ). (139) frequency of the bottom disk. The amplitudes of these dτ ˆj Fourier components depend sensitively on the number of masses used, on the geometry of the experiment, and This represents the proper time rate of change of the on the particular harmonic considered. This indicates spatial spin vector as measured in a comoving but non- that a careful analysis involving a detailed model of the rotating frame. actual experiment is necessary for definitive predictions. To display the dependence of the spin precession on the However, the results suggest that Lorentz-violating ef- underlying geometry, which includes Lorentz-violating fects from certain coefficients sˆjkˆ of order 10−3 to 10−5 terms in the metric, the right-hand side of Eq. (139) can could be discernable at a separation of about 100 microns be expressed in the Sun-centered frame. This requires in experiments of this type, at the currently attainable expressing the components of S and ∇ueˆj to the appro- torque sensitivities of about 10−17 Nm. Other experi- priate post-newtonian order. Of particular use for this ments studying deviations from short-range newtonian purpose are the Sun-centered frame components of eˆj , 30 which are given by be converted into ones involving V J by using the identity (30). Also, contributions from the potential Y JKL can T J J 1 2 e ˆj = δ ˆj [v (1 + hT T + 2 v ) be converted into ones involving only the other potentials by virtue of the antisymmetrization in Eq. (143) and the + 1 h vK + h ]+ O(4), 2 JK TJ identity (33). The five necessary Earth potentials can J K 1 1 J K ~ e ˆj = δ ˆj [δJK − 2 hJK + 2 v v ]+ O(4). (140) be written as functions of the spatial position X in the Sun-centered frame, given by We can then establish that GM⊕ ST = vJ SJ + O(2), U = , |X~ − R~| J J ˆj S = δ S + O(2), J K ˆj JK GM⊕(X − R) (X − R) T J J U = ue ~ ~ 3 (∇ ˆj ) = δ ˆj a + O(3), |X − R| K J 1 K 1 J (J K) GI u ⊕ J K (∇ eˆj) = δ ˆ[ 4 v ∂J hT T − 4 v ∂K hT T + a v − [3(X − R) (X − R) j ~ ~ 5 L 3|X − R| +∂[J hK]T + v ∂[J hK]L]+ O(4). (141) − δJK |X~ − R~|2], Combining Eqs. (140) and (141) via the expression J JKL K L Ξ Π J GM⊕V⊕ Gǫ J⊕ (X − R) S · (∇ueˆ)= S (∇ueˆ) gΞΠ (142) V = + , j j |X~ − R~| 2|X~ − R~|3 J K KJ yields the general expression for the rate of change of W = V⊕ U + ..., the spatial spin vector of a gyroscope, valid to post- XJKL = V J U KL + ..., (145) newtonian O(3). It is given by ⊕

dS ~ ˆ ˆj kˆ K J [J K] 1 K 1 J where R = RR is the Earth’s location in the Sun-centered = S δ ˆδ ˆ v a + v ∂J hT T − v ∂K hT T J dτ k j 4 4 frame and V⊕ is its velocity. Explicit expressions for these quantities are given in Eq. (A9) of Appendix A + ∂ h + vL∂ h . (143) [J K]T [J K]L and in the associated discussion. In the limit of the usual general-relativistic metric, this The ellipses in Eqs. (145) represent omitted pieces of agrees with existing results [30, 62]. the potentials arising from the purely rotational component of the Earth’s motion, which generate gravitomagnetic effects involving Lorentz violation. A crude 2. Gyroscope in Earth orbit estimate reveals that these gravitomagnetic terms are suppressed relative to the associated geodetic terms displayed in Eqs. (145) by about two orders of magnitude. In this subsection, we specialize to a gyroscope in a As a result, although gravitomagnetic frame-dragging near-circular orbit around the Earth. For this case, the terms are crucial for tests of conventional general relativ- acceleration term vanishes. ity [30, 62], they can be neglected for studies of the dom- The first step is to determine the contributions to the inant Lorentz-violating contributions to the spin preces- metric potentials (28) from the various relevant bodies in sion. For simplicity, the frame-dragging precession effects the solar system. For simplicity, we focus on the poten- involving Lorentz violation are omitted in what follows. tials due to the Earth, neglecting any contributions from the Sun, Moon, or other sources, and we take the Earth For use in Eq. (143), these potentials must be eval- as a perfect sphere rotating uniformly at frequency ω and uated along the worldline of the gyroscope. At post- newtonian order, this worldline is adequately described with spherical moment of inertia I⊕ give by Eq. (120). For convenience in what follows, we introduce a quantity by the expressions for an arbitrary satellite orbit given in Appendix A. Thus, the value of (X~ − R~) along the ˜i⊕ related to I⊕ along with corresponding quantities ˜i(α) defined for any real number α by the equations worldline can be written as ~ ~ ˜ I⊕ (X − R)gyro = r0ρ,ˆ (146) i⊕ = 2 , M⊕r0 ˜ ˜ whereρ ˆ is given by Eq. (A6). To post-newtonian order, i(α) = 1+ αi⊕, (144) J J J the gyroscope velocity vgyro can be taken as vgyro = V⊕ + J J where r0 is the mean orbital distance to the gyroscope. v0 , where v0 is the mean orbital velocity defined by ~v0 = The explicit form is required for five of the six d(r0ρˆ)/dT = v0τˆ. The normal to the plane of the orbit types of metric potentials in Eq. (28). Some general- isσ ˆ =ρ ˆ × τˆ. relativistic contributions from the potential W J involv- Inserting the potentials (145) into Eq. (143) and eval- K K ing the Earth’s angular momentum J⊕ = 2I⊕ω /3 can uating along the worldline yields a result that can be 31 separated into two pieces, terms can be shown to be unobservable in principle by a consideration of the physical precession referenced to the E s dSˆ dSˆ dSˆ fixed stars [30, 69]. It is an open question whether this j = j + j . (147) dτ dτ dτ stronger result remains true in the presence of Lorentz violation. The first piece is the standard result from general rela- Combining the above results yields the vector equation tivity, given by for the secular evolution of the gyroscope spin. It can be written in the form dSE ˆj J K [K J] [K J] = gδ ˆj S [(−3v0 ρˆ + V⊕ ρˆ ) dS~ dτ = gv0Ω~ × S.~ (153) 2J L dt − ⊕ (ǫJKL +3ǫLM[J ρˆK]ρˆM )], M⊕r0 The secular precession frequency Ω~ is comprised of two (148) pieces, J J J where the reference gravitational acceleration g is now Ω = ΩE +Ωs . (154) J g = GM /r2. (149) The first term ΩE contains precession due to conventional ⊕ 0 effects in general relativity, and it is given by

