Hamiltonian formulation of general relativity and post-Newtonian dynamics of compact binaries

Hamiltonian formalisms provide powerful tools for the computation of approximate analytic solutions of the Einstein field equations. The post-Newtonian computations of the explicit analytic dynamics and motion of compact binaries are discussed within the most often applied Arnowitt–Deser–Misner formalism. The obtention of autonomous Hamiltonians is achieved by the transition to Routhians. Order reduction of higher derivative Hamiltonians results in standard Hamiltonians. Tetrad representation of general relativity is introduced for the tackling of compact binaries with spinning components. Configurations are treated where the absolute values of the spin vectors can be considered constant. Compact objects are modeled by use of Dirac delta functions and their derivatives. Consistency is achieved through transition to d-dimensional space and application of dimensional regularization. At the fourth post-Newtonian level, tail contributions to the binding energy show up. The conservative spin-dependent dynamics finds explicit presentation in Hamiltonian form through next-to-next-to-leading-order spin–orbit and spin1–spin2 couplings and to leading-order in the cubic and quartic in spin interactions. The radiation reaction dynamics is presented explicitly through the third-and-half post-Newtonian order for spinless objects, and, for spinning bodies, to leading-order in the spin–orbit and spin1–spin2 couplings. The most important historical issues get pointed out.


Introduction
Before entering the very subject of the article, namely the Hamiltonian treatment of the dynamics of compact binary systems within general relativity (GR) theory, some historical insight will be supplied. The reader may find additional history, e.g., in Damour (1983aDamour ( , 1987b, Futamase and Itoh (2007), Blanchet (2014), and Porto (2016).

Early history (1916-1960)
The problem of motion of many-body systems is an important issue in GR (see, e.g., Damour 1983aDamour , 1987b. Earliest computations were performed by Droste, de Sitter, andLorentz in the years 1916-1917, at the first post-Newtonian (1PN) order of approximation of the Einstein field equations, i.e., at the order n = 1, where (1/c 2 ) n corresponds to the nth post-Newtonian (PN) order with n = 0 being the Newtonian level. Already in the very first paper, where Droste calculated the 1PN gravitational field for a many-body system (Droste 1916), there occurred a flaw in the definition of the rest mass m of a self-gravitating body of volume V (we follow the Dutch version; the English version contains an additional misprint), reading, in the rest frame of the body, indicated in the following by=, m Droste 1916 where the "Newtonian" mass density * = √ −g u 0 /c [g = det(g μν ), u 0 is the time component of the four-velocity field u μ , u μ u μ = −c 2 ] fulfills the metric-free continuity equation where v = (v i ) is the Newtonian velocity field (with v i = cu i /u 0 ). The Newtonian potential U is defined by with the usual boundary condition for U at infinity: lim |r|→∞ U (r, t) = 0. Let us stress again that the definition (1.1) is not correct. The correct expression for the rest mass contrarily reads, at the 1PN level, with specific internal energy Π . For pressureless (dust-like) matter, the correct 1PN expression is given by The error in question slept into second of two sequential papers by de Sitter (1916ade Sitter ( , b, 1917 when calculating the 1PN equations of motion for a many-body system. Luckily, that error had no influence on the de Sitter precession of the Moon orbit around the Earth in the gravitational field of the Sun. The error became identified (at least for dusty matter) by Eddington and Clark (1938). On the other side, Levi-Civita (1937b) used the correct rest mass formula for dusty bodies. Einstein criticized the calculations by Levi-Civita because he was missing pressure for stabilizing the bodies. Hereupon, Levi-Civita argued with the "effacing principle", inaugurated by Brillouin, that the internal structure should have no influence on the external motion. The 1PN gravitational field was obtained correctly by Levi-Civita but errors occurred in the equations of motion including self-acceleration and wrong periastron advance (Levi-Civita 1937a;Damour and Schäfer 1988). Full clarification was achieved by Eddington and Clark (1938), letting aside the unstable interior of their dusty balls. Interestingly, in a 1917 paper by Lorentz and Droste (in Dutch), the correct 1PN Lagrangian of a self-gravitating many-body system of fluid balls was obtained but never properly recognized. Only in 1937, for the edition of the collected works by Lorentz, it became translated into English (Lorentz and Droste 1937). A full-fledged calculation made by Einstein et al. (1938)-posed in the spirit of Hermann Weyl by making use of surface integrals around field singularities-convincingly achieved the 1PN equations of motion, nowadays called Einstein-Infeld-Hoffmann (EIH) equations of motion. Some further refining work by Einstein andInfeld appeared in the 1940s. Fichtenholz (1950) computed the Lagrangian and Hamiltonian out of the EIH equations. A consistent fluid ball derivation of the EIH equations has been achieved by Fock (1939) and Petrova (1949) (delayed by World War II), and Papapetrou (1951a) (see also Fock 1959).
In the 1950s, Infeld and Plebański rederived the EIH equations of motion with the aid of Dirac δ-functions as field sources by postulating the properties of Infeld's "good" δ-function (Infeld 1954(Infeld , 1957Infeld and Plebański 1960;see Sect. 4.2 of our review for more details). Also in the 1950s, the Dirac δ-function became applied to the post-Newtonian problem of motion of spinning bodies by Tulczyjew (1959), based on the seminal work by Mathisson (1937Mathisson ( , 2010, with the formulation of a general relativistic gravitational skeleton structure of extended bodies. Equations of motion for spinning test particles had been obtained before by Papapetrou (1951b) and Corinaldesi and Papapetrou (1951). Further in the 1950s, another approach to the equations-of-motion problem, called fast-motion or post-Minkowskian (PM) approximation, which is particularly useful for the treatment of high-speed scattering problems, was developed and elaborated by Bertotti (1956) and Kerr (1959a, b, c), at the 1PM level. First results at the 2PM level were obtained by Bertotti and Plebański (1960).

