First-post-Newtonian quadrupole tidal interactions in binary systems
aa r X i v : . [ g r- q c ] O c t First-post-Newtonian quadrupole tidal interactions in binary systems
Justin E. Vines and ´Eanna ´E. Flanagan
Department of Physics, Cornell University, Ithaca, NY 14853, USA. (Dated: October 23, 2018)We consider tidal coupling in a binary stellar system to first-post-Newtonian order. We derivethe orbital equations of motion for bodies with spins and mass quadrupole moments and show thatthey conserve the total linear momentum of the binary. We note that spin-orbit coupling must beincluded in a 1PN treatment of tidal interactions in order to maintain consistency (except in thespecial case of adiabatically induced quadrupoles); inclusion of 1PN quadrupolar tidal effects whileomitting spin effects would lead to a failure of momentum conservation for generic evolution of thequadrupoles. We use momentum conservation to specialize our analysis to the system’s center-of-mass-energy frame; we find the binary’s relative equation of motion in this frame and also presenta generalized Lagrangian from which it can be derived. We then specialize to the case in which thequadrupole moment is adiabatically induced by the tidal field (in which case it is consistent to ignorespin effects). We show how the adiabatic dynamics for the quadrupole can be incorporated intoour action principle and present the simplified orbital equations of motion and conserved energy forthe adiabatic case. These results are relevant to gravitational wave signals from inspiralling binaryneutron stars.
I. INTRODUCTION AND SUMMARYA. Background and motivation
Inspiralling and coalescing compact binaries presentone of the most promising sources for ground-based grav-itational wave (GW) detectors [1]. A primary goal in themeasurement of GW signals from neutron star-neutronstar (NSNS) and black hole-neutron star (BHNS) bina-ries is to probe the neutron star matter equation of state(EoS), which is currently only loosely constrained by elec-tromagnetic observations in the relevant density range ρ ∼ × g/cm [2]. The EoS will leave its imprinton the GW signal via the effects of tidal coupling, asthe neutron star is distorted by the non-uniform field ofits companion. Many recent studies of the effects of theneutron star EoS on binary GW signals have been basedon numerical simulations of the fully relativistic hydro-dynamical evolution of NSNS and BHNS binaries (seee.g. the reviews [3, 4]). These simulations have largely fo-cused on the binaries’ last few orbits and merger, at GWfrequencies &
500 Hz, and have investigated constrainingneutron star structure via (for example) the GW energyspectrum [5], effective cutoff frequencies at merger [6],and tidal disruption signals [7].As recently investigated by Flanagan and Hinderer [8],neutron star internal structure may also have a measur-able influence on the GW signal from the earlier inspiralstage of a binary’s orbital evolution, at GW frequencies .
500 Hz. While tidal coupling will produce only a smallperturbation to the GW signal in this low-frequency adi-abatic regime, the tidal signal should be relatively clean,depending (at leading order) on a single parameter per-taining to the neutron star structure. This tidal de-formability parameter λ is the proportionality constantbetween the applied tidal field and the star’s inducedquadrupole moment and is sensitive to the neutron starEoS. The measurement scheme proposed in Ref. [8] is based on an analytical model for the tidal contributionto the GW signal, giving a linear perturbation to the GWphase proportional to λ . At GW frequencies .
400 Hz,the model should be sufficiently accurate to constrain λ to ∼ Tidal Deformability Computations:
While computingtidal deformations of stars is a well studied problem inNewtonian gravity [11], it has recently been re-examinedin the context of fully relativistic stars by Hinderer etal. [9, 12], Damour and Nagar [13], and Binnington andPoisson [14]. These authors used fully relativistic mod-els for the star’s interior, with various candidate equa-tions of state, to calculate the perturbations to the star’sequilibrium configuration produced by a static exter-nal tidal field. In Refs. [9, 12], the results were usedto determine the (electric-type) quadrupolar tidal de-formability λ , while Refs. [13, 14] corrected computa-tional errors in Ref. [9] and extended the analysis andincluded all higher-multipolar electric-type response co-efficients (for the mass multipole moments) as well astheir magnetic-type analogues (for the current multipolemoments). While these results concern a star’s responseto a static tidal field, they are still applicable for inspi-ralling binaries at sufficiently low orbital frequencies; astar will approximately maintain equilibrium with the in-stantaneous tidal field if the field changes adiabatically. Orbital-tidal conservative dynamics:
In their treat-ment of tidal effects in binary GW signals in Ref. [8],Flanagan and Hinderer used Newtonian gravity to treat(the conservative part of) the system’s orbital dynamics;they estimated that 1PN corrections to the orbits wouldmodify the calculated tidal signal by ∼ Gravitational Wave Emission:
To calculate the emit-ted GW signal and radiation reaction effects, Flanaganand Hinderer [8] used the quadrupole radiation approxi-mation and associated 2.5PN radiation reaction forces.To consistently generalize their calculation in Ref. [8]to (relative) 1PN order, it is necessary to compute the3.5PN corrections to the GW generation, in additionto the 1PN orbital corrections considered in this paper.These calculations have been carried out by Vines, Hin-derer and Flanagan in Ref. [24], which builds off of thework presented here. The issue of tidal effects in inspiral-stage NSNS binary GW signals has also been recentlyaddressed by means of numerical relativity simulationsin Refs. [25, 26].We now turn to a more detailed overview of the prob-lem of the conservative orbital-tidal dynamics, at Newto-nian order in Sec. I B and at 1PN order in Sec. I C, beforesummarizing the results of this paper in Sec. I D.
B. Newtonian tidal coupling
Gravitational tidal coupling arises from the interac-tion of the non-spherical components of a body’s matterdistribution with a non-uniform gravitational field. Thenon-sphericity is characterized at leading order by thebody’s mass quadrupole moment, Q ij ( t ) = Z d x ρ (¯ x i ¯ x j − δ ij | ¯ x | ) , (1.1)with ρ ( t, x ) being the mass density and ¯ x i ( t ) = x i − z i ( t )being the displacement from the body’s center of massposition x i = z i ( t ). The quadrupole is (at most) on theorder of Q ij ∼ M R , with M being the body’s mass and R its radius. Higher-order deformations are describedby the octupole, Q ijk ∼ R d x ρ ¯ x i ¯ x j ¯ x k ∼ M R , andhigher-order multipole moments. The non-uniform fieldcan be characterized by derivatives of an external Newto-nian potential φ ext ( t, x ); in a binary system, the leading-order potential is φ ext = − GM/r , where G is Newton’sconstant, M is the mass of the companion, and r thedistance between the body and its companion.The non-uniform field of the companion produces tidalforces on the non-spherical body (in addition to the usual1 /r force) according to M ¨ z i = − (cid:18) M ∂ i + 12 Q jk ∂ ijk + 16 Q jkl ∂ ijkl + . . . (cid:19) φ ext ∼ GM (cid:20) Mr + | Q ij | r + | Q ijk | r + . . . (cid:21) ∼ GM r (cid:20) R r + O (cid:18) R r (cid:19)(cid:21) , (1.2)where M is the mass of the body or its companion, as-sumed here to be roughly equal, and the derivatives ofthe external potential are evaluated at the body’s centerof mass, x i = z i ( t ). The contributions to the net forcefrom the quadrupole and higher-order multipoles are thusseen to take the form of an expansion in R/r , the ratioof the size of the body to the orbital separation. Whenthis finite-size parameter is small, as in the early stagesof binary inspiral, the tidal force is well approximatedby the quadrupolar term alone. A more detailed accountof Newtonian tidal forces and related results is given inSec. II below.In neutron star binaries, a quadrupole is induced bydifferential forces resulting from the non-uniform field ofthe companion. In the adiabatic limit, when the responsetime scale of the body is much less than the time scaleon which the tidal field changes, the induced quadrupolewill be given (to linear order in the tidal field) by Q ij ( t ) = − λ∂ i ∂ j φ ext ( t, z ) , (1.3)with the constant λ being the tidal deformability. This isrelated to the more often used dimensionless Love num-ber k by λ = 2 k R / G , where R is the body’s radius[11].Using Newtonian gravity to describe the orbital dy-namics and the adiabatic approximation to model thestars’ induced quadrupoles, Flanagan and Hinderer [8]calculated the effect of tidal interactions on the phaseof the gravitational waveform emitted by an inspirallingneutron star binary and analyzed the measurability of thetidal effects. They found that Advanced LIGO should beable to constrain the neutron stars’ tidal deformability to λ ≤ (2 . × g cm s )(D /
50 Mpc) with 90% confidence,for a binary of two 1.4 M ⊙ neutron stars at a distance D from the detector, using only the portion of the signalwith GW frequencies less than 400 Hz. The calculationsof λ for a 1.4 M ⊙ neutron star in Refs. [9, 12–14], us-ing several different equations of state, give values in therange 0.03–1.0 × g cm s , so that nearby events mayallow Advanced LIGO to place useful constraints on can-didate equations of state. Reference [10] discusses howthe EOB formalism can be used to extend the range ofvalidity of analytic waveforms up to merger, and arguesthat Advanced LIGO should be able to detect and mea-sure the tidal deformability of neutron stars.Refs. [8, 9] estimate the fractional corrections tothe tidal signal due to several effects neglected by themodel of the GW phasing used in Ref. [8], namely,non-adiabaticity ( . . . . . . λ usedin the Newtonian treatment and thus can be easily in-corporated into the data analysis methods outlined inRefs. [8, 9]. C. First-post-Newtonian corrections
For inspiralling neutron star binaries with a total massof ∼ M ⊙ at orbital frequencies of ∼
200 Hz (GW frequen-cies of ∼
400 Hz), the post-Newtonian expansion param-eter v /c ∼ GM/c r is ∼ /c correction terms to (1.2)which depend not only on the bodies’ positions and massmultipole moments, M , Q ij , Q ijk , etc., but also on theirvelocities and current multipole moments, S i , S ij , etc.Expanding in both the post-Newtonian parameter v /c and the finite size parameter R/r , the 1PN equations of motion can be written schematically as M ¨ z i ∼ GM (cid:20) Mr + v c Mr + O (cid:18) v c Mr (cid:19) + | Q ij | r + v c | Q ij | r + O (cid:18) | Q ijk | r ∼ R r Mr (cid:19) + v | S i | c r + O (cid:18) v | S ij | c r ∼ v c R r Mr (cid:19) (cid:21) (1.4)In the top line, the first term gives the usual point-particle (monopole) force of Newtonian gravity, and thesecond term represents its 1PN corrections. The lastterm of the top line denotes 2PN and higher post-Newtonian order corrections, which we neglect in thispaper. Point-particle EoMs are in fact currently knownup through 3PN order [27].In the second line of Eq. (1.4), we have first the New-tonian quadrupole-tidal term, followed by its 1PN cor-rections (which are the subject of this paper), and finallycontributions from the octupole and higher mass multi-poles (and their post-Newtonian corrections) which aresuppressed by higher powers of R/r and which we neglectin our analysis [19].The first term in the bottom line represents the 1PNspin-orbit coupling [16], which will be included in ouranalysis, and the final term of the bottom line denotes1PN contributions from the bodies’ current quadrupolesand higher current multipoles, which will not be included.The DSX treatment of 1PN celestial mechanics [15–18]provides a framework for calculating the orbital EoMs forbodies with arbitrarily high-order mass and current mul-tipole moments. In Ref. [16], DSX applied their formal-ism to rederive the explicit 1PN EoMs for bodies withmass monopoles and current dipoles (spins). The calcu-lation was then extended to include the effects of the bod-ies’ mass quadrupoles by Xu et al [20, 21]. Racine andFlanagan (RF) [19] later reworked the DSX formalismand presented explicit 1PN EoMs for bodies with arbi-trarily high-order multipoles, which can be specialized tothe case of bodies with only spins and mass quadrupoles.We have found, however, that the final results of Xu et aland those of RF are in disagreement with each other andwith our recent calculations. Some typos and omissionsleading to errors in RF have been identified and are out-lined in an upcoming erratum; the results given in thispaper agree with the corrected results of RF. In Sec. IIIbelow, we review the essential ideas of the DSX formal-ism, following the notations and conventions of RF, andwe outline the full procedure by which our results for the1PN EoMs are derived.Though it would be convenient to be able to spe-cialize to the case of non-spinning bodies when study-ing tidal interactions, this would lead to inconsistenciesat 1PN order when considering generic behavior of thequadrupole moments. A body with a quadrupole in atidal field will generically experience tidal torques [ac-cording to Eq. (1.7) below] which will spin up the bodyeven if it started with no spin; this is a Newtonian-ordereffect. The resultant spin affects the orbital dynamics viathe 1PN spin-orbit coupling. For this reason, if one wereto work through the DSX formalism and simply dropall spin terms while keeping mass quadrupole terms, onewould arrive at inconsistencies. In particular, one wouldfind that momentum is not conserved at 1PN order (seeSec. IV C below). However, for the special case of adia-batically induced quadrupoles, the tidal torques vanish,and it is then consistent to ignore all spin terms.The relation between the work of DN [22] and BDF [23]on the 1PN conservative orbital-tidal dynamics and theresults of this paper is described in the next subsectionalong with the summary of our results, and in more detailin Appendix A. We also refer the reader to Refs. [10, 28–36] for other recent work on tidal effects in inspirallingbinaries.
