Strong field effects on binary systems in Einstein-aether theory
aa r X i v : . [ g r- q c ] S e p Strong field effects on binary systems in Einstein–aether theory
Brendan Z. Foster ∗ Institute for Theoretical Physics, Utrecht University,Leuvenlaan 4, NL-3584 CE Utrecht, The Netherlands (Dated: September 23, 2008)
Abstract “Einstein–aether” theory is a generally covariant theory of gravity containing a dynamical pre-ferred frame. This article continues an examination of effects on the motion of binary pulsarsystems in this theory, by incorporating effects due to strong fields in the vicinity of neutron starpulsars. These effects are included through an effective approach, by treating the compact bodiesas point particles with nonstandard, velocity dependent interactions parametrized by dimension-less “sensitivities”. Effective post-Newtonian equations of motion for the bodies and the radiationdamping rate are determined. More work is needed to calculate values of the sensitivities for agiven fluid source; therefore, precise constraints on the theory’s coupling constants cannot yet bestated. It is shown, however, that strong field effects will be negligible given current observationaluncertainties if the dimensionless couplings are less than roughly 0 .
01 and two conditions thatmatch the PPN parameters to those of pure general relativity are imposed. In this case, weakfield results suffice. There then exists a one-parameter family of Einstein–aether theories with“small-enough” couplings that passes all current observational tests. No conclusion can be reachedfor larger couplings until the sensitivities for a given source can be calculated.
PACS numbers: 04.50.+h, 04.30.Db, 04.25.Nx, 04.80.Cc ∗ [email protected] . INTRODUCTION This article examines the motion of stellar systems in “Einstein–aether” theory—an al-ternative theory of gravity that permits breaking of Lorentz symmetry through a dynamicalpreferred frame. The general theory contains four dimensionless couplings whose values canbe constrained by comparing the predictions of the theory with observations—in particular,observations of binary pulsar systems. It will be demonstrated that all current tests willbe passed by a one-parameter family whose couplings are “small-enough”—that is, on theorder of 0 .
01 or less. Identifying whether there is a viable extension of this family to largecoupling values requires additional work beyond this article.The question of whether the physical world is exactly Lorentz invariant has receivedincreasing attention in recent years. This interest is sourced largely by hints of Lorentz vio-lation in popular candidates for theories of quantum gravity—for instance, string theory [1],loop quantum gravity [2], and noncommutative field theory [3]. More broadly, challengingthe rule of Lorentz symmetry means challenging the fundamentals of all of modern physics,and doing that is just plain exciting.The review [4] discusses a wide variety of theoretical models that feature Lorentz-symmetry violating effects, and observational searches for violations. So far no conclusivesign of Lorentz variance has been identified, and very strong bounds exist on the size ofcouplings for Lorentz-violating effects in standard model extensions [4, 5]. The effects ofLorentz violation in a gravitational context, however, are not covered by these bounds.Einstein–aether theory—or “ae-theory” for short—is a classical metric theory of gravitythat contains an additional dynamical vector field. The vector field “aether” is constrainedto be timelike everywhere and of fixed norm. The aether can be thought of as a remnantof unknown, Planck-scale, Lorentz-violating physics. It defines a preferred frame, while itsstatus as a dynamical field preserves diffeomorphism invariance. The fixed norm, which canalways be scaled to unity, ensures that the aether picks out just a spacetime direction andremoves instabilities in the unconstrained theory [6].Much of past work on ae-theory has focussed on placing observational bounds on thevalues of the four free parameters c n appearing in the ae-theory action, Eqn. (7). Constraintshave been derived from the rate of primordial nucleosynthesis [7], the rate of ˆCerenkovradiation [6], the requirements of stability and energy positivity of linearized wave modes [8],2nd the parameterized post-Newtonian (PN) form of the theory [8, 9, 10]. A summary ofthese constraints was presented in [8], where it was shown that they are met by a largetwo-parameter subset of the original four-parameter class of theories.Additional constraints on the c n come from observations of binary pulsar systems. Studyof the predictions of ae-theory for binary pulsars was begun in [11]. There, an expression forthe rate of radiation damping in N -body systems was derived to lowest non-trivial PN orderand neglecting effects due to strong fields in the vicinity of the bodies . It was shown that a one-parameter subset of the two-parameter family allowed by the collected constraints discussedin [8] would pass tests from binary pulsar systems if there were justification for ignoringstrong