History on Hamiltonian results
Hamiltonian frameworks are powerful tools in theoretical physics because of their capacity of full-fledged structural exploration and efficient application of mathematical theories (see, e.g., Holm 1985;Alexander 1987;Vinti 1998). Most importantly, Hamiltonians generate the time evolution of all quantities in a physical theory. For closed systems, the total Hamiltonian is conserved in time. Together with the other conserved quantities, total linear momentum and total angular momentum, which are given by very simple universal expressions, and the boost vector, which is connected with the Hamiltonian density and the total linear momentum, the total Hamiltonian is one of the generators of the globally operating Poincaré or inhomogeneous Lorentz group. A natural ingredient of a Hamiltonian formalism is the (3 + 1)-splitting of spacetime in space and time. Consequently Hamiltonian formalisms allow transparent treatments of both initial value problems and Newtonian limits. Finally, for solving equations of motion, particularly in approximation schemes, Hamiltonian frameworks naturally fit into the powerful Lie-transform technique based on action-angle variables (Hori 1966;Kinoshita 1978;Vinti 1998;Tessmer et al. 2013).
Additionally we refer to an important offspring of the Hamiltonian framework, the effective-one-body (EOB) approach, which will find its presentation in an upcoming Living Reviews article by Thibault Damour. References in the present article referring to EOB are particularly Damour (1999, 2000), Damour et al. (2000aDamour et al. ( , 2008bDamour et al. ( , 2015, Damour (2001Damour ( , 2016. The focus of the present article is on the Hamiltonian formalism of GR as developed by Arnowitt, Deser, and Misner (ADM) (Arnowitt et al. 1959(Arnowitt et al. , 1960a, with its Routhian modification Schäfer 1998, 2000c) (where the matter is treated in Hamiltonian form and the field in the Lagrangian one) and classical-spin generalization (Steinhoff and Schäfer 2009a;Steinhoff 2011), and with application to the problem of motion of binary systems with compact components including proper rotation (spin) and rotational deformation (quadratic in the spin variables); for other approaches to the problem of motion in GR, see the reviews by Futamase and Itoh (2007), Blanchet (2014), and Porto (2016). The review article by Arnowitt et al. (1962) gives a thorough account of the ADM formalism (see also Regge and Teitelboim 1974 for the discussion about asymptotics). In this formalism, the final Hamiltonian, nowadays called ADM Hamiltonian, is given in form of a volume integral of the divergence of a vector over three-dimensional spacelike hypersurface, which can also naturally be represented as surface integral at flat spatial infinity i 0 .
It is also interesting to give insight into other Hamiltonian formulations of GR, because those are closely related to the ADM approach but differently posed. Slightly ahead of ADM, Dirac (1958Dirac ( , 1959 had developed a Hamiltonian formalism for GR, and slightly afterwards, Schwinger (1963a, b). Schwinger's approach starts from tetrad representation of GR and ends up with a different set of canonical variables and, related herewith, different coordinate conditions. Dirac has developed his approach with some loose ends toward the final Hamiltonian (see Sect. 2.1 below and also, e.g., Deser 2004), but the coordinate conditions introduced by him-nowadays called Dirac gauge-are often used, mainly in numerical relativity. A subtle problem in all Hamiltonian formulations of GR is the correct treatment of surface terms at spacelike infinity which appear in the asymptotically flat spacetimes. In 1967, this problem has been clearly addressed by De Witt (1967) and later, in 1974, full clarification has been achieved by Regge and Teitelboim (1974). For a short comparison of the three canonical formalisms in question, the Dirac, ADM, and Schwinger ones, see Schäfer (2014).
The first authors who had given the Hamiltonian as two-dimensional surface integral at i 0 on three-dimensional spacelike hypersurfaces were ADM. Of course, the representation of the total energy as surface integral was known before, particularly through the Landau-Lifshitz gravitational stress-energy-pseudotensor approach. Schwinger followed the spirit of ADM. He was fully aware of the correctness of his specific calculations modulo surface terms only which finally became fixed by asymptotic Lorentz invariance considerations. He presented the Hamiltonian (as well as the other generators of the Lorentz group) as two-dimensional surface integrals. Only one application of the Schwinger approach by somebody else than Schwinger himself is known to the authors. It is the paper by Kibble in 1963 in which the Dirac spin-1/2 field found a canonical treatment within GR (Kibble 1963). This paper played a crucial role in the implementation of classical spin into the ADM framework by Steinhoff and Schäfer (2009a) and Steinhoff (2011) (details can be found in Sect. 7 of the present article).
The ADM formalism is the most often used Hamiltonian framework in the analytical treatment of the problem of motion of gravitating compact objects. The main reason for this is surely the very well adapted coordinate conditions for explicit calculations introduced by Arnowitt et al. (1960c) (generalized isotropic coordinates; nowadays, for short, often called ADMTT coordinates, albeit the other coordinates introduced by Arnowitt et al. 1962, are ADMTT too), though also in Schwinger's approach similar efficient coordinate conditions could have been introduced (Schäfer 2014). Already Kimura (1961) started application of the ADM formalism to gravitating point masses at the 1PN level. In 1974, that research activity culminated in a 2PN Hamiltonian for binary point masses obtained by Hiida and Okamura (1972), Ohta et al. (1974a, b). However, one coefficient of their Hamiltonian was not correctly calculated and the Hamiltonian as such was not clearly identified, i.e., it was not clear to which coordinate system it referred to. In 1985, full clarification has been achieved in a paper by Damour and Schäfer (1985) relying on the observation by Schäfer (1984) that the perturbative use of the equations of motion on the action level implies that coordinate transformations have been applied; also see O'Connell (1984, 1986). In addition, Damour and Schäfer (1985) showed how to correctly compute the delicate integral (U TT ) which had been incorrectly evaluated by Hiida and Okamura (1972), Ohta et al. (1974a, b), and made contact with the first fully correct calculation of the 2PN dynamics of binary systems (in harmonic coordinates) by Damour and Deruelle (1981) and  In Schäfer (1983b), the leading-order 2.5PN radiation reaction force for n-body systems was derived by using the ADM formalism. The same force expression had already been obtained earlier by Schäfer (1982) within coordinate conditions closely related to the ADM ones-actually identical with the ADM conditions through 1PN and at 2.5PN order-and then again by Schäfer (1983a), as quoted in Poisson and Will (2014), based on a different approach but in coordinates identical to the ADM ones at 2.5PN order. The 2PN Hamiltonian shown by Schäfer (1982) and taken from Ohta et al. (1974b), apart from the erroneous coefficient mentioned above, is the ADM one as discussed above (the factor 7 in the static part therein has to be replaced by 5), and in the definition of the reaction force in the centre-of-mass system, a misprinted factor 2 is missing, i.e. 2F = F 1 − F 2 . The detailed calculations were presented in Schäfer (1985Schäfer ( , 1986, a further ADM-based derivation by use of a PM approximation scheme has been performed. At 2PN level, the genuine 3-body potential was derived by Schäfer (1987). However, in the reduction of a 4-body potential derived by Ohta et al. (1973Ohta et al. ( , 1974a to three bodies made by Schäfer (1987) some combinatorical shortcomings slept in, which were identified and corrected by Lousto and Nakano (2008), and later by Galaviz and Brügmann (2011) in different form. The n-body 3.5PN non-autonomous radiation reaction Hamiltonian 1 was obtained by the authors in Jaranowski and Schäfer (1997), confirming energy balance results in Blanchet and Schäfer (1989), and the equations of motion out of it were derived by Königsdörffer et al. (2003).
Additionally within the ADM formalism, for the first time in 2001, the conservative 3PN dynamics for compact binaries has been fully obtained by Damour and the authors, by also for the first time making extensive use of the dimensional regularization technique ) (for an earlier mentioning of application of dimensional regularization to classical point particles, see Damour 1980Damour , 1983aand for an earlier n-body static result, i.e. a result valid for vanishing particle momenta and vanishing reduced canonical variables of the gravitational field, not based on dimensional regularization, see Kimura and Toiya 1972). Only by performing all calculations in a d-dimensional space the regularization has worked out fully consistently in the limit d → 3 (later on, a d-dimensional Riesz kernel calculation has been performed too, Damour et al. 2008a). In purely 3-dimensional space computations two coefficients, denoted by ω kinetic and ω static , could not be determined by analytical three-dimensional regularization. The coefficient ω kinetic was shown to be fixable by insisting on global Lorentz invariance and became thus calculable with the aid of the Poincaré algebra (with value 24/41) (Damour et al. 2000c, d). The first evaluation of the value of ω static (namely ω static = 0) was obtained by Schäfer (1999, 2000b) by assuming a matching with the Brill-Lindquist initial-value configuration of two black holes. The correctness of this value (and thereby the usefulness of considering that the Brill-Lindquist initial-value data represent a relevant configuration of two black holes) was later confirmed by dimensional regularization . Explicit analytical solutions for the motion of compact binaries through 2PN order were derived by Damour and Schäfer (1988) and Schäfer and Wex (1993b, c), and through 3PN order by Memmesheimer et al. (2005), extending the seminal 1PN post-Keplerian parametrization proposed by Damour and Deruelle (1985).
Quite recently, the 4PN binary dynamics has been successfully derived, using dimensional regularization and sophisticated far-zone matching (Jaranowski and Schäfer 2012, 2013, 2015Damour et al. 2014). Let us remark in this respect that the linear in G (Newtonian gravitational constant) part can be deduced to all PN orders from the 1PM Hamiltonian derived by Ledvinka et al. (2008). For the first time, the contributions to 4PN Hamiltonian were obtained by the authors in Jaranowski and Schäfer (2012) through G 2 order, including additionally all log-terms at 4PN going up to the order G 5 . Also the related energy along circular orbits was obtained as function of orbital frequency. The application of the Poincaré algebra by Jaranowski and Schäfer (2012) clearly needed the noncentre-of-mass Hamiltonian, though only the centre-of-mass one was published. By Jaranowski and Schäfer (2013), all terms became calculated with the exception of terms in the reduced Hamiltonian linear in ν ≡ m 1 m 2 /(m 1 +m 2 ) 2 (where m 1 and m 2 denote the masses of binary system components) and of the orders G 3 , G 4 , and G 5 . Those terms are just adding up to the log-terms mentioned above. However, taking a numerical self-force solution for circular orbits in the Schwarzschild metric into account, already the innermost (or last) stable circular orbit could be determined numerically through 4PN order by Jaranowski and Schäfer (2013). The complete 4PN analytic conservative Hamiltonian has been given for the first time by Damour et al. (2014), based on Jaranowski and Schäfer (2015), together with the results of Le Tiec et al. (2012) and Bini and Damour (2013). Applications of it for bound and unbound orbits were performed by Damour et al. (2015) and Bini and Damour (2017).
For spinning bodies, counting spin as 0.5PN effect, the 1.5PN spin-orbit and 2PN spin-spin Hamiltonians were derived by O'Connell (1975, 1979), where the given quadrupole-moment-dependent part can be regarded as representing spinsquared terms for extended bodies (notice the presence of the tensor product of two unit vectors pointing each to the spin direction in the quadrupole-moment-dependent Hamiltonians). For an observationally important application of the spin-orbit dynamics, see Damour and Schäfer (1988). In 2008, the 2.5PN spin-orbit Hamiltonian was successfully calculated by Damour et al. (2008c), and the 3PN spin1-spin2 and spin1-spin1 binary black-hole Hamiltonians by Steinhoff et al. (2008a, b, c). The 3PN spin1-spin1 Hamiltonian for binary neutron stars was obtained by Hergt et al. (2010). The 3.5PN spin-orbit and 4PN spin1-spin2 Hamiltonians were obtained by Hartung and Steinhoff (2011a, b) (also see Hartung et al. 2013;Levi and Steinhoff 2014). The 4PN spin1-spin1 Hamiltonian was presented in Levi and Steinhoff (2016a). Based on the Dirac approach, the Hamiltonian of a spinning test-particle in the Kerr metric has been obtained by Barausse et al. ( , 2012. The canonical Hamiltonian for an extended test body in curved spacetime, to quadratic order in spin, was derived by Vines et al. (2016). Finally, the radiation-reaction Hamiltonians from the leadingorder spin-orbit and spin1-spin2 couplings have been derived by Steinhoff and Wang (2010) and Wang et al. (2011).