The second piece is proportional to the coefficients for K JK J 3 J J⊕ JK J K Lorentz violation s , and it takes the form ΩE = 2 σˆ + (δ − 3ˆσ σˆ ). (155) 2M⊕r0v0 dSs ˆj J K T [K J] ˜ L[J K] L The second term ΩJ contains contributions from the co- = gδ ˆS 2s ρˆ +4i(−1)s v ρˆ s dτ j 0 efficients for Lorentz violation sJK , and it has the form [K J] h 5˜ T T 9˜ LM L M +v0 ρˆ (− 2 i(3/5)s − 2 i(−5/3)s ρˆ ρˆ ) J 9 ˜ T T ˜ KL K L J Ωs = 8 (i(−1/3)s − i(−5/3)s σˆ σˆ )ˆσ ˜ L[K J] L L[K J] L ˜ L[K J] L +i(1)s ρˆ v0 − 2s ρˆ V⊕ + i(−1)s V⊕ ρˆ 5˜ JK K + 4 i(−3/5)s σˆ . (156) +V [K ρˆJ](− 1 sT T˜i + 3˜i sLM ρˆLρˆM ) . ⊕ 2 (−1) 2 (−5/3) In the limit of no Lorentz violation, Eq. (154) reduces (150)i as expected to the conventional result (155) of general relativity involving the geodetic precession arising Among the effects described by the result (147) are os- from the spacetime curvature near the Earth and the cillations of the orientation of the gyroscope spin vector frame-dragging precession arising from the rotation of that occur at frequencies ω, ω ± Ω⊕, and higher harmon- the Earth. ics. At the frequency ω, these oscillations change the 2 angular orientation of a gyroscope by roughly δθ ≈ 2πv0. For a gyroscope in low Earth orbit, this is δθ ≈ 8 × 10−4 3. Gravity Probe B arcseconds, well below observable levels for existing sensitivity to nonsecular changes. Gravity Probe B (GPB) [70] is an orbiting-gyroscope We therefore focus instead on the dominant secular experiment that has potential sensitivity to certain com- contributions to the motion of the gyroscope spin. Aver- binations of coefficients contained in Eq. (156). Our gen- aging various key quantities over an orbital period yields eral results (153) for the spin precession can be special- the following results: ized to this experiment. In terms of the angles in Fig. 4, ◦ ◦ J K 1 2 JK J K the GPB orbit is polar with α = 163 and β = 90 . One < r0 r0 > = 2 r0(δ − σˆ σˆ ), consequence of the polar orbit is that the normal vector J K 1 JKL L σˆJ to the orbital plane appearing in Eq. (156) reduces to < r0 v0 > = 2 r0v0ǫ σˆ , K [L M] N 1 3 LMP P KN K N the simpler form < r0 r0 v0 r0 > = 4 r0v0ǫ σˆ (δ − σˆ σˆ ), J J K L M σˆ = (sin α, − cos α, 0). (157) < r0 r0 V⊕ r0 > ≈ 0, J K ~ < r0 V⊕ > ≈ 0, The terms in Eq. (153) proportional toσ ˆ×S act to pre- < rJ > = 0. (151) cess the gyroscope spin in the plane of the orbit. They 0 include the geodetic precession in Eq. (155) predicted In these equations, the vectorσ ˆJ is given explicitly as by general relativity, along with modifications due to Lorentz violation. The magnitude of the precession is σˆJ = (sin α sin β, − cos α sin β, cos β), (152) determined by the projectionσ ˆ · Ω.~ The explicit form of the relevant projection of Ω~ arising due to Lorentz viola- where the angles α and β are those defined in Sec. VB tion is found to be and Fig. 4. Note that the averaging process eliminates J σ 9˜ T T 1˜ JK J K the terms involving V⊕ . In the conventional case, such Ωs = 8 i(−1/3)s + 8 i(9)s σˆ σˆ . (158) 32

The terms in Eq. (153) proportional to Jˆ × S~, where tional properties of the spacetime regions close to the J~ is the Earth’s angular momentum, cause precession of pulsar and its companion. Nonetheless, provided the typ- the gyroscope spin in the plane perpendicular to J~. Since ical radius of the pulsar and companion are much smaller J~ lies along Zˆ, this is the XY plane of Fig. 4. These than the average orbital distance between them, the or- terms include the frame-dragging effect in Eq. (155) pre- bital dynamics can be modeled with an effective post- dicted by general relativity, together with effects due to newtonian description. Lorentz violation. These have magnitude determined by One standard approach is to construct a modified the projection Zˆ·Ω.~ For the contributions due to Lorentz Einstein-Infeld-Hoffman (EIH) lagrangian describing the violation, we find the explicit form post-newtonian dynamics of the system [30]. In principle, an analysis of the effects of Lorentz violation could Z 5˜ ZK K Ωs = 4 i(−3/5)s σˆ . (159) be performed via a modified EIH approach based on the In addition to the above, however, Lorentz violation action of the pure-gravity sector of the minimal SME. induces a third type of precession that is qualitatively This would involve generalizing Eq. (54) to allow for different from those predicted by general relativity. This the usual EIH effects, along with modifications to the is a precession of the gyroscope spin in the plane with Lorentz-violating behavior arising from the structure of normaln ˆ alongσ ˆ × Jˆ. With the definitions shown in the pulsar and its companion. A detailed analysis along Fig. 4, this normal is given byn ˆ = (cos α, sin α, 0). The these lines would be of interest but lies beyond the scope magnitude of the associated precession is given byn ˆ · Ω.~ of the present work. We obtain In fact, a simpler approach suffices to extract key features arising from Lorentz violation. Instead of a full n 5˜ JK J K Ωs = 4 i(−3/5)s nˆ σˆ . (160) EIH-type treatment, we adopt in what follows a sim- The GPB experiment is expected to attain sensitivity plified picture and consider only the effects arising in a to secular angular precessions of the order of 5 × 10−4 point-mass approximation for each of the two bodies in arcseconds per year. If the spin precession in the three the system. To retain some generality and an EIH flavor orthogonal directions can be disentangled, GPB could in the treatment, we interpret the gravitational constant, produce the first measurements of three combinations of the masses, and the coefficients for Lorentz violation as the sJK coefficients, including one involving precession effective quantities [74]. However, the results disregard about then ˆ direction. The attainable sensitivity to sJK details of possible strong-field gravitational effects or con- coefficients is expected to be at the 10−4 level. sequences of nonzero multipole moments and tidal forces across the pulsar and companion. The reader is cautioned that the corresponding conventional effects from E. Binary pulsars newtonian theory can be crucial for studies of binary pulsars, so they should be included in a complete analysis. Binary pulsars form a useful testing ground for general Nonetheless, the results obtained here offer a good base- relativity [71, 72]. This subsection examines the possi- line for the sensitivities to Lorentz violation that might bility of probing SME coefficients for Lorentz violation be achieved through observations of binary pulsars. using pulsar timing data [73]. We find that Lorentz vi- The analysis that follows makes use of the standard olation induces secular variations in the orbital elements keplerian characterization of an elliptical orbit [75]. Some of binary-pulsar systems, along with modifications to the of the key quantities are displayed in Fig. 6. In this figure, standard pulsar timing formula. The discussion is sep- the point P is the location of the pulsar at time t, and arated into five parts: one establishing the basic frame- it lies a coordinate distance r from the focus F of the work and assumptions, a second presenting the secular elliptical orbit. The line segments CA and CB are the changes, a third describing the timing formula, a fourth semimajor axes, each of length a. The eccentric anomaly determining some experimental effects, and a final part E is defined as the angle ACQ, where Q is the point on explaining the transformation between the frame used to the enclosing circle with the same abcissa as P. The true study the properties of the binary pulsar and the stan- anomaly f is defined as the angle AFP, while ω is the dard Sun-centered frame. angle subtended between the line of ascending nodes FN The reader is cautioned that the notation for the and the major axis of the ellipse. binary-pulsar systems in this subsection is chosen to The general solution of the elliptic two-body problem match the literature and differs in certain respects from is given in terms of the time t and six integration con- the notation adopted elsewhere in the paper. This nota- stants called the orbital elements. Three of the latter tion is described in Sec. VE1. are associated with inherent properties of the ellipse: the semimajor axis length a, the eccentricity e, and a phase variable l0 called the mean anomaly at the epoch. The 1. Basics other three describe the position of the ellipse in the reference coordinate system: the inclination i, the longitude The general problem of describing the dynamics of bi- of the ascending node Ω, and the angle ω defined above. nary pulsars is complicated by the strong-field gravita- The orbital frequency is denoted n, and it is related as 33

where the definitions of ~r, r, ~v, M and δm parallel those adopted in Table 1 of Sec. V B. The ellipses here represent corrections from multipole moments, from general relativistic effects at O(4), and from Lorentz-violating effects at O(4). Note that enhancements of secular effects stemming from general relativistic corrections at O(4) are known to occur under some circumstances [74], but the issue of whether O(4) enhancements can occur in the SME context is an open question lying beyond our present scope. To establish the observational effects of Lorentz violation that are implied by Eq. (162), we adopt the method of osculating elements [75]. This method assumes the relative motion of the binary pulsar and companion is instantaneously part of an ellipse. The subsequent motion is obtained by letting the orbital elements vary with time while keeping the equations of the ellipse fixed. In FIG. 6: Definitions for the elliptical orbit. the present instance, the unperturbed ellipse solves the equation d2rj GM usual to the period Pb = 2π/n. Standard expressions = − (1 + 3 s00)rj . (163) relate these various quantities. For example, the eccen- dt2 r3 2 tric anomaly E is related to the mean anomaly l0 at the Note that this implies the frequency and the semimajor epoch t = t0 by axis of the orbit are related by