field effects. That neglect is dangerous, though, since the fields inside neutron starpulsars should be very strong. Justification requires an unclear assumption on the values ofthe c n .In this article, I will incorporate strong field effects on binary pulsar systems, calculatingthe PN equations of motion and the rate of radiation damping of a system of stronglyself-gravitating bodies. The effects will be handled via an effective approach in which thecompact bodies are treated as point particles whose action contains nonstandard couplingsthat depend on the velocity of the particles in the preferred frame. The effective approachto N -body dynamics in relativistic gravity theories has previously been employed in puregeneral relativity (GR) and other alternative theories; see for example [12, 13, 14, 15, 16, 17].The new interactions are parametrized by dimensionless coefficients, or “sensitivities”, whosevalues can be calculated for a given stellar source by matching the effective theory onto theexact, perfect fluid theory. Prior work [11] reveals just the form of the “first” sensitivity atlowest order in the self-potential of a body.The expressions obtained can be used to constrain the allowed class of ae-theories. Obser-vations of binary pulsar systems allow for measurement of “post-Keplerian” (PK) parametersthat describe perturbations of the binary’s Keplerian orbit due to relativistic effects. Theseparameters are mostly “quasi-static” ones, whose expressions can be derived from the non-radiative parts of the PN forms of the gravitational fields and the effective equations ofmotion for the bodies. In addition, there is the radiation damping rate, whose expressiondepends on the radiative parts of the fields. The ae-theory expressions for the PK param-eters differ from those of pure GR in that they depend on the c n , the sensitivities, andthe center-of-mass velocity of the system of bodies. Stating precise constraints for general3 n values will require work beyond the scope of this article—specifically, what is needed isa method for dealing with dependence on the unmeasurable center-of-mass velocity and acalculation of the values of the sensitivities of a given source.For the time being, a few comments can be made, which will be defended below. A crucialpiece of information learned by comparing the weak field limit of the effective theory withthe weak field limit of the perfect fluid theory [11] is that the sensitivities will be “small”.That is, they will be at least as small as ( G N m/d ) , where m is the body’s mass and d itssize, times a c n dependent coefficient that must scale at least linearly with c n in the small c n limit. For neutron stars in pure GR, ( G N m/d ) ∼ (0 . ∼ . c n limit, to GR plus a non-dynamical vector field.It then follows that bounds on the magnitude of violations of the strong equivalenceprinciple [19] constrain the c n dependent factor to be less than (0 . G N m/d ) − . It furtherfollows that the strong field corrections fall below the level of current observational uncer-tainties when | c n | . .
01 and the two conditions that match the ae-theory PPN parametersto those of GR are imposed. Thus, weak field analysis [11] suffices for small enough c n , andimplies the existence of a one-parameter family of theories that passes all current tests frombinary pulsar systems.I will now present the strong field formulas. First, the effective particle action is con-structed, and the exact field equations are defined in Sec. II. The PN expansions of themetric and aether fields are then given in Sec. III A, and used to express the PN equationsof motion for a binary system in Sec. III B. The rate of radiation damping is then determinedin Sec. IV. Comments on dealing with center-of-mass velocity and sensitivity dependenceare given in Sec. V, along with the argument for the viability of the weakly coupled familyof theories.I follow the conventions of Wald [20]. In particular, I use units in which the flat spacespeed-of-light c = 1, and I use metric signature ( − , + , + , +). This signature is opposite tothat employed in [11], but it is much more convenient for calculations involving a time-space decomposition. The ae-theory action is defined here in such a way as to permiteasy comparison between [11] and this article. The following shorthand conventions for4ombinations of the c n will be used: c = c + c , (1) c = c + c + c , (2) c ± = c ± c . (3)When covariant equations are expanded in Minkowskian coordinates, the following con-ventions are observed. Spatial indices will be indicated by lowercase Latin letters from themiddle of the alphabet: i, j, k, . . . . One exception is when the coefficients c , , , are referredto collectively as c n . Indices will be raised and lowered with the flat metric η ab . Repeatedspatial indices will be summed over, regardless of vertical position: T ii = P i =1 ... T ii . Timeindices will be indicated by a 0; time derivatives will be denoted by an overdot: ˙ f ≡ ∂ f . II. EFFECTIVE ACTION AND FIELD EQUATIONSA. Particle action