More recent history on non-Hamiltonian results
At the 2PN level of the equations of motion, the Polish school founded by Infeld succeeded in getting many expressions whereby the most advanced result was obtained by Ryteń in her MSc thesis from 1961 using as model for the source of the gravitational field Infeld's "good δ-function". Using the same source model as applied by Fock and Petrova, Kopeikin (1985) and Grishchuk and Kopeikin (1986) derived the 2PN and 2.5PN equations of motion for compact binaries. However, already in 1982, Damour and Deruelle had obtained the 2PN and 2.5PN equations of motion for compact binaries, using analytic regularization techniques [Damour , 1983a (for another such derivation see Blanchet et al. 1998)]. Also Ohta and Kimura (1988) should be mentioned for a Fokker action derivation of the 2PN dynamics. Regarding the coordinate conditions used in the papers quoted in the present subsection, treating spinless particles, all are based on the harmonic gauge with the exceptions of the ones with a Hamiltonian background and those by Ryteń or Ohta and Kimura. Using the technique of Einstein, Infeld, and Hoffmann (EIH), Itoh and Futamase (2003) and Itoh (2004) succeeded in deriving the 3PN equations of motion for compact binaries, and Blanchet et al. (2004) derived the same 3PN equations of motion based on dimensional regularization. The extended Hadamard regularization, developed and applied before Faye 2000a, b, 2001a, b), is incompatible 2 with distribution theory and with the method of dimensional regularization . The 3.5PN equations of motion were derived within several independent approaches: by Pati and Will (2002) using the method of direct integration of the relaxed Einstein equations (DIRE) developed by Pati and Will (2000), by Nissanke and Blanchet (2005) applying Hadamard self-field regularization, by Itoh (2009) using the EIH technique, and by Galley and Leibovich (2012) within the effective field theory (EFT) approach. Radiation recoil effects, starting at 3.5PN order, have been discussed by Bekenstein (1973), Fitchett (1983), Junker and Schäfer (1992), Kidder (1995), and Blanchet et al. (2005). Bernard et al. (2016) calculated the 4PN Fokker action for binary point-mass systems and found a nonlocal-in-time Lagrangian inequivalent to the Hamiltonian obtained by Damour et al. (2014). On the one hand, the local part of the result of Bernard et al. (2016) differed from the local part of the Hamiltonian of Damour et al. (2014) only in a few terms. On the other hand, though the nonlocal-in-time part of the action in Bernard et al. (2016) was the same as the one in Damour et al. (2014Damour et al. ( , 2015, Bernard et al. (2016) advocated to treat it (notably for deriving the conserved energy, and deriving its link with the orbital frequency) in a way which was inequivalent to the one in Damour et al. (2014Damour et al. ( , 2015. It was then shown by Damour et al. (2016) that: (i) the treatment of the nonlocal-in-time part in Bernard et al. (2016) was not correct, and that (ii) the difference in local-in-time terms was composed of a combination of gauge terms and of a new ambiguity structure which could be fixed either by matching to Damour et al. (2014Damour et al. ( , 2015 or by using the results of self-force calculations in the Schwarzschild metric. In their recent articles (Bernard et al. 2017a, b) Blanchet and collaborators have recognized that the criticisms of Damour et al. (2016) were founded, and, after correcting their previous claims and using results on periastron precession first derived by Damour et al. (2015Damour et al. ( , 2016, have obtained full equivalence with the earlier derived ADM results. Let us also mention that Marchand et al. (2018) has presented a self-contained calculation of the full 4PN dynamics (not making any use of self-force results), which confirms again the correctness of the 4PN dynamics first obtained by Damour et al. (2014). The computation of Marchand et al. (2018) can be viewed as a 4PN analog of the 3PN derivation presented in Damour et al. (2001), in which the power of dimensional regularization in post-Newtonian calculations has been established for the first time. An application of the 4PN dynamics for bound orbits was performed by Bernard et al. (2017b).
The application of EFT approach to PN calculations, devised by Goldberger and Rothstein (2006a, b), has also resulted in PN equations of motion for spinless particles up to the 3PN order (Gilmore and Ross 2008;Kol and Smolkin 2009;Foffa and Sturani 2011). At the 4PN level, Foffa and Sturani (2013a) calculated a quadratic in G higher-order Lagrangian, the published version of which was found in agreement with Jaranowski and Schäfer (2012). The quintic in G part of the 4PN Lagrangian was derived within the EFT approach by Foffa et al. (2017) (with its 2016 arXiv version corrected by Damour and Jaranowski 2017). Galley et al. (2016) got the 4PN nonlocal-Footnote 2 continued can be clearly seen from the paper by Blanchet et al. (2004)  in-time tail part. Recently, Porto and Rothstein (2017) and Porto (2017) performed a deeper analysis of IR divergences in PN expansions.
The 1.5PN spin-orbit dynamics was derived in Lagrangian form by Tulczyjew (1959) and . The 2PN spin-spin equations of motion were derived by D'Eath (1975a, b), and Thorne and Hartle (1985), respectively, for rotating black holes. The 2.5PN spin-orbit dynamics was successfully tackled by Tagoshi et al. (2001), and by Faye et al. (2006), using harmonic coordinates approach. Within the EFT approach, Porto (2010) and Levi (2010a) succeeded in determining the same coupling (also see Perrodin 2011). The 3PN spin1-spin2 dynamics was successfully tackled by Porto and Rothstein (2008b, 2010b) (based on Porto 2006Porto and Rothstein 2006) and by Levi (2010b), and the 3PN spin1-spin1 one, again by Porto and Rothstein (2008a), but given in 2010 only in fully correct form (Porto and Rothstein 2010a). For the 3PN spin1-spin1 dynamics, also see Bohé et al. (2015). The most advanced results for spinning binaries can be found in Levi (2012), Marsat et al. (2013), Bohé et al. (2013), Marsat (2015), and Levi and Steinhoff (2016a, b, c), reaching 3.5PN and 4PN levels (also see Steinhoff 2017). Finally, the radiation-reaction dynamics of the leading-order spin-orbit and spin1-spin2 couplings have been obtained by Wang and Will (2007) and Zeng and Will (2007), based on the DIRE method ) (see also Maia et al. 2017a, b, where the EFT method became applied).

Notation and conventions
In this article, Latin indices from the mid alphabet are running from 1 to 3 (or d for an arbitrary number of space dimensions), Greek indices are running from 0 to 3 (or d for arbitrary space dimensions), whereby x 0 = ct. We denote by x = (x i ) (i ∈ {1, . . . , d}) a point in the d-dimensional Euclidean space R d endowed with a standard Euclidean metric defining a scalar product (denoted by a dot). For any spatial here for the Euclidean length of a vector, δ i j = δ i j denotes Kronecker delta. The partial differentiation with respect to x μ is denoted by ∂ μ or by a comma, i.e., ∂ μ φ ≡ φ ,μ , and the partial derivative with respect to time coordinates t is denoted by ∂ t or by an overdot, ∂ t φ ≡φ. The covariant differentiation is generally denoted by ∇, but we may also write ∇ α (·) ≡ (·) ||α for spacetime or ∇ i (·) ≡ (·) ;i for space variables, respectively. The signature of the (d + 1)-dimensional metric g μν is +(d − 1). The Einstein summation convention is adopted. The speed of light is denoted by c and G is the Newtonian gravitational constant.
We use the notion of a tensor density. The components of a tensor density of weight w, k times contravariant and l times covariant, transform, when one changes one coordinate system to another, by the law [see, e.g., p. 501 in Misner et al. (1973) or, for more general case, Sects. 3.7-3.9 and 4.5 in Plebański and Krasiński (2006), where however definition of the density weight differs by sign from the convention used by us] is the Jacobian of the transformation x → x (x). E.g., determinant of the metric g ≡ det(g μν ) is a scalar density of weight +2. The covariant derivative of the tensor density of weight w, k times contravariant and l times covariant, is computed according to the rule For the often used case when ..β l is a tensor k times contravariant and l times covariant), Eq. (1.7) implies that the covariant derivative of T α 1 ...α k β 1 ...β l can be computed by means of the rule, Letters a, b (a, b = 1, 2) are particle labels, so x a = (x i a ) ∈ R d denotes the position of the ath point mass. We also define r a ≡ x − x a , r a ≡ |r a |, n a ≡ r a /r a ; and for a = b, r ab ≡ x a − x b , r ab ≡ |r ab |, n ab ≡ r ab /r ab . The linear momentum vector of the ath particle is denoted by p a = ( p ai ), and m a denotes its mass parameter. We abbreviate Dirac delta distribution δ(x − x a ) by δ a (both in d and in 3 dimensions); it fulfills the condition d d x δ a = 1.
Thinking in terms of dimensions of space, d has to be an integer, but whenever integrals within dimensional regularization get performed, we allow d to become an arbitrary complex number [like in the analytic continuation of factorial n! = Γ (n + 1) to Γ (z)].