D. Summary of results
1. The M - M - S - Q system Our results concern the 1PN gravitational interactionsin a system of two bodies, which we label “1” and “2”.We model body 1 as an effective point particle, witha mass monopole moment only, while we take body 2to have additionally a spin and a mass quadrupole mo-ment. We consistently work to linear order in the spinand quadrupole, and our results can thus be easily gener-alized to the case of two spinning, deformable bodies byinterchanging particle labels. We initially assume noth-ing about the bodies’ internal structure or dynamics. Ourprimary assumption is the validity of the 1PN approxima-tion to general relativity in a vacuum region surroundingthe bodies.The system’s orbital dynamics can be formulatedin terms of the bodies’ center-of-mass worldlines z i ( t )and z i ( t ) and their multipole moments: the massmonopoles M ( t ) and M ( t ), the spin S i ( t ), and the massquadrupole Q ij ( t ).The worldlines x i = z i ( t ) and x i = z i ( t ) parametrizethe bodies’ positions to 1PN accuracy in a (confor-mally Cartesian and harmonic) ’global’ coordinate sys-tem ( t, x i ), which tends to an inertial coordinate systemin Minkowski spacetime as | x | → ∞ . The global coor-dinates and center-of-mass worldlines are defined moreprecisely in Sec. III D. We use the following notation forthe relative position and velocity: z i = z i − z i , r = | z | = p δ ij z i z j , n i = z i /r,v i = ˙ z i , v i = ˙ z i , v i = v i − v i , ˙ r = v i n i , with dots denoting derivatives with respect to the globalframe time coordinate t .The multipole moments— M ( t ), M ( t ), S i ( t ), and Q ij ( t ) for our truncated system—are defined inSecs. III B and III E via a multipole expansion of the 1PNmetric in a vacuum region surrounding each body [19]. In the case of weakly self-gravitating bodies, these momentscan be defined as integrals of the stress-energy tensor overthe volume of the bodies, as in Eqs. (3.12); these defini-tions coincide with those of the Blanchet-Damour multi-pole moments introduced in Ref. [37] and used by DSX[15–18]. The mass multipole moments (like M , M , and Q ij ) are defined with 1PN accuracy, while the currentmultipole moments (like S i ) appear only in 1PN-orderterms and thus need only be defined with Newtonian ac-curacy. We will often denote the spin and quadrupole ofbody 2 by S i and Q ij , dropping the “2” labels.
2. General equations of motion and orbital Lagrangian
The equations of motion for the monopoles M ( t ) and M ( t ), the spin S i ( t ), and the worldlines z i ( t ) and z i ( t )are determined by Einstein’s equations alone, while thatfor the quadrupole Q ij ( t ) will depend on the details ofbody 2’s internal dynamics and can initially be left un-specified. The mass monopole of body 1, the effectivepoint particle, is found to be conserved to 1PN order:˙ M = O ( c − ) , (1.5)while that of body 2 is not. As discussed in Sec. III F,the 1PN-accurate mass monopole M can be decomposedaccording to M = n M + c − (cid:0) E int2 + 3 U Q (cid:1) + O ( c − ) . (1.6a)Here, n M is the Newtonian-order (rest mass) contribu-tion, which is conserved: n ˙ M = 0 . (1.6b)The 1PN contributions to (1.6a) involve the Newtonianpotential energy of the quadrupole-tidal interaction, U Q = − M r Q ij n i n j , (1.6c)and the Newtonian internal energy of body 2, E int2 , whoseevolution is governed by the rate at which the tidal fielddoes work on the body [38]:˙ E int2 = 3 M A r ˙ Q ij n i n j , (1.6d)(cf. Sec. II F). Equations (1.6) ensure that M satisfies the1PN evolution equation (3.28,3.29). The decompositionof the monopole M in (1.6a) is essential for properlyformulating an action principle for the orbital dynam-ics. The evolution of the spin S i is determined by the(Newtonian-order) tidal torque formula:˙ S i = 3 M r ǫ ijk Q ja n a n k + O ( c − ) , (1.7)as in (2.45). Finally, the 1PN translational equations ofmotion, which govern the evolution of the worldlines z i and z i , are of the form¨ z i = F i ( z j , v j , v j , M , M , S j , Q jk , ˙ Q jk , ¨ Q jk ) , ¨ z i = F i ( z j , v j , v j , M , M , S j , Q jk , ˙ Q jk ) , and are given explicitly by Eqs. (3.33).In Sec. IV, we define and calculate the 1PN-accuratemass dipole moment of the entire system M i sys ( t ), givenin Eq. (4.8). We find that the condition ¨ M i sys = O ( c − ),required by Einstein’s equations and reflecting the con-servation of the system’s total momentum, is satisfiedas a consequence of the orbital EoMs (3.33); this servesas a non-trivial consistency check for our results. Theconservation of momentum also allows us to specializethe EoMs to the system’s center-of-mass(-energy) (CoM)frame, which can be defined by the condition M i sys = 0as in Sec. V A. The two EoMs (3.33) for the worldlines z i and z i can then be traded for the single EoM forthe CoM-frame relative position z i = z i − z i , given inEq. (5.9).Our results can be most compactly summarized by giv-ing a Lagrangian formulation of the CoM-frame orbitaldynamics, as discussed in Sec. (V B). We find that the CoM-frame orbital EoM (5.9) can be derived from thegeneralized Euler-Lagrange equation (cid:18) ∂∂z i − ddt ∂∂v i + d dt ∂∂a i (cid:19) L orb = 0 , (1.8)with a generalized Lagrangian L orb given by L orb = L M + L S + L Q , (1.9a)with the monopole part, L M = µv µMr + µc (cid:20) − η v (1.9b)+ M r (cid:18) η ˙ r + (3 + η ) v − Mr (cid:19) (cid:21) + O ( c − ) , the spin part, L S = χ c ǫ abc S a v b (cid:20) Mr n c + χ a c (cid:21) + O ( c − ) , (1.9c)and the quadrupole part, L Q = 3 M r Q ab n a n b + 1 c ( Mr Q ab (cid:20) n a n b (cid:18) A v + A ˙ r + A Mr (cid:19) + A v a v b + A ˙ rn a v b (cid:21) + Mr ˙ Q ab (cid:2) A n a v b + A ˙ rn a n b (cid:3) + E int2 (cid:20) A v + A Mr (cid:21) ) + O ( c − ) . (1.9d)We use here the notation M = M + n M , χ = M /M, χ = n M /M,µ = M n M /M, η = χ χ = µ/M, with M being the total (conserved) Newtonian rest mass, µ the reduced mass, and η the symmetric mass ratio. Thedimensionless coefficients A – A appearing in (1.9d) arefunctions only of the mass ratios χ and χ given by(5.10e).
3. Adiabatic approximation for the induced quadrupole
The above results concerning the orbital EoM and itsLagrangian formulation are valid regardless of the inter-nal structure of body 2, i.e. for arbitrary evolution of itsquadrupole Q ij ( t ). In Sec. VI, we discuss a simple adi-abatic model for the evolution of Q ij . In the adiabaticlimit, the (body-frame) quadrupole responds to the in-stantaneous tidal field according to Q ij ( t ) = λG ij ( t ) , (1.10) where G ij ( t ), given by Eq. (6.6b), is the (body-frame)quadrupolar gravito-electric tidal moment of body 2, a1PN generalization of the derivatives of the Newtonianpotential in Eq. (1.3), and λ is the (constant) tidal de-formability. With the quadrupole given by (1.10), thetidal torque (1.7) vanishes (cf. Eq. (2.46)), so that thespin S i is constant; thus, in the adiabatic limit, we canspecialize to the case S i = 0 without generating incon-sistencies. In Sec. VI A, we show that the adiabatic evo-lution for the quadrupole (1.10) can be derived from aLagrangian that adds to the orbital Lagrangian an inter-nal elastic potential energy term which is quadratic inthe 1PN-accurate quadrupole: L = L orb [ z i , Q ij ] − λ Q ab Q ab + O ( c − ) , (1.11)with L orb given by Eq. (1.9) with S i = 0, and with E int2 = (1 / λ ) Q ab Q ab + O ( c − ). (Note that any addi-tional constant contribution to the internal energy E int2 can be included as a 1PN contribution to the constant n M in Eq. 1.6a.) Substituting the solution (1.10) for Q ij ( t ) into this Lagrangian, we obtain a simplified La-grangian involving only the CoM-frame relative position z i ( t ): L [ z i ] = µv µMr (cid:18) r (cid:19) + µc (cid:26) θ v + Mr (cid:20) v (cid:18) θ + ξ Λ r (cid:19) + ˙ r (cid:18) θ + ξ Λ r (cid:19) + Mr (cid:18) θ + ξ Λ r (cid:19)(cid:21)(cid:27) (1.12)with Λ = (3 χ / χ ) λ , and with the dimensionless coeffi-cients θ = (1 − η ) / ,θ = (3 + η ) / ,θ = η/ ,θ = − / ,ξ = ( χ / χ ) ,ξ = − − χ + χ ) ,ξ = − χ . (1.13)This Lagrangian represents one of the primary resultsof this paper. Since the first appearance of this work,an analogous Lagrangian (in a different gauge) has been derived by Bini, Damour and Faye in Ref. [23], usingeffective action techniques. In fact, BDF have greatlyextended the analysis by carrying the calculation to 2PNorder. Also (prior to this work), Damour and Nagar [22]presented a 1PN Hamiltonian valid for circular orbits.These works both incorporate their results into the EOBformalism. In Appendix A, we demonstrate the completeequivalence of those results with ours at 1PN order.The orbital EoM resulting from the Lagrangian (1.12),which can also be found by substituting the adiabaticsolution (1.10) for Q ij ( t ) directly into the general orbitalEoM (5.9), is given by Eq. (6.10). The conserved energy E ( z , v ) derived from the Lagrangian (1.12) is given byEq. (6.13). In the case of circular orbits, we find thegauge-invariant energy-frequency relationship E ( ω ) = µ ( M ω ) / (cid:20) −
12 + 9 χ χ λω / M / + (9 + η )24 ( M ω ) / c + 11 χ χ (3 + 2 χ + 3 χ ) λω M c (cid:21) . (1.14)This result, along with others from this paper, are usedin calculating the phasing of GW signals from inspirallingneutron star binaries in Ref. [24]. E. Notation and Conventions
We use units where Newton’s constant is G = 1, butretain factors of the speed of light c , with 1 /c serving asthe formal expansion parameter for the post-Newtonianexpansion. We use lowercase Latin letters a, b, i, j, . . . for indices of (three-dimensional) spatial tensors. Spa-tial indices are contracted with the Euclidean metric, v i w i = δ ij v i w j , with up or down placement of the indiceshaving no meaning. We use uppercase Latin letters to de-note multi-indices: L denotes the l indices a a . . . a l , K denotes the k indices b b . . . b k , etc. For a given vector v i or for the partial derivative operator ∂ i , we use multi-indices or explicit sequences of indices to denote theirtensorial powers: v L = v a a ...a l = v a v a . . . v a l ,∂ K = ∂ b b ...b k = ∂ b ∂ b . . . ∂ b k , (1.15) and, for example, v ij = v i v j . We also use v = v ii and ∇ = ∂ ii . Multi-indices are also used for sets ofdistinct tensors of varying rank, { M, M a , M ab , . . . } , with M L = M a a ··· a l denoting the tensor of rank l . We usethe Einstein summation convention for both individualindices and multi-indices. Derivatives with respect to atime coordinate t are denoted by ∂ t or by overdots.We use angular brackets to denote the symmetric,trace-free (STF) projection of tensors [15]: T
In this section, we review the standard treatment oftidal coupling in Newtonian theory [11]; the first-post-Newtonian treatment given subsequent sections makesextensive use of these Newtonian-order results. We definethe multipole moments and tidal moments of an extendedobject and use them to derive the orbital (or transla-tional) equations of motion for systems of gravitatingbodies. We consider in particular the case of a binarysystem containing a point particle (body 1) and an ex-tended deformable star (body 2), working to quadrupolarorder in the star’s multipole series. We also discuss anaction principle formulation of the orbital dynamics, theprocess of energy transfer between the gravitational fieldand the deformable body, and the evolution of the body’sspin due to tidal torques. Finally, we discuss the evolu-tion of the body’s tidally induced quadrupole moment inthe adiabatic limit.