The aim of this work is to treat within ae-theory a system of compact bodies that poten-tially possess strong internal gravitational fields. The complicated internal workings of thebodies will be dealt with via an effective approach that reproduces the bulk motion of thebodies and the fields far from them. Each body will be treated as a point particle with thecomposition dependent effects encapsulated in nonstandard couplings in the particle action.The form of the effective action can be deduced from the following considerations. Theone-particle action S A will have the rough form S A = − ˜ m R dt O , where the integral is alongthe particle worldline parametrized by t , ˜ m has dimensions of mass, and O is a sum ofdimensionless local scalar quantities. The fundamental theory has only one dimensionfulparameter G . For a first approximation, the spin of the body can be neglected. Derivativecouplings in the particle theory are then suppressed by powers of ( d/R ), where d is thesize of the underlying finite-sized body and R is the radius of curvature of the backgroundspacetime. In addition, S A presumably reduces to the standard free particle action if theparticle is comoving with the local aether and must be invariant under reparametrization ofthe particle worldline. 5hese considerations imply the following one-particle action: S A = − ˜ m A Z dτ A (cid:0) σ A ( u a v a + 1) + σ ′ A u a v a + 1) + · · · (cid:1) , (4)where A labels the body, τ A is the proper time along the body’s curve, v a is the body’s unitfour-velocity, and u a is the aether. The quantity u a v a expressed in a PN expansion with theaether purely timelike at lowest order, is of order v , the square of the velocity of the bodyin the aether frame. By assumption, v is first PN order (1PN). The 1PN corrections toNewtonian equations of motion will follow from the part of the action that is m A × (2PN),so only the terms in S A written above are needed for current purposes. For a system of N particles, the action is given by the sum of N copies of S A .This action can be thought of as a Taylor expansion of the standard worldline action,but with a mass that is a function of γ ≡ − u a v a : S A = − Z dτ ˜ m A [ γ ] . (5)The expansion is made about γ = 1. The parameters σ, σ ′ are then defined as σ A = − d ln ˜ m A d ln γ | γ =1 , σ ′ A = σ A + σ A + ¯ σ A , ¯ σ A = d ln ˜ m A d (ln γ ) | γ =1 . (6)This form of S A suggests that that σ A , ¯ σ A can be determined by considering asymptoticproperties of perturbations of static stellar solutions. B. Field equations
The full action is the four-parameter ae-theory action SS = 116 πG Z d x p | g | (cid:16) R − K abcd ∇ a u c ∇ b u d + λ ( u a u b g ab + 1) (cid:17) , (7)plus the sum of N copies of S A (4), retaining only the terms explicitly written above. Here, K abcd = (cid:0) c g ab g cd + c δ ac δ bd + c δ ad δ bc − c u a u b g cd (cid:1) . (8)While the sign of the c term looks awkward, it permits easier comparison with the resultsof [11].The field equations are then as follows. There are the Einstein equations G ab − S ab = 8 πGT ab , (9)6here G ab = R ab − Rg ab , (10) S ab = ∇ c (cid:0) K c ( a u b ) − K c ( a u b ) − K ( ab ) u c (cid:1) + c (cid:0) ∇ a u c ∇ b u c − ∇ c u a ∇ c u b (cid:1) + c ( u c ∇ c u a )( u d ∇ d u b )+ λu a u b + 12 g ab ( K cd ∇ c u d ) , (11)with K ac = K abcd ∇ b u d , (12)and T ab is the particle stress tensor T ab = X A ˜ m A ˜ δ A (cid:2) A A v aA v bA + 2 A A u ( a v b ) A (cid:3) , (13)with a covariant delta-function ˜ δ A = δ ( ~x − ~x A ) v A p | g | , (14)and A A = 1 + σ A − σ ′ A (cid:0) ( u c v cA ) − (cid:1) , (15) A A = − σ A − σ ′ A ( u c v cA + 1) . (16)The aether field equation is ∇ b K ba = c ( u c ∇ c u b ) ∇ a u b + λu a + 8 πGσ a , (17)where σ a = X A ˜ m A ˜ δ A A A v aA . (18)Varying λ gives the constraint g ab u a u b = −
1. Eqn. (17) can be used to eliminate λ , giving λ = − u a (cid:16) ∇ b K ba − c ( ∇ a u b )( u c ∇ c u b ) − πGσ a (cid:17) . (19)The covariant equation of motion for a single particle has the form ∇ b T abA − ∇ b (cid:0) ( σ A ) a u b (cid:1) − ( σ A ) b ∇ a u b = 0 , (20)where T abA and ( σ A ) a are the one-particle summands in (13) and (18). This can be writtenmore explicitly as v bA ∇ b ( A A v aA + A A u a ) − A A v Ab ∇ a u b = 0 . (21)7 II. POST-NEWTONIAN EXPANSIONA. Fields
The PN expansion of the fields can be determined by iteratively solving the field equationsin a weak field, slow motion approximation [8, 21]. A background of a flat metric andconstant aether is assumed, and a Lorentzian coordinate system with the time directiondefined by the background aether is chosen. Following the procedures of [8] gives g = − X A G N ˜ m A r A − X A,B G N ˜ m A ˜ m B r A r B − X A,B = A G N ˜ m A ˜ m B r A r AB + 3 X A G N ˜ m A r A v A (1 + σ A ) ,g ij = (cid:16) X A G N ˜ m A r A (cid:17) δ ij ,g i = X A B − A G N ˜ m A r A v iA + X A B + A G N ˜ m A r A ( v jA r jA ) r iA , (22)where r iA = x i − x iA , r iAB = x iA − x iB , B ± A = ± ±
14 ( α − α ) (cid:16) − c )(2 c + − c ) σ A (cid:17) −