Hamiltonian formalisms of GR
The presented Hamiltonian formalisms do all rely on a (3 + 1) splitting of spacetime metric g μν in the following form: here γ i j is the inverse metric of γ i j (γ ik γ k j = δ j i ), γ ≡ det(γ i j ); lowering and raising of spatial indices is with γ i j . The splitting (2.1), and the associated explicit 3 + 1 decomposition of Einstein's equations, was first introduced by Fourès-Bruhat (1956). The notations N and N i are due to Arnowitt et al. (1962) and their names, respectively "lapse" and "shift" functions, are due to Wheeler (1964). Let us note the useful relation between the determinants g ≡ det(g μν ) and γ : We restrict ourselves to consider only asymptotically flat spacetimes and we employ quasi-Cartesian coordinate systems (t, x i ) which are characterized by the following asymptotic spacelike behaviour (i.e., in the limit r → ∞ with r ≡ √ x i x i and t = const) of the metric coefficients: De Witt (1967) and later, in a more refined way, Regge and Teitelboim (1974) explicitly showed that the Hamiltonian which generates all Einsteinian field equations can be put into the form, where R is the intrinsic curvature scalar of the spacelike hypersurfaces of constantin-time slices t = x 0 /c = const; the ADM canonical field momentum is given by the denote Christoffel symbols; π ≡ γ i j π i j ; ∇ k denotes the three-dimensional covariant derivative (with respect to γ i j ). The given densities are densities of weight one with respect to three-dimensional coordinate transformations. Let us note the useful formula for the density of the three-dimensional scalar curvature of the surface t = const: (2.10) The matter densities H m and H mi are computed from components of the matter energymomentum tensor T μν by means of formulae where n μ = (−N , 0, 0, 0) is the timelike unit covector orthogonal to the spacelike hypersurfaces t = const. Opposite to what the right-hand sides of Eqs. (2.11)-(2.12) seem to suggest, the matter densities must be independent on lapse N and shift N i and expressible in terms of the dynamical matter and field variables q A , π A , γ i j only (π i j does not show up for matter which is minimally coupled to the gravitational field). The variation of (2.6) with respect to N and N i yields the constraint equations The most often applied Hamiltonian formalism employs the following coordinate choice made by ADM (which we call ADMTT gauge), where the TT piece h TT i j is transverse and traceless, i.e., it satisfies ∂ j h TT i j = 0 and h TT ii = 0. The TT piece of any field function can be computed by means of the TT projection operator defined as follows where Δ −1 denotes the inverse of the flat space Laplacian, which is taken without homogeneous solutions for source terms decaying fast enough at infinity (in 3-dimensional or, if not, then in generalized d-dimensional space). The nonlocality of the TT-operator δ TTkl i j is just the gravitational analogue of the well-known nonlocality of the Coulomb gauge in the electrodynamics.
Taking into account its gauge condition as given in Eq. (2.14), the field momentum c 3 16π G π i j can be split into its longitudinal and TT parts, respectively, where the TT part π i j TT fulfills the conditions ∂ j π i j TT = 0 and π ii TT = 0 and where the longitudinal partπ i j can be expressed in terms of a vectorial function V i , It is also convenient to parametrize the field function ψ from Eq. (2.14) in the following way (2.18) The independent field variables are π i j TT and h TT i j . Already Kimura (1961) used just this presentation for applications. The Poisson bracket for the independent degrees of freedom reads where δ F(x)/(δ f (z)) denotes the functional (or Frèchet) derivative. ADM gave the Hamiltonian in fully reduced form, i.e., after having applied (four) constraint equations (2.13) and (four) coordinate conditions (2.14). It reads (2.20) The reduced Hamiltonian generates the field equations of the two remaining metric coefficients (eight metric coefficients are determined by the four constraint equations and four coordinate conditions combined with four otherwise degenerate field equations for the lapse and shift functions). By making use of (2.18) the reduced Hamiltonian (2.20) can be written as (2.21)

Hamiltonian formalisms of Dirac and Schwinger
Dirac had chosen the following coordinate system, called "maximal slicing" because of the field momentum condition, The reason for calling the condition π = 2K γ 1/2 = 0 "maximal slicing" is because the congruence of the timelike unit vectors n μ normal to the t = const hypersurfaces (slices)-as such irrotational-is free of expansion (notice that ∇ μ n μ = −K ). Hereof it immediately follows that a finite volume in any slice gets unchanged by a small timelike deformation of the slice which vanishes on the boundary of the volume, i.e. an extremum principle holds (see, e.g., York 1979). The corresponding independent field variables are (no implementation of the three differential conditions!) with the algebraic properties γ i jπ i j = 0 and det(g i j ) = 1. To leading order linear in the metric functions, the Dirac gauge coincides with the ADM gauge. The reduction of the Dirac form of dynamics to the independent tilded degrees of freedom has been performed by Regge and Teitelboim (1974), including a fully satisfactory derivation of the Hamiltonian introduced by Dirac. The Poisson bracket for the Dirac variables reads The Hamiltonian proposed by Dirac results from the expression (2.29) If Schwinger would have chosen coordinate conditions corresponding to those introduced above in Eqs. (2.14) (ADM also introduced another set of coordinate conditions to which Schwinger adjusted), namely  Kibble (1963) delivers an application of Schwinger's formalism, apart from Schwinger himself, namely a Hamiltonian formulation of the Dirac spinor field in gravity. Much later, Nelson and Teitelboim (1978) completed the same task within the tetrad-generalized Dirac formalism (Dirac 1962).

Derivation of the ADM Hamiltonian
The ADM Hamiltonian was derived via the generator of field and spacetimecoordinates variations. Let the generator of general field variations be defined as (it corresponds to the generator G ≡ p i δx i of the point-particle dynamics in classical mechanics with the particle's canonical momentum p i and position x i ) Let the coefficients of three space-metric γ i j be fixed by the relations (2.14), then the only free variations left are or, modulo a total variation, It is consistent with the Einstein field equations in space-asymptotically flat spacetime with quasi-Cartesian coordinates to put [the mathematically precise meaning of this equation is detailed in the Appendix B of Arnowitt et al. (1960a)] which results in, dropping total space derivatives, Hereof the Hamiltonian easily follows in the form which can also be written, using the form of the three-metric from Eq. (2.14), This expression is valid also in case of other coordinate conditions (Arnowitt et al. 1962). For the derivation of the generator of space translations, the reader is referred to Arnowitt et al. (1962) or, equivalently, to Schwinger (1963a).

Reduced Hamiltonian for point-mass systems
In this section we consider the ADM canonical formalism applied to a system of self-gravitating nonrotating point masses (particles). The energy-momentum tensor of such system reads where m a is the mass parameter of ath point mass (a = 1, 2, . . . labels the point is the four-velocity along the worldline x μ = x μ a (τ a ) of the ath particle. After performing the integration in (3.1) one gets where x a = (x i a ) is the position three-vector of the ath particle. The linear fourmomentum of the ath particle equals p α a ≡ m a u α a , and the three-momentum canonically conjugate to the position x a comes out to be p a = ( p ai ), where p ai = m a u ai .
The action functional describing particles-plus-field system reads whereẋ i a ≡ dx i a /dt. The asymptotic value 1 of the lapse function enters as prefactor of the surface integral in the Hamiltonian H 0 , which takes the form is the finite part of the inverse metric evaluated at the particle position, which can be perturbatively and, using dimensional regularization, unambiguously defined (see Sects. 4.2, 4.3 below and Appendix A4 of Jaranowski and Schäfer 2015).
The evolutionary part of the field equations is obtained by varying the action functional (3.3) with respect to the field variables γ i j and π i j . The resulting equations read jl a p al γ mn a p am p an + m 2 a c 2 −1/2 δ a . (3.8) The constraint part of the field equations results from varying the action (3.3) with respect to N and N i . It has the form The variation of the action (3.3) with respect to x a and p a leads to equations of motion for the particles,ṗ 2 m 2 a c 2 + γ mn a p am p an (3.11) Notice the involvement of lapse and shift functions in the equations of motion. Both the lapse and shift functions, four functions in total, get determined by the application of the four coordinate conditions (2.14) to the field equations (3.7) and (3.8).
The reduced action, which is fully sufficient for the derivation of the dynamics of the particles and the gravitational field, reads (only the asymptotic value 1 of the shift function survives) where both the constraint equations (3.9) and the coordinate conditions (2.14) are taken to hold. The reduced Hamilton functional H red is given by (3.13) The remaining field equations read (3.14) and the equations of motion for the point masses take the forṁ Evidently, there is no involvement of lapse and shift functions in the equations of motion and in the field equations for the independent degrees of freedom (Arnowitt et al. 1960b;Kimura 1961).