2 3 3 00 E − e sin E = l0 + n(t − t0). (161) n a = GM(1 + 2 s ). (164) The reader is referred to Refs. [75] for further details. The remaining terms are then viewed as a perturbing acceleration α′j , given in this case by

′j GM jk k 3 GM kl k l j 2. Secular changes α = s r − s rˆ rˆ r r3 2 r3 2Gδm + [s0kvkrj − s0j vkrk]. (165) We begin the analysis with the derivation of the sec- r3 ular changes in the orbital elements of a binary-pulsar system that arise from Lorentz violation. Adopting the The idea is to extract from this equation the variations point-mass limit as described above, Eq. (52) can be used in the six orbital elements a, e, l0, i, Ω, ω describing the to determine the relative coordinate acceleration of the instantaneous ellipse for the relative motion. pulsar and companion, which are taken to be located at The elliptical orbits of the pulsar and companion, referred to the chosen post-newtonian frame, are illustrated ~r1 and ~r2, respectively. The result can be written in the form of Eq. (99), with the quadrupole moment set to zero in Fig. 7. The relative motion can also be represented by and the effects from external bodies omitted. an ellipse in the same plane. To specify the orientation It is convenient for the analysis to choose a post- of the orbit, three linearly independent unit vectors are ~ newtonian frame with origin at the center of mass of needed. A natural set includes one vector k normal to the binary-pulsar system. The newtonian definition of the orbital plane, a second vector P~ pointing from the the center of mass suffices here because the analysis is focus to the periastron, and a third vector Q~ = ~k × P~ perturbative. The frame is taken to be asymptotically perpendicular to the first two. This triad of vectors can inertial as usual, so that the coefficients for Lorentz vi- be expressed in terms of the spatial basis ej for the post- olation obey Eqs. (11). The coordinates in this frame newtonian frame. Explicitly, these vectors are given by are denoted xµ = (t, xj ), and the corresponding basis P~ = (cosΩcos ω − cos i sin Ω sin ω)~e1 is denoted eµ. The issue of matching the results to the Sun-centered frame is addressed in Sec. V E 5 below. +(sinΩcos ω + cos i cosΩsin ω)~e2

In this post-newtonian binary-pulsar frame, the rel- + sin i sin ω~e3, ative acceleration of the pulsar and companion can be Q~ = −(cos Ω sin ω + cos i sinΩcos ω)~e written 1 +(cos i cosΩcos ω − sin Ω sin ω)~e2 d2rj GM 3 00 j jk k 3 kl k l j + sin i cos ω~e3, 2 = − 3 [(1 + 2 s )r − s r + 2 s rˆ rˆ r ] dt r ~ 2Gδm k = sin i sinΩ~e1 − sin i cosΩ~e2 + [s0kvkrj − s0j vkrk]+ ..., (162) r3 +cos i~e3. (166) 34

post-newtonian frame are dω n ε h i = − s sin ω dt tan i(1 − e2)1/2 e2 kP h (e2 − ε) δm 2naε + s cos ω − s cos ω e2 kQ M e k (e2 − 2ε) i −n (s − s ) 2e4 P P QQ h δm 2na(e2 − ε) + s , M e3(1 − e2)1/2 Q di nε nε i h i = s cos ω − s sin ω dt e2(1 − e2)1/2 kP e2 kQ δm 2n2aeε + s sin ω, M e2(1 − e2)1/2 k dΩ n ε h i = s sin ω dt sin i(1 − e2)1/2 e2 kP h (e2 − ε) δm 2naε + s cos ω − s cos ω . e2 kQ M e k FIG. 7: Elliptical orbits in the post-newtonian frame of the i (170) binary pulsar. Finally, the secular evolution of the mean anomaly at the epoch is

dl0 1 In terms of these vectors, the unperturbed elliptical orbit h i = 2 n(sP P + sQQ) can be expressed as dt n(e2 − 2ε)(1 − e2 − ε) + 4 (sP P − sQQ) 2 2e a(1 − e ) ~ ~ δm 2n2a(e2 − ε) ~r = (P cos f + Q sin f). (167) + s . (171) 1+ e cos f M e3 Q In all these expressions, the coefficients for Lorentz vio- Following standard methods, we insert the solution ~r lation with subscripts P , Q, and k are understood to be for the elliptic orbit given by Eq. (167) into the expres- appropriate projections of sµν along the unit vectors P~ , 0j j sion (165) for the perturbative acceleration, project the Q~ , and ~k, respectively. For example, sP ≡ s P , while jk j k result as appropriate to extract the orbital elements, and sQQ ≡ s Q Q . average over one orbit. The averaging is performed via The analysis of the timing data from binary-pulsar sys- integration over the true anomaly f, which serves as the tems permits the extraction of information about the sec- phase variable. Details of this general procedure are dis- ular changes in the orbital elements. In the present con- cussed in Refs. [75]. text, performing this analysis for a given binary pulsar After some calculation, we find the secular evolutions would permit measurements of the projected combina- of the semimajor axis and the eccentricity are given by tions of coefficients for Lorentz violation appearing in the equations above. This is discussed further in Sec. V E 4. Matching the projections of the coefficients to standard da h i = 0, components in the Sun-centered frame can be performed dt with a Lorentz transformation, described in Sec. V E 5. de (e2 − 2ε) δm naε h i = 2n(1 − e2)1/2 s − s , dt 2e3 P Q M e2 P h (168)i 3. Timing formula

Accompanying the above secular changes in the or- where the eccentricity function ε(e) is given by bital elements are modiﬁcations to the standard pulsar timing formula arising from Lorentz violation in the ε(e) = 1 − (1 − e2)1/2. (169) curved spacetime surrounding the binary pulsar. The basic structure of the pulsar timing formula can be derived by considering the trajectory of a photon traveling from The secular evolutions of the three orbital elements as- the emitting pulsar to the Earth. Together with the rela- sociated with the orientation of the ellipse relative to the tion between proper time at emission and arrival time on 35