A. Field equations
In Newtonian physics, the scalar potential φ ( t, x )obeys the Poisson equation, ∇ φ = 4 πρ, (2.1)with ρ ( t, x ) being the rest mass density of matter. Theinfluence of the gravitational field on matter is describedby the test particle acceleration ¨ x i = − ∂ i φ , or more gen-erally, by Euler’s equation supplemented by the continu-ity equation (the conservation of mass), ∂ t ( ρv i ) + ∂ j ( ρv i v j + t ij ) = − ρ∂ i φ, (2.2)˙ ρ + ∂ i ( ρv i ) = 0 , (2.3)with v i ( t, x ) being the matter’s velocity field, and t ij ( t, x ) the material stress tensor. Together, Eqs. (2.1-2.3) provide a complete description of Newtonian grav-itational interactions. However, they do not in generalform a closed set of evolution equations for the fields φ , ρ , v i , and t ij ; one needs also to specify the matter’s inter-nal dynamics, in particular concerning the stress tensor t ij . (In the simplest cases, one can fix t ij by an algebraicequation, like t ij = 0 for ‘dust’ or t ij = pδ ij with p ( ρ )being the pressure for an isentropic perfect fluid.) Still,one can derive many useful results, like the form of thetranslational equations of motion for a system of gravitat-ing bodies, while leaving the matter’s internal dynamicsunspecified. B. Multipole moments
We consider N isolated celestial bodies, i.e. regions ofspace containing matter ( ρ = 0) surrounded by regionsof vacuum ( ρ = 0), and label the bodies by an index A ,with 1 ≤ A ≤ N . The potential that is locally generatedby body A , which we will call the internal potential φ int A , is given by the standard solution to (2.1) as an integralover the volume of the body: φ int A ( t, x ) = − Z A d x ′ ρ ( t, x ′ ) | x − x ′ | . (2.4)We can express this potential as a multipole series arounda (moving) point x i = z iA ( t ) by using the Taylor series1 | x − x ′ | = ∞ X l =0 ( − l l ! ( x ′ − z A ) L ∂ L | x − z A | , (2.5)where L = a a . . . a l is a spatial multi-index, denotinghere tensorial powers of the vector ( x ′ − z A ) i and of theoperator ∂ i (cf. Eq. (1.15)). Using the Taylor series(2.5) in (2.4) allows us to write the internal potential,for points x i exterior to the body, in the form φ int A ( t, x ) = − ∞ X l =0 ( − l l ! M LA ( t ) ∂ L | x − z A ( t ) | , (2.6)with M LA ( t ) = Z A d x ρ ( t, x ) [ x − z A ( t )]
Having described the potential generated by an iso-lated body with its multipole moments, we can similarlydescribe the potential felt by the body with tidal mo-ments. Given a collection of several bodies indexed by B , each giving rise its own intrinsic potential of the form(2.6), we define the external potential felt by a givenbody A to be the sum of the potentials due to all theother bodies: φ ext A = X B = A φ int B (2.11)The body’s tidal moments G L g ,A ( t ) are then defined ascoefficients in the Taylor expansion of the external po-tential about the center-of-mass position z iA : φ ext A ( t, x ) = − ∞ X l =0 l ! G L g ,A ( t ) [ x − z A ( t )] L , (2.12) G L g ,A ( t ) = − ∂ L φ ext A ( t, x ) (cid:12)(cid:12) x = z A ( t ) . (2.13) The subscript g, standing for ‘global,’ has been included hereto avoid confusion with tidal moments introduced in our post-Newtonian treatment below. We introduce there a set of body-frame tidal moments G LA , defined in an accelerated referenceframe attached to the body, and a set of global-frame tidal mo-ments G L g ,A , defined in an (asymptotically) inertial frame. Thetidal moments defined in (2.13) coincide with the latter at New-tonian order. While we have chosen to work exclusively in aninertial frame in our Newtonian treatment here, an analogousNewtonian treatment that uses accelerated frames can be foundin Sec. III of Ref. [16]. Like the multipole moments, the tidal moments are STFtensors, G L g ,A = G
2, the G L g ,A are higher-order derivatives of the potential that will giverise to tidal forces on a non-spherical body.The tidal moments of a body A can be expressed interms of the multipole moments of the other bodies B = A by combining (2.6), (2.11), and (2.13), with the result G L g ,A = − X B = A ∂ L φ int B ( t, x ) (cid:12)(cid:12) x = z A ( t ) = X B = A ∞ X k =0 ( − k k ! M KB ∂ ( A ) KL | z A − z B | , (2.14)where ∂ ( A ) i = ∂/∂z iA , and ∂ KL = ∂ b . . . ∂ b k ∂ a . . . ∂ a l . D. Translational equations of motion
A primary advantage of the language of multipole andtidal moments is that it allows one to take the PDEs (2.1-2.3) governing the evolution of the fields ρ , v i , t ij , and φ and extract from them ODEs for the center-of-massworldlines z iA ( t ) of a collection of gravitating bodies. Tothis end, we consider a body A with multipole moments M LA defined by (2.7), in the presence of an external poten-tial φ ext A generated by other bodies B = A according to(2.12) and (2.14). The body’s translational EoM can befound by applying two time derivatives to the definitionof z iA in (2.10), using the Euler and continuity equations(2.3) and (2.2), and integrating by parts. The result isan expression for the body’s center-of-mass acceleration, M A ¨ z iA = − Z d x ρ ∂ i φ ext A , (2.15)which can be rewritten in terms of the body’s multipoleand tidal moments by using (2.12) and (2.7): M A ¨ z iA = ∞ X l =0 l ! M LA G iL g ,A (2.16)= M A G i g ,A + 12 Q jkA G ijk g ,A + . . . The first term in the second line represents the force thatwould act on a freely falling test mass M A , while thesecond term gives the leading-order tidal force.To render the EoM (2.16) fully explicit, one can use theexpressions for the tidal moments (2.14) in terms of themultipole moments and worldlines of the other bodies.Considering the case of a two-body system A = 1 ,
2, withbody 1 having only a monopole moment M , and body 2having a monopole M and a quadrupole Q ij ≡ Q ij , wefind the following EoMs: M ¨ z i = M M ∂ (1) i | z − z | + 12 M Q jk ∂ (1) ijk | z − z | M ¨ z i = M M ∂ (2) i | z − z | + 12 Q jk M ∂ (2) ijk | z − z | Defining the radius and unit vector associated with therelative position, z i = z i − z i , r = | z | , n i = z i /r, (2.17)and using the general identity, ∂ L r = ( − l (2 l − n
As for any Newtonian system, a Lagrangian for a col-lection of gravitating bodies can be constructed from L = T − U , with T being the kinetic energy and U the po-tential energy. The total kinetic energy receives separatecontributions from each body, T = P A T A , and each T A can be split into a contribution from the center-of-massmotion of the body and an internal contribution: T A = 12 Z A d x ρ v = 12 M A ˙ z A + T int A , (2.21) T int A = 12 Z A d x ρ ( v − ˙ z A ) . (2.22) Here, we have used the fact that R A d x ρ v i = M A ˙ z iA , im-plied by the continuity equation (2.3) and the definitionof the center-of-mass position z iA in (2.10). The gravi-tational potential energy U = P A U A can be similarlysplit into external and internal parts: U A = 12 Z A d x ρ φ = U ext A + U int A , (2.23) U int A = 12 Z A d x ρ φ int A , (2.24) U ext A = 12 Z A d x ρ φ ext A = − ∞ X l =0 l ! M LA G L g ,A . (2.25)In the last line, we have used (2.12) to express φ ext A interms of the tidal moments and (2.7) for the definition ofthe multipole moments.While the system’s total potential energy will also re-ceive non-gravitational contributions from the internalstructure of each body, we can lump these contributions,along with T int A and U int A as defined above, into an inter-nal Lagrangian L int A for each body. We can then writethe total Lagrangian for an N -body system as L = X A M A ˙ z A + 12 ∞ X l =0 l ! M LA G L g ,A + L int A ! . (2.26)The bodies’ center-of-mass worldlines z iA ( t ) enter this La-grangian through the translational kinetic energy termsand through the external gravitational potential energyterms (via the tidal moments); the internal Lagrangians L int A , however, are independent of the worldlines z iA , byconstruction. The L int A will be functions of some set ofinternal configuration variables q αA (and their time deriva-tives) for each body, which will include e.g. Euler anglesfor the orientation of the body, vibrational mode ampli-tudes, etc., depending on the model of the body’s internalstructure. (In full generality, the proper internal configu-ration variables q αA are the fields ρ , v i and t ij , subject to(2.10) as a constraint.) The bodies’ multipole moments M LA , for l ≥
2, appearing in the gravitational potentialenergy terms in (2.26), will be functions of these sameinternal variables q αA . Together, the z iA and q αA , for allbodies A , form a complete set of dynamical variables forthe N -body system. Varying the action S = R L dt withrespect to the worldlines z iA reproduces the translationalEoMs (2.16). Determining the evolution of the variables q αA , and hence the moments M LA for l ≥
2, will require amodel for L int A ( q αA , ˙ q αA ) and M LA ( q αA ).Specializing to the two-body M - M - Q case and using(2.14), (2.17), and (2.18), the Lagrangian (2.26) becomes L = 12 M ˙ z + 12 M ˙ z + M M r − U Q + L int2 . (2.27)where U Q is the potential energy of the quadrupole-tidalinteraction: U Q = − Q ij G ij g , , G ij g , = 3 M r n
Continuing to specialize to the two-body M - M - Q case, we can construct a conserved energy for the systemfrom E = T + U = ∂ L ∂ ˙ z i ˙ z i + ∂ L ∂ ˙ z i ˙ z i + ∂ L ∂ ˙ q α ˙ q α − L , (2.31)with summation over α implied. Using the (CoM-frame)Lagrangian (2.29), we find E = µ ˙ z − µMr + U Q + E int2 , (2.32)where U Q is given by (2.28), and the internal energy ofbody 2 is given by E int2 ( q α , ˙ q α ) = ∂ L int2 ∂ ˙ q α ˙ q α − L int2 (2.33)This internal energy will generally have several contribu-tions: internal gravitational potential energy, rotationalkinetic energy, vibrational kinetic and potential energy,thermal energy, etc. Nonetheless, the rate at which en-ergy is exchanged between the interior of body 2 and itssurroundings, via the gravitational tidal interaction, isa function only of the orbital separation, M , and Q ij .Differentiating (2.33) with respect to time and using theEoM (2.30) for the internal variables q α , we find˙ E int2 = ˙ q α (cid:18) ddt ∂ L int2 ∂ ˙ q α − ∂ L int2 ∂q α (cid:19) = 12 G ij g , ∂Q ij ∂q α ˙ q α = 12 G ij g , ˙ Q ij = 3 M r n ij ˙ Q ij (2.34) The energy transfer described by (2.34) is often referredto as tidal heating (see e.g. [39]). This expression for thepower delivered to the body is valid (in the quadrupolarapproximation) regardless of the body’s internal dynam-ics. Using (2.34) and the orbital EoM (2.20), one canconfirm that the binary system’s total energy (2.32) isconserved. G. Adiabatic approximation
While we have thus far left unspecified the internaldynamics for the deformable body 2, which determinesthe evolution of the quadrupole, we now specialize ouranalysis to the case where Q ij ( t ) is adiabatically inducedby the tidal field. This will lead to a closed system ofevolution equations for the binary. In the adiabatic limit,when the body’s internal dynamical time scales are muchless than the orbital period, the quadrupole will respondto the instantaneous tidal field according to Q ij ( t ) = λG ij g , ( t ) , (2.35)where λ is the tidal deformability, and G ab g , ( t ) is the tidaltensor given in Eq. (2.28). As discussed in Sec. I B and inmore detail in Refs. [8, 9], the relation (2.35) should bevalid to ∼
1% for neutron star binaries at GW frequencies .
400 Hz.The adiabatic evolution of the quadrupole can be in-corporated into our action principle (2.29) by taking Q ij ( t ) to be our lone internal configuration variable( q α ), and by taking the internal Lagrangian L int2 to con-tain a simple quadratic potential energy cost for thequadrupole: L [ z i , Q ij ] = µ ˙ z µMr − U Q + L int2 (2.36)= µ ˙ z µMr + 12 G ab g , Q ab − λ Q ab Q ab . Varying the action with respect to the orbital separation z i still gives the orbital EoM (2.20), and varying withrespect to the quadrupole Q ij reproduces the adiabaticevolution equation (2.35). Using Eq. (2.35) to replace Q ij and Eq. (2.28) for G ab g , , the Lagrangian can can writtensolely in terms of z i as L [ z i ] = µ ˙ z µMr (cid:18) λM r M (cid:19) . (2.37)This Lagrangian leads to the orbital EoM (2.20) with Q ij replaced by its adiabatic value (2.35), a i = − M n i r (cid:18) λM r M (cid:19) , (2.38)which shows that the tidal coupling results in an attrac-tive force. For circular orbits, with a i = − rω n i , we findthe radius-frequency relationship r ( ω ) = M / ω / (cid:18) − λM ω / M M / (cid:19) , (2.39)1to linear order in the tidal deformability λ .The internal energy E int2 (2.33), in the adiabatic ap-proximation, is given by E int2 = 14 λ Q ab Q ab , (2.40)up to a constant, and satisfies the tidal heating equa-tion (2.34) by virtue of Eq. (2.35). (We should note thatthere is actually no ‘heating’ taking place here, as thismodel neglects dissipative effects and is completely con-servative.) Using Eq. (2.35), the binary system’s totalenergy E (2.32) can be written as E = µ ˙ z − µMr (cid:18) λM r M (cid:19) , (2.41)in the adiabatic model, and is conserved by the orbitalEoM (2.38). Then, using ˙ z = r ω and Eq. (2.39), wefind E ( ω ) = − µ M ω ) / (cid:18) − λM ω / M M / (cid:19) , (2.42)for the circular-orbit energy-frequency relationship. H. Spin
