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Canonical formulation of gravitating spinning objects at 3.5 post-Newtonian order

The 3.5 post-Newtonian (PN) order is tackled by extending the canonical formalism of Arnowitt, Deser, and Misner to spinning objects. This extension is constructed order by order in the PN setting by utilizing the global Poincare invariance as the important consistency condition. The formalism is valid to linear order in the single spin variables. Agreement with a recent action approach is found. A general formula for the interaction Hamiltonian between matter and transverse-traceless part of the metric at 3.5PN is derived. The wave equation resulting from this Hamiltonian is considered in the case of the constructed formalism for spinning objects. Agreement with the Einstein equations is found in this case. The energy flux at the spin-orbit level is computed.


I. INTRODUCTION
Several laser interferometric gravitational-wave (GW) detectors, including LIGO, VIRGO, GEO600, and TAMA300 are currently searching for the GWs emitted by inspiraling compact binaries, which consist of black holes and/or neutron stars. Because the data analysis method used by these experiments, namely, the matched filtering technique, requires the detector's output signal to be compared with a large amount of theoretical waveforms (templates), the post-Newtonian (PN) calculation for both of the binary's motion and the gravitational waves emitted has to be performed. For nonspinning binaries, the PN expansion has been successfully carried out through 3.5PN order. However, since astrophysical observations suggest that most astrophysical objects carry a certain amount of spin angular momentum, and compact objects like black holes are usually rapidly rotating, the effect of spin is too large to be ignored.
Both the dynamics of a spinning binary and the GWs emitted by such a system are very different from those of a nonspinning system. The coupling between the orbital angular momentum L and the individual spins S a leads to precession of the individual spins and the orbital plane, which in turn leads to additional amplitude modulation of the GWs emitted by the system. The detailed calculation by Kidder at 1PN order [1] showed that spin itself can also directly contribute to the gravitational waveforms and the emission of energy and angular momentum. Including spin as an intrinsic parameter of the source also increases the dimension of the parameter space to be used in the data analysis process, which not only requires more computational resources but could also affect the accuracy on parameter estimation [2,3].
Thus, it is desired to carry the PN approximation for spinning binary systems to a sufficiently high PN order.
The spin effect to the motion and to the gravitational field have been a long-standing problem in general relativity (GR). Papapetrou and Corinaldesi in the 1950s [4,5] calculated the leading order spin effects to the motion of a spinning test body in a given gravitational field. The leading order spin-orbit (SO) and spin(1)-spin(2) (S 1 S 2 ) contributions to the equations of motion for a system of two spinning black holes were derived by D'Eath [6], Barker and O'Connell [7,8] in the 1970s, and later by various authors (for references and reviews, see, e.g., [9,10]). Using Thorne's multipole expansion formalism [11] in terms of symmetric trace-free (STF) radiative multipoles in the harmonic gauge, the group of Kidder, Will, and Wiseman derived the leading order SO and S 1 S 2 contribution to the gravitational radiation flux [1,12] for a general spinning binary system and the polarized gravitational waveform emitted by a spinning binary system with quasicircular orbit [1]. The leading order SO and S 1 S 2 radiation reaction effects to the equations of motion (EOM) were derived by Wang and Will in the harmonic gauge [13,14]. A general form of the SO contributions in arbitrary coordinates was also derived by Zeng and Will using an energy and angular momentum balance approach [15].
The (conservative) next-to-leading order (NLO) spin effects were only tackled recently. The first derivation of the NLO SO EOM was attempted by Tagoshi, Ohashi, and Owen [16], their result was essentially confirmed by Faye, Blanchet, and Buonanno in the harmonic gauge [17], and later also by Damour, Jaranowski, and Schäfer using a Hamiltonian approach in the Arnowitt, Deser, and Misner (ADM) gauge [18] (see also [19]). The corresponding energy flux (and formulas for a phasing) is given in [20]. The first attempt to compute the NLO S 1 S 2 contributions to the EOM of a spinning binary system was made by Porto and Rothstein [21] using an effective field theory technique, namely, an extension of nonrelativistic general relativity [22] to systems with spin [23]. The first complete NLO S 1 S 2 Hamiltonian was presented by Steinhoff, Schäfer, and Hergt [19,24], and agrees with [25,26].