Earth, this trajectory is used to determine the number All effects from the pulsar itself contribute to constant of pulses as a function of arrival time. For simplicity, the unobservable shifts and have been neglected in this result. 3 00 derivation that follows neglects any effects on the photon Note that the factors of (1+ 2 s ) appearing in Eq. (175) trajectory from gravitational influences of bodies in the merely act to scale G and hence are unobservable in the solar system and effects from the motion of the detector present context. For simplicity in what follows, we absorb during measurement. them into the definition of G. The photon trajectory is determined by the null condi- To integrate Eq. (175), it is convenient to change vari- tion ds2 = 0. The relevant metric is the post-newtonian ables from the coordinate time to the eccentric anomaly result of Eqs. (35), (36), (37) taken at O(2) and incor- E. This makes it possible to insert the expression porating effects of the perturbing companion of mass m2 at ~r2. In practice, the time-delay effects from the pulsar ~r = a(cos E − e)P~ + a(1 − e2)1/2 sin E Q~ (176) itself produce only a constant contribution to the result and can be disregarded. The null condition can then be into Eq. (175) and perform the integration over E. Up solved for dt and integrated from the emission time tem to constants, we obtain at the pulsar located at ~r1 to the arrival time tarr at the Earth at ~rE . Assuming |~rE | ≫ |~r1| and |~rE | ≫ |~r2|, we Gm P Gm P s − s (1 − e2) find the time delay t − t for the photon is given by τ = t − 2 b E − 2 b P P QQ E arr em 2πa 4πa e2 jk j k 2rE Gm2Pb t − t = |~r − ~r | + Gm ∆ nˆ nˆ ln . − (sP P − sQQ)F1(E) arr em E 1 2 r +ˆn · ~r 4πa Gm P (172) − 2 b (1 − e2)1/2s F (E). (177) 2πa P Q 2 This result is a simplification of the general time-delay formula, which is derived in Sec. V F 2 below. In Eq. The functions F1 and F2 appearing in the last two terms (172), rE = |~rE| is the coordinate distance to the Earth, are given by ~r = ~r1 −~r2 is the pulsar-companion separation as before, andn ˆ is a unit vector along ~r − ~r . The quantity ∆jk (1 − e2) sin E 2(1 − e2)1/2 E 1 F (E) = − contains combinations of coefficients for Lorentz violation 1 e(1 − e cos E) e2 andn ˆ, given explicitly in Eq. (198) of Sec. V F 2. 1+ e 1/2 We seek an expression relating the photon arrival time × arctan tan 1 E , 1 − e 2 to the proper emission time. The differential proper time h i dτ as measured by an ideal clock on the surface of the e − cos E ln(1 − e cos E) F2(E) = − . (178) pulsar is given in terms of the differential coordinate time e(1 − e cos E) e2 dt by the expression Note that in the circular limit e → 0 the result (177) 3 00 1 jk jk 1 2 dτ = dt[1 − (1 + 2 s )U − 2 s U − 2~v ]. (173) remains finite, despite the seeming appearance of poles in e. The constant in the definition (161) is fixed by The potentials appearing in this expression include those requiring from both the pulsar and its companion, and they are to be evaluated along the worldline of the ideal clock. It suffices here to display the explicit potentials for the E − e sin E = n(tarr − rE ). (179) companion, so that The integration assumes that the positions of the pulsar Gm U = 2 + U , and its companion are constant over the photon travel r 1 time near the binary-pulsar system. Gm rj rk The desired expression for the proper time τ is ob- U jk = 2 + U jk, (174) r3 1 tained by combining the result (172) with Eq. (177) to jk yield an expression as a function of the arrival time. It where U1 and U1 are the contributions from the pulsar. turns out that the Lorentz-violating corrections to the In Eq. (173), ~v is the velocity of the clock, which can Shapiro time delay, which arise from the last term of Eq. be identified with the velocity ~v1 of the pulsar. Using (172), are negligibly small for the cases of interest and so the time derivative of Eq. (167) to obtain ~v1 and explic- can be disregarded. We find itly evaluating Eq. (173) along the pulsar trajectory, we obtain τ = tarr − A(cos E − e) − (B + C) sin E 2 dτ 3 00 Gm2 2π (A sin E −B cos E) = 1 − (1 + 2 s ) − [A(cos E − e)+ B sin E] dt Mr Pb (1 − e cos E) jk j k πaa 3 00 1 1 s r r 1 2 −Gm2 (1 + s ) + . − [(sP P − sQQ)F1(E)+2 (1 − e ) sP QF2(E)]. 2 r 2 r3 Pb h i (175) p (180) 36

In this expression, a1 = am2/M and the quantities A, B, from the variations of the orbital elements are considered, and C are given by the number of independent coefficients for Lorentz violation to which binary-pulsar systems are sensitive can A = a1 sin i sin ω, be quantified. Through the measurement ofω ˙ ande ˙, a 2 B = (1 − e ) a1 sin i cos ω, given binary-pulsar system is sensitive to at least 2 combinations of coefficients. These are 2πaa s − s (1 − e2) C = p 1 e + P P QQ . (181) P 2e δm 2naeε b s = s − s , h i e P Q M (e2 − 2ε) P The reader is reminded that all these equations are to 2 1/2 be used in conjunction with the method of osculating sω = skP sin ω + (1 − e ) skQ cos ω elements, for which the orbital elements vary in time. δm − 2nae s cos ω In practice, the application of the above results in- M k volves inserting the result (180) into an appropriate (1 − e2)1/2(e2 − 2ε) model for the number N(τ) of emitted pulses as a func- + tan i 2 (sP P − sQQ) tion of pulsar proper time τ. The model for N(τ) is 2e ε δm (e2 − ε) usually of the form + 2na tan i s . (184) M eε Q 2 N(τ)= N0 + ντ +ντ ˙ /2+ ..., (182) In principle, sensitivities to the secular changes in i and where ν is the frequency of the emitted pulses. The re- Ω might increase the number of independent coefficients sult of the substitution is an expression representing the that could be measured in a given system. µν desired timing formula for the number of pulses N(tarr) Since there are 9 independent coefficients in s , it as a function of arrival time. is worth studying more than one binary-pulsar system. Typically, distinct binary-pulsar systems are oriented differently in space. This means the basis vectors P~, Q~ , 4. Experiment ~k differ for each system, and hence the dependence on the coefficients for Lorentz violation in the Sun-centered We are now in a position to consider some experimental frame also differs. Examination of Eq. (184) reveals that implications of our results. Consider first the timing for- the combination sP P + sQQ is inaccessible to any orien- mula (182) with the substitution of the expression (180) tation and that 5 independently oriented binary-pulsar for the proper time. Modifications to this formula aris- systems can in principle measure 8 independent coeffi- ing from Lorentz violation can be traced to two sources. cients in sµν . If information from i and Ω can also be First, the terms in Eqs. (180) and (181) proportional to extracted, then fewer than 5 binary-pulsar systems might sP P , sQQ, and sP Q represent direct corrections to the suffice. proper time arising from Lorentz violation. Second, the The error bars from existing timing analyses offer structural orbital elements entering the expression for the a rough guide to the sensitivities of binary pulsars to proper time τ acquire secular Lorentz-violating correc- Lorentz violation. As an explicit example, consider the tions, given in Eqs. (168) and (170). Both these sources binary pulsar PSR 1913+16. The errors one ˙ andω ˙ are of Lorentz violation affect the predictions of the timing roughly 10−14 and 10−7, respectively [77]. These imply −9 formula. the following sensitivities to Lorentz violation: se <∼ 10 In applications, the timing formula N can be regarded −11 and sω <∼ 10 . Interesting sensitivities could also be as a function of a set of parameters achieved by analysis of data from other pulsar systems [78, 79]. N = N(Pb, P˙b, e, e,˙ ω, ω,...˙ ). (183)

One approach to measuring the coefficients for Lorentz violation would be to fit the observed number of pulses 5. Transformation to Sun-centered frame N as a function of arrival time tarr using a least-squares method. In principle, an estimate of the sensitivity of In the preceding discussions of tests with binary pul- this approach could be performed along the lines of the sars, all the coefficients for Lorentz violation enter as pro- analysis for the conventional timing formula in Ref. [76]. jections along the triad of vectors P~, Q~ , ~k defining the However, we adopt here a simpler method that nonethe- elliptical orbit. These projected coefficients involve the less provides estimates of the sensitivities that might be post-newtonian frame of the binary-pulsar system. The attainable in a detailed fitting procedure. explicit construction of the transformation that relates To obtain these simple estimates, we limit attention to these projections to the standard coefficients sΞΠ in the the secular changes in the orbital elements e and ω given Sun-centered frame is given next. in Eqs. (168) and (170). Values fore ˙ andω ˙ obtained The first step is to express the various projections of from the timing analysis can be used to probe combina- the coefficients directly in terms of the orbital angles i,Ω tions of coefficients in sµν . Assuming only measurements and ω and the coordinate system xµ of the binary pulsar. 37

This involves substitution for the projection vectors using F. Classic Tests Eqs. (166). For example, for the projection sP P we have j k jk Among the classic tests of general relativity are the sP P ≡ P P s 11 2 precession of the perihelia of the inner planets, the time- = s (cosΩcos ω − cos i sin Ω sin ω) delay effect, and the bending of light around the Sun. +s22(sinΩcos ω + cos i cosΩsin ω)2 The sensitivities to Lorentz violation attainable in ex- +s33 sin2 i sin2 ω isting versions of these tests are typically somewhat less +2s12(cosΩcos ω − cos i sin Ω sin ω) than those discussed in the previous sections. Nonethe- less, an analysis of the possibilities reveals certain sen- ×(sinΩcos ω + cos i cosΩsin ω) sitivities attaining interesting levels, as described in this +2s13 sin i sin ω(cosΩcos ω − cos i sin Ω sin ω) subsection. +2s23 sin i sin ω(sinΩcos ω + cos i cosΩsin ω). (185) 1. Perihelion shift The resulting expressions give the projected coefficients in terms of the coefficients sjk, with components in the To obtain an estimate of the sensitivity to Lorentz vi- basis eµ of the binary pulsar. olation offered by studies of perihelion shifts, the results The conversion to coefficients sΞΠ in the Sun-centered from Sec. VE can be adapted directly. The reference frame can be accomplished with a Lorentz transforma- plane can be chosen as the ecliptic, so the inclination i is tion. We have small for the inner planets. We seek the perihelion shift with respect to the sµν =Λµ Λν sΞΠ. (186) Ξ Π equinox. The angular locationω ˜ of the perihelion with The Lorentz transformation consists of a rotation to align respect to the equinox is given byω ˜ = ω + Ωcos i. The the axes of the binary-pulsar and Sun-centered frames, change inω ˜ per orbit can be obtained by appropriate use along with a boost to correct for the relative velocity. The of Eqs. (170) with δm/M ≈ 1. Neglecting terms propor- latter is taken as approximately constant over the time tional to sin i, we find that the shift in perihelion angle scales of relevance for the experimental observations. per orbit is The components of the Lorentz transformation are (e2 − 2ε) 2na(e2 − ε) given by ∆˜ω = −2π (s − s )+ s . 2e4 P P QQ e3(1 − e2)1/2 Q Λ0 = 1, h i T (189) 0 J Λ J = −β , Λj = −(R · β~)j , Substituting the values of e and na for Mercury and T the Earth in turn, we find the corresponding perihelion j jJ Λ J = R . (187) shifts in arcseconds per century (C) are given by