14 (8 + α ) (cid:16) c − c σ A (cid:17) , (23)and G N = 22 − c G, (24) α = − c + c c )2 c − c + c − , (25) α = α − ( c + 2 c − c )(2 c + 3 c + c + c )(2 − c ) c . (26)The numerical values of the PPN parameters α and α are constrained to be very smallby weak field experiments, via analysis that allows for a possible lack of Lorentz symmetryin the underlying theory [21]. There are two independent pairs of conditions on the c n thatwill set α and α to zero. One pair is c = − c + c c − c c , c = − c c . (27)The other is c + = c = 0. With this second pair, the spin-1 and spin-0 wave speeds diverge(Sec. IV); also, the spin-0 linearized energy density vanishes while that of spin-1 remains8nite [22]. Observational signatures of this behavior have not been worked out, and I willnot consider these conditions further here. Hence, the first pair of conditions is assumedbelow whenever attention is restricted to the case of vanishing α and α .The aether to order of interest is u =1 + X A G N ˜ m A r A ,u i = X A C − A G N ˜ m A r A ( v A ) i + X A C + A G N ˜ m A r A ( v jA r jA ) r iA , (28)where C ± A = (cid:0) α c (cid:1)(cid:0) c − − (1 − c − ) σ A (cid:1) ± (2 − c )2 (cid:16) ( α − α )( c + 2 c − c ) + 1 c σ A (cid:17) . (29)The results of this section are equivalent to the weak field expressions obtained in [8] when σ A is set to zero. B. Post-Newtonian equations of motion
The equations of motion for the system of compact bodies follow by expressing the exactresult (20) in a PN expansion using the forms of the fields given above. The Newtonian orderresult can be used to define the effective two-body coupling G and the “active” gravitationalmass m : ˙ v iA = X B = A − G N ˜ m B (1 + σ A ) r AB r iAB ≡ X B = A −G AB m B r AB r iAB , (30)with the two-body coupling G AB = G N (1 + σ A )(1 + σ B ) , (31)and the active gravitational mass m B = (1 + σ B ) ˜ m B . (32)These definitions arise by requiring that G AB = G BA and that m B / ˜ m B depend on just σ B .Using the Newtonian result and continuing with the expansion leads to the 1PN equationsof motion, expressed here just for the case of a binary system:9 v i = G m r ˆ r i h − m r + (cid:16) −
21 + σ D (cid:17) ˜ m r − (cid:16) σ + σ ′ σ (cid:17) v − (cid:16)
32 (1 + σ ) + ( E − D ) (cid:17) v − Dv j v j + 3( E − D )( v j ˆ r j ) i + G m r h v i (cid:16) v j ˆ r j (cid:0) σ − σ ′ σ (cid:1) − σ ) v j ˆ r j (cid:17) + v i (2 Dv j ˆ r j − Ev j ˆ r j ) i , (33)where G = G , r i = r i − r i , and D = −
14 (8 + α ) (cid:16) c − c ( σ + σ ) + (1 − c − )2 c σ σ (cid:17) , (34) E = − −
14 ( α − α ) (cid:16) − c )( c + 2 c − c ) ( σ + σ ) + (2 − c )2 c σ σ (cid:17) (35)The expression for ˙ v i is obtained by exchanging all body-1 quantities and body-2 quantities,including the switch r i → − r i .The “Einstein–Infeld–Hoffman” Lagrangian [12]—that is, the effective Lagrangian ex-pressed purely in terms of particle quantities—can be determined by working backwardsfrom the equations of motion. It is L = − ( m + m ) + 12 ( m v + m v )+ 18 (cid:18)(cid:16) − σ ′ σ (cid:17) v + (cid:16) − σ ′ σ (cid:17) v (cid:19) + G m m r (cid:20) (cid:16) (1 + σ ) v + (1 + σ ) v (cid:17) − (cid:16) G m r (1 + σ ) + G m r (1 + σ ) (cid:17) + D ( v j v j ) + E ( v j ˆ r j v k ˆ r k ) (cid:21) . (36)This Lagrangian is not Lorentz invariant unless σ A = σ ′ A = 0. This follows from the analysisof Will [23] and the list of criteria therein. In particular, the action and the equations ofmotion depend on the velocity of the system’s center of mass in the aether frame. IV. RADIATION DAMPING RATE
The radiation damping rate is the rate at which the particle system loses energy viagravity-aether radiation. This energy loss manifests as a change in the orbital period of a10inary system, equating the energy radiated to minus the change in mechanical energy. Theexpression for the rate in the effective particle theory can be determined by adapting themethods of [11], which were used to find the rate for a system of weakly self-gravitatingperfect fluid bodies in ae-theory. It will be convenient to introduce the parameter s A s A = σ A / (1 + σ A ) , (37)and to work with the active gravitational mass (32) m A = (1 + σ A ) ˜ m A = ˜ m A / (1 − s A ) . (38) A. Wave forms
The method of [11] begins by assuming a background of a flat metric and constant aether,with a coordinate system with respect to which the background metric is the Minkowskimetric η ab and the background aether is aligned with the time direction. The metric andaether perturbations are then decomposed into irreducible transverse and longitudinal pieces.The spatial vectors u i and h i are written as: h i = γ i + γ ,i u i = ν i + ν ,i , (39)with γ i,i = ν i,i = 0. The spatial metric h ij is decomposed into a transverse, trace-free tensor,a transverse vector, and two scalar quantities giving the transverse and longitudinal traces: h ij = φ ij + 12 P ij [ f ] + 2 φ ( i,j ) + φ ,ij , (40)where 0 = φ ij,j = φ jj = φ i,i , (41)and P ij [ f ] = δ ij f ,kk − f ,ij ; (42)hence, P ij [ f ] ,j = 0, and h ii = ( f + φ ) ,ii . Further, define F = f ,jj . (43)The list of variables then consists of a transverse-traceless spin-2 tensor φ ij , transverse spin-1 vectors γ i , ν i , φ i , and spin-0 scalars