In anticipation of the 1PN treatment of the binary’sorbital dynamics, in which a body’s angular momentum(or spin) has a direct influence on the orbit, it will beuseful to discuss the evolution of an extended body’s spinat Newtonian order. The spin of a body A about its CoMworldline z iA is defined by S aA ( t ) = ǫ abc Z d x ρ ( t, x ) (cid:2) x b − z bA ( t ) (cid:3) v c ( t, x ) , (2.43)with ρ being the mass density and v c the velocity field.Taking a time derivative of this equation, using the Eulerand continuity equations (2.2) and (2.3) and the defini-tion of z iA in (2.10), and integrating by parts, we find˙ S aA = − ǫ abc Z d x ρ ( x b − z bA ) ∂ c φ ext A = ǫ abc ∞ X l =0 l ! M bLA G cL g ,A . (2.44)In the second line, we have used the definitions of M LA and G L g ,A in (2.7) and (2.13). This formula gives the torqueon the body due to tidal forces. As it is not directlyrelevant to our purposes, we will not discuss a Lagrangianformulation of the Newtonian rotational dynamics.Applying Eq. (2.44) to the M - M - Q system, we findthat the tidal torque on body 2 is given by˙ S a = ǫ abc Q bd G cd g , (2.45)with the tidal tensor G cd g , given by (2.28). Eq. (2.45) isvalid (in the quadrupolar approximation) regardless of the internal dynamics of body 2. In the special case ofan adiabatically induced quadrupole, as in Eq. (2.35), wefind ˙ S a = λǫ abc G bd g , G cd g , = 0 , (2.46)so that the spin is conserved. III. POST-NEWTONIAN TIDALINTERACTIONSA. Overview
The Newtonian theory of gravity arises as a limit-ing case of general relativity (GR). In the limit of smallsource velocities and weak gravity, the spacetime metricof GR takes the form ds = − (cid:18) φc (cid:19) c dt + δ ij dx i dx j + O ( c − ) , (3.1)with φ ( t, x ) being the Newtonian potential. This expres-sion represents a perturbation expansion of the theorywith 1 /c playing the role of a formal expansion param-eter. At leading order in 1 /c , Einstein’s equation andcovariant stress-energy conservation for the metric (3.1)reproduce the Poisson, Euler, and continuity equations(2.1-2.3)—the basic equations of Newtonian gravity.The first post-Newtonian (1PN) approximation to GRcontinues this perturbation expansion to next-to-leadingorder in 1 /c . The 1PN metric can be written as ds = − (cid:20) φc + 2 c ( φ + ψ ) (cid:21) c dt + 2 ζ i c dt dx i + (cid:18) − φc (cid:19) δ ij dx i dx j + O ( c − ) , (3.2)(cf. Weinberg [40]), with two new degrees of freedom ap-pearing: a 1PN (three-)vector potential ζ i ( t, x ) (oftencalled the gravito-magnetic potential), and a 1PN scalarpotential ψ ( t, x ). Following Refs. [15, 37] (apart froma change of sign) we shall work with the single scalarpotential single scalar potential Φ( t, x ) which has φ and ψ as its Newtonian- and 1PN-order parts (and hence ahidden c -dependence),Φ = φ + c − ψ + O ( c − ) , (3.3)so that the metric can be written as ds = − (cid:20) c + 2Φ c (cid:21) c dt + 2 ζ i c dt dx i + (cid:18) − c (cid:19) δ ij dx i dx j + O ( c − ) . (3.4)We choose to work here in conformally Cartesian co-ordinates (see e.g. [15]), which is already implicit in the2form (3.4) of the metric, and to adopt the harmonic gaugecondition: ∂ µ ( √− gg µν ) = 0 ⇔ ∂ i ζ i = O ( c − ) . (3.5)In this gauge, one finds that Einstein’s equation for themetric, at next-to-leading order in 1 /c , is equivalent tothe following linear field equations for the potentials: ∇ Φ = 4 πT tt + c − (cid:16) πT ii + ¨Φ (cid:17) + O ( c − ) , (3.6a) ∇ ζ i = 16 πT ti + O ( c − ) , (3.6b)where T µν are the contravariant components of thestress-energy tensor in the ( t, x i ) coordinate system.Note that the T tt component must include both O ( c )Newtonian and O ( c − ) post-Newtonian contributions,while the components T ti and T ij and the quantity ¨Φare needed only to Newtonian order. The influence ofthe gravitational field on matter is governed by covari-ant stress-energy conservation: ∇ µ T µν = 0; the 1PN-expanded form of this equation can be found in AppendixD of RF [19].In the remainder of this section, we review thetreatment of tidal interactions within this first-post-Newtonian framework. We employ a formalism, origi-nally developed by DSX [15, 16] and later expoundedupon by RF [19], that uses multiple coordinate systemsto describe the global motion and local structure of ex-tended bodies. We attempt to present here the pri-mary ingredients and broad logical flow of this formalism,which are essential for properly interpreting the resultsstated in Sec. I D above.We begin in Sec. III B by presenting a general solutionto the 1PN-order Einstein equations (3.6) which givesthe spacetime metric in a vacuum region surrounding anastronomical body A . The solution (3.8) is parametrizedby and defines the body’s multipole moments and tidalmoments. The mass and current multipole moments M LA and S LA characterize the body’s internal structure, andthe gravito-electric and -magnetic tidal moments G LA and H LA characterize the external gravitational fields felt bythe body.In Sec. III C we discuss the gauge freedom in the 1PNmetric, summarized by the parametrization of a general1PN coordinate transformation in Eq. (3.13). In 1PNcelestial mechanics, it is advantageous to use and trans-form between two types of coordinate systems: global co-ordinates ( t, x i ) used to describe the motion of multiplebodies, and body-adapted coordinates ( s A , y iA ) used inthe local description of a given body A . We discuss howto fix all 1PN coordinate freedom in the body-adaptedcoordinates ( s A , y iA ) by enforcing the body-frame gaugeconditions (3.15). The body-frame multipole moments M LA ( s A ) and S LA ( s A )—the moments defined by the mul-tipole expansions in Sec. III B using the body-adaptedcoordinates—then become unique and meaningful de-scriptors of a body’s internal structure.In Sec. III D, we discuss the form of the metric in theglobal coordinates ( t, x i ). It is written in terms of a set of global-frame multipole moments M L g ,A ( t ) and Z iL g ,A ( t )and tidal moments G L g ,A ( t ) and Y iL g ,A ( t ) for each body A ,which differ from the body-frame moments. The relation-ship between the global- and body-frame moments is de-termined by the transformation (3.13) between the globaland body-adapted coordinates; the moment transforma-tion formulae are presented in full detail in Appendix B.The functions parameterizing the coordinate transforma-tion, or the worldline data D A (3.14), are seen to take onthe role of configuration variables for body A with respectto the global frame. Among other things, they determinethe body’s center-of-mass worldline, x i = z iA ( t ).In Sec. III E, we discuss the 1PN single-body laws ofmotion (3.22) which govern the evolution of a body’smass monopole M A , mass dipole M iA , and current dipole(or spin) S iA . These laws reflect the conservation of en-ergy, momentum, and angular momentum and can be de-rived from stress-energy conservation at 1PN order [16],or equivalently, from Einstein’s equation at 2PN order[19]. A body’s translational equation of motion—an ODEfor its global-frame center-of-mass worldline z iA ( t )—canbe deduced from the law of motion for its body-framemass dipole. The result is an expression for the accelera-tion ¨ z iA , for each member A of an N -body system, writtenin terms of the body-frame multipole moments M LA and S LA and global-frame worldlines z iA of all the bodies A .Finally, in Sec. III F, we specialize our discussion tothe case of a two-body system with a body 1 havingonly a mass monopole M , and a body 2 having a massmonopole M , a mass quadrupole Q ij ≡ Q ij , and a spin S i ≡ S i . We present and discuss the explicit forms ofthe evolution equations for the moments M , M , and S i and the worldlines z i and z i , which depend only on thesequantities and Q ij . B. 1PN multipole and tidal moments
In Sec. II, we defined the Newtonian mass multipolemoments n M LA (called simply M LA there) as integralsover the body’s mass distribution (2.7). In so doing,we implicitly assumed that the Newtonian Poisson equa-tion (2.1) was valid in all space, including the interiorof the body. A similar approach can be taken at 1PNorder, defining 1PN-accurate multipole moments as inte-grals over a body’s stress-energy distribution (as in (3.12)below), assuming that the 1PN field equations (3.6) arevalid in all space. This was the approach taken in theoriginal DSX formalism.As stressed by RF, one can also define a body’s 1PNmultipole moments without requiring the validity of the1PN field equations in the interior of the body. Instead,one need only impose the field equations in a vacuumbuffer region B A , a region of finite extent enclosed be-tween two coordinate spheres centered on the body; themoments can then be defined through the multipole ex-pansion of the 1PN metric in the region B A . This allowsone to consider objects with strong internal gravity, like3neutron stars and black holes, as long as there exists aregion B A exterior to the object where gravity is suffi-ciently weak and quasi-static for the 1PN field equationsto be valid.Taking the latter approach, we assume the existence ofa local coordinate system ( s A , y iA ), in the vicinity of thebody A , having the following properties: (i) The rangeof the coordinates includes the product of the open ball | y A | < r , for some finite radius r , with an open inter-val of time ( s A , s A ). (ii) There exists a spatial region W A (the worldtube) of the form | y A | < r that contains allthe body’s stress-energy and/or regions of strong gravity. (iii) In the buffer region B A ( r < | y A | < r ),the coordi-nates ( s A , y iA ) are conformally Cartesian and harmonic,and the metric takes the 1PN form (3.4), with potentialsΦ A ( s A , y A ) and ζ iA ( s A , y A ) satisfying the 1PN vacuumfield equations: ∇ Φ A = c − ¨Φ + O ( c − ) , ∇ ζ iA = O ( c − ) . (3.7)Under these assumptions, RF showed that the generalsolution for the potentials in B A is of the formΦ A ( s A , y A ) = − ∞ X l =0 l ! (cid:26) ( − l M LA ( s A ) ∂ L | y A | + G LA ( s A ) y LA (3.8a)+ 1 c (cid:20) ( − l (2 l + 1)( l + 1)(2 l + 3) ˙ µ LA ( s A ) ∂ L | y A | + ( − l M LA ( s A ) ∂ L | y A |− ˙ ν LA ( s A ) y LA + 12(2 l + 3) ¨ G LA ( s A ) y jjLA (cid:21)(cid:27) + O ( c − ) , (3.8b) ζ iA ( s A , y A ) = − ∞ X l =0 l ! (cid:26) ( − l Z iLA ( s A ) ∂ L | y A | + Y iLA ( s A ) y LA (cid:27) + O ( c − ) , (3.8c)with Z iLA ( s A ) = 4 l + 1 ˙ M iLA ( s A ) − ll + 1 ǫ jijA ( s A ) + 2 l − l + 1 δ iA ( s A ) + O ( c − ) , (3.9a) Y iLA ( s A ) = ν iLA ( s A ) + ll + 1 ǫ jijA ( s A ) − l − l + 1 ˙ G
0, are thebody’s mass multipole moments, which are defined with1PN-accuracy. Next are the current multipole moments, S LA ( s A ), with l ≥
1, needed only to Newtonian accuracy.Together, the mass and current multipole moments con-tain all the information about the body’s internal struc-ture that is encoded in the gravitational field it produces(at 1PN order). They are associated with the contribu-tions to the potentials that appear to diverge as | y A | → µ LA ( l ≥ | y A | → ∞ (the external parts) arethe tidal moments: the gravito-electric tidal moments, G LA ( s A ), with l ≥
0, are defined with 1PN accuracy (like M LA ), and the gravito-magnetic tidal moments H LA ( s A ),with l ≥
1, are defined with Newtonian accuracy (like S LA ). The tidal moments contain information about grav-itational fields generated by external sources and aboutinertial effects associated with the motion of the local co-ordinate system. Finally, there are the tidal gauge mo-ments ν LA , defined for l ≥ Z iLA ( s A ) and Y iLA ( s A ) appearing in thegravito-magnetic potential (3.8c) have been defined asuseful shorthands for the expressions in (3.9). Unlike allthe other moments just introduced, they are not STF onall their indices, but they are STF on their last l indices(i.e. on all but the first index). Eqs. (3.9) in fact repre-sent their unique decompositions in terms of fully STFtensors; the ‘inverse’ relations are S LA = − Z jk
The 1PN metric (3.4) harbors residual coordinate free-dom not fixed by the conformally Cartesian and harmonicgauge conditions. As a result, the multipole and tidalmoments defined in the last section (not just the ‘gaugemoments’ µ LA and ν LA , but rather all of the moments)are not unique and will vary with the choice of coordi-nates. To define a unique set of multipole moments for agiven body, one must further specialize the body-framecoordinates. Thus we turn now to a discussion of 1PNcoordinate transformations.In RF [19], it was shown that the most general transfor-mation between two harmonic coordinate systems ( s, y i )and ( t, x i ) in which the metric takes the 1PN form (3.4)can be written as x i ( s, y ) = y i + z i ( s ) + 1 c (cid:26) (cid:20)
12 ˙ z kk ( s ) δ ij − ˙ α ( s ) δ ij + ǫ ijk R k ( s ) + 12 ˙ z ij ( s ) (cid:21) y j + (cid:20)