Though the above mentioned results are useful for the creation of templates, further work needs to be done. In general, a parametrization of the orbits must be obtained by solving the EOM. It is common to describe the conservative dynamics in terms of certain orbital elements, see, e.g., [27]. Spin precession and dissipative effects can then be described by secular EOM of the orbital elements. For explicit solutions including spin see [28,29]. It is also possible to obtain the dissipative orders of these secular EOM with the help of the conservative parts as well as the energy and angular momentum flux. In this way secular EOM corresponding to the LO radiation-reaction EOM mentioned above have already been obtained in [30][31][32][33].
In this paper, we extend the canonical formalism of ADM [34][35][36] to n-body systems with n spinning objects up to a PN order sufficient for the computation of the SO and S 1 S 2 contribution to the next-to-next-to-leading order (NNLO) conservative Hamiltonian H con ≤3P N and the leading order dissipative Hamiltonian H diss ≤3.5P N . It is important to mention that we count PN orders in a rather formal way; see Appendix A. The canonical framework in the present paper is constructed order by order in the PN setting by utilizing the global Poincaré invariance as the important consistency condition, similar to [19]. Further a general formula for the interaction Hamiltonian between matter and the transverse-traceless part of the metric h TT ij at 3.5PN is derived. From this Hamiltonian a wave equation for h TT ij can be obtained by canonical methods. For the canonical formalism presented in this paper, this wave equation agrees with a corresponding one that can be followed directly from the Einstein equations. This provides a thorough check of the canonical formalism. Further the obtained formulas are the basis for applications. Using the wave equation, we are able to derive, in the radiation zone of a system with two spinning objects, the leading order SO contribution to h TT ij and the energy flux (see also [1,37]).
A canonical framework for spinning test-particles valid to any PN order and linear in the spin of the particle was given very recently in [38]. The Hamiltonian of a spinning test-particle in Kerr spacetime was given explicitly. This includes parts of the conservative Hamiltonian H con ≤3P N mentioned above (as well as contributions of cubic and higher order in spin; see also [39,40]). An action approach to the canonical formulation of self-gravitating spinning objects valid to all orders linear in the single spin variables was recently given in [41] and is shown to agree with the present paper up to 3.5PN. The order by order construction performed in this paper gives an independent derivation of the results in [41] up to 3.5PN. Further the method developed in this paper to construct a canonical formalism might have some advantages over an action approach at higher orders in spin. Knowledge of the formalism in [19] and its extension given in the present paper was important information to succeed with the action approach in [41]. The consistency checks and applications performed in this paper also apply to the action approach. Further, the present paper provides more details than [41]. In particular, the source terms of the constraints depending on canonical variables are given explicitly.
The paper is organized as follows. In Sec. II it is shown how the ADM formalism can be extended to spinning objects order by order in a PN setting. In Sec. III this approach is applied to 3.5PN and linear in the spin. A comparison with the action approach in [41] is given. Section IV gives the PN expansion of the constraints, including the matter source terms in canonical variables. In Sec. V a general formula for the interaction Hamiltonian is derived. Further it is shown that the evolution equations given by this Hamiltonian, specialized to our canonical formalism, coincide with the ones following from the Einstein equations. In Sec. VI the SO energy flux is computed. Section VII gives conclusions and outlook.
Our units are c = 1 and G = 1, where G is the Newtonian gravitational constant. Greek indices will run over 0, 1, 2, 3, Latin indices from the middle of the alphabet over 1, 2, 3. Latin indices from the beginning of the alphabet label the individual objects. For the signature of spacetime we choose +2. The short-cut notation ab (= a µ b µ = a µ b µ ) for the scalar product of two vectors a µ and b µ will be used. Square brackets denote index antisymmetrization and round brackets index symmetrization, i.e., a (µ b ν) = 1 2 (a µ b ν + a ν b µ ). The spatial part of a 4-vector x is x. Round brackets around an index denote a local basis, while round brackets around a number denote the formal order in c −1 , as in [19].