Routh functional
The Routh functional (or Routhian) of the system is defined by This functional is a Hamiltonian for the point-mass degrees of freedom, and a Lagrangian for the independent gravitational field degrees of freedom. Within the post-Newtonian framework it was first introduced by Schäfer (1998, 2000c). The evolution equation for the gravitational field degrees of freedom reads The Hamilton equations of motion for the two point masses take the forṁ For the following treatment of the conservative part of the dynamics only, we will make now a short model calculation revealing the structure and logic behind the treatment. Let's take a Routhian of the form R (q, p; ξ,ξ ). Then the action reads Its variation through the independent variables gives Going on-shell with the ξ -dynamics yields (3.21) The vanishing of the last term means-thinking in terms of h TT i j andḣ TT i j , i.e. considering the term ( d 3 x π i j TT δh TT i j ) +∞ −∞ on the solution space of the field equations ("on-field-shell")-that as much incoming as outgoing radiation has to be present, or time-symmetric boundary conditions have to be applied. Thus in the Fokker-type procedure no dissipation shows up. This, however, does not force the use of the symmetric Green function, which would exclude conservative tail contributions at 4PN and higher PN orders. Assuming a leading-order-type prolongation of the form R = R (q, p,q,ṗ), the autonomous dynamics can be deduced from the variation where the Euler-Lagrange derivative δ A/δz ≡ ∂ A/∂z − d(∂ A/∂ż)/dt has been introduced.
Having explained that, the conservative part of the binary dynamics is given by the higher-order Hamiltonian equal to the on-field-shell Routhian, where the field variables h TT i j ,ḣ TT i j were "integrated out", i.e., replaced by their solutions as functionals of particle variables. The conservative equations of motion defined by the higher-order Hamiltonian (3.23) reaḋ where the functional derivative is given by with z = x i a or z = p ai . Schäfer (1984) and Damour and Schäfer (1991) show that time derivatives of x a and p a in the higher-order Hamiltonian (3.23) can be eliminated by the use of lower-order equations of motion, leading to an ordinary Hamiltonian, (3.26) Notice the important point that the two Hamiltonians H con and H ord con do not belong to the same coordinate system. Therefore, the Hamiltonians H con and H ord con and their variables should have, say, primed and unprimed notations which usually however does not happen in the literature due to a slight abuse of notation. A This equation must now be solved step by step using either retarded integrals for getting the whole dynamics or time-symmetric ones for only the conservative dynamics defined by H con , which themselves have to be expanded in powers of 1/c. In higher orders, however, non-analytic in 1/c log-terms do show up (see, e.g., Damour et al. 2014Damour et al. , 2016. To calculate the reduced Hamiltonian of Eq. (2.21) for a many-particle system one has to perturbatively solve for φ andπ i j the constraint equations H = 0 and H i = 0 with the densities H, H i defined in Eqs.
The expansion of the momentum constraint equation up to c −7 reads In the Eqs. (3.29) and (3.30) dynamical field variables h TT i j and π i j TT are counted as being of the orders 1/c 4 and 1/c 5 , respectively [cf. Eq. (3.28)].

Poincaré invariance
In asymptotically flat spacetimes the Poincaré group is a global symmetry group. Its generators P μ and J μν are realized as functions P μ (x a , p a ) and J μν (x a , p a ) on the many-body phase-space. They are conserved on shell and fulfill the Poincaré algebra relations for the Poisson bracket product (see, e.g., Regge and Teitelboim 1974), where the Poisson brackets are defined in an usual way, (3.34) The meaning of the components of P μ and J μν is as follows: the time component P 0 (i.e., the total energy) is realized as the Hamiltonian H ≡ cP 0 , , J kl = J kl , and J i j = ε i jk J k ] is angular momentum, and Lorentz boost vector is K i ≡ J i0 /c. The boost vector represents the constant of motion associated to the centre-of-mass theorem and can further be decomposed as In terms of three-dimensional quantities the Poincaré algebra relations read (see, e.g., Damour et al. 2000c, d) The Hamiltonian H and the centre-of-mass vector G i have the integral representations where n r 2 dΩ (n is the radial unit vector) is the two-dimensional surface-area element at i 0 . The two quantities H and G i are the most involved ones of those entering the Poincaré algebra.
The Poincaré algebra has been extensively used in the calculations of PN Hamiltonians for spinning binaries (Hergt and Schäfer 2008a, b). Hereby the most useful equation was (3.38), which tells that the total linear momentum has to be a total time derivative. This equation was also used by Damour et al. (2000c, d) to fix the so called "kinetic ambiguity" in the 3PN ADM two-point-mass Hamiltonian without using dimensional regularization. In harmonic coordinates, the kinetic ambiguity got fixed by a Lorentzian version of the Hadamard regularization based on the Fock-de Donder approach (Blanchet and Faye 2001b).
The explicit form of the generators P μ (x a , p a ) and J μν (x a , p a ) (i.e., P, J, G, and H ) for two-point-mass systems is given in Appendix C with 4PN accuracy.
The global Lorentz invariance results in the following useful expressions (see, e.g., Rothe and Schäfer 2010;Georg and Schäfer 2015). Let us define the quantity M through the relation (3.43) and let us introduce the canonical centre of the system vector X (with components Then the following commutation relations are fulfiled: The commutation relations clearly show the complete decoupling of the internal dynamics from the external one by making use of the canonical variables. The equations (3.43) additionally indicate that M 2 is simpler (or, more primitive) than M, cf. Georg and . A centre-of-energy vector can be defined by This vector, however, is not a canonical position vector, see, e.g., Hanson and Regge (1974).
In view of our later treatment of particles with spin, let us decompose the total angular momentum J μν of a single object into orbital angular momentum L μν and spin S μν , both of them are anti-symmetric tensors, (3.48) The orbital angular momentum tensor is given by where Z μ denotes 4-dimensional position vector (with Z 0 = ct). The splitting in space and time results in Remarkably, relativity tells us that any object with mass M, spin length S, and positive energy density must have extension orthogonal to its spin vector of radius of at least S/(Mc) (see, e.g., Misner et al. 1973). Clearly then, the position vector of such an object is not given a priori but must be defined. As the total angular momentum should not depend on the fixation of the position vector, the notion of spin must depend on the fixation of the position vector and vice versa. Thus, imposing a spin supplementary condition (SSC) fixes the position vector. We enumerate here the most often used SSCs (see, e.g., Fleming 1965;Hanson and Regge 1974;Barker and O'Connell 1979).
(i) Covariant SSC (also called Tulczyjew-Dixon SSC): The variables corresponding to this SSC are denoted in Sect. 7 by Z i = z i , S i j , and P i = p i . (ii) Canonical SSC (also called Newton-Wigner SSC): where n μ = (−1, 0, 0, 0), n μ n μ = −1. The variables corresponding to this SSC are denoted in Sect. 7 byẑ i ,Ŝ i j , and P i . (iii) Centre-of-energy SSC (also called Corinaldesi-Papapetrou SSC): (3.53) Here the boost vector takes the form of a spinless object,

Poynting theorem of GR
Let us start with the following local identity, having structure of a Poynting theorem for GR in local form, Integrating this equation over whole space gives, assuming past stationarity, where V ∞ is just another expression for R 3 . Notice that the far zone is understood as area of the t = const slice where gravitational waves are decoupled from their source and do freely propagate outwards, what means that the relation h TT (3.56) with V fz as the volume of the space enclosed by the outer boundary of the far (or, wave) zone (fz) and ds k = n k r 2 dΩ surface-area element of the two-surface of integration with dΩ as the solid-angle element and r the radial coordinate, it follows Dropping the left side of this equation as negligibly small, assuming the source term for h TT i j , which follows from the Routhian field equation (3.17), to decay at least as 1/r 3 for r → ∞ (for isolated systems, all source terms for h TT i j decay at least as 1/r 4 if not TT-projected; the TT-projection may raise the decay to 1/r 3 , e.g. TT-projection of Dirac delta function), results in with meaning that the energy flux through a surface in the far zone equals the growth of gravitational energy beyond that surface.