12 ¨ z i ( s ) δ jk − ¨ z k ( s ) δ ij (cid:21) y jk (cid:27) + O ( c − ) , (3.13a) t ( s, y ) = s + 1 c (cid:2) α ( s ) + ˙ z j ( s ) y j (cid:3) + 1 c (cid:20) β ( s, y ) + 16 ¨ α ( s ) y jj + 110 ... z jA ( s ) y jkk (cid:21) + O ( c − ) , (3.13b)being parametrized by the following functions. The vec-tor z i ( s ) provides a time-dependent translation betweenthe spatial coordinates and is defined with 1PN accuracy.Each defined with Newtonian accuracy are the rotationvector R i ( s ), and the functions α ( s ) and β ( s, y ) whichtransform the time coordinate. All of these may be arbi-trary functions of their arguments (within the bounds oftheir post-Newtonian scaling), except that β ( s, y ) mustbe harmonic, ∇ β = 0, in order to preserve the harmonicgauge condition.In the treatment of the N -body problem, we will make Though an O ( c − ) contribution to α ( s ) would contribute at O ( c − ) in Eq. (3.13b), this contribution can be absorbed intothe function β ( s, y ). use of one global coordinate system ( t, x i ) and one body-adapted coordinate system ( s A , y iA ) for each body A .The global coordinates (described further in the next sec-tion) are used to track the bulk motion of all the bodies,while the body-adapted coordinates are used in the localdescription of each body—in particular, to define theirbody-frame multipole and tidal moments. The transfor-mation between the ( t, x i ) and ( s A , y iA ) coordinates willtake the form (3.13), with different ’worldline data’ func-tions, D A = { z iA ( s A ) , R iA ( s A ) , α A ( s A ) , β A ( s A , y iA ) } (3.14)for each body A . These functions may be viewed as con-figuration variables for the body-adapted frame, speci-fying its position, orientation, etc. relative to the globalframe.In order to uniquely define the body-frame multipole5and tidal moments, we must fix some of the remaininggauge freedom in the body-adapted coordinates ( s A , y iA ).Here, we will fix all remaining gauge freedom in the body-adapted coordinates (in the bodies’ buffer regions), whichwill also uniquely determine the gauge moments. It wasshown in RF that this can be always be accomplishedby imposing the following conditions, which define thebody-adapted gauge: M iA ( s A ) = 0 (3.15a) R iA ( s A ) = 0 (3.15b) G A ( s A ) = µ A ( s A ) = 0 (3.15c) µ LA ( s A ) = ν LA ( s A ) = 0 , l ≥ M iA tozero, fixes the body’s center of mass-energy to the originof the spatial coordinates y iA = 0 to 1PN order. Settingthe rotation vector R iA to zero in Eq. (3.15b) fixes theorientation of the body-frame spatial axes to those of theglobal frame. If the extended body A were replaced by afreely falling observer at y iA = 0 (assuming an extensionof the body-frame coordinates to y iA = 0), Eq. (3.15c)would ensure that the time coordinate s A measures theirproper time. Finally, the fact that all the gauge momentscan always be set to zero by a coordinate transformation,as in (3.15d), shows that they are pure gauge degrees offreedom.We can think of the body-adapted coordinates as defin-ing the body’s local asymptotic rest frame [41], in whichthe effects of external gravitational fields and inertialeffects have been removed as much as possible . Thebody-frame moments—the multipole and tidal moments defined by (3.8) in the body-adapted coordinates—thentake on the values that would be measured by a localcomoving observer in the body frame. The body-framemultipole moments M LA and S LA are the quantities de-scribing the bodies’ internal structure that will appearin the final form of the translational equation of motionsfor an N -body system. D. The global frame
To treat the orbital dynamics of a collection of sev-eral bodies A = 1 . . . N , we consider N + 1 separate co-ordinate systems: one body-adapted coordinate system( s A , y iA ) for each body A , and one global coordinate sys-tem ( t, x i ). We take these coordinate systems to havethe following properties: (i) For each body A , the body-adapted coordinates ( s A , y iA ) cover the body’s buffer re-gion B A and satisfy all the assumptions and gauge con-ditions outlined in Secs. III B and III C. (ii) The bodies’buffer regions B A are non-overlapping. (iii) The globalcoordinates ( t, x i ) cover the buffer regions of all the bod-ies as well as the intervening space; i.e. they cover theregion B g = M\ S A W A , the entire spacetime mani-fold M except for the worldtubes. (iv) In the region B g , the coordinates ( t, x i ) are conformally Cartesian andharmonic, and the metric takes the form (3.4), with po-tentials Φ g ( t, x ) and ζ i g ( t, x ) satisfying the PN vacuumfield equations (Eqs. (3.7) with A → g).The final assumption allows us to write down the fol-lowing multipole expansion of the global-frame potentialsin B g : In place of the condition (3.15b), RF chose to set the gravito-magnetic dipole tidal moment H iA ( s A ) to zero, which cancelsleading-order Coriolis forces in the body-adapted frame and re-quires a non-zero value of the rotation vector R iA ( s A ). Whilethis more completely effaces external gravitational and inertial effects in the body frame, Eq. (3.15b) leads to more simplifica-tions in calculations. The effects of these differing gauge choicescancel in all final results. Φ g ( t, x ) = − N X A =1 ∞ X l =0 ( − l l ! (cid:26) M L g ,A ( t ) ∂ L | x − z A ( t ) | + 12 c ∂ t h M L g ,A ( t ) ∂ L | x − z A ( t ) | i(cid:27) + O ( c − ) , (3.16a) ζ i g ( t, x ) = − N X A =1 ∞ X l =0 ( − l l ! Z iL g ,A ( t ) ∂ L | x − z A ( t ) | + O ( c − ) , (3.16b)This expansion is analogous to that for the body-framepotentials (3.8) but has several important differences.Firstly, the potentials are written as a sum of contribu-tions from each body A ; this is justified by the linearityof the field equations (3.6). Each such contribution isparametrized by the body’s global-frame multipole mo-ments: the mass multipole moments M L g ,A ( t ) are fullySTF and 1PN-accurate, and the tensors Z iL g ,A ( t ) are STFon all but the first index and Newtonian-accurate. Both sets of tensors are defined for l ≥
0. As these global-frame moments will only appear in intermediate stagesof our calculations, we will not bother decomposing thetensors Z iL g ,A in terms of fully STF current and gauge mo-ments as in the body-frame case (3.9a). We have takenthe moments Z iL g ,A to satisfy Z
0) and nottidal pieces (which would appear to diverge as | x − z A | →∞ ). This makes the global-frame metric tend to theMinkowski metric as | x | → ∞ , thus eliminating any tidalor inertial forces on the N -body system as a whole. Eachbody will still experience local tidal fields, but they willarise from the contributions to the potentials generatedby the other bodies.We can introduce a set of global-frame tidal momentsfor each body A by rewriting the global-frame potentials,in the body’s buffer region B A , as The second condition in (3.17), along with the fact that theglobal-frame potentials all vanish as | x | → ∞ , reduces theresidual gauge freedom in the global-frame metric to the groupof post-Galilean transformations (the post-Newtonian Poincar´e group) [19], which are the coordinate transformations given by(3.13) with ¨ z i = ˙ R i = β = 0 and ˙ α = ˙ z / Φ g ( t, x ) = − ∞ X l =0 l ! (cid:26) ( − l M L g ,A ( t ) ∂ L | x − z A ( t ) | + G L g ,A ( t )[ x − z A ( t )] L (3.18a)+ 12 c ∂ t (cid:20) ( − l M L g ,A ( t ) ∂ L | x − z A ( t ) | + 12 l + 3 G L g ,A ( t )[ x − z A ( t )] jjL (cid:21) (cid:27) + O ( c − ) ζ i g ( t, x ) = − ∞ X l =0 l ! (cid:20) ( − l Z iL g ,A ( t ) ∂ L | x − z A ( t ) | + Y iL g ,A [ x − z A ( t )] L (cid:21) + O ( c − ) , (3.18b)Here, we have absorbed the contributions to the poten-tials from the other bodies B = A into tidal terms forbody A . This defines the global-frame tidal moments G L g ,A ( t ) and Y iL g ,A ( t ). They can be expressed in terms ofthe global-frame multipole moments of the other bodies B = A and the worldlines z iA of all the bodies A by equat-ing the expressions for the potentials in (3.18) with thosein (3.16); these relations are given in Appendix B 2.Now, as the global coordinates x µ = ( t, x i ) and thebody-adapted coordinates y µA = ( s A , y iA ) are related bythe coordinate transformation (3.13), the metrics in theglobal and body frames must be related by the tensortransformation law: g Aµν = ∂x ρ ∂y µA ∂x σ ∂y νA g gρσ (3.19)This requirement allows one to determine both the pa-rameters of the coordinate transformation between thetwo coordinate systems (3.13) and the relationship be-tween the global- and body-frame multipole and tidalmoments. Making use of the form (3.4) for the metric interms of the potentials (in both coordinate systems) andthe expressions for the body-frame potentials (3.8) andthe global-frame potentials (3.18), as detailed in RF [19],Eq. (3.19) yields expressions for the body-frame moments in terms of the global-frame moments and the worldlinedata (or the inverse relations):( M LA , S LA ) D A ←→ ( M L g ,A , Z iL g ,A )( G LA , H LA ) D A ←→ ( G L g ,A , Y iL g ,A ) (3.20)These moment transformations are presented in full de-tail in Appendices B 1 and B 3.By combining the transformation formulae for the tidalmoments with the body-frame gauge conditions (3.15),one can solve for and eliminate the worldline data func-tions α A ( s A ) and β A ( s A , y jA ). The only remaining pieceof the worldline data D A (3.14) is the translation vec-tor z iA ( s A ). Recall that the body-frame gauge condition M iA = 0 (3.15a), setting the mass dipole to zero, fixesthe body’s center of mass-energy to the body-frame spa-tial origin y iA = 0. Setting y iA = 0 in the coordinatetransformation (3.13) and eliminating s A , we see that x i = z iA ( t ) encodes the body’s global-frame center-of-mass worldline, where z iA ( t ) = z iA ( s A ) (cid:12)(cid:12)(cid:12) s A = s A ( t ) (3.21)is the quantity z Ai ( s A ) expressed as a function of t , withthe function s A ( t ) found by setting y iA = 0 in Eq. (3.13b)7(see Eq. (B1) and discussion thereabouts). The transla-tional equation of motion for a body A , discussed in thenext subsection, can be written in the form of a second-order ODE for the global-frame CoM worldline z iA ( t ). E. Single-body laws of motion and translationalequations of motion
The single-body laws of motion are constraints on thelowest-order multipole moments of any body which gov-ern the exchange of energy, momentum, and angular mo-mentum between the body and the gravitational field.The laws of motion at 1PN order were first found by DSX, who derived them by using covariant stress-energyconservation at 1PN order in the interior of the body.The same laws of motion were later rederived by RFby using the 2PN (next-to-next-to-leading order in 1 /c )vacuum Einstein equation in a buffer region surroundingthe body, thus extending their range of validity to includebodies with strong internal gravity.The laws of motion are written in terms of the body’smultipole and tidal moments as defined by the expansionof the 1PN potentials (3.8) and are valid in any coordi-nate system in which the spacetime metric takes the formgiven by (3.4) and (3.8)—not just in body-adapted coor-dinates. The results are˙ M A = − c ∞ X l =0 l ! h ( l + 1) n M LA n ˙ G LA + l n ˙ M LA n G LA i + O ( c − ) , (3.22a)¨ M iA = ∞ X l =0 l ! (cid:26) M LA G LA + 1 c (cid:20) l + 2 ǫ ijk M jLA ˙ H kLA + 1 l + 1 ǫ ijk ˙ M jLA H kLA − l + 7 l + 15 l + 6( l + 1)(2 l + 3) M iLA ¨ G LA − l + 5 l + 12 l + 5( l + 1) ˙ M iLA ˙ G LA − l + l + 4 l + 1 ¨ M iLA G LA + ll + 1 S LA H iLA − l + 1)( l + 2) ǫ ijk S jLA ˙ G kLA − l + 2 ǫ ijk ˙ S jLA G kLA (cid:21)(cid:27) + O ( c − ) . (3.22b)˙ S iA = ∞ X l =0 l ! ǫ ijk M jLA G kLA + O ( c − ) , (3.22c)Eq. (3.22a) shows that the mass monopole M A is con-served at Newtonian order ( O ( c )), but not at 1PN or-der. As discussed further in Sec. III F below, M A con-tains O ( c − ) contributions from the internal energy ofthe body, which can vary as tidal forces do work on thebody. The law of motion (3.22c) for the the spin S iA isthe same Newtonian-order tidal torque formula found inEq. (2.45).The law of motion (3.22b) for the mass dipole M iA governs the evolution of the body’s total linear momen-tum. The body’s translational equation of motion canbe derived by applying (3.22b) in the body frame, i.e. byapplying it to the body-frame dipole moment, as follows.At Newtonian order, the O ( c ) part of ¨ M iA in (3.22b)gives the net force acting on the body, as ˙ M iA is thebody’s total momentum (cf. (3.12a)). Since the body-adapted coordinates are chosen to be mass-centered( M iA = 0), this net force must vanish in the body frame.This apparent equilibrium in the body frame is achievedby the balancing of gravitational forces from the otherbodies with inertial forces, which are due to the factthat the body frame is accelerating with respect to the(asymptotically) inertial global frame, along the world-line z iA ( t ). Both of these effects are accounted for by the body-frame tidal moments G LA ; from the O ( c ) part of[ref], we have G iA = G i g ,A − ¨ z iA + O ( c − ) ,G LA = G L g ,A + O ( c − ) , ( l ≥ ,G L g ,A = X B = A ∞ X k =0 ( − k k ! M KB ∂ ( A ) KL | z A − z B | + O ( c − )Using these relations in (3.22b), the requirement of equi-librium in the body frame, ¨ M iA = 0, determines the equa-tion of motion for the worldline z iA ( t ): M A ¨ z Ai = X B = A ∞ X l =0 l ! M AL G g,AiL + O ( c − ) (3.23)= X B = A ∞ X k,l =0 k ! l ! M AL M BK ∂ ( A ) iKL | z A − z B | + O ( c − ) . This matches the Newtonian equation of motion foundabove in (2.16).At 1PN order, the procedure is essentially the same,but more involved. One begins by setting ¨ M iA = 0 in thebody frame, with ¨ M iA given by (3.22b). To arrive at a8suitable form for the final equation of motion, one mustthen rewrite the body-frame tidal moments G LA and H LA of body A in terms of the body-frame multipole moments M LB and S LB of the other bodies B = A and the worldlinedata D C for all the bodies C ; the details of this procedureare presented in Appendix B.In the end, one arrives at an expression for the accel-eration ¨ z iA ( t ) of the 1PN-accurate global-frame center-of-mass worldline z iA ( t ), defined by (3.21). (As in theNewtonian case, the acceleration term, describing iner-tial forces in the body frame, emerges from the trans-formation laws for the body-frame tidal moments.) Theexpression depends only on the body-frame mass and cur-rent multipole moments M LB ( t ) and S LB ( t ), the global-frame worldlines z iB ( t ), and the time derivatives of thesequantities, for all bodies B :¨ z iA ( t ) = F iA ( z iB , ˙ z iB , M LB , ˙ M LB , ¨ M LB , S LB , ˙ S LB ) . (3.24a)Similar (though simpler) manipulations applied to thelaws of motion (3.22a) and (3.22c) allow one to writeequations of motion for the mass monopole M A ( t ) andspin S iA ( t ) in terms of the same variables:˙ M A ( t ) = F A ( z iB , ˙ z iB , M LB , ˙ M LB ) , (3.24b)˙ S iA ( t ) = ˜ F iA ( z iB , M LB ) . (3.24c)The explicit forms of the translational equations of mo-tion (3.24a) and the mass and spin evolution equations(3.24b) and (3.24c) will be given for the M - M - S - Q case in Sec. III F, and are given in the fully general casein RF [19] as corrected by an upcoming erratum.To arrive at a closed set of evolution equations forthe quantities z iA ( t ), M LA ( t ), and S LA ( t ), for all bodies A ,the equations of motion (3.24) must be supplemented byequations for the multipole moments M AL ( t ) and S AL ( t )for l ≥
2. Finding such equations will require a model forthe bodies’ internal dynamics, which will be addressed inSec. VI. F. M - M - S - Q truncation We have presented above the formalism for treatingthe 1PN dynamics of a collection of many bodies, eachwith arbitrarily high-order multipole moments. Here, weapply that formalism to the two-body system discussedin Sec. I D, with a body 1 having only a mass monopolemoment M , and a body 2 having a mass monopole M ,a current dipole, or spin, S i ≡ S i , and a mass quadrupole M ij ≡ Q ij ≡ Q ij . More precisely, we truncate the in-ternal parts of body-frame multipole series (3.8) for each Here, and throughout, M LA ( t ) and S LA ( t ) are the body-framemoments M LA ( s A ) and S LA ( s A ) expressed as functions of t at y iA = 0, i.e. the same physical quantities expressed as functionsof different variables; cf. Eq. (B1) and surrounding discussion. body according toΦ , int = − M | y | + O ( c − ) ,ζ i , int = O ( c − ) , neglecting the moments M L for l ≥ S L for l ≥ , int = − M | y | − Q ij ∂ ij | y |− c ¨ Q ij ∂ ij | y | + O ( c − ) ,ζ i , int = 2 (cid:16) ˙ Q ij − ǫ ijk S k (cid:17) ∂ j | y | + O ( c − ) , neglecting the moments M L for l ≥ S L for l ≥
2. (The external parts of these potentials will be justas in (3.8), with arbitrarily higher-order tidal moments.)These expressions for the body-frame potentials definethe body-frame moments M ( s ), M ( s ), Q ij ( s ), and S i ( s ) as functions of the body-frame time coordinates s and s . These moments can be expressed as functions ofthe global time coordinate, written M ( t ), M ( t ), S i ( t )and Q ij ( t ), using the coordinate transformation (3.13)with y iA = 0 (cf. Eq. (B1)). For the bodies’ global-frameCoM worldlines z i ( t ) and z i ( t ), we will use the definitions z i = z i − z i , r = | z | , n i = z i /r, (3.25)as similar to (2.17), except that the worldlines are nowdefined with 1PN accuracy, and v i = ˙ z i , v i = ˙ z i , v i = v i − v i . (3.26)With these conventions in place, we can apply thelaws of motion presented in Sec. III E to find the evolu-tion equations for the moments M ( t ), M ( t ), and S i ( t )and the global-frame center-of-mass worldlines z i ( t ) and z i ( t ). The results involve only these quantities and thequadrupole Q ij ( t ).As body 1 has no higher-order multipole moments, thelaw of motion (3.22a) requires that its mass monopole M be constant in time: ˙ M = O ( c − ) . (3.27)The same law of motion applied to body 2 gives˙ M = − c (cid:18) Q ij ˙ G ij − ˙ Q ij G ij (cid:19) + O ( c − ) , (3.28)where the body-frame gravito-electric tidal moment G ij is given by (B7) as G ij = G ij g , + O ( c − ) = 3 M A r n
32 ( n a v a ) − M r − M r (cid:21) + v i n a (4 v a − v a ) (cid:27) + O ( c − ) , (3.33c) F i ,M = − M r n i − c M r (cid:26) n i (cid:20) v − v −
32 ( n a v a ) − M r − M r (cid:21) + v i n a (4 v a − v a ) (cid:27) + O ( c − ) , (3.33d)the spin contributions, F i ,S = 1 c M r ǫ abc S c h δ ai (4 v b − n bd v d ) − n ai v b i + O ( c − ) , (3.33e) F i ,S = 1 c M r ǫ abc S c h δ ai ( n bd v d − v b ) + 6 n ai v b i + O ( c − ) , (3.33f)0and the quadrupole contributions, F i ,Q = 3 M r Q ab (cid:0) n abi − n a δ bi (cid:1) + 1 c M r Q ab (cid:26) n abi (cid:20) v − v −
72 ( n c v c ) − M r − M r (cid:21) − n a δ bi (cid:20) v − v −
52 ( n c v c ) − M r − M r (cid:21) + n a v bi + (5 n ai − δ ai ) v bc n c + v i (5 n abc − n a δ bc )(4 v c − v c ) (cid:27) + 3 M r ˙ Q ab h n ab (5 v c n ci + 3 v i ) − v a n bi − δ ai n bc (2 v c − v c ) i − M r ¨ Q ab (cid:0) n abi + 2 n a δ bi (cid:1)! + O ( c − ) , (3.33g) F i ,Q = − M r Q ab (cid:0) n abi − n a δ bi (cid:1) + 1 c M r Q ab (cid:26) − n abi (cid:20) v − v −
72 ( n c v c ) − M r − M r (cid:21) +2 n a δ bi (cid:20) v − v − n c v c ) −
52 ( n c v c ) − M r − M r (cid:21) + n i v ab + 5 n aci (2 v b v c − v bc )+ v i (5 n abc − n a δ bc )(4 v c − v c ) + n a v b ( v i − v i ) + δ bi n c [(5 v a − v a ) v c − v a v c ] (cid:27) + 3 M r ˙ Q ab h v b (2 n ai − δ ai ) + δ ai n bc v c − n ab v i i! + O ( c − ) . (3.33h)(It should be noted that occurrences of ˙ S i in the equa-tions of motion have been replaced by (3.32) and includedin the quadrupole contributions.)The monopole contributions (3.33c,3.33d) give thewell-known Lorentz-Droste-Einstein-Infeld-Hoffmann ac-celerations, and the spin contributions (3.33e,3.33f)give the well-known 1PN spin-orbit terms [16]. Thequadrupole contributions (3.33g,3.33h) have been derivedpreviously by Xu, Wu, and Schafer [20], though our re-sults disagree with theirs in several terms; we have notbeen able to pin down the source of the disagreement.Our results also disagree with the final results of RF [19],but agree with their corrected results given in an upcom-ing erratum. The strongest indication of the correctnessof our expressions for the EoMs is the fact that, unlikethe results in [19, 20], they are consistent with the conser-vation of the binary system’s total linear momentum, asdiscussed in Sec. IV C below. We can also note that theaction derived below from these results agrees with therecent work (using a rather different method) by Damourand Nagar [22] and Bini, Damour, and Faye [23]. IV. SYSTEM MULTIPOLE MOMENTS ANDCONSERVATION LAWS
In Sec. III B, we defined the multipole moments of asingle body through the multipole expansion of the met- ric in a vacuum buffer region surrounding the body. Thesame procedure can be applied to a collection of severalbodies to define multipole moments for the entire sys-tem. Applying the general laws of motion discussed inSec. III E to these system multipole moments will allowus to formulate conservation laws for the energy, momen-tum, and angular momentum of an isolated N -body sys-tem, expressed as constraints on the worldlines and mul-tipole moments of the constituent bodies. These conser-vation laws can serve both as a consistency check for theequations of motion given in Sec. III F and as a means tospecialize the equations of motion to the system’s center-of-mass frame. A. General formulae
We have already discussed, in Sec. III D, a form forthe metric generated by a system of N bodies. Using theglobal coordinate system ( t, x i ), we expressed the poten-tials parameterizing the metric as a sum of multipoleexpansions for each body A , written in terms of the bod-ies’ global-frame multipole moments M L g ,A and Z iL g ,A andNewtonian-order worldlines z iA :1Φ g = − X A ∞ X l =0 ( − l l ! (cid:20) M L g ,A ∂ L | x − z A | + 12 c ∂ t (cid:0) M L g ,A ∂ L | x − z A | (cid:1)(cid:21) + O ( c − ) , (4.1a) ζ i g = − X A ∞ X l =0 ( − l l ! Z iL g ,A ∂ L | x − z A | + O ( c − ) , (4.1b)with Z iL g ,A satisfying (3.17). This solution for the met-ric was constructed to be valid in the region B g , whichextends out to spatial infinity.In a region far outside the system, we can rewrite theseexpressions for the global-frame potentials to mirror the forms (3.8) used to define the multipole moments of asingle body, with multipole expansions about the global-frame origin x i = 0:Φ g = − ∞ X l =0 ( − l l ! (cid:26) M L sys ∂ L | x | + 1 c (cid:20) (2 l + 1)( l + 1)(2 l + 3) ˙ µ L sys ∂ L | x | + 12 n ¨ M L sys ∂ L | x | (cid:21)(cid:27) + O ( c − ) , (4.2a) ζ i g = − ∞ X l =0 ( − l l ! Z iL sys ∂ L | x | + O ( c − ) , (4.2b)with Z iL sys = 4 l + 1 n ˙ M iL sys − ll + 1 ǫ jij sys + 2 l − l + 1 δ i sys + O ( c − ) . (4.3)These expansions define the system multipole moments M L sys and S L sys and the gauge moments µ L sys . The tidalterms present in (3.8) are absent here, as the global-framepotentials vanish as | x | → ∞ (cf. Eq. (4.1)). To comparethe metric (4.2) here to the metric (4.1) above, we mustexpress them in the same gauge. We have chosen thegauge that enforces Z jjL g ,A = 0 in (4.1), which will resultin nonzero values for the system gauge moments µ L sys in(4.2).Since the potentials given by (4.1) and by (4.2) rep-resent the same metric in the same gauge, they shouldbe explicitly equal. This condition will allow us to solvefor the system multipole moments appearing in (4.2) interms of the individual bodies’ global-frame multipolemoments and worldlines appearing in (4.1).Considering first the vector potential ζ i g , we can usethe Taylor series | x − z A | n = ∞ X k =0 ( − k k ! z KA ∂ K | x | n (4.4) to rewrite (4.1b) in the form ζ i g = − X A ∞ X l,k =0 ( − l + k l ! k ! Z iL g ,A z KA ∂ LK | x | = − X A ∞ X p =0 p X k =0 ( − p p ! p ! k !( p − k )! Z A ∂ P | x | . In the second line, we have relabeled the multi-indicesaccording to LK → P , adjusted the summations accord-ingly, and used the fact that ∂ P | x | − is STF. Renaming P → L and comparing this with (4.2b) gives an expres-sion for the tensors Z iL sys : Z iL sys = X A l X k =0 l !( l − k )! k ! Z i Specializing the general formula (4.6) for the systemmass multipoles to the monopole ( l = 0) case, and usingthe M - M - S - Q truncation, we find the binary system’stotal 1PN-accurate mass monopole to be M sys = M g , + M g , + 1 c (cid:20) − 13 ˙ µ sys + 16 d dt (cid:0) M g , z + M g , z (cid:1) (cid:21) + O ( c − ) . Using (4.5) for ˙ µ sys and the formulae in Appendix B 1relating the global- and body-frame multipole moments, we can rewrite this expression in terms of the body-framemultipole moments and the CoM worldlines: M sys = M + M + 1 c (cid:18) M v M v − M M r − U Q (cid:19) + O ( c − ) , where the tidal potential energy U Q , as in (2.28), is U Q = − M r Q ij n ij . If we rewrite the mass monopole of body 2 as M = n M + c − (cid:0) E int2 + 3 U Q (cid:1) , as in (3.31), we find that the1PN contribution to the system mass monopole is exactlythe system’s total Newtonian energy E given by (2.32): M sys = M + n M + c − E + O ( c − ) (4.7) E = 12 M v + 12 M v − M M r + U Q + E int2 . This is a further validation of the decomposition of thetotal mass monopole M in Eq. (3.31). The constancy of M sys , required by the law of motion (3.22a) as appliedto the entire system in the global frame (for which thereare no tidal moments), then follows from the constancyof E . C. System mass dipole Taking the l = 1 case in the general formula (4.6) givesthe system’s 1PN-accurate mass dipole: M i sys = M g , z i + M g , z i + M i g , + M i g , + 1 c (cid:20) − 310 ˙ µ i sys + 110 ∂ t (cid:16) M ij g , z j + M g , z ijj + M g , z ijj (cid:17) (cid:21) + O ( c − )Using (4.5) for ˙ µ i sys , (3.32) to replace an occurrence of˙ S i , (3.31) to replace M , (2.28) for U Q , and the momenttransformations from Appendix B 1, this becomes M i sys = M z i + n M z i + 1 c " z i (cid:18) M v − M M r + U Q (cid:19) + z i (cid:18) M v − M M r + U Q E int2 (cid:19) + 3 M r Q ij n j + ǫ ijk v j S k + O ( c − ) (4.8)3From the law of motion (3.22b) as applied to the en-tire system in the global frame, for which there are