II. FROM ADM ENERGY TO ADM HAMILTONIAN
In this section we outline how the ADM canonical formalism [34][35][36] can be extended to self-gravitating spinning objects order by order in a PN setting. We only consider a fully reduced canonical framework here where the gauge is fixed and all constraints are eliminated. Then the ADM energy can be used as a Hamiltonian, the ADM Hamiltonian, if it is expressed in terms of variables with standard canonical meaning. The transformation to such variables can be found from consistency considerations. At the 3.5PN SO and S 1 S 2 orders the global Poincaré algebra and the constant Euclidean length of the canonical spin uniquely fixes this transformation to standard canonical variables.
The approach outlined here is a natural generalization of the one in [19]. It was suggested in Appendix B of [19], by considering the algebra of the gravitational constraints, that at higher orders spin corrections to the canonical field momentum might be necessary, and that the gauge structure needs to be extended. Indeed, the former is an important ingredient of the approach in this paper [see Eq. (2.6)] as well as of the action approach in [41]. Further, the action approach is based on tetrad gravity, which has more gauge freedom than metric gravity. In this paper, however, the gauge is always fixed and the original gauge structure is less important.

A. Field constraints
Most important for an explicit calculation of the generators of the global Poincaré algebra, including the Hamiltonian, are the constraint equations of the gravitational field. They can be written as with the definitions and arise as certain projections of the Einstein equations with respect to a timelike unit 4-vector n µ with components n µ = (−N, 0, 0, 0) or n µ = (1, −N i )/N . Here γ ij is the induced three-dimensional metric of the hypersurfaces orthogonal to n µ , γ its determinant, R the three-dimensional Ricci scalar, K ij the extrinsic curvature, N the lapse function, N i the shift vector, √ γT µν the stress-energy tensor density of the matter system, and ; denotes the three-dimensional covariant derivative. Partial derivatives are indicated by a comma. For nonspinning objects 1 16π π ij is the canonical momentum conjugate to γ ij before gauge fixing. For spinning objects we now make an ansatz for the canonical field momentum of the form where π ij matter shall be linear in the spins and will be fixed later on. In the ADM transverse-traceless (ADMTT) gauge defined by which will be used throughout this paper, one has the decompositions where h TT ij and π ijTT can are transverse-traceless, e.g, h TT ii = h TT ij,j = 0, andπ ij can is related to vector potentials V i can andπ i can bỹ with the inverse Laplacian ∆ −1 , the partial spacecoordinate derivatives ∂ i and (2.15) See Sec. IV for details on the decompositions for γ ij and π ij can . Notice that the form of the trace term in (2.8) is adapted to the Schwarzschild metric in isotropic coordinates, with obvious advantages for perturbative expansions.
Now the four field constraints can be solved for the four variables φ andπ i can in terms of h TT ij , π ijTT can and matter variables, which enter through the source terms H matter and H matter i . An analytic solution for φ andπ i can , however, can in general only be given in some approximation scheme.
In the ADMTT gauge the momentum constraint (2.2) can exactly be written as with the definitions The global Poincaré algebra is a consequence of the asymptotic flatness and is represented by Poisson brackets of the corresponding conserved quantities. These quantities are the ADM energy E, total linear momentum P i , total angular momentum J i = 1 2 ijk J jk , and the boost vector K i . They are given by surface integrals at spatial infinity. The boosts have an explicit dependence on the time t and can be decomposed as K i = G i − tP i , where X i = G i /E is the coordinate of the center-of-mass. G i will be called center-of-mass vector in the following. The corresponding surface integrals read, with spatial coordinates denoted x i , (2.20) See, e.g., [35]. Using the gauge conditions and also the momentum constraint in the form (2.16), these surface integrals can be transformed into the volume integrals with the matter parts (2.28) Here we used the fact that π ij matter has a compact support. Now we require that the matter parts of total linear and angular momentum are of the form whereẑ i a , P aj , and S a(i)(j) = ijk S a(k) are the canonical position, momentum, and spin of the particles, because this is the expected form for standard canonical variables with equal-time Poisson brackets . Equations (2.29) and (2.30) ensure that a great part of the Poincaré algebra is fulfilled; see Appendix B. With the definition At linear order in spin and 3.5PN, the first equation defines the canonical momentum, while the second one fixes the canonical position, the triad (i.e., the local basis of the canonical spin) and π ij a , up to canonical transformation. Further the Euclidean spin length s a given by (2.38) This is the ADM energy depending on the canonical variables. It arises from solving the constraints for φ, once the source terms of the constraints, H matter and H matter i , are expressed in terms of the canonical variables. An action corresponding to H ADM is given by (4.33) in [19] or (51) in [41].
Total linear and angular momentum could, of course, also be represented on the phase space in a more complicated way than given by (2.25), (2.26), (2.29), and (2.30). However, the ADMTT gauge manifestly respects the Euclidean group in its standard representation, which implies that its generators P i and J ij are also in its standard representation on the phase space; see Appendix B and also [42].