γ, ν, F, φ, h , and u . The Lorentz gauge, or any11bvious extension of it, does not usefully simplify the ae-theory field equations. Instead, thefollowing convenient conditions will be imposed:0 = u i,i = h i,i = h i [ j,k ] i , (44)or equivalently, 0 = ν = γ = φ i . (45)Because φ i is transverse, these constitute just four conditions.Following [11], the field equations can then be linearized and expressed in terms of theabove variables, and sorted to obtain a set of wave equations with matter terms and nonlinearterms as sources. Having done this, the linear contributions can be seen by inspection tosatisfy a conservation law. This fact implies the existence of a conserved source τ ab τ ab = T ab − σ a δ b + ˜ τ ab , (46)where T ab and σ a are as defined in Eqns. (13) and (18), and ˜ τ ab is constructed from nonlinearterms—its precise form will not be needed. The non-symmetric τ ab satisfies the conservationlaw with respect to the right-index only: τ ab,b = 0. The corresponding conserved total energy E and momentum P i to lowest PN order are E = Z d x τ = X A ˜ m A = X A (1 − s A ) m A , (47) P i = Z d x τ i = X A m A v iA . (48)Conservation of P i means that the system center-of-mass X i defined via m A X i = P A m A x iA P A m A , (49)is unaccelerated to lowest order.The field equations reduce to the following. For spin-2,1 w ¨ φ ij − φ ij,kk = 16 πGτ TT ij , (50)where TT signifies the transverse, trace-free components, and w = 11 − c + . (51)12or spin-1, 1 w (cid:0) ¨ ν i + ¨ γ i (cid:1) = 16 πG c − c + c − ( c + τ i + (1 − c + ) σ i ) T , (52)( c + ν i + γ i ) ,kk = − πGτ T i , (53)where T signifies the transverse components, and w = 2 c − c + c − − c + ) c . (54)For the spin-0 variables, the constraint gives to linear order u = 1 + 12 h . (55)Non-linear corrections to this are of uninteresting order, as explained in more detail in [11].The other equations are1 w ¨ F − F ,kk = 16 πGc − c (cid:0) τ kk − c + c + c τ L kk + 2 c τ (cid:1) , (56)( F − c h ) ,kk = − πGτ , (57)(1 + c ) ˙ F ,i + c ˙ φ ,kki = − πGτ L i , (58)where L signifies the longitudinal component, and w = (2 − c ) c (2 + 3 c + c + )(1 − c + ) c . (59)All these equations can be solved formally via Greens function methods, and the resultingintegrals expanded in a far field, slow motion approximation. The expressions can be furthersimplified using the conservation of τ ab . A result that holds within the approximation schemeis that for a field ψ satisfying a wave equation with speed w evaluated at field point x i ≡ | x | ˆ n i with only outgoing waves, wψ ,i = − ˙ ψ ˆ n i . (60)Also, differentially transverse becomes equivalent to geometrically transverse to ˆ n i .The results to lowest PN order and ignoring static contributions are as follows. For spin-2, φ ij = 2 G | x | ¨ Q TT ij , (61)13here the right-hand side is evaluated at time ( t − | x | /w ) and the quadrupole moment Q ij is the trace-free part of the system’s second mass moment I ij : I ij = X A m A x iA x jA . (62)For spin-1 variables, ν i = − G | x | c − c + c − (cid:16) ˆ n j w ( c + − c + ¨ Q ij + ¨ Q ij ) − i (cid:17) T , (63) γ i = − c + ν i , (64)where the right-hand side of the first equation is evaluated at time ( t − | x | /w ), Q ij is thetrace-free part of the rescaled mass moment I ij : I ij = X A s A m A x iA x jA , (65)and Σ i = − X A s A m A v iA . (66)For spin-0 variables, F = − G | x | c − c h(cid:16)(cid:0) α − α c + − c ) + 3 (cid:1) ¨ Q ij + 2 w c ¨ Q ij (cid:17) ˆ n i ˆ n j + 2 α − α c + − c ) ¨ I + 23 w c ¨ I − w c ˆ n i Σ i i , (67) h = 1 c F, (68)˙ φ ,i = − c c ˙ f ,i , (69)where the right-hand side of the first equation is evaluated at time ( t − | x | /w ), and I = I ii , I = I ii .At this point, the expected smallness of the sensitivities, mentioned in the introduction,can be explained. One should take the weak field limit ( s A → “small”) of the above waveforms and compare them with the perfect-fluid theory wave forms determined in [11]. Theonly s A -dependence that remains at potentially leading order is in Σ i . Comparing (66) withEqn. (85) of [11] indicates that in the small s A limit, s A = ( α − α ) Ω A m A + O ( G N md ) , (70)14here Ω A is the binding energy of the body: Ω /m ∼ ( G N m/d ), where d is the characteristicsize of the body. The implication is that when α = α = 0, s must scale as ( G N m/d ) ,times a c n dependent coefficient. This coefficient should scale at least linearly in c n , in the c n → B. Damping rate expression
For the next step, the wave forms are inserted into an expression for the rate of changeof energy ˙ E . This expression can be derived via the Noether charge method of Iyer andWald [24, 25], using the ae-theory Noether charges derived in [26], with the result:˙ E = − πG Z d Ω R (cid:16) w ˙ φ ij ˙ φ ij + (2 c − c + c − )(1 − c + ) w ˙ ν i ˙ ν i + 2 − c w c ˙ F ˙ F (cid:17) + ˙ O, (71)where ˙ O is a total time-derivative that will be argued away in a moment.Using the above results for the wave forms, performing the angular integral, and ignoring˙ O gives˙ E = − G N ( A Q ij ... Q ij + A Q ij ... Q ij + A Q ij ... Q ij + B ... I ... I + B ... I ... I + B ... I ... I + C ˙Σ i ˙Σ i ) , (72)where A = (cid:0) − c (cid:1)(cid:18) w + 2 c c (2 c − c + c − ) w + c − c ) (cid:16) α − α c + − c ) (cid:17) w (cid:19) , (73) A = (cid:18) (2 − c ) c + c − c + c − w + (cid:16) α − α c + − c ) (cid:17) w (cid:19) , (74) A = 1 c (cid:16) − c w −