notidal moments, we see that ¨ M i sys should vanish; this isa statement of total momentum conservation. By differ-entiating (4.8), order reducing as appropriate, using thefull 1PN translational equations of motion (3.33) in theNewtonian terms and their Newtonian parts in the 1PNterms, and also using (2.34) and (3.32) for ˙ E int2 and ˙ S i ,we find that indeed ¨ M i sys = 0. This is an important checkof the correctness of our expressions for the equations ofmotion and of the consistency of the formalism (which isnot satisfied by the EoMs given in Ref. [19, 20]).Note that the inclusion of the spin term in Eq. (4.8) isessential for this consistency check, reflecting the neces-sity of including spin terms when working with genericmass quadrupoles at 1PN order. Anecdotally, at thebeginning of this work, we tried working through theDSX formalism dropping all spin terms while keepingquadrupoles, and we obtained certain EoMs. To checkour results, we wanted to show that the momentum givenby the time derivative of Eq. (4.8) (without the finalterm, the spin term) vanished as a result of our EoMs,which it did not. Only with the inclusion of the spin termin (4.8) and with all spin terms in the EoMs were we ableto ensure momentum conservation for completely genericevolution of the quadrupoles. V. ORBITAL DYNAMICS IN THE SYSTEM’SCENTER-OF-MASS FRAMEA. Equation of motion of the relative position The conservation of momentum allows us to reduce theproblem of solving for the two worldlines z i ( t ) and z i ( t )to solving for just their separation z i ( t ) = z i ( t ) − z i ( t )in the binary system’s center-of-mass (CoM) frame. Wecan define the CoM frame to be that in which the 1PN-accurate mass dipole vanishes, M i sys ( t ) = 0 , (5.1)so that the system’s center-of-mass(-energy) is at rest atthe global-frame spatial origin. This fixes all remaining (post-Galilean) coordinate freedom in the global-framemetric. Using Eq. (4.8), this condition can be used tosolve for the worldlines z i and z i in the global CoM framein terms of the relative position z i (working perturba-tively in c − ); one finds z i = − χ z i + c − (cid:0) P z i − D i (cid:1) + O ( c − ) , (5.2) z i = χ z i + c − (cid:0) P z i − D i (cid:1) + O ( c − ) , (5.3)with P = η ( χ − χ ) (cid:18) v − M r − χ r Q ij n ij (cid:19) − χ M E int2 , D i = 3 χ i r Q ij n j + χ M ǫ ijk v j S k , (5.4)and with the new notation M = M + n M , χ = M /M, χ = n M /M,µ = M n M /M, η = χ χ = µ/M. (5.5)To find the acceleration of the relative position in theCoM frame, we can simply subtract our above results(3.33) for the individual accelerations: a i ≡ ¨ z i = ¨ z i − ¨ z i . (5.6)The resulting expression depends only on z i , v i , v i , M , M , Q ij , S i , and E int2 . As v i and v i appear only in 1PNterms, we can replace them with their Newtonian valuesin the CoM frame, v i = − χ v i + O ( c − ) , v i = χ v i + O ( c − ) , (5.7)from differentiating the O ( c ) parts of (5.2,5.3). Afterdefining one last shorthand,˙ r = n a v a , (5.8)we can write our result for the 1PN-accurate CoM-framerelative acceleration as follows: a i = a iM + a iS + a iQ , (5.9a)with the monopole contribution, a iM = − Mr n i − c Mr (cid:26) n i (cid:20) (1 + 3 η ) v − η r − η ) Mr (cid:21) − − η ) ˙ rv i (cid:27) + O ( c − ) , (5.9b)the spin contribution, a iS = ǫ abc S c c χ r (cid:2) (3 + χ ) v a δ bi − χ ) ˙ rn a δ bi + 2 n ai v b (cid:3) + O ( c − ) , (5.9c)4and the quadrupole contribution, a iQ = − Q ab χ r (cid:2) n abi − n a δ bi (cid:3) + 1 c ( Q ab r (cid:20) n abi (cid:18) B v + B ˙ r + B Mr (cid:19) + n a δ bi (cid:18) B v + B ˙ r + B Mr (cid:19) + B ˙ rn ab v i + B n a v bi + B ˙ rn ai v b + B v ab n i + B ˙ rv a δ bi (cid:21) + ˙ Q ab r (cid:2) B n ab v i + B ˙ rn abi + B n ai v b + B v a δ bi + B ˙ rn a δ bi (cid:3) + ¨ Q ab r (cid:2) B n abi + B n a δ bi (cid:3) − B E int2 r n i ) + O ( c − ) , (5.9d)with coefficients B = − χ (1 + 3 η ) , B = 105 χ , B = 12 χ (5 − χ ) , B = 3 χ (2 + 2 χ − χ ) ,B = − χ (2 − χ − χ ) , B = − χ (8 − χ − χ ) , B = 15 χ (2 − η ) , B = − χ (7 − χ + 3 χ ) ,B = − χ χ (1 + χ ) , B = 3 χ χ , B = 32 χ (5 − χ − χ ) , B = − χ (4 − χ ) , B = − χ ,B = 6 χ , B = − χ χ , B = 3 χ (1 − χ − χ ) , B = 34 , B = 32 , B = 1 . (5.9e)In this form for the CoM-frame orbital EoM, wehave used (3.31) to write the total 1PN-accurate massmonopole M in terms of the (constant) Newtonian mass n M , the internal energy E int2 , and the tidal potentialenergy U Q (giving a contribution to B ). This decompo-sition is useful in formulating an action principle for theorbital dynamics (as in the next subsection), as E int2 isindependent of the orbital degrees of freedom, while M is not. B. Generalized Lagrangian for the orbital dynamics The monopole contributions (5.9b) to the 1PN CoM-frame orbital EoMs are known to be derivable from theLagrangian L M = µv µMr + µc (cid:20) − η v (5.10a)+ M r (cid:18) (3 + η ) v + η ˙ r − Mr (cid:19) (cid:21) + O ( c − ) , (see e.g. [42]). The spin contributions (5.9c) can also bederived from an action principle, but with a generalizedLagrangian (one depending not only on the relative posi-tion z i and velocity v i = ˙ z i , but also on the acceleration a i = ¨ z i ) given by adding L S = χ c ǫ abc S a v b (cid:20) Mr n c + χ a c (cid:21) + O ( c − ) (5.10b)to (5.10a) (see e.g. [43]). Applying the generalized Euler-Lagrange equation, (cid:18) ∂∂z i − ddt ∂∂v i + d dt ∂∂a i (cid:19) L = 0 , (5.10c)to L = L M + L S , and using ˙ M = ˙ M = ˙ S i = 0 (whichreplaces (3.27,3.28,3.32) in the case with no quadrupole),one recovers the EoM a i = a iM + a iS from (5.9b,5.9c).We have found that the quadrupole contributions tothe orbital EoM (5.9d) can also be encoded in a general-ized Lagrangian. To determine the necessary additions tothe Lagrangian, one can proceed by guesswork, using theknown Newtonian Lagrangian (2.27), and writing downall possible 1PN-order scalars that can be formed fromthe relative position z i and velocity v i , the total (Newto-nian) mass M , and linear factors of the quadrupole Q ij ,its time derivative ˙ Q ij , and the internal energy E int2 ; in-cluding dimensionless coefficients A – A for each suchterm, we have5 L Q = 3 M r Q ab n ab + 1 c ( Mr Q ab (cid:20) n ab (cid:18) A v + A ˙ r + A Mr (cid:19) + A v ab + A ˙ rn a v b (cid:21) + Mr ˙ Q ab (cid:2) A n a v b + A ˙ rn ab (cid:3) + E int2 (cid:20) A v + A Mr (cid:21) ) + O ( c − ) . (5.10d)Since the spin-orbit terms (5.10b) require the accelera-tion a i , one might expect that terms with factors of a i and also ¨ Q ij should be included here; we find, however,that such terms are not necessary. The only further termallowed by general considerations but not included hereis E int2 ˙ r , as recovering the EoM (5.9) requires its coeffi-cient to be zero.By applying the Euler-Lagrange equation (5.10c) tothe generalized Lagrangian L = L M + L S + L Q , using theevolution equations (3.32) and (2.34) for time derivativesof S i and E int2 , one finds an EoM of the same form as (5.9)but with coefficients B – B in (5.9d) given as functionsof the Lagrangian coefficients A – A and the mass ratios χ and χ . Setting these coefficients equal to the valuesfor B – B given (5.9e) gives a system of 19 equations forthe 9 unknowns A – A , which has the unique solution A = 3 χ η ) , A = 15 ηχ ,A = − χ χ ) , A = 3 χ ,A = − χ χ ) , A = − η ,A = − η , A = χ , A = χ . (5.10e)Thus, the action principle (5.10) reproduces the 1PNCoM-frame equation of motion (5.9) for the relative po-sition z i —if we also make use of the evolution equations(3.32) and (2.34) for the spin S i and internal energy E int of body 2. In the next section, we discuss an action prin-ciple that leads to a closed set of evolution equations forthe binary system in the adiabatic approximation. VI. INTERNAL DYNAMICS IN THEADIABATIC APPROXIMATIONA. Euler-Lagrange equation for Q ab We have just seen that the CoM-frame orbital EoM(5.9), the evolution equation for the binary’s 1PN-accurate relative position z i ( t ) = z i ( t ) − z i ( t ), can bederived from the action principle (5.10). We now seek toextend this action principle to incorporate the internaldynamics of the deformable body 2 in the case where thequadrupole moment is adiabatically induced by the tidalfield. In Sec. II G, we saw how the adiabatic evolution of thequadrupole can encoded in a Newtonian action principle;varying the action (2.36) with respect to the quadrupole Q ij gives Q ij = λ n G ij + O ( c − )= λ M r n By substituting the solution (6.6) for the quadrupoleinto the Lagrangian (6.2), we find a reduced Lagrangianfor the orbital dynamics involving only the CoM-frameorbital separation z i ( t ): L [ z i ] = µv µMr (cid:18) r (cid:19) + µc (cid:26) θ v + Mr (cid:20) v (cid:18) θ + ξ Λ r (cid:19) + ˙ r (cid:18) θ + ξ Λ r (cid:19) + Mr (cid:18) θ + ξ Λ r (cid:19) (cid:21)(cid:27) (6.7)with Λ = 3 χ χ λ, (6.8) and with the dimensionless coefficients θ = (1 − η ) / ,θ = (3 + η ) / ,θ = η/ ,θ = − / ,ξ = ( χ / χ ) ,ξ = − − χ + χ ) ,ξ = − χ . (6.9)While this form for the Lagrangian has been derived inharmonic gauge, we note that (some) other gauge choiceslead to a Lagrangian with the same terms as in (6.7)but with different values of the θ and ξ coefficients. Inparticular, the Lagrangian derived by BDF [23] (whenspecialized to 1PN accuracy and to the center-of-massframe) has this form, as does that obtained (via a Legen-dre transformation) from the EOB Hamiltonian includ-ing 1PN tidal effects proposed by Damour and Nagar[22] (except that their work originally did not provide avalue for the coefficient ξ ). In Appendix A, we derivethe canonical transformation relating the EOB Hamilto-nian to the harmonic-gauge Hamiltonian, which fixes thevalue of ξ in the EOB Hamiltonian, and we present aseparate transformation relating our results to those ofBDF. We thus demonstrate the complete equivalence ofall of these results at 1PN order.The orbital EoM resulting from the Lagrangian (6.7)is given by a i = − M n i r (cid:18) r (cid:19) (6.10)+ Mc r (cid:20) v n i (cid:18) φ + ζ Λ r (cid:19) + ˙ r n i (cid:18) φ + ζ Λ r (cid:19) + Mr n i (cid:18) φ + ζ Λ r (cid:19) + ˙ rv i (cid:18) φ + ζ Λ r (cid:19) (cid:21) , with coefficients φ = 4 θ − θ − θ ,φ = 3 θ ,φ = 2( θ + θ − θ ) ,φ = 2(4 θ + θ ) ,ζ = 2(12 θ − ξ − ξ ) ,ζ = 8 ξ ,ζ = 12 θ + 12 θ + 2 ξ + 2 ξ − ξ ,ζ = 12(4 θ + ξ ) , (6.11)7for general values of the Lagrangian coefficients, and with φ = − − η,φ = 3 η/ ,φ = 2(2 + η ) ,φ = 2(2 − η ) ,ζ = − − χ )(1 + 6 χ ) ,ζ = 24(1 − χ + χ ) ,ζ = 66 + 9 χ − χ ,ζ = 6(2 − χ )(3 − χ ) , (6.12)in harmonic gauge.Finally, from the Lagrangian (6.7), we can constructthe conserved energy, E = v i ∂ L /∂v i − L = µv − µMr (cid:18) r (cid:19) + µc (cid:26) θ v + Mr (cid:20) v (cid:18) θ + ξ Λ r (cid:19) + ˙ r (cid:18) θ + ξ Λ r (cid:19) − Mr (cid:18) θ + ξ Λ r (cid:19) (cid:21)(cid:27) , (6.13)which is a constant of motion of the orbital EoM (6.10).As an application of these results, we can compute thegauge-invariant energy-frequency relationship for circularorbits. Using the relations ˙ r = 0, v = r ω , and a i = − rω n i for a circular orbit, the orbital EoM (6.10) canbe solved perturbatively, working to linear order bothin the post-Newtonian parameter 1 /c and in the tidaldeformability parameter Λ (6.8), to find the radius r asa function of the orbital frequency ω . Combining thisresult with a similar treatment of the energy (6.13), wecan eliminate r to find E ( ω ): E ( ω ) = µ ( M ω ) / (cid:20) − 12 + 3Λ ω / M / + f M ( M ω ) / c + f Q Λ ω M c (cid:21) ,f M = 13 ( θ + θ + θ ) = 9 + η ,f Q = 113 (8 θ + 2 θ − θ + ξ + ξ )= 116 (3 + 2 χ + 3 χ ) . (6.14)While the θ and ξ coefficients may take different values indifferent gauges, their combinations appearing here mustbe gauge-invariant. As E ( ω ) is independent of ξ , we cancheck this result against those obtained from the EOBHamiltonian of Damour and Nagar [22] (see AppendixA), and we find that they agree. VII. CONCLUSION We have derived the first-post-Newtonian orbital equa-tions of motion for binary systems of bodies with spinsand mass quadrupole moments, at linear order in thespin and quadrupole, and shown that they conserve thetotal linear momentum of the binary. After specializingthese results to the binary’s center-of-mass-energy frame,we have found an action principle from which the orbitalequations of motion can be derived. Finally, we consid-ered the case in which the quadrupole moment is adiabat-ically induced by the tidal field, giving a simplified La-grangian and equation of motion for this case, as well asthe conserved energy function and the energy-frequencyrelationship for circular orbits. These results are use-ful for the calculation of tidal effects in the gravitationalwave signals from inspiralling neutron star binaries. Acknowledgments The authors would like to thank Tanja Hinderer andEtienne Racine for many helpful discussions and com-ments. This work was supported by the New York StateNASA Space Grant Consortium and by NSF Grant PHY-0757735. Appendix A: Hamiltonian for the adiabatic orbitaldynamics, canonical transformations, andcomparison