III. THE SOURCE A. (3+1)-split
The stress-energy tensor density to linear order in spin is given by [43][44][45] Here m a is the mass, u a the 4-velocity, τ the proper time parameter, S µν a the spin tensor, || denotes the fourdimensional covariant derivative, and δ (4)a = δ(x − z a ) with normalization d 4 x δ (4)a = 1. z µ a is the coordinate of the a-th object. The matter EOM, i.e., the Mathisson-Papapetrou equations [4,45,46], in covariant SSC and at linear order in spin can be followed from T µν ||ν = 0 as Here (4) R µ ρβα is the four-dimensional Riemann tensor and D the four-dimensional covariant parameter derivative. The spin length s a is given by 2s 2 a = S µν a S aµν and obviously is a constant of motion due to (3.3a).
The (3+1)-split of u 2 a = −1, the SSC and the spin length reads Notice that nS i a = γ ij nS aj . The components of the stress-energy tensor density are given by, with δ a = δ(x − z a ), where T ij = √ γT ij . After transition to Newton-Wigner (NW) variables where Γ k ij are the three-dimensional Christoffel symbols, the source expressions of the constraints read [now δ a = δ(x −ẑ a )] Obviously (3.12) is now of the from (2.35), which uniquely fixed the relation between covariant linear momentum p ai and canonical momentum P ai , Eq. (3.10c). The spin redefinition (3.10a) transforms the spin length (3.6) into whereŜ a(i)(j) are the components ofŜ aij in some local Euclidean basis. Comparing with (2.37) suggests that S a(i)(j) is equal to the canonical spin S a(i)(j) . However, a local Euclidean basis is only unique up to a rotation. Fortunately Eq. (2.36) will be seen to uniquely fix a basis such thatŜ a(i)(j) and S a(i)(j) can be identified in that basis. (Notice that S a(i)(j) are not the components of the covariant spin S aij in a local basis here, in contrast to [41].) The redefinition of the position (3.10b) consists of a term known from flat-space and a yet unknown quantity δz i a , which will also be fixed by Eq. (2.36) later on.

B. Triad
The relation betweenŜ aij andŜ a(i)(j) can be written with the help of a triad e i(j) aŝ S a(i)(j) = e k(i) e l(j)Ŝ akl . (3.14) This spin has a constant Euclidean length for all choices of e k(i) . Notice that the triad is needed only on the worldlines and not as a field over the entire spacetime here. The triad can be split into symmetricẽ i(j) and antisymmetricê i(j) parts as e i(j) =ẽ i(j) +ê i(j) . Perturbative expansion of e i(k) e j(k) = γ ij leads to 2 e i(j) where γ ij (0) = δ ij and e i(k) (0) = δ ik was assumed. For example, the leading order results are: 2 Notice that in this formula the subscripts in round brackets denote the formal order in c −1 , not an index in a local basis.
The symmetric part is thus fixed. The antisymmetric partê i(j) , however, must be imposed, as it represents the three rotational degrees of freedom left in the definition of the local basis. Thusê i(j) represents the degrees of freedom left in the definition of the canonical spin variable.

C. Fixation of the NW variables
Whereas the canonical momentum was already unambiguously fixed by (2.35) as (3.10c), δz i a ,ê i(j) , and π ij a are still unknown. We will see now that these can be fixed up to a canonical transformation with the help of (2.36). For our source one gets for the leading orders of s and thus from (2.36) one concludeŝ e i(k) It is crucial thatê (2) must be antisymmetric. At the next order it holds (3.21) Further π ij a(3) = 0 because π ij a is linear in spin, symmetric, and contains an even number of momentum variables (because of parity). Equation (2.36) then leads tô e i(k) (3.23) The most general (sensible) solution of (2.36) at this order is with an arbitrary constant C. Notice that π ii (5)a = 0. Now we can remove the ambiguity C by a canonical transformation generated by which transforms an arbitrary phase space function A as to the required order. For our fundamental variables this means This indeed removes all terms depending on C from the source expressions in (3.11) and (3.12) at the considered order. We can therefore choose C = 0. This choice has the nice properties that the triad fulfills the gauge condition e i(j) = e j(i) and that the transition to the NW positionẑ i a can be expressed in terms of a Lie-shift; see Appendix C. Further this choice agrees with the action approach in [41].