J In this expression, β is the boost of the binary-pulsar ω˜˙ ≈ 7 × 107′′ s C−1, system as measured in the Sun-centered frame. The ma- ' ' jJ 7′′ −1 trix with components R is the inverse of the rotation ω˜˙ ⊕ ≈ 2 × 10 s⊕ C . (190) taking the basis eΞ for the Sun-centered frame to the basis eµ for the binary-pulsar frame. Explicitly, these In these expressions, the perihelion combinations of SME components are coeﬃcients for Mercury and the Earth are deﬁned by

xX −3 R = cos γ cos α − sin γ sin α cos β, s' ≈ (sP P − sQQ) − 6 × 10 sQ, xY R = cos γ sin α + sin γ cos α cos β, −2 s⊕ ≈ (sP P − sQQ) − 5 × 10 sQ. (191) RxZ = sin γ sin β, RyX = − sin γ cos α − cos γ sin α cos β, The reader is cautioned that the vector projections of µν ~ ~ RyY = − sin γ sin α + cos γ cos α cos β, s along P and Q differ for Mercury and the Earth. yZ Indeed, Eqs. (166) reveal that they differ by the values of R = cos γ sin β, i, ω and Ω, once again indicating the value of considering RzX = sin β sin α, more than one system for measurements separating the RzY = − sin β cos α, independent values of sµν . RzZ = cos β. (188) The error bars in experimental data for the perihelion shifts of Mercury and the Earth provide an estimated In these expressions, the angles α, β, and γ are Euler attainable sensitivity to Lorentz violation. For example, angles defined for the rotation of eJ to ej as follows: adopting the values from Refs. [30, 72], the error bars on first, rotate about eZ by γ; next, rotate about the new the perihelion shift of Mercury and the Earth are roughly Y axis by α; finally, rotate about the new Z axis by γ. given by 0.043′′ C−1 and 0.4′′ C−1, respectively. Taking 38 these values as an upper bound on the size of the observ- The terms ∆1 and ∆2 on the right-hand side are given able shifts in Eq. (190), we obtain attainable sensitivities by −9 −8 of s < 10 and s⊕ < 10 . ' ∼ ∼ ∆ = Gm ∆jknˆj nˆk For simplicity in the above, attention has been focused 1 2 on Mercury and the Earth. However, the varying ori- |ntˆ + ζ~(t )| + t +ˆn · ζ~(t ) × ln arr arr arr arr entations i, ω, Ω of other bodies in the solar system ~ ~ " |ntˆ em + ζ(tem)| + tem +ˆn · ζ(tem) # suggests examining the corresponding perihelion shifts might yield interesting sensitivities on independent coef- (195) ficients for Lorentz violation. For example, observations and of the asteroid Icarus have been used to verify to within ′′ −1 jk 20 percent the general-relativistic prediction of 10 C Gm2∆ j k 2 2 2 ∆2(t) = nˆ nˆ [((ˆn · ζ~) − d~ )t +ˆn · ζ~ζ~ ] for the perihelion shift [80]. Translated into measure- d~2|ntˆ + ζ~| ments of Lorentz-violating effects, this would correspond h to a sensitivity at the 10−7 level for the combination − 2ˆnj ζk(ζ~2 +ˆn · ζt~ )+ ζj ζk(t +ˆn · ζ~) , s = (s − s ) − 7 × 10−4s . Ic P P QQ Q (196)i

where ζ~ and d~ are functions of a time argument and are 2. Time-delay eﬀect deﬁned as