Near-zone energy loss and far-zone energy flux
The change in time of the matter Routhian reads, assuming R to be local in the gravitational field, The equation for d R/dt is valid provided the equations of motioṅ (3.62) The canonical field momentum is given by (3.63) Performing the Legendre transformation yields, assuming past stationarity [meaning that at any finite time t no radiation can have reached spacelike infinity, so the first (surface) term in the right-hand side of Eq.
(3.58) shows that the Eq. (3.64) infers, employing the leading-order quadratic field structure of is the well known total energy flux (or luminosity) of gravitational waves. The Eq.
(3.68) can be put into the energy form, again employing the leading-order quadratic field structure of R, Taking into account the Eqs. (3.29) and (3.41) we find that the second term in the parenthesis of the left side of Eq. (3.70) exactly subtracts the corresponding terms from pure (h TT i j,k ) 2 and (π i j TT ) 2 expressions therein. This improves, by one order in radial distance, the large distance decay of the integrand of the integral of the whole left side of Eq. (3.70), which runs over the whole hypersurface t = const. We may now perform near-and far-zone PN expansions of the left and right sides of the Eq. (3.70), respectively. Though the both series are differently defined-on the left side, expansion in powers of 1/c around fixed time t of an energy expression which is time differentiated; on the right side, expansion in powers of 1/c around fixed retarded time t − r /c-the expansions cannot contradict each other as long as they are not related term by term. For the latter relation we must keep in mind that PN expansions are instantaneous expansions so that the two times, t and t − r /c, are not allowed to be located too far apart from each other. This means that we have to read off the radiation right when it enters far zone. Time-averaging of the expressions on the both sides of Eq. (3.70) over several wave periods makes the difference between the two times negligible as it should be if one is interested in a one-to-one correspondence between the terms on the both sides. The Newtonian and 1PN wave generation processes were explicitly shown to fit into this scheme by Königsdörffer et al. (2003).

Radiation field
In the far zone, the multipole expansion of the transverse-traceless (TT) part of the gravitational field, obtained by algebraic projection with where n ≡ x/r (r ≡ |x|) is the unit vector in the direction from the source to the far away observer, reads (see, e.g., Thorne 1980;Blanchet 2014) ..i l denote the lth time derivatives of the symmetric and tracefree (STF) radiative mass-type and current-type multipole moments, respectively. The term with the leading mass-quadrupole tensor takes the form (see, e.g., Schäfer 1990) showing the leading-order tail term of the quadrupole radiation (the gauge dependent relative phase constant κ between direct and tail term was not explored by Schäfer 1990; for more details see, e.g., Blanchet and Schäfer 1993;Blanchet 2014). Notice the modification of the standard PN expansion through tail terms. This expression nicely shows that also multipole expansions in the far zone do induce PN expansions. The mass-quadrupole tensor M i j is just the standard Newtonian one. Higher-order tail terms up to "tails-of-tails-of-tails" can be found in Marchand et al. (2016). Leadingorder tail terms result from the backscattering of the leading-order outgoing radiation, the "tails-of-tails" from their second backscattering, and so on. Through 1.5PN order, the luminosity expression (3.69) takes the form (3.76) On reasons of energy balance in asymptotically flat space, for any coordinates or variables representation of the Einstein theory, the time-averaged energy loss has to fulfill a relation of the form where the time averaging procedure takes into account typical periods of the system. Generalizing our considerations after Eq. (3.70) we may take the observation time t much larger than the time, say t bfz , the radiation enters the far or wave zone, even larger than the damping time of the radiating system, by just freely transporting the radiation power along the null cone with tacitly assuming L(t) = L(t bfz ) . Coming back to Eq. (3.70), time averaging on the left side of Eq. (3.70) eliminates total time derivatives of higher PN order, so-called Schott terms, and transforms them into much higher PN orders. The both sides of the equation (3.77) are gauge (or, coordinate) invariant. We stress that the Eq. (3.77) is valid for bound systems. In case of scattering processes, a coordinate invariant quantity is the emitted total energy. The energy flux to nPN order in the far zone implies energy loss to (n+5/2)PN order in the near zone. Hereof it follows that energy-loss calculations are quite efficient via energy-flux calculations (Blanchet 2014). In general, only after averaging over orbital periods the both expressions do coincide. In the case of circular orbits, however, this averaging procedure is not needed.

Applied regularization techniques
The most efficient source model for analytical computations of many-body dynamics in general relativity are point masses (or particles) represented through Dirac delta functions. If internal degrees of freedom are come into play, derivatives of the delta functions must be incorporated into the source. Clearly, point-particle sources in field theories introduce field singularities, which must be regularized in computations. Two aspects are important: (i) the differentiation of singular functions, and (ii) the integration of singular functions, either to new (usually also singular) functions or to the final Routhian/Hamiltonian. The item (ii) relates to the integration of the field equations and the item (i) to the differentiation of their (approximate) solutions. On consistency reasons, differentiation and integration must commute.
The most efficient strategy developed for computation of higher-order PN pointparticle Hamiltonians relies on performing a 3-dimensional full computation in the beginning (using Riesz-implemented Hadamard regularization defined later in this section) and then correcting it by a d-dimensional one around the singular points, as well the local ones (UV divergences) as the one at infinity (IR divergences). A d-dimensional full computation is not needed. At higher than the 2PN level 3dimensional computations with analytical Hadamard and Riesz regularizations show up ambiguities which require a more powerful treatment. The latter is dimensional regularization. The first time this strategy was successfully applied was in the 3PN dynamics of binary point particles ; IR divergences did not appear therein, those enter from the 4PN level on only, the same as the nonlocal-in-time tail terms to which they are connected. At 4PN order, using different regularization methods for the treatment of IR divergences , an ambiguity parameter was left which, however, got fixed by matching to self-force calculations in the Schwarzschild metric (Le Tiec et al. 2012;Bini and Damour 2013;Damour et al. 2014).
The regularization techniques needed to perform PN calculations up to (and including) 4PN order, are described in detail in Appendix A of Jaranowski and Schäfer (2015).

Distributional differentiation of homogeneous functions
Besides appearance of UV divergences, another consequence of employing Diracdelta sources is necessity to differentiate homogeneous functions using an enhanced (or distributional) derivative, which comes from standard distribution theory (see, e.g., Sect. 3.3 in Chapter III of Gel'fand and Shilov 1964).
Let f be a real-valued function defined in a neighbourhood of the origin of R 3 . f is said to be a positively homogeneous function of degree λ, if for any number a > 0 (4.1) Let k := −λ − 2. If λ is an integer and if λ ≤ −2 (i.e., k is a nonnegative integer), then the partial derivative of f with respect to the coordinate x i has to be calculated by means of the formula where ∂ i f on the lhs denotes the derivative of f considered as a distribution, while ∂ i f on the rhs denotes the derivative of f considered as a function (which is computed using the standard rules of differentiation), Σ is any smooth close surface surrounding the origin and dσ i is the surface element on Σ.
The distributional derivative does not obey the Leibniz rule. It can easily be seen by considering the distributional partial derivative of the product 1/r a and 1/r 2 a . Let us suppose that the Leibniz rule is applicable here: The right-hand side of this equation can be computed using standard differential calculus (no terms with Dirac deltas), whereas computing the left-hand side one obtains some term proportional to ∂ i δ a . The distributional differentiation is necessary when one differentiates homogeneous functions under the integral sign. For more details, see Appendix A5 in Jaranowski and Schäfer (2015).