D. Comparison with the action approach
For the action approach in [41] the gauge condition e i(j) = e j(i) = e ij holds to all orders, which fixes e ij as the matrix square-root of the three-dimensional metric e ij e jk = γ ik , or e ij = (γ kl ) . (3.32) Further, the transition to NW variables in [41] agrees with (3.10) for δz i a = 0 to all orders. Notice that Eq.
(31) in [41] is written in terms of the covariant spin and one has to use Eq. (45) in [41] and (3.33) on the triad terms. It is shown in [41] that a suitable choice of π ij a extends the ADM formalism for spinning objects to all PN orders linear in spin. According to [41], it holds where the quantity B kl ij is defined by This can also be written as which must be evaluated perturbatively using Eq. (4.22) in [19], e.g., which follows from B kl ij δ kl = 0. This leads to a deviation of our gauge condition π ii can = 0 from the original ADM one π ii = 0 at the formal 5PN order.
We will now show that (2.36) exactly holds for the action approach. This gives a check of the action approach to all orders, for any approximation scheme. The second term on the right-hand side is symmetric in i and j and cancels from (3.39) with (3.36) inserted. It is then easy to see that (3.39) and thus (2.36) are fulfilled. The action approach shows that (3.11) and (3.12) can be applied to all orders, if δz i a = 0,ê i(j) = 0, and (3.33) are inserted. Further, if the spin correction to the field momentum π ij a is neglected, (3.11) and (3.12) coincide with (4.23) and (4.25) in [19]. In [19] a Lie-shift was used to redefine the position instead of a Taylor expansion here; see Appendix C.

IV. PN EXPANSION
In this section we give the PN expansion of the field constraints relevant for the ADM Hamiltonian up to and including 3.5PN. Here and in Sec. VI we made use of xTensor [47] (a free package for Mathematica [48]), especially of its fast index canonicalizer based on the package xPerm [49].
For π ij we use the general decomposition with This can be shown by inserting (4.2) and (2.15) into (4.1), which then turns into an identity. We further introduce an alternative vector potential V i by for which it holds Notice that this decomposition reduces to the one for π ij can , asπ ij can = 0 follows from the gauge condition (2.7b). This gauge condition can also be inserted into (4.2d) in the form π ii = −π ii matter . This immediately yieldsπ ij , which is zero at the considered order due to π ii (5)matter = 0; also see (3.38) for higher orders. Similarly the decomposition of γ ij can be derived using (2.7a).
Now we have to decide whether to use π ijTT can andπ ij can , or π ijTT ,π ij , andπ ij for the expansion of the field constraints. We choose the latter option as it simplifies the calculation at the considered PN order. Then one has to go over to π ijTT can later on using Notice that it holdsπ ij can,j =π ij ,j + π ij matter,j and thus π i can =π i + ∂ j ∆ −1 π ij matter . The expansion of the momentum constraint immediately follows from the exact formulã

6)
which is analogous to (2.16). With the help of Notice that s ij a (9) is not needed for the Hamiltonian at the considered order, it only contributes to total linear and angular momentum.
The expansion of the Hamilton constraint (2.1) reads , (4.13) ∆φ (10) = H matter where (td) denotes a total divergence. These equations can be solved iteratively for φ by applying an inverse Laplacian to them. The ADM Hamiltonian (2.38) results from an integration over the right-hand sides of these equations. The source expressions are given by H matter H matter H matter H matter H matter Although the source terms given in this section seem to be SO couplings only, the expressions given here are enough to give all S 1 S 2 contributions, too. All S 1 S 2 terms in the Hamiltonian come in from the nonlinearities on the right-hand sides of the expanded constraints, as in [19,24]. At higher orders in the single spin variables, however, more contributions are needed in the source of the constraints (i.e., in the stress-energy tensor).

V. FIELD EVOLUTION
In this section we derive the wave equation for h TT ij from the ADM Hamiltonian and compare it with the corresponding one that follows directly from the Einstein equations. Agreement is found, which proves that the constructed ADM Hamiltonian gives the correct time evolution for the gravitational field up to and including 3.5PN. This provides a thorough check of the canonical formalism derived in this paper and also of Ref. [41].

A. Interaction Hamiltonian and wave equation
The field EOM can be obtained from the ADM Hamiltonian by where the dot denotes a partial time derivative. It is suitable to introduce an interaction Hamiltonian H int between matter and gravitational field as where H TT-parts ADM denotes the parts of the ADM Hamiltonian depending on h TT ij and π ijTT can . The field EOM then read with = ∆ − ∂ 2 t and the partial time derivative ∂ t . Notice that φ (6) , φ (8) , andπ ij (7) depend on h TT ij and/or π ijTT , and at a first look it seems that one has to explicitly solve the constraints for these functions in order to get the interaction Hamiltonian. However, one can use the expanded constraints to eliminate φ (6) , φ (8) , and π ij (7) by performing certain partial integrations. With the definitions for which φ (4) = φ 1(4) + φ 2(4) holds, these partial integrations read H matter H matter where dots denote total divergences and/or terms independent of h TT ij and π ijTT can . The interaction Hamiltonian then results as 14) The field EOM finally result from (5.4) as with B (6)ij =B (6)ij +π ij (5)matter . For our source, one gets Here the φ (6) terms in H matter (10) were rewritten as − 1 2 H matter (4) φ (6) + (td) and handled by (5.8). The time derivativeπ ij (5)matter was calculated using leading-order Hamiltonians. Notice that the formula given for the interaction Hamiltonian (and thus the wave equation) is quite general and in principle applicable not only to linear order in spin. Further, for nonspinning objects the result in [50] is reproduced.
One can remove π ij (5)matter from the interaction Hamil-tonian by a canonical transformation generated by (5.19) corresponding to the choice C = 1 in Eqs. (3.24).