For the binary pulsar, the Lorentz-violating corrections ζ~(t) = ~r1(tem) − ~r2(t) − ntˆ em, to the Shapiro time-delay effect are negligible compared d~(t)=n ˆ × (ζ~ × nˆ). (197) to other effects, as described in Sec. VE. However, solar- system tests involving the time-delay effect have the po- The combination of coefficients ∆jk, which includes the tential to yield interesting sensitivities. To demonstrate effects due to Lorentz violation, is given by this, we derive next the generalization of the usual time- jk jk jk delay result in general relativity to allow for Lorentz vi- ∆ = 2δ + ∆s , (198) olation. We seek an expression for the time delay of a light where signal as it passes from an emitter to a receiver in the jk 00 0l l jk presence of a massive body such as the Sun. Assuming ∆s = (s − s nˆ )δ 00 j k jk 1 0j k 1 0k j that light travels along a null geodesic in spacetime, the +s nˆ nˆ + s − 2 s nˆ − 2 s nˆ 2 photon trajectory satisfies the condition ds = 0. In − 1 sjlnˆlnˆk − 1 sklnˆlnˆj . (199) terms of the underlying post-newtonian coordinates, this 2 2 can be written in the form Note that the trace part of ∆jk cancels in Eq. (196). j j 1/2 j j k The result (194) is the generalization to the pure- dx dx 1 dx 1 dx dx · = 1 − h00 − h0j − hjk . gravity minimal SME of the standard time-delay formula dt dt 2 dt 2 dt dt of general relativity, which can be recovered as the limit (192) sµν = 0. In principle, Eq. (194) could be applied to solar- The metric components are taken to be the O(2) forms system experiments including, for example, reflection of of Eqs. (35), (36), and (37) in the presence of Lorentz a signal from a planet or spacecraft. A detailed analysis along these lines would be of definite interest but lies be- violation and with a perturbing point mass m2 located yond our present scope. However, the form of Eq. (194) at ~r2. Solving Eq. (192) for dt, we integrate from an initial indicates that differently oriented experiments would be sensitive to different coefficients for Lorentz violation. emission spacetime point (tem, ~r1) to a final arrival space- At present, the most sensitive existing data on the time point (tarr, ~rE ). At order O(2), the zeroth-order straight-line solution time-delay effect within the solar system are obtained from tracking the Cassini spacecraft [81]. Assuming a ~r(t)=ˆn(t − tem)+ ~r1(tem) (193) detailed analysis of these data, we estimate that sensitivities to various combinations of the coefficients sJK can be inserted into the metric components in Eq. (192). could attain at least the 10−4 level. This is compara- Here,n ˆ is a unit vector in the direction ~rE − ~r1. In performing the integration, the positions of the emitter ble to the sensitivities of GPB discussed in Sec. V D 3, but the differences between the two spacecraft trajecto- at ~r1, receiver at ~rE, and perturbing body at ~r2 are all taken to be constant during the travel time of the light ries makes it likely that the attainable sensitivities would involve different combinations of the coefficients sJK . pulse. The resulting expression for the time difference t − We remark in passing that a generalization of the stan- arr dard formula for the bending of light should also exist, t is found to be em following a derivation along lines similar to those lead- tarr − tem = |~rE − ~r1| + ∆1 + ∆2(tarr) − ∆2(tem). (194) ing to Eq. (194). Current measurements of the deflection 39 of light by the Sun have attained the 10−4 level [72], implications for experiments with torsion pendula are de- which suggests sensitivities to Lorentz violation at this scribed in Sec. V C 3. The latter include some proposed level might already be attainable with a complete data high-sensitivity tests to the coefficients sΞΠ based on the analysis. Future missions such as GAIA [82] and LATOR local anisotropy of gravity. [83] may achieve sensitivities to coefficients for Lorentz Possible sensitivities to Lorentz violation in measure- −6 −8 violation at the level of 10 and 10 , respectively. ments of the spin precession of orbiting gyroscopes are considered in Sec. V D. The generalized formula describing the spin precession is found in Eq. (143), and the VI. SUMMARY secular effects are isolated in Eq. (153). The results are applied to the Gravity Probe B experiment [70]. This ex- In this work, we investigated the gravitational sector of periment is shown to have potential sensitivity to effects the minimal SME in the Riemann-spacetime limit, with involving spin precession in a direction orthogonal to the specific focus on the associated post-newtonian physics. usual geodetic and frame-dragging precessions predicted The relevant basics for this sector of the SME are re- in general relativity. viewed in Sec. II A. There are 20 independent back- Timing studies of signals from binary pulsars offer fur- ground coefficient fields u, sµν , and tκλµν , with corre- κλµν ther opportunities to measure certain combinations of sponding vacuum values u, sµν , and t . coefficients for Lorentz violation. Section V E derives the The primary theoretical challenge to the study of post- secular evolution of the six orbital elements of the binary- newtonian physics is the extraction of the linearized field pulsar motion and calculates the modifications to the pul- equations for the metric while accounting for the dynam- sar timing formula arising from Lorentz violation. Fits to ics of the coefficients for Lorentz violation. At leading observational data for several binary-pulsar systems have order, this challenge is met in Sec. II B. Under a few mild the potential to achieve high sensitivity to coefficients of conditions, the linearized effective field equations appli- the type sΞΠ. cable for any theory with leading-order Lorentz violation Section V F discusses the attainable sensitivities in are obtained in the practical form (23). some of the classic tests of general relativity, involving the Section III is devoted to the post-newtonian metric. perihelion precession, time-delay, and bending of light. Its derivation from the effective linearized field equations Perihelion shifts for the Earth, Mercury, and other bodies is described in Sec. III A. Under the simplifying assump- are considered in Sec. V F 1, while versions of the time- tion u = 0, the result is given in Eqs. (35)-(37). Section delay and light-bending tests are studied in Sec. V F 2. III B obtains the equations of motion (42) and (43) describing the corresponding post-newtonian behavior of a Table 6 collects the estimated attainable sensitivities for all the experiments considered in Sec. V. Each hori- perfect fluid. The results are applied to the gravitating many-body system, with the point-mass equations of zontal line contains estimated sensitivities to a particular motion and lagrangian given by Eqs. (52) and (54). The combination of coefficients for Lorentz violation relevant post-newtonian metric obtained here is compared and for a particular class of experiment. Most experiments contrasted with other existing post-newtonian metrics in are likely to have some sensitivity to Lorentz violation, but the table includes only the better estimated sensitiv- Sec. III C. Finally, to complete the theoretical discussion, illustrative examples of the practical application of our ities for each type of experiment. methodologies and results to specific theories are given Relatively few sensitivities at these levels have been in Sec. IV in the context of various bumblebee models. achieved to date. In Table 6, estimated sensitivities that The largest part of the paper, Sec. V, is devoted to might be attainable in principle but that as yet remain exploring the implications of these theoretical results for to be established are shown in brackets. Values without a variety of existing and proposed gravitational exper- brackets represent estimates of bounds that the theoreti- iments. It begins with a generic discussion of coordi- cal treatment in the text suggests are already implied by nate frames and existing bounds in Sec. V A. Section existing data. All estimates are comparatively crude and V B considers laser ranging both to the Moon and to should probably be taken to be valid only within about artificial satellites, which offers promising sensitivity to an order of magnitude. They are based on existing tech- certain types of Lorentz violation controlled by the co- niques or experiments, and some planned or near-future efficients sΞΠ in the Sun-centered frame. The key result experiments can be expected to supersede them. for the relative acceleration of the Earth-satellite system In conclusion, both astrophysical observations and ex- is given in Eq. (99) and the associated definitions. The periments on the Earth and in space have the potential technical details of some of the derivations in this section to offer high sensitivity to signals for Lorentz violation of are relegated to Appendix A. purely gravitational origin. A variety of tests is required Section VC considers Earth-based laboratory exper- to span the predicted coefficient space for the dominant iments. The expression for the local terrestrial accel- Lorentz-violating effects. These tests are currently fea- eration as modified by Lorentz violation is obtained as sible, and they represent a promising direction to follow Eq. (126) in Sec. V C 1. Applications of this result to in the ongoing search for experimental signals of Lorentz gravimeter measurements are treated in Sec. V C 2, while violation from the Planck scale. 40

Coeﬃcient Lunar Satellite Gravimeter Torsion Gravity Binary Time Perihelion Solar-spin combinations ranging ranging pendulum Probe B pulsar delay precession precession s12 [10−11] [10−10] ------s11 − s22 [10−10] [10−9] ------s01 [10−7] [10−8] ------s02 [10−7] [10−8] ------−7 sΩ⊕c [10 ] ------−7 sΩ⊕s [10 ] ------sXX − sY Y - - [10−11] [10−15] - - - - - sXY - - [10−11] [10−15] - - - - - sXZ - - [10−11] [10−15] - - - - - sY Z - - [10−11] [10−15] - - - - - sT Y - - [10−7] [10−11] - - - - - sT X - - [10−7] [10−11] - - - - - sT Z - - [10−7] [10−11] - - - - - σ −4 Ωs - - - - [10 ] - - - - n −4 Ωs - - - - [10 ] - - - - Z −4 Ωs - - - - [10 ] - - - - −9 se - - - - - [10 ] - - - −11 sω - - - - - [10 ] - - - jk −4 ∆s ------[10 ] - - −9 s' ------10 - −8 s⊕ ------10 - −7 sIc ------[10 ] - −13 sSSP ------10

Table 6. Crude estimates of attainable experimental sensitivities.

Acknowledgments in the text. The second and third parts are devoted to forced and unforced oscillations, respectively. The ﬁnal This work is supported in part by DOE grant DE- part extracts the radial oscillations needed for the anal- FG02-91ER40661 and by NASA grant NAG3-2914. ysis in Sec. VB.

1. Generalities APPENDIX A: SATELLITE OSCILLATIONS The procedure adopted here involves perturbation This appendix is devoted to calculating the Lorentz- techniques similar to those presented in Refs. [52, 84]. violating corrections to the observable relative Earth- The equation (99) for the coordinate acceleration of the satellite distance r = |~r1 −~r2| in the Sun-centered frame. Earth-satellite distance is expanded in a vector Taylor J These corrections are needed for the analysis of lunar series about an unperturbed circular-orbit solution r0 . and satellite laser ranging in Sec. V B. For simplicity, This unperturbed circular orbit satisfies the only perturbing body is taken to be the Sun. Also, J J we limit attention to the dominant frequencies of direct r¨0 = α0 (r0), (A1) relevance for the text. A more detailed analysis could reveal other frequencies that would permit sensitivity to where different combinations of coefficients for Lorentz viola- GMrJ Ω2 rJ 3QrJ αJ (r)= − + ⊕ − , (A2) tion. 0 r3 2 2r5 The derivation is separated into four parts. The first discusses generalities of the perturbative expansion and with establishes notational conventions. Some of the results GM Ω2 = ⊙ . (A3) in this part are also applied in other contexts elsewhere ⊕ R3 41

The frequency ω of the circular orbit is deﬁned as where the oscillating basis vectors ~ρν , ~τν , ~σν are given by