Riesz-implemented Hadamard regularization
The usage of Dirac δ-functions to model point-mass sources of gravitational field leads to occurrence of UV divergences, i.e., the divergences near the particle locations x a , as r a ≡ |x − x a | → 0. To deal with them, Infeld (1954Infeld ( , 1957, and Infeld and Plebański (1960) introduced "good" δ-functions, which, besides having the properties of ordinary Dirac δ-functions, also satisfy the condition for some positive integer p (in practical calculations one takes p large enough to take all singularities appearing in the calculation into account). They also assumed that the "tweedling of products" property is always satisfied where "reg" means regularized value of the function at its singular point (i.e., x 0 in the equation above) evaluated by means of the rule (4.4). A natural generalization of the rule (4.4) is the concept of "partie finie" value of function at its singular point, defined as Defining, for a function f singular at x = x 0 , the "tweedling of products" property (4.5) can be written as The above property is generally wrong for arbitrary singular functions f 1 and f 2 . In the PN calculations problems with fulfilling this property begin at the 3PN order. This is one of the reasons why one should use dimensional regularization. The Riesz-implemented Hadamard (RH) regularization was developed in the context of deriving PN equations of motion of binary systems by Jaranowski and Schäfer (1997, 1998 to deal with locally divergent integrals computed in three dimensions. The method is based on the Hadamard "partie finie" and the Riesz analytic continuation procedures. The RH regularization relies on multiplying the full integrand, say i(x), of the divergent integral by a regularization factor, (4.10) and studying the double limit 1 → 0, 2 → 0 within analytic continuation in the complex 1 and 2 planes (here s 1 and s 2 are arbitrary three-dimensional UV regularization scales). Let us thus consider such integral performed over the whole space R 3 and let us assume than it develops only local poles (so it is convergent at spatial infinity). The value of the integral, after performing the RH regularization in three dimensions, has the structure Let us mention that in the PN calculations regularized integrands i(x)(r 1 /s 1 ) 1 (r 2 /s 2 ) 2 depend on x only through x − x 1 and x − x 2 , so they are translationally invariant. This explains why the regularization result (4.11) depends on x 1 and x 2 only through x 1 − x 2 . In the case of an integral over R 3 developing poles only at spatial infinity (so it is locally integrable) it would be enough to use a regularization factor of the form (r /r 0 ) (where r 0 is an IR regularization scale), but it is more convenient to use the factor and study the limit → 0. Let us denote the integrand again by i(x). The integral, after performing the RH regularization in three dimensions, has the structure (4.13) Let us remark here that the extended Hadamard regularization procedure developed by Faye (2000a, b, 2001a, b) is both incompatible with standard distribution theory and with the method of dimensional regularization; for more details see Jaranowski and Schäfer (2015) (see also the footnote 2 above).
Many integrals appearing in PN calculations were computed using a famous formula derived in Riesz (1949) (4.14) To compute the 4PN-accurate two-point-mass Hamiltonian one needs to employ a generalization of the three-dimensional version of this formula for integrands of the form r α 1 r β 2 (r 1 +r 2 +r 12 ) γ . Such formula was derived by Schäfer (1998, 2000c) and also there an efficient way of implementing both formulae to regularize divergent integrals was proposed (it employs prolate spheroidal coordinates in three dimensions). See Appendix A1 of Jaranowski and Schäfer (2015) for details and Appendix A of Hartung et al. (2013) for generalization of this procedure to d space dimensions.

Dimensional regularization
It was first shown by Damour et al. (2001), that the unambiguous treatment of UV divergences in the current context requires usage of dimensional regularization (see, e.g., Collins 1984). It was used both in the Hamiltonian approach and in the one using the Einstein field equations in harmonic coordinates (Damour et al. , 2014Blanchet et al. 2004;Schäfer 2013, 2015;Bernard et al. 2016). The dimensional regularization preserves the law of "tweedling of products" (4.9) and gives all involved integrals, particularly the inverse Laplacians, a unique definition.

D-dimensional ADM formalism
Dimensional regularization (DR) needs the representation of the Einstein field equation for arbitrary space dimensions, say d for the dimension of space and D = d +1 for the spacetime dimension. In the following, G D = G N d−3 0 will denote the gravitational constant in D-dimensional spacetime and G N the standard Newtonian one, 0 is the DR scale relating both constants.
The unconstraint Hamiltonian takes the form The Hamiltonian and the momentum constraint equations written for many-point-particle systems are given by (4.17) The gauge (or coordinate) ADMTT conditions read The field momentum π i j splits into its longitudinal and TT parts, respectively, where the longitudinal partπ i j can be expressed in terms of a vectorial function V i , and where the TT part satisfies the conditions, The reduced Hamiltonian of the particles-plus-field system takes the form (4.23) The equations of motion for the particles reaḋ (4.24) and the field equations for the independent degrees of freedom are given by where the d-dimensional TT-projection operator is defined by Finally, the Routh functional is defined as (4.27) and the fully reduced matter Hamiltonian for the conservative dynamics reads (4.28)

Local and asymptotic dimensional regularization
The technique developed by Damour et al. (2001) to control local (or UV) divergences boils down to the computation of the difference where H RH loc (3) is the "local part" of the Hamiltonian obtained by means of the threedimensional RH regularization [it is the sum of all integrals of the type Damour et al. (2001) showed that to find the DR correction to the integral I RH (3; 1 , 2 ) of Eq. (4.11) related with the local pole at, say, x = x 1 , it is enough to consider only this part of the integrand i(x) which develops logarithmic singularities in three dimensions, i.e., which locally behaves like 1/r 3 1 , Then the pole part of the integral (4.11) related with the singularity at x = x 1 can be recovered by RH regularization of the integral ofc 1 (n 1 ) r −3 1 over the ball B(x 1 , 1 ) of radius 1 surrounding the particle x 1 . The RH regularized value of this integral reads where c 1 /(4π) is the angle-averaged value of the coefficientc 1 (n 1 ). The expansion of the integral I RH 1 (3; 1 ) around 1 = 0 equals The idea of the technique developed by Damour et al. (2001) relies on replacing the RH-regularized value of the three-dimensional integral I RH 1 (3; 1 ) by the value of its d-dimensional version I 1 (d). One thus considers the d-dimensional counterpart of the expansion (4.30). It reads (4.33) Let us note that the specific exponent 6 − 3d of r 1 visible here follows from the r 1 → 0 behaviour of the (perturbative) solutions of the d-dimensional constraint equations (4.16)-(4.17). The number k in the exponent of is related with the momentum-order of the considered term [e.g., at the 4PN level the term with k is of the order of O( p 10−2k ), for k = 1, . . . , 5; such term is proportional to G k D ]. The integral I 1 (d) is defined as (Ω d−1 stands for the area of the unit sphere in R d ) is the angle-averaged value of the coefficientc 1 (d; n 1 ), One checks that always there is a smooth connection between c 1 (d) and its threedimensional counterpart c 1 , Let us note that the coefficient c 1 (3) usually depends on ln r 12 and it has the structure . Therefore the DR correction also changes the terms ∝ ln r 12 .
The DR correction to the RH-regularized value of the integral I RH (3; 1 , 2 ) relies on replacing this integral by Then one computes the double limit Note that all poles ∝ 1/ 1 , 1/ 2 and all terms depending on radii 1 , 2 or scales s 1 , s 2 cancel each other. The result (4.41) is as if all computations were fully done in d dimensions.
In the DR correcting UV of divergences in the 3PN two-point-mass Hamiltonian performed by Damour et al. (2001), after collecting all terms of the type (4.41) together, all poles ∝ 1/(d − 3) cancel each other. This is not the case for the UV divergences of the 4PN two-point-mass Hamiltonian derived by Jaranowski and Schäfer (2015). As explained in Sect. VIII D of Jaranowski and Schäfer (2015), after collecting all terms of the type (4.41), one has to add to the Hamiltonian a unique total time derivative to eliminate all poles ∝ 1/(d − 3) (together with 0 -dependent logarithms).
The above described technique of the DR correcting of UV divergences can easily be transcribed to control IR divergences. This is done by the replacement of the integrals where B(0, R) means a large ball of radius R (with the centre at the origin 0 of the coordinate system), and by studying expansion of the integrand i(x) for r → ∞. This technique was not used to regularize IR divergences in the computation of the 4PN two-point-mass Hamiltonian by Damour et al. (2014) and Jaranowski and Schäfer (2015). This was so because this technique applied only to the instantaneous part of the 4PN Hamiltonian is not enough to get rid of the IR poles in the limit d → 3. For resolving IR poles it was necessary to observe that the IR poles have to cancel with the UV poles from the tail part of the Hamiltonian (what can be achieved e.g. after implementing the so-called zero-bin subtraction in the EFT framework, see Porto and Rothstein 2017). Another two different approaches were employed by Damour et al. (2014) and Jaranowski and  to regularize IR divergences in the instantaneous part of the 4PN Hamiltonian (see Appendix A3 in Jaranowski and Schäfer 2015): (i) modifying the behavior of the function h TT (6)i j at infinity, 4 (ii) implementing a d-dimensional version of Riesz-Hadamard regularization. Both approaches were developed in d dimensions, but the final results of using any of them in the limit d → 3 turned out to be identical with the results of computations performed in d = 3 dimensions. Moreover, the results of the two approaches were different in the limit d → 3, what indicated the ambiguity of IR regularization, discussed in detail by Jaranowski and Schäfer (2015) and fixed by Damour et al. (2014). This IR ambiguity can be expressed in terms of only one unknown parameter, because the results of two regularization approaches, albeit different, have exactly the same structure with only different numerical prefactors. This prefactor can be treated as the ambiguity parameter. The full 4PN Hamiltonian was thus computed up to a single ambiguity parameter and it was used to calculate, in a gauge invariant form, the energy of two-body system along circular orbits as a function of frequency. The ambiguity parameter was fixed by comparison of part of this formula [linear in the symmetric mass ratio ν, see Eq. (6.3) below for the definition] with the analogous 4PN-accurate formula for the particle in the Schwarzschild metric which included self-force corrections.
Analogous ambiguity was discovered in 4PN-accurate calculations of two-body equations of motion done by Bernard et al. (2016) in harmonic coordinates, where also analytic regularization of the IR divergences of the instantaneous part of the dynamics was performed. However, the computations made by Bernard et al. (2016) faced also a second ambiguity Bernard et al. 2017b), which must come from their different (harmonic instead of ADMTT) gauge condition and the potentiality of analytic regularization not to preserve gauge (in contrast to dimensional regularization). The first method of analytic regularization applied by Damour et al. (2014) and Jaranowski and Schäfer (2015) is manifest ADMTT gauge preserving. Finally, Marchand et al. (2018) and Bernard et al. (2017a) successfully applied in harmonic-coordinates approach d-dimensional regularization all-over.