B. Comparison with the Einstein equations
The time evolution parts of the Einstein equations in the variables used here reaḋ whileγ ij,j − 1 3γ jj,i = 0 leads to with the solution Using the Hamilton constraint and these expressions for lapse and shift, the PN expansion of the TT-projected evolution equations readṡ This can be written as a wave equation identical to (5.15), but now with different expressions for B (4)ij and B (6)ij . Also, Eq. (5.28) obviously is the same as (5.16). The question thus is if the results for B (4)ij and B (6)ij coincide with the ones from the last section. To see this we need the expressions for T (4)ij and T (6)ij . These follow from (3.9) after the transition to NW variables by (3.10) and PN expansion as 4m 3 a ∂ k P 2 a P ai S a(k)(j) δ a + P 2 a P aj S a(k)(i) δ a + P ai P ak P al S a(l)(j) δ a + P aj P ak P al S a(l)(i) δ a , For our source it holds T (4)ii = 2H matter (4) and thus ∆ −1 T (4)ii = − 1 8π φ 1 (4) . The quantities B (4)ij and B (6)ij , and thus the evolution equations of the gravitational field, can now be seen to coincide with the result of the last section.

VI. ENERGY FLUX
In this section we reproduce, within the ADMTT gauge, the 1PN energy flux at the SO level, obtained in [1,12] within the harmonic gauge. This should be seen as a further check for the wave equation (with source terms depending on standard canonical variables) derived in the last section, which is most important for the calculation of the 3.5PN Hamiltonian. It also gives a check of the applied regularization techniques. Notice that the Newtonian flux in the SO and S 1 S 2 cases vanishes. We will restrict to two objects here.

A. Far zone expansion of the wave equation
The retarded solution of the wave equation reads where f is some field and t ret = t − c −1 |x − x |. We explicitly show the speed of light c here in order to simplify the discussion. f shall not change much during time intervals significantly smaller than some time interval T , e.g., f describes a binary system with an orbital period T and the internal dynamics of the individual objects does not introduce a significantly smaller time scale relevant for f . This allows Taylor expansion in time of f in certain cases. The first case is the near zone defined by |x − x | cT where f (x , t ret ), and thus −1 ret f , can formally be expanded in c −1 . The near zone expansion is important for the calculation of the Hamiltonian, as the metric at the position of the spinning objects is needed there. The second case is the far zone (or wave zone) defined by |x | cT |x| ≡ R (if the support of f is centered around the origin). Again we can formally expand in c −1 , but the quantity t fz ret = t − R c must be held constant. Both near and far zone expansion thus fit well into the PN scheme, as they can be seen as expansions in c −1 . Useful formulas for the far zone expansion are 3) where n i = x i /R and L is a multi-index with l even. Notice that the spacial derivative of t fz ret does not vanish. Equation (6.4) can be shown using (6.5) and the Fourier transform of f (x , t fz ret ) with respect to time, This Fourier transform can also be used to show We can replace S kl by its STF part with the multipole moments Here M is a multi-index and the moments I klM are STF with respect to k and l. At higher orders it is better to work with multipole moments which are STF in all indices; see, e.g., [20,51] and references therein. The TT-projector δ TTij kl was replaced by P ijkl , P ij = δ ij − n i n j , (6.14) using (6.4). The integrations needed for the multipole moments also appear in the calculation of the Hamiltonian. Details on the calculation and the applied regularization techniques will be given in [52], we only show the results here. Though the expressions for the multipole moments are quite long, after extracting total time derivatives as (6.17) they can be written in the compact form (6.20) Here the subscript STF ij means to take the STF part in i and j, and (1 ↔ 2) denotes an exchange of the particle labels. Only terms needed for the 1PN SO part of the flux are shown. As the source of the wave equation is expressed in terms of variables with a standard canonical meaning, leading order Hamiltonians are used to calculate the time derivatives appearing here.

B. 1PN energy flux
The energy flux L results from the formula where t fz ret is held constant in the limit R → ∞. Using (6.11) we can express this in terms of I kl , I klm , and I klmn as I ijkk .