2 GM 1 2 3Q ω = 3 − 2 Ω⊕ + 5 . (A4) ~ρν =ρ ˆcos(νT + θν ), r0 2r0 ~τν =τ ˆ sin(νT + θν ), Contributions from perturbing bodies and the Earth’s ~σν =σ ˆ cos(νT + θν ). (A11) quadrupole potential are included to mute resonant effects [48, 51, 52], arising in this instance from Lorentz- violating accelerations. With respect to this oscillating basis, Eq. (A5) can be Denoting the perturbation by ǫJ , we find expressed as a matrix equation. For example, ~ǫν takes the column-vector form: J K J 1 K L J ǫ¨ − (ǫ ∂K )α0 (r0) = 2 (ǫ ǫ ∂K ∂L)α0 (r0)+ . . . J K J +αT′ (r0) + (ǫ ∂K )αT′ (r0)+ . . . Xν J K J ~ǫ = . (A12) +αQ′ (r0) + (ǫ ∂K )αQ′ (r0)+ . . . ν,θν  Yν  J K J Zν +αLV(r0) + (ǫ ∂K )αLV(r0)+ ....   (A5)   The types of oscillations contained in Eq. (A5) can be The ellipses here represent higher-order terms in the vec- J J split into two categories: forced or inhomogeneous oscilla- tor Taylor series. The quantities αT′ and αQ′ are given tions, and unforced or homogeneous oscillations [52, 84]. by modifications of Eqs. (101) and (102), respectively, in The forced oscillations arise primarily from the terms which the tidal and quadrupole terms for Eq. (A2) are ~α ′ , ~α ′ , ~α . The unforced oscillations are the nat- J T Q LV removed. The term αLV(r) is defined in Eq. (103). ural oscillations of the Earth-satellite system in the pres- It is convenient to decompose Eq. (A5) using a basis of ence of the perturbative forces. They are controlled by 3 orthonormal unit vectorsρ ˆ,τ ˆ,σ ˆ. These vectors are, re- the matrix structure of the derivative terms in Eq. (A5), spectively, normal, tangential, and perpendicular to the K L J such as ǫ ǫ ∂K ∂Lα0 (r0)/2. Next, we study each of these unperturbed orbit. In the basis for the Sun-centered categories of oscillations in turn. frame, the explicit form forρ ˆ is

ρˆ = cos α cos(ωT + θ) − sin α cos β sin(ωT + θ), sin α cos(ωT + θ)+cos α cos β sin(ωT + θ), 2. Forced oscillations sin β sin(ωT + θ) , (A6) where θ is the phase of the orbit with respect to the line The forced oscillations due to Lorentz violation em- of ascending nodes. Explicit forms for the other two basis anate primarily from the terms in ~αLV. For oscillations vectors can be obtained from the equations near the resonant frequency ω, however, the tidal and quadrupole derivative terms in Eq. (A5) can play a sig- ωτˆ = dρ/dTˆ (A7) niﬁcant role. Keeping all the relevant terms, the lowest- order contributions arise from the equation and J K J J K J σˆ =ρ ˆ × τ.ˆ (A8) ǫ¨ − (ǫ ∂K )α0 (r0) = αLV(r0) + (ǫ ∂K )αT′ (r0) K J +(ǫ ∂K )α ′ (r0). (A13) J J Q Note also that r0 = r0ρˆ , where r0 is the radius of the circular orbit. In the oscillating basis of Eq. (A12), the relevant part of For our purposes, it suﬃces to assume that m2 ≫ m1 and to take RJ as the vector pointing to the Earth’s this equation takes the matrix form location. The Earth’s motion can also be approximated as a circular orbit around the Sun. This means that ↔ ↔ ↔ K (ν) · ~ǫν,θν = (α T + αQ) · ~ǫν,θν + ~αLV(ν,θν )

Rˆ = − cosΩ⊕T, − cos η sinΩ⊕T, − sin η sinΩ⊕T (A14) (A9) ↔ J J for each frequency ν. Here, we deﬁne K (ν) to be the main the Sun-centered frame, so we can write R = RRˆ J 2 2 J J J trix operator with components δK (d /dT ) − ∂K α0 (r0) and V⊕ = dR /dT . ↔ ↔ J in the Sun-centered frame, while αT and αQ denote the We write the perturbative oscillations ǫ in the general J J diagonal parts of ∂ α ′ and ∂ α ′ , respectively. form K T K Q By evaluating all terms in Eq. (A13) in the oscillat-

~ǫν,θν = Xν ~ρν + Yν~τν + Zν~σν , (A10) ing basis, we ﬁnd that the components of these matrix 42 operators for a given frequency ν are given by by the expressions

2 2 3 2 1 ν +3ω + Ω˜ 2ων 0 2 ↔ 2 ⊕ GMs2ω K (ν) = − 2ων ν2 0 , ~αLV(2ω, 2θ − σ2ω) = − 2  1  ,   r0 2 2 0 0 0 ν − ω        1  1 a1 − 0 0 ↔ 2 2Gδmsω,1v0 2 1 ~αLV(ω,θ + σω,1) = 0 , αT = 3Ω⊕ 0 a1 − 0 , 2    2  r0 1 0 0 a2 − 0  2        21 sin2 β 0 0 0 ↔ GMsω,2 Q 21 2 ~α (ω,θ − σ ) = , αQ = 0 − sin β 0 . LV ω,2 2  0  r5  4  r0 0 0 0 −3 − 3 sin2 β 1 4        (A15) 1 −3GMV⊕sΩ⊕,1 ~αLV(Ω⊕, σΩ⊕,1) = 2  0  , r0 Here, the quantity Ω˜ 2 is defined by 0 ⊕     1 −GMV⊕sΩ ,2 2Q ~α (Ω , σ ) = ⊕ . (A19) Ω˜ 2 = Ω2 + , (A16) LV ⊕ Ω⊕,2 2  0  ⊕ ⊕ 5 2r0 r0 0     while a1 and a2 are given by The various combinations of the coefficients for Lorentz violation have magnitudes and phases defined by 1 2 2 2 2 2 a1 = 4 [cos α(1 + cos β cos η) + sin β sin η s cos σ = 1 (s11 − s22), + sin2 α(cos2 β + cos2 η) 2ω 2ω 2 s sin σ = s12, +2 cos α cos β sin β cos η sin η], 2ω 2ω 02 1 2 2 2 2 2 sω,1 cos σω,1 = s , a2 = 2 (sin α sin β + cos α sin β cos η s sin σ = s01, +cos2 β sin2 η) − cos α cos β sin β cos η sin η. ω,1 ω,1 13 (A17) sω,2 cos σω,2 = s , 23 sω,2 sin σω,2 = s , T Y T Z ↔ sΩ ,1 cos σΩ ,1 = cos ηs + sin ηs , Inspection of the form of α given in Eq. (103) re- ⊕ ⊕ LV T X veals that the dominant oscillations forced by the Lorentz sΩ⊕,1 sin σΩ⊕,1 = s , 01 violation occur at the frequencies 2ω, ω, and Ω⊕. The sΩ⊕,2 cos σΩ⊕,2 = sin α cos ηs amplitudes at these frequencies are determined by the +(sin β sin η + cos α cos β cos α)s02, following matrix equations: 01 02 sΩ⊕,2 sin σΩ⊕,2 = cos αs − sin α cos βs . (A20)

↔ ↔ ↔ µν K (2ω) · ~ǫ2ω,2θ−σ2ω = (α T + αQ) · ~ǫ2ω,2θ−σ2ω In these expressions, the quantities s are combinations ΞΠ +~αLV(2ω, 2θ − σ2ω), of the coeﬃcients for Lorentz violation s in the Sun- 11 22 12 01 02 ↔ ↔ ↔ centered frame. For s − s , s , s , and s , these (ω) · ~ǫ = (α + α ) · ~ǫ K ω,θ+σω,1 T Q ω,θ+σω,1 combinations can be found in Eqs. (107) of Sec. V B, +~αLV(ω,θ + σω,1), while for s13 and s23 they are given by ↔ ↔ ↔ K (ω) · ~ǫω,θ−σω,2 = (α T + αQ) · ~ǫω,θ−σω,2 s13 = 1 sin β sin2α(sXX − sY Y ) − sin β cos2αsXY +~αLV(ω,θ − σω,2), 2 XZ Y Z ↔ ↔ ↔ +cos β(cos αs + sin αs ), α α K (Ω⊕) · ~ǫΩ⊕,σΩ⊕,1 = ( T + Q) · ~ǫΩ⊕,σΩ⊕,1 23 1 1 XX Y Y ZZ s = − 2 sin2β[ 2 (s + s ) − s ] +~α (Ω , σ ), LV ⊕ Ω⊕,1 + 1 sin2β cos2α(sXX − sY Y ) ↔ ↔ ↔ 2 α α 1 XY K (Ω⊕) · ~ǫΩ⊕,σΩ⊕,2 = ( T + Q) · ~ǫΩ⊕,σΩ⊕,2 + 2 sin2β sin2αs +~α (Ω , σ ). (A18) XZ Y Z LV ⊕ Ω⊕,2 − sin α cos2βs + cos α cos2βs . (A21)