with DN and BDF Our aim here is to demonstrate the equivalence of theresults for the orbital-tidal conservative dynamics givenby Damour and Nagar [22], Bini, Damour and Faye [23],and the present work. We begin with a discussion ofHamiltonians and canonical transformations, then relateour results to the EOB Hamiltonian given by DN (alsorevisited and completed by BDF), and finish by relatingthe Lagrangian given by BDF to ours.As in Sec. VI B, we consider a Lagrangian for the CoM-frame orbital separation z i = z i − z i of the formˆ L = v r (cid:18) r (cid:19) + 1 c (cid:20) θ v + v r (cid:18) θ + ξ Λ r (cid:19) + ˙ r r (cid:18) θ + ξ Λ r (cid:19) + 1 r (cid:18) θ + ξ Λ r (cid:19) (cid:21) + O ( c − ) , (A1)in units that set M = 1, and with the Lagrangian havingbeen rescaled by the symmetric mass ratio, ˆ L = L /η . Wehave shown that the harmonic-gauge values of the θ and ξ coefficients are given by Eq. (6.9).The (rescaled) momentum canonically conjugate to z i p i = ∂ ˆ L ∂v i = v i + 1 c (cid:20) θ v v i + 2 v i r (cid:18) θ + ξ Λ r (cid:19) + 2 ˙ rn i r (cid:18) θ + ξ Λ r (cid:19) (cid:21) + O ( c − ) ≡ v i + 1 c δp i ( z , v ) + O ( c − ) . (A2)From a Legendre transformation of the Lagrangianˆ L ( z , v ), we can construct the Hamiltonian:ˆ H ( z , p ) = p j v j − ˆ L ( z , v )= p − c p j δp j − ˆ L ( z , p ) + 1 c ∂ ˆ L ∂v j δp j + O ( c − )= p − ˆ L ( z , p ) + O ( c − ) , (A3)having used δp i ( z , v ) = δp i ( z , p ) + O ( c − ). This givesˆ H = v − r (cid:18) r (cid:19) − c (cid:20) θ v + v r (cid:18) θ + ξ Λ r (cid:19) + ˙ r r (cid:18) θ + ξ Λ r (cid:19) + 1 r (cid:18) θ + ξ Λ r (cid:19) (cid:21) + O ( c − ) , (A4)Note that the effect of the Legendre transformation hasbeen simply to flip the sign of all terms except the first.In Ref. [22], Damour and Nagar consider an EOBHamiltonian of the formˆ H EOB = 1 η h η ( ˆ H eff − i / , (A5)with ˆ H eff = " AB p r c + A p φ c r ! / , (A6)Here, p r and p φ are the momenta conjugate to the po-lar coordinates ( r, φ ) in the plane of motion, related tothe Cartesian momenta used above by p r = n · p and p φ /r = p − p r . The functions A ( r ) and B ( r ) are coeffi-cients in the EOB effective metric [45]; A ( r ) completelyencodes the energetics of circular orbits, while B ( r ) haseffects only when p r = 0. Damour and Nagar proposedto incorporate Newtonian and 1PN tidal effects into thisEOB Hamiltonian by adding tidal terms to the radialpotential A : A ( r ) = 1 − c r − c r (cid:16) α c r (cid:17) + O ( c − ) , (A7)for the quadrupole l = 2 case. They have computed the1PN tidal coefficient to be α = 52 χ . (A8) While they did not propose to modify the potential B ( r )from its point-particle value of B = 1 + 2 /c r + O ( c − ),we find that such a modification, B ( r ) = 1 + 2 c r (cid:18) β Λ r (cid:19) + O ( c − ) (A9)for some coefficient β , is necessary to match our results.Expanding the EOB Hamiltonian with these values forthe potentials, we find a Hamiltonian of the form (A4)with coefficients ¯ θ = (1 + η ) / , ¯ θ = (1 − η ) / , ¯ θ = 1 , ¯ θ = (1 + η ) / , ¯ ξ = (1 − η ) / , ¯ ξ = β , ¯ ξ = 1 + η + α , (A10)instead of the harmonic-gauge coefficients in Eq. (6.9).Without tidal effects, the EOB Hamiltonian and theharmonic-gauge Hamiltonian (which coincides with theADM Hamiltonian at 1PN order) are known to be relatedby a canonical transformation [45]. Considering a 1PN-order canonical transformation with generating function G , z i → z i + 1 c ∂∂p i G ( z , p ) , p i → p i − c ∂∂z i G ( z , p ) , (A11)the Hamiltonian changes byˆ H → ˆ H + 1 c { ˆ H, G } + O ( c − ) , (A12)with only the Newtonian part of the Hamiltonian con-tributing in the Poisson bracket. We find that the mostgeneral generating function G that preserves the form ofthe Hamiltonian including tidal effects (A4), changing itscoefficients but adding no new terms, is of the form G = ( z · p ) (cid:18) γ p + γ r + γ Λ r (cid:19) (A13)with arbitrary constant γ coefficients. The changes in theHamiltonian coefficients induced by the canonical trans-formation are ∆ θ = − γ , ∆ θ = γ − γ , ∆ θ = 2 γ + γ , ∆ θ = γ , ∆ ξ = 6 γ − γ , ∆ ξ = 12 γ + 6 γ , ∆ ξ = 6 γ + γ , (A14)If we set the EOB Hamiltonian coefficients (A10) equalto the harmonic Hamiltonian coefficients (6.9) plus the9transformation parameters (A14), we find that this (re-dundant) system of equations has a unique solution. Thecoefficients in the canonical transformation (A13) mustbe γ = − η/ ,γ = (2 + η ) / ,γ = 2 − χ / χ , (A15)with the values for γ and γ matching those computedin Ref. [45], and the parameters in the EOB potentialsmust be α = 5 χ / ,β = 3(3 − η ) . (A16)The value for α matches that given by Damour and Na-gar, and the value for β can be used to extend the rangeof validity of their EOB Hamiltonian to non-circular or-bits.We turn now to the more recent work of BDF [23].They have derived a Lagrangian for the orbital-tidalconservative dynamics including terms up to 2PN or-der. Here, we restrict attention to the 0PN and 1PNterms, specialize their results to the center-of-mass frame,and translate their notation into ours. Their Lagrangian[whose tidal part can be found from their Eqs. (2.12),(4.3), (4.4) and (4.10)] is of the form (A1) with the samepoint-mass ( θ ) coefficients coefficients, but with tidal ( ξ )coefficients ˜ ξ = − χ / ξ = − − χ ˜ ξ = − χ . (A17)The corresponding Hamiltonian is again given byEq. (A4). One can then verify that a canonical trans-formation of the form (A13) with γ = γ = 0 , γ = 3 χ , (A18)using (A14), will transform their ξ coefficients fromEq. (A17) into ours from Eq. (6.9). This demonstratesthe complete equivalence of our respective results. Appendix B: Moment transformations andtranslational equations of motion We present here the formulae that relate the body-frame multipole and tidal moments and the global-frame multipole and tidal moments. Such formulae were firstderived by Damour, Soffel, and XU [15–18] (see, in par-ticular, Sec. VD of Ref. [16]). They are derived by requir-ing the equivalence of the body- and global-frame met-rics in the body’s buffer region B A . More specifically,one substitutes the expansions of the body-frame poten-tials (3.8) and the global-frame potentials (3.18), alongwith the coordinate transformation (3.13), into the tensortransformation law for the metric (3.19), using (3.4) toexpress the metrics in terms of the potentials in both co-ordinate systems. Matching coefficients of the resultantmultipole expansions gives the moment transformationformulae. Having used the body-frame gauge conditions(3.15) to eliminate the worldline-data functions α A and β A , one finds that the transformation formulae involveonly the various moments and the CoM worldlines z iA .A more detailed account of the procedure is given in RF[19] Sec. V.After presenting the various moment transformationformulae in Secs. B 1, B 2, and B 3, we show in Sec. (B 4)how to use them to arrive at the translational equationsof motion for an N -body system.Note that all the quantities appearing below (momentsand worldlines) are treated as functions of the global timecoordinate t (with the argument suppressed). Any quan-tities originally defined as functions of the body-frametime coordinate s A (like the body-frame moments or theworldlines z iA ) can be converted to functions of t by usingthe coordinate transformation (3.13) at y iA = 0, whichgives s A = s A ( t ) = t − c − α A ( s A ) (cid:12)(cid:12) s A = t + O ( c − ) . (B1)This change of variables does affect the forms of any equa-tions presented here (or elsewhere in the paper). It is im-portant to note that our use of f ( s A ) and f ( t ) to denotethe same physical quantity differs from the convention of(e.g.) RF. There, symbols are used to denote functions ofa particular variable, not physical quantities, and f ( s A )and f ( t ) would be different physical quantities. 1. Body-frame multipole moments → global-framemultipole moments First, via the metric transformation law, the global-frame multipole moments M L g ,A and Z iL g ,A [defined by(3.16)] are given in terms of the body-frame moments M LA and S LA [defined by (3.8)] and the CoM worldlines z iA , by0 M L g ,A = M LA + 1 c (cid:20) (cid:18) v A − ( l + 1) G g ,A (cid:19) M LA − l + 5 l − l + 1)(2 l + 3) v jA ˙ M jLA (B2) − l + 7 l + 16 l + 7( l + 1)(2 l + 3) a jA M jLA − l + 17 l − l + 1) v jjA + 4 ll + 1 v jA ǫ jkk (cid:21) + O ( c − ) ,Z iL g ,A = 4 l + 1 ˙ M iLA + 4 v iA M LA − l − l + 1 v jA M j 2. Global-frame multipole moments → global-frame tidal moments Next, one can express the global-frame tidal moments G L g ,A ( t ) and Y iL g ,A ( t ) (def) a body A in terms of the global- frame multipole moments M L g ,A and Z iL g ,A (def) of all theother bodies B = A and the all the bodies’ worldlines z iC . For this purpose, rather than working directly with G L g ,A , it is easier to work with the tensors F L g ,A definedbyΦ g , ext = − ∞ X l =0 l ! (cid:26) G L g ,A ( x − z A ) L + 12(2 l + 3) c ∂ t (cid:2) G L g ,A ( x − z A ) jjL (cid:3)(cid:27) + O ( c − )= − ∞ X l =0 l ! (cid:26) F L g ,A ( x − z A ) L + 12(2 l + 3) c J L g ,A ( x − z A ) jjL (cid:27) + O ( c − ) , (B6)1cf. (3.18). The tensor J L g ,A will not be needed in ourcalculations (and contains no extra information). Notethat F L g ,A and G L g ,A agree at Newtonian order, F L g ,A = G L g ,A + O ( c − ) . The Newtonian part of G L g ,A is also given in Eq. (2.14).By equating the Eq. (B6) and the external part ofEq. (3.16), one finds that F L g ,A is given by F L g ,A = X B = A ∞ X k =0 ( − k k ! (cid:20) N K g ,B ∂ ( A ) KL | z A − z B | (B7)+ 12 c P K g ,B ∂ ( A ) K 1) casesof (B7) and the l = 0 , , l = 0) cases of (B8)for A = 2, B = 1 (resp. A = 1, B = 2). The deriva-tives appearing here can be easily expressed in terms of r = | z − z | and n i = ( z i − z i ) /r via the identities ∂ (1) L r = ( − l ∂ (2) L r = 1(2 l + 1)!! n 3. Global-frame tidal moments → body-frame tidalmoments Finally, the metric transformation law gives the body-frame tidal moments G LA and H LA (defined by (3.8)) interms of the global-frame tidal moments F L g ,A and Y L g ,A (defined by (B6),(3.18)). First, the gravito-magnetictidal moments can be found from H LA = Y jk 43 ˙ G g ,A δ ij , Λ ijkζ = − δ i 4. Translational equations of motion The results of Secs. B 1, B 2, and B 3 allow one to ex-press the body-frame tidal moments G LA and H LA for agiven body A in terms of the body-frame multipole mo-ments M LB and S LB and CoM worldlines z iB of all bodies B in an N -body system (eliminating all reference to theglobal frame moments). With this done, one can find thebody’s translational equation of motion, written only in2terms of the M LB , S LB , and z iB , by using the law of motion(3.22b) for the body-frame mass dipole M iA .As the body-frame gauge condition (3.15a) requires M iA = 0 (fixing the body’s center of mass-energy to thebody-frame origin), one proceeds by setting the right-hand side of (3.22b) to zero. This yields an expres- sion for the acceleration a iA = ¨ z iA of the body’s 1PN-accurate global-frame CoM worldline z iA ( t ). To see thismore clearly, we can explicitly evaluate the l = 0 case ofthe first term on the RHS of (3.22b) using the momenttransformation formulae presented above. The result is M A a iA = M A F g ,A + ∞ X l =2 l ! M LA G LA + 1 c g iA (B11)+ 1 c (cid:20) ˙ Y i g ,A − v jA Y ji g ,A + (2 v A − G g ,A ) G i g ,A − v ijA G j g ,A − ( v A + 3 G g ,A ) a iA − v ijA a jA − G g ,A v iA (cid:21) + O ( c − ) , where g iA represents the terms on the RHS of Eq. (3.22b)except for the first. Appendix C: Derivation of system mass multipolemoment formulae We derive here Eq. (4.6), which gives the mass multi-pole moments M L sys of an N -body system in terms of its global frame multipole moments M L g ,A and Z L g ,A . Thisparallels the derivation given in Sec. (IV A) for the mo-ments Z iL sys . We begin by using the Taylor series (4.4) torewrite the global-frame scalar potential Φ g as given inEq. (4.1a) in the formΦ g = − X A ∞ X l,k =0 ( − l + k l ! k ! (cid:20) M L g ,A z KA ∂ LK | x | + 12 c ∂ t (cid:0) M L g ,A z KA ∂ LK | x | (cid:1)(cid:21) = − X A ∞ X p =0 p X k =0 ( − p p ! p ! k !( p − k )! (cid:20) M A ∂ P | x | + 12 c ∂ t (cid:16) M ( P − K g ,A z K ) A (cid:17) ∂ P | x | (cid:21) . (C1)To bring this into the form (4.2a), which gives Φ g interms of the system moments, we must decompose ∂ P | x | into its STF and trace parts. Using the identity ∂ ijL | x | = ∂ A ∂ P | x | + 12 c ∂ t (cid:16) M A (cid:17) ∂ | x | (cid:21) − c X A ∞ X l =0 l +2 X k =0 ( − l ( l + 2)! ( l + 2)! k !( l + 2 − k )! ∂ t (cid:16) M ( ijL − K g ,A z K ) A (cid:17) ( l + 1)( l + 2)2 l + 3 δ ij ∂ L | x | (C3)Though the sum over l should start at l = − P → ijL , the l = − , − l + 1)( l + 2) factor. In the P l +2 k =0 term, the k indices K are to be3chosen from the l + 2 indices ijL . This term can besimplified by explicitly performing the symmetrization over all l +2 indices; using the fact that M L g ,A and ∂ L | x | − are STF, we have M ( ijL − K g ,A z K ) A δ ij ∂ L | x | = (cid:20) k ( l + 2 − k )( l + 2)( l + 1) M j