(6.22)
A symbol (n) on top of a multipole denotes the n-th time derivative. Plugging in our expressions for the multipole moments we get for the SO part where S a has components S a(i) , after going to the centerof-mass frame (see Appendix D), where it holds (6.26) r 12 =ẑ 1 −ẑ 2 , v =ż 1 −ż 2 (6.27) r = r 12 , n 12 = r 12 r , (6.28) Our result for L SO exactly coincides with the one in [1,12]. Similarly, we could get the total angular momentum flux. This requires one to keep all O (R −2 ) terms.

VII. CONCLUSIONS AND OUTLOOK
In the present paper we extend the ADM canonical formalism to spinning objects up to and including 3.5PN and linear in the single spin variables. Further, general formulas for the interaction Hamiltonian and the wave equation for h TT ij are derived. This is the foundation for the calculation of the 3.5PN SO and S 1 S 2 radiationreaction Hamiltonians in [52] (which, respectively, are 4PN and 4.5PN for maximally rotating black holes), as they result from the general interaction Hamiltonian by utilizing a near zone expansion of the wave equation [50]. The important difference to the formalism in [19] is the spin-dependent correction to the canonical field momentum.
The conservative NNLO SO and S 1 S 2 Hamiltonians (which are formally at 3PN or, respectively, at 3.5PN and 4PN for maximally rotating black holes) can also be obtained from the results of the present paper, once they have been rederived in arbitrary dimension. This is needed for dimensional regularization, which is the only one known to work consistently at 3PN in the ADM formalism; see [53,54]. The NNLO SO Hamiltonian is the last missing piece to complete the EOM for maximally rotating binary black holes up to and including 3.5PN.
As the calculation of the mentioned NNLO Hamiltonians is quite involved (comparable to the 3PN point-mass Hamiltonian), it is a good idea to thoroughly check the used formalism before starting such a calculation. The present paper provides several such checks. The wave equation for h TT ij was compared with the Einstein equations and applied to the leading order SO energy flux. Further, agreement with the action approach in [41] up to the considered order was shown.
The method to construct a canonical formalism in the present paper is formulated in a quite general way and could be applied also in other situations. In particular, our method makes no use of a covariant generalization of flat-space expressions, in contrast to [19]. It is thus applicable to nonminimal couplings, which appear at the S 2 1 level. One can use the four-dimensional stress-energy tensor with S 2 1 quadrupole terms [55,56] to derive the NLO S 2 1 Hamiltonian with the method of the present paper. This Hamiltonian was already obtained in [40,55], and agreement with the spin EOM from [57] was shown in [58]. The LO conservative dynamics was obtained in [7,8]. For more LO results at the S 2 1 level oriented toward application in GW astronomy, see, e.g., [59][60][61][62][63].

Acknowledgments
We gratefully acknowledge many useful discussions with G. Schäfer. We further thank P. Jaranowski for sharing his insight in the calculation of the 3.5PN pointmass Hamiltonian. JS wishes to thank S. Hergt and M. Tessmer for useful discussions. HW thanks J. Zeng for helpful discussions. This work is supported by the Deutsche Forschungsgemeinschaft (DFG) through SFB/TR7 "Gravitational Wave Astronomy" and GRK 1523.