13 1 w (cid:17) , (75) B = c (cid:16) α − α c + − c ) (cid:17) w , (76) B = 2 α − α c + − c ) 1 w , (77) B = 16 c w , (78) C = 23 c (cid:16) − c w + 1 w (cid:17) . (79)The coefficients A , B , and C are respectively identical to A , B , and C of [11]. Takingthe weak field limit corresponds to retaining only the A , B , and C terms, and invoking15he relation (70) for s A in the C term. In the case that α = α = 0, B vanishes, as does s A in the weak field limit. The weak field damping rate in this case then contains only aquadrupole contribution and is identical to the GR rate when A = 1. This remaining curveof c n values intersects the range of values allowed by collected constraints considered in [8],as illustrated in Figure 1. Thus, this curve gives a one-parameter family of viable ae-theoriesif the weak field results alone are sufficient. + c - + c - FIG. 1: Class of allowed ae-theories, if strong field effects in binary pulsar systems can be ignored.The four-dimensional c n space has been restricted to the ( c + , c − ) plane by setting the PPN pa-rameters α and α to zero via the conditions (27). The shaded region is the region allowed byprimordial nucleosynthesis, ˆCerenkov radiation, linearized stability and energy positivity, and PPNconstraints, demarcated in [8]. The dashed curve is the curve along which binary pulsar tests willbe satisfied, assuming ae-theory weak field expressions. Specifically, it is the curve along which A = 1 in the α = α = 0 case, so that the damping rate (72) is identical to the quadrupole for-mula of general relativity. Along both this curve and the boundary of the allowed region, c − → ∞ as c + →
1. The curve remains within the allowed region for all c + between 0 and 1. As explainedin Sec. V, strong field effects may lead to system dependent corrections to the binary pulsar curvefor large c n ; however, all such curves will coincide with the weak field curve for | c n | . .
01 givencurrent observational uncertainties.
16o simplify the expression (72), it is crucial to note that the damping rate is calculated tolowest PN order using the Newtonian results for the motion of the system. Thus, the system’smotion can be decomposed into a uniform center-of-mass motion—recall the conservationof P i —and a fixed Keplerian orbit in the center-of-mass frame. Since the motion is steady-state, the damping rate must have no secular time dependence. This observation impliesthat secular terms in ˙ E arising from ... I ij , see below, must cancel with secular terms in ˙ O .In addition, the non-secular portion of ˙ O will average to zero when a time average of thedamping rate over an orbital period is taken, since it is a total time derivative. Thus, ˙ O canbe discarded.Hence restricting attention to a binary system, and taking a time average over an orbitalperiod, the expression reduces as follows. First, define the quantities m = m + m , µ A = m A /m, µ = m m /m, (80)and the vectors r i = x i − x i , v i = ˙ r i , (81) X i = µ x i + µ x i , V i = ˙ X i . (82)To Newtonian order, ˙ v i = − ( G m/r )ˆ r i , and ˙ V i = 0. I ij can be diagonlized I ij = µr i r j + mX i X j , (83)so that ... I ij = 2 G µmr (3ˆ r i ˆ r j ˙ r − v ( i ˆ r j ) ) . (84)As for I ij , I ij = µ ( s µ + s µ ) r i r j + m ( s µ + s µ ) X i X j + 2 µ ( s − s ) r ( i X j ) , (85)and ... I ij = S ... I ij − V ( i ˙Σ j ) + 2 µ ( s − s )... r ( i X j ) , (86)where S = s µ + s µ , (87)and ˙Σ i = ( s − s ) G µmr r i . (88)17erms in ... I ij with X i dependence are secular; following the discussion above, they can bediscarded.Substituting into Eqn. (72) and imposing the time average gives the final expression˙ E = − G N D(cid:16) G µmr (cid:17) × h
815 ( A + SA + S A )(12 v −
11 ˙ r )+ 4( B + SB + S B ) ˙ r + ( s − s ) (cid:16) C + 65 (3 A V + ( A + 30 B )( V i ˆ r i ) ) (cid:17) + ( s − s ) (cid:16)
85 ( A + 2 SA )(3 v i V i − V i ˆ r i v j ˆ r j ) + 12( B + 2 SB ) V i ˆ r i v j ˆ r j (cid:17)iE , (89)where the angular brackets denote the time average. V. OBSERVATIONAL CONSTRAINTSA. Center-of-mass velocity dependence