In these equations, the relevant terms in ~αLV are given Using the appropriate inverse matrix, the solutions for 43

2ω and Ω⊕ can be written as 3. Unforced oscillations

−2 s2ωr0 Among the unforced oscillations satisfying Eq. (A5) are ~ǫ2ω,θ−σ2 = 5 , ω 12   the natural eccentric oscillations of the system occurring 0 in the absence of Lorentz violation. These satisfy a set of     coupled matrix equations that can be derived from Eq. b1/b2 (A5). Here, we are interested in unforced oscillations at ~ǫΩ⊕,σΩ 1 = −3V⊕r0sΩ⊕,1 2ω/Ω b , ⊕,  ⊕ 2  the anomalistic frequency ω0. 0 In a simplified scenario, where contributions from the     right-hand side of Eq. (A5) can be ignored, the anomal- b1/b2 V⊕r0 istic oscillation satisfies ~ǫΩ⊕,σΩ⊕,2 = − sΩ⊕,2 2ω/Ω⊕b2 , (A22) 2   ↔ ↔ ↔ 0 K (ω0) · ~ǫω0 = (α T + αQ) · ~ǫω0 . (A28)     where Requiring the vanishing of the determinant of the matrix ′ 2 K = K − αT − αQ allows the extraction of approximate 1 21Q sin β b1 = 2 − 3a1 + 2 5 , solutions for ω0. This gives three solutions, two of which 4Ω⊕r0 are physical. One is denoted ω0 and corresponds to ec- 2 11 63Q sin β centric motion in the plane of the orbit. The other is b2 = − 2 +9a1 − 5 2 , (A23) ′ 4r0Ω⊕ denoted ω0 and corresponds to eccentric motion perpendicular to the orbit. Explicitly, we find the approximate with a1 defined in Eq. (A17). solutions Special consideration is required for the near-resonant 2 frequency ω. The matrix to be inverted for this case is Ω1 ′ ω − ω0 ≈ ω, K (ω)= K(ω) − αT − αQ. To sufficient approximation, 2ω2 the determinant of this matrix can be expanded in terms Ω2 ′ ′ 2 of the anomalistic frequencies ω and ω . These are as- ω − ω0 ≈ ω, (A29) 0 0 2ω2 sociated with eccentric motions in the plane of the orbit and perpendicular to the orbit, respectively. The motion where involving the phase θ+σω,1 is confined to the plane of the 2 2 3Q orbit and hence is at frequency ω0, while that involving Ω = Ω [15a − 6]+ , 1 ⊕ 1 r5 the phase θ − σω,2 is perpendicular to the orbit and is at 0 ′ 3Q frequency ω0. We make use of the approximation 2 2 3 1 2 Ω2 = Ω⊕[3a2 − 2 ] − 5 (1 + 4 sin β). (A30) ′ ′ r0 det K (ω) = det K (ω0 + ω − ω0) ′ ′ d det K (ω0) More accurate solutions require a detailed study of a ≈ det K (ω0) + (ω − ω0) , dω0 system of coupled equations involving other frequencies. (A24) For example, the eccentric oscillations in the lunar case are at frequencies ω0 and 2(ω − Ω) − ω0 [52]. Also, in the ′ along with the similar relation for the frequency ω0. This case of a generic satellite orbit, the Earth’s quadrupole permits the inverse of K′(ω) to be determined as terms can be expected to play a significant role. For our purposes, the primary interest is in the eigen- −12 0 vector for ω0, which is given by ′ −1 1 [K (ω)] ≈  2 −4 0  , (A25) 2ω0(ω − ω0) 0 0 −γ 1     ~ǫω0 ≈ er0 −2 . (A31) where   0 ω (ω − ω )   0 0   γ = ′ ′ . (A26) ω0(ω − ω0) The overall normalization of this eigenvector is defined in terms of the eccentricity of the orbital motion. In consequence, the ω amplitudes now read Using Eq. (A31), the leading unforced oscillation due −1 to Lorentz violation can be established. Typically, the ω δmsω,1v0r0 dominant terms occur at frequencies near the resonant ~ǫω,θ+σω,1 = 2 , (ω − ω0) M   frequency, which here is the circular orbit frequency ω. 0   In this case, we find that the oscillation at frequency 2ω−   0 ω0 is particularly enhanced. In the Sun-centered frame, ω sω,2r0 the dominant terms contributing to the amplitude are ~ǫω,θ−σω,2 = ′  0  . (A27) K L J (ω − ω0) 2 the frequency-mixing terms in ǫ ǫ ∂K ∂Lα0 (r0)/2 and −1 K J   ǫ ∂K αLV that lower the frequency by 2ω. The former is   44 obtained by substitution of the forced-oscillation solution 4. Radial oscillations K for ǫ2ω,θ−σ2ω in Eq. (A22), while the latter is obtained from

KL L LM L M K At this stage, we can extract the information appearing −2ω K GMs r 3GMs r r r (α ) = ∂ − . in Table 2 for radial oscillations at the frequencies ω,Ω⊕, s J J r3 2r5 2ω, 2ω − ω0. The procedure is to take theρ ˆ component (A32) of each column vector ~ǫν,θν , to multiply by cos(νT + θν ), and to expand the result in terms of sines and cosines The resulting matrix equation takes the form using the phase deﬁnitions in Eq. (A20). ↔′ K (2ω − ω0) · ~ǫ2ω−ω0,2θ−σ2ω = After some calculation, we ﬁnd the following radial os- ↔−2ω ↔−2ω cillations: 2 α0 + αs · ~ǫ2ω−ω0,2θ−σ2ω , (A33)

1 11 22 where the operators on the right-hand side are deﬁned in (ǫ2ω)ρ = − 12 (s − s )r0 cos(2ωT +2θ) the oscillating basis by 1 12 − 6 s r0 sin(2ωT +2θ),

5 ω (δm)v0r0 02 2 1 4 0 ↔−2ω s2ωω (ǫω)ρ = − [s cos(ωT + θ) 2 α ≈ 5 1 , (A34) (ω − ω0) M 0  4 2 0  2 01 0 0 − 1 − s sin(ωT + θ)],  2   1  b1 1 0 (ǫΩ⊕ )ρ = VEr0[sΩ⊕c cosΩ⊕T + sΩ⊕s sinΩ⊕T ], ↔−2ω 2 b α ≈ s ω2 5 . (A35) 2 s 2ω  1 4 0  ω er0 0 0 − 3 (ǫ2ω−ω0 )ρ = −  4  (ω − ω0) 8   1 11 22 × (s − s ) cos[(2ω − ω0)T +2θ] Since the frequency 2ω − ω0 is near resonant, the pro- 2 ′ 12 cedure to ﬁnd the inverse of K (2ω − ω0) can be modeled + s sin[(2ω − ω0)T +2θ] . (A38) on that used for the frequency ω in Eqs. (A24)-(A25). We ﬁnd

−12 0 The new combinations of coefficients for Lorentz viola- ′ −1 1 [K (2ω − ω0)] ≈ 2 −4 0 , tion appearing in these equations are defined by 4ω0(ω − ω0)   0 0 −γ    (A36) s = −3s cos σ − 1 s cos σ , where γ is defined in Eq. (A26). It then follows that the Ω⊕c Ω⊕,1 Ω⊕,1 2 Ω⊕,2 Ω⊕,2 1 solution for the oscillations at frequency 2ω − ω0 is given sΩ⊕s = −3sΩ⊕,1 sin σΩ⊕,1 − 2 sΩ⊕,2 sin σΩ⊕,2. by (A39)

1 − 2 s2ωωer0 ~ǫ2ω−ω0,2θ−σ2ω =  1  . (A37) 4(ω − ω0) 0    

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