Appendix A: PN orders and spin
In the present paper, PN orders (orders in c −2 ) are counted in terms of the velocity of light c originally present in the Einstein equations. We call this formal counting. This has some computational advantages in our method, e.g., similarities to calculations for nonspinning objects are more manifest. Further, this formal way of counting best reflects the computational demands, e.g., the difficulty of the integrations and the regularization techniques that need to be applied.
On the other hand, it makes sense to assume that the spin variables possess a numerical value of the order c −1 , which holds for maximally rotating black holes. This way of counting best reflects the relevance of the spin corrections to the motion of rapidly rotating objects. Compared to the formal counting, it adds half a PN order for each spin variable appearing in a specific expression. For example, the NLO SO and S 1 S 2 Hamiltonians are of the orders 2.5PN and 3PN for rapidly rotating objects, respectively, but are both of the order 2PN in the formal counting.
To conclude, the formal counting overestimates the importance of the spin corrections, while the other way of counting overestimates the computational complexity. The action of an element of the three-dimensional Euclidean group, a subgroup of the Poincaré group, on the coordinates of the three-dimensional hypersurfaces can be written as with a i a constant infinitesimal vector describing a translation and a rotation matrix R ij (ω). It holds R(ω) = e ω , where ω ij = ω ji is a constant antisymmetric matrix describing the axis and angle of the rotation. This is the standard representation of the Euclidean group on the coordinates. On a field, e.g., the metric, the standard representation of the Euclidean groups acts as Obviously, the ADMTT gauge conditions (2.7) manifestly respect the Euclidean group in its standard representation. Thus the global Euclidean group, as a part of the global Poincaré group, is given by its standard representation in the ADMTT gauge. Now we restrict to infinitesimal transformations, i.e., a i and ω ij shall be small. Then it holds Thus the matrices M ij form a representation of the Liealgebra so (3), namely, the vector representation. Further from (B2) we have which can be written as where L denotes the Lie-derivative. [This would of course be valid for any infinitesimal coordinate transformation, not only for (B3).] As the Euclidean group is given by its standard representation in the ADMTT gauge, its generators in phase space are also given by its usual representations, Eqs. (2.25), (2.26), (2.29), and (2.30). Indeed, the transformation rule for an arbitrary phase space function A, applied to our fundamental variables then readŝ Thus the generators J ij and P i in its standard representation give the transformation induced by (B3) on the fundamental variables, as expected. In (B9d) and (B9e) it was used that, e.g., δ TTkl ij L δx m h TT kl = L δx m h TT ij which again reflects the compatibility of the ADMTT gauge with the standard representation of the Euclidean group.
Further, the canonical action given by Eq. (51) in [41] (see also (4.33) in [19]) is invariant under and the transformations (B9). The corresponding conserved quantities can be obtained in the standard Noether manner and result as (2.25) and (2.26) with (2.29) and (2.30) inserted, as expected.
A straightforward (3+1)-split of the Poincaré algebra leads to In consideration of (B8) the first two equations reflect the translation invariance of P j and E, while the third one requires E to be a scalar under rotations. Similarly, the next equations state that P k and G k transform as vectors under rotations, while J kl transforms as a bivector. Equation (B17) means that the center-of-mass X k = G k /E has the expected transformation property under translations, {X i , P j } = δ ij . Thus all except the last two equations are fulfilled by construction if J ij and P i are given by its standard representation 3 . For the calculation in [40] the fulfillment of (B19) implied that (B18) also holds. However, a generalization of this fact is not known to the authors. It is instructive to give a physical interpretation of (B18) and (B19). Equation (B19) can be written aṡ and states that the center-of-mass is moving with constant velocity. If we define a total spin of the system as 3 If G i is determined by an ansatz instead of the integral (2.24), then one should also check (B17).
we get {S total ij , P k } = 0 , (B23) where M 2 = E 2 − P 2 ; see, e.g., [64]. This transforms (B21) into and finally leads to the standard Poisson brackets It still holdsŻ i = P i /E = const andṠ total ij = 0. Equation (B18) was transformed into (B31). Thus (B18) reflects the fact that by a straightforward (3+1)-split of the Poincaré algebra one arrives at a center X i associated with the (noncanonical) SSC S total 0i = 0. Notice that the structure of the total angular momentum in (B21) and (B29) is the same, but the variables in (B21) are not standard canonical. It is thus astonishing that the condition (2.30) uniquely fixes the canonical spin and position variables in this paper. This is due to the important additional requirement of having a constant Euclidean spin-length. In this section, however, even the individual components of S total ij andŜ total ij are constant due to the fact that the total system does not interact, e.g., with an external field.
with the shift δx µ on the a-th worldline given by It holds δx 0 a = 0 and p ai δx i a = 0. Although spatial derivatives of p ai are not defined (the linear momentum is only known on the worldline), p ai is treated as a vector field for the Lie-shift. Therefore we must have δx k a;i p ak = −δx k a (p ai;k + p ak,i − p ai,k ) = 0 , which precisely cancels the spatial derivatives of p ai introduced by the Lie-shift. Thus p ai , as a vector field, must be parallel transported to the new worldline without rotation. The transition to the canonical momentum now has to read p ai = P ai − nS k a K ik − π jk a γ jk,i +

2Ŝ
ajk × γ lj γ kp γ il,p − P am P aq nP a (m a − nP a ) γ mj γ kl γ qp γ lp,i , in order to satisfy Eq. (2.35). For π ij a = 0 the momentum redefinition from [19] is obtained. Further, this leads to the results (3.11) and (3.12) for the case δz i a = 0. However, this formulation is not so useful for variable redefinitions in an action approach. Notice that (C5) is missing a term when compared to (3.10c), however, here p ai is treated as a vector field and is not held constant for the redefinition of the position.