While the aether frame center-of-mass velocity V i of a binary system is not directlymeasurable, dependence of a binary systems’s motion on V i should actually be beneficialfor constraining the theory. This is because constraints arise from a failure to observe V i dependent effects. It may be possible to formulate such constraints without having todetermine the physical frame, as in the manner of bounds on the PPN parameter α . Thepresence of alignment between the sun’s spin axis and the ecliptic plane signals the absenceof frame dependent effects, and leads to a strong bound of | α | < × − [21]. Thisargument does require the assumption that the component of the preferred frame in thesun’s rest frame is not conveniently aligned with the sun’s spin axis; such an assumptionmay generally be required for similar arguments. For example, V i dependence should causea binary’s orbital plane to precess, but not if V i happens to be normal to the plane.An assumption on the order of magnitude of the norm V is necessary to justify the useof just the leading PN order expressions for the PK parameters when applied to observedbinary systems. The validity of the 1PN expressions depends on whether corrections ofrelative order v and ( V /v ) are smaller than observational uncertainties. Terms of order v are negligible for all observed systems, for now, although the “double pulsar” [27] is18ushing this limit. For all but the double pulsar, v ∼ − , and uncertainties are at leasta thousand times this [19]. The double pulsar PSR J0737-3039A/B is the so-far uniquebinary containing two pulsars. The orbital velocity is high, v ∼ − , and the presence oftwo pulsars happens to make measurement of system parameters much easier and thus moreprecise—the smallest relative uncertainty is 10 − on the rate of periastron advance. The v corrections are therefore small enough for now, but it is expected that precision will increaseto probe the next PN order within the next 10-20 years [19].The V i dependent terms must feature c n dependent factors, since it is known that thereis no center-of-mass velocity dependence at next PN order in pure GR [21]. Ignoring thosefactors for the moment, validity of leading PN order for the double pulsar requires that( V /v ) . − , giving V . − . , or ( V /v ) . . ≈
3. For other systems, givenuncertainties ranging from (10 − ∼ − ), the conditions are ( V /v ) . (10 − ∼ − ),giving V . (10 − . ∼ − . ), or ( V /v ) . (10 . ∼ . ) ≈ (300 ∼ c n dependent factor actually goes to zero as some positive power of c n , so V can be largerin the small c n limit. A reasonable first guess for the aether frame is the rest frame of thecosmic microwave background. A typical velocity for compact objects in our galaxy in thisframe is V ∼ − , so the restriction on V is met. B. Constraints in the small coupling regime
A formula for the sensitivities for a given source should be obtainable by comparing thestrong field results of this article with analogous results in the exact perfect fluid theory.Higher order terms in the exact theory must be calculated, though, since the leading orderresults of [11] only give the O ( G N m/d ) part of s expressed in (70). The calculation canbe done in the case of a single body that is static except for a constant aether framevelocity, by, for example, continuing the iterative procedure used to determine the PPNparameters [8, 23]. The process may be lengthy, but straightforward.I have shown that the sensitivity of a body will scale with the body’s self-potential like β [ c n ]( G N m/d ) , where β is some c n -dependent coefficient that scales at least as fast as c n inthe small c n limit. Even in the absence of a formula for the sensitivities and precise knowledgeof center-of-mass velocities, two useful comments can be derived. First, a constraint can beroughly stated: | β | . (0 . ∼ c n with magnitude less than roughly0 . | β | . (0 . ∼
1) follows from constraints [19] on the magnitude of vio-lations of the strong equivalence principle—that is, that a body’s acceleration is independentof its composition. A violation would lead to a polarization of the orbit of pulsar systemsdue to unequal acceleration of the binary bodies in the gravitational field of the galaxy. Theobserved lack of polarization in neutron star–white dwarf systems leads to a constraint thatcan be stated here as s < .
01, where here s is the sensitivity of the neutron star in theconsidered pulsars. Assuming that ( G N m/d ) ≈ (0 . ∼ .
3) for the pulsar, as it is in GR,the constraint on the size of β arises. It is possible that when the weak field conditions areimposed, β will automatically satisfy the above inequality; certainly it will in the small c n regime when | c n | < . c n freedom satisfies | c n | . .
01, can be derived by consideringthe battery of binary pulsar tests. First, consider tests that probe only the quasi-staticPK parameters—that is, all but the damping rate. The tightest quasi-static test comesfrom the double pulsar [27]. The relative size of the strong field corrections to the weak fieldexpressions will be O ( s A ), while the prediction of GR has been confirmed to within a relativeobservational uncertainty of 0 . s . − and assuming that ( G N m/d ) ≈ ( . ∼ .
3) for the pulsars, the condition | c n | . .
01 arises. Given this and the two conditionsthat set the PPN parameters α and α to zero, all current quasi-static tests will be passed.Tests that incorporate the damping rate will also be satisfied by the small- c n condition andthe weak field conditions. I note first that for systems in which the damping rate is probed,uncertainty on its measurement dominates uncertainties on quasi-static parameters [19, 21].Thus, it is conventional to use the measurements of the quasi-static parameters to solvefor the mass values of the binary bodies. When α = α = 0, and | c n | . .
01, so thatthe expressions for the quasi-static parameters are close to those of GR, the predicted massvalues will also be close.Now, the dipole contribution to ˙ E can be significant in asymmetric systems where the20ensitivity of one body is much larger than the other. The dipole contribution is˙ E Dipole = − G N (cid:10) ( G µmr ) (cid:11) C ( s − s ) , (90)which is of order ( C s / v ) compared to the quadrupole and monopole contribution, where s is the dominant sensitivity. An applicable system is a neutron star–white dwarf binary,since for a typical white dwarf, ( G N m/d ) ∼ − . Constraints have been derived [19] on themagnitude of dipole radiation from neutron star–white dwarf binaries PSR B0655+64 andPSR J1012+5307 by requiring that the dipole radiation rate be no larger than the observedrate. The analysis applied here leads to the condition C s . − , where s is the sensitivityof the neutron star. In the small c n regime, this translates again to the condition | c n | . . .
2% [19, 21]. In the small c n regime, the condition A = 1matches the leading order damping rate to that of GR. The strong field corrections are ofrelative order s ; to be smaller than the uncertainty again requires | c n | . . | c n | will decrease as observational uncertainties decrease. Themost promising candidate for lowering the limit is the double pulsar: 2PN-order and spin-dependent effects should be observable within the next ten or twenty years [27]. Anothertype of system, yet undetected, for which high levels of accuracy could be obtained is aneutron star–black hole binary, as the structureless black hole would decrease noise due tofinite-size effects and mass transfer between the bodies.For | c n | > .
01, strong field contributions to the expressions for the PK parameters maybe significant. Those contributions for a given source cannot yet be calculated, so the theorycannot be checked against observations. Thus, there is no conclusion yet on the viability oflarge c n values. If it were possible to calculate precise predictions for a given binary system,then each observed system would imply an extension from small to large c n of the curve ofallowed values. The only physically viable values would be those for which the curves forall observed systems overlapped within error.21 cknowledgments I wish to thank Alessandra Buonanno, Cole Miller, Ira Rothstein, Clifford Will, andespecially Ted Jacobson for fruitful discussions. This research was supported in part by theNSF under grant PHY-0601800 at the University of Maryland. [1] V. A. Kostelecky and S. Samuel, Phys. Lett.
B207 , 169 (1988).[2] R. Gambini and J. Pullin, Phys. Rev.
D59 , 124021 (1999), gr-qc/9809038.[3] J. L. Hewett, F. J. Petriello, and T. G. Rizzo, Phys. Rev.
D64 , 075012 (2001), hep-ph/0010354.[4] D. Mattingly, Living Rev. Rel. , 5 (2005), gr-qc/0502097.[5] R. Bluhm, Overview of the sme: Implications and phenomenology of Lorentz violation , talkgiven at 339th WE Heraeus Seminar, Potsdam, Germany, 13-18 Feb 2005, hep-ph/0506054.[6] J. W. Elliott, G. D. Moore, and H. Stoica, JHEP , 066 (2005), hep-ph/0505211.[7] S. M. Carroll and E. A. Lim, Phys. Rev. D70 , 123525 (2004), hep-th/0407149.[8] B. Z. Foster and T. Jacobson, Phys. Rev.
D73 , 064015 (2006), gr-qc/0509083.[9] C. Eling and T. Jacobson, Phys. Rev.
D69 , 064005 (2004), gr-qc/0310044.[10] M. L. Graesser, A. Jenkins, and M. B. Wise, Phys. Lett.
B613 , 5 (2005), hep-th/0501223.[11] B. Z. Foster, Phys. Rev.
D73 , 104012 (2006); Erratum: Phys. Rev.
D75 ,129904 (E) (2007);Beware typos, see latest version, gr-qc/0602004.[12] A. Einstein, L. Infeld, and B. Hoffmann, Annals Math. , 65 (1938).[13] D. Eardley, Astrophys. J. , L59 (1975).[14] C. Will and D. Eardley, Astrophys. J. , L91 (1977).[15] T. Damour and G. Esposito-Farese, Phys. Rev. D53 , 5541 (1996), gr-qc/9506063.[16] W. D. Goldberger and I. Z. Rothstein, Phys. Rev.
D73 , 104029 (2006), hep-th/0409156.[17] R. A. Porto and I. Z. Rothstein, Phys. Rev. Lett. , 021101 (2006), gr-qc/0604099.[18] C. Eling and T. Jacobson, Class. Quant. Grav. , 5625 (2006), gr-qc/0603058.[19] I. H. Stairs, Living Rev. Rel. , 5 (2003), astro-ph/0307536.[20] R. M. Wald, General Relativity (Univ. Pr., Chicago, 1984).[21] C. M. Will, Living Rev. Rel. , 4 (2001), gr-qc/0103036.[22] C. Eling, Phys. Rev. D73 , 084026 (2006), gr-qc/0507059.
23] C. M. Will,
Theory and Experiment in Gravitational Physics (Univ. Pr., Cambridge, UK,1993).[24] R. M. Wald, Phys. Rev.
D48 , R3427 (1993), gr-qc/9307038.[25] V. Iyer and R. M. Wald, Phys. Rev.
D50 , 846 (1994), gr-qc/9403028.[26] B. Z. Foster, Phys. Rev.
D73 , 024005 (2006), gr-qc/0509121.[27] M. Kramer et al., Science , 97 (2006), astro-ph/0609417., 97 (2006), astro-ph/0609417.