Distributional differentiation in d dimensions
One can show that the formula (4.2) for distributional differentiation of homogeneous functions is also valid (without any change) in the d-dimensional case. It leads, e.g., to equality To overcome the necessity of using distributional differentiations it is possible to replace Dirac δ-function by the class of analytic functions introduced in Riesz (1949), resulting in the Dirac δ-function in the limit (4.46) On this class of functions, the inverse Laplacian operates as and instead of (4.44) one gets There is no need to use distributional differentiation here, so no δ-functions are involved.
Though the replacements in the stress-energy tensor density of δ a through δ a (with a = 1, 2) do destroy the divergence freeness of the stress-energy tensor and thus the integrability conditions of the Einstein theory, the relaxed Einstein field equations (the ones which result after imposing coordinate conditions) do not force the stressenergy tensor to be divergence free and can thus be solved without problems. The solutions one gets do not fulfill the complete Einstein field equations but in the final limits a → 0 the general coordinate covariance of the theory is manifestly recovered. This property, however, only holds if these limits are taken before the limit d = 3 is performed (Damour et al. 2008a).

Point-mass representations of spinless black holes
This section is devoted to an insight of how black holes, the most compact objects in GR, can be represented by point masses. On the other side, the developments in the present section show that point masses, interpreted as fictitious point masses (analogously to image charges in the electrostatics), allow to represent black holes. Later on, in the section on approximate Hamiltonians for spinning binaries, neutron stars will also be considered, taking into account their different rotational deformation. Tidal deformation will not be considered in this review; for information about this topic the reader is referred to, e.g., Damour and Nagar (2010) and Steinhoff et al. (2016).
The simplest black hole is a Schwarzschildian one which is isolated and nonrotating. Its metric is a static solution of the vacuum Einstein field equations. In isotropic coordinates, the Schwarzschild metric reads (see, e.g., Misner et al. 1973) where M is the gravitating mass of the black hole and (x 1 , x 2 , x 3 ) are Cartesian coordinates in R 3 with r 2 = (x 1 ) 2 +(x 2 ) 2 +(x 3 ) 2 and dx 2 = (dx 1 ) 2 +(dx 2 ) 2 +(dx 3 ) 2 . The origin of the coordinate system r = 0 is not located where the Schwarzschild singularity R = 0, with R the radial Schwarzschild coordinate, is located, rather it is located on the other side of the Einstein-Rosen bridge, at infinity, where space is flat. The point r = 0 does not belong to the three-dimensional spacelike curved manifold, so we do have an open manifold at r = 0, a so-called "puncture" manifold (see, e.g., Brandt and Brügmann 1997;Cook 2005). However, as we shall see below, the Schwarzschild metric can be contructed with the aid of a Dirac δ function with support at r = 0, located in a conformally related flat space of dimension smaller than three. Distributional sources with support at the Schwarzschild singularity are summarized and treated by Pantoja and Rago (2002) and Heinzle and Steinbauer (2002). A two black hole initial value solution of the vacuum Einstein field equations is the time-symmetric Brill-Lindquist one (Brill and Lindquist 1963;Lindquist 1963), where r a ≡ x − x a and r a ≡ |r a | (a = 1, 2), the coefficients α a and β a can be found in Jaranowski and Schäfer (2002) (notice that h TT i j = 0, π i j = 0, and, initially, ∂ t r a = 0). Its total energy results from the ADM surface integral [this is the reduced ADM Hamiltonian from Eq. (2.20) written for the metric (5.2)] where dS i = n i r 2 dΩ is a two-dimensional surface-area element (with unit radial vector n i ≡ x i /r and solid angle element dΩ) and Introducing the inversion map x → x defined by Brill and Lindquist (1963) r 1 ≡ r 1 α 2 1 G 2 4c 4 r 2 1 ⇒ r 1 = r 1 α 2 1 G 2 4c 4 r 2 1 , (5.5) where r 1 ≡ x − x 1 , r 1 ≡ |x − x 1 |, the three-metric dl 2 = Ψ 4 dx 2 transforms into where r 2 = r 1 α 2 1 G 2 /(4c 4 r 2 1 ) + r 12 with r 12 ≡ x 1 − x 2 . From the new metric function Ψ the proper mass of the throat 1 results in, where i 1 0 denotes the black hole's 1 own spacelike infinity. Hereof the ADM energy comes out in the form, This construction, as performed by Brill and Lindquist (1963), is a purely geometrical (or vacuum) one without touching singularities. Recall that this energy belongs to an initial value solution of the Einstein constraint equations with vanishing of both h TT i j and particle together with field momenta. In this initial conditions spurious gravitational waves are included. In the following we will show how the vacuum Brill-Lindquist solution can be obtained with Dirac δ-function source terms located at r 1 = 0 and r 2 = 0 in a conformally related three-dimensional flat space. To do this we will formulate the problem in d space dimensions and make analytical continuation in d of the results down to d = 3. The insertion of the stress-energy density for point masses into the Hamiltonian constraint equation yields, for p ai = 0, h TT i j = 0, and π i j = 0, where Ψ and φ parametrize the space metric, If the lapse function N is represented by an equation for χ results of the form (using the initial-data conditions p ai = 0, With the aid of the relation it is easy to show that for 1 < d < 2 the equations for Ψ and χ do have well-defined solutions. To obtain these solutions we employ the ansatz where α 1 and α 2 are some constants. After plugging the ansatz (5.15) into Eq. (5.10) we compare the coefficients of the Dirac δ-functions on both sides of the equation.
For point mass 1 we get After taking 1 < d < 2, one can perform the limit r 1 → 0 for the coefficient of δ 1 in the left-hand-side of the above equation, Going over to d = 3 by arguing that the solution is analytic in d results in the relation where b = a and a, b = 1, 2. The ADM energy is again given by, in the limit d = 3, Here we recognize the important aspect that although the metric may describe close binary black holes with strongly deformed apparent horizons, the both black holes can still be generated by point masses in conformally related flat space. This is the justification for our particle model to be taken as model for orbiting black holes. Obviously black holes generated by point masses are orbiting black holes without spin, i.e., Schwarzschild-type black holes. The representation of a Schwarzschild-type black hole in binary-black-hole systems with one Dirac δ-function seems not to be the only possibility. As shown by Jaranowski and Schäfer (2000a), binary-black-hole configurations defined through isometry-conditions at the apparent horizons (Misner 1963) need infinitely many Dirac δ-functions per each one of the black holes. Whether or not those black holes are more physical is not known. It has been found by Jaranowski and Schäfer (1999) that the expressions for ADM energy of the two kinds of binary black holes do agree through 2PN order, and that at the 3PN level the energy of the Brill-Lindquist binary black holes is additively higher by G 4 m 2 1 m 2 2 (m 1 + m 2 )/(8c 6 r 4 12 ), i.e. the Misner configuration seems stronger bound. The same paper has shown that the spatial metrics of both binary-black-hole configurations coincide through 3PN order, and that at least through 5PN order they can be made to coincide by shifts of black-hole position variables.

Post-Newtonian Hamilton dynamics of nonspinning compact binaries
In this section we collect explicit results on Hamilton dynamics of binaries made of compact and nonspinning bodies. Up to the 4PN order the Hamiltonian of binary point-mass systems is explicitly known and it can be written as the sum This Hamiltonian is the PN-expanded reduced ADM Hamiltonian of point-masses plus field system; the nontrivial procedure of reduction is described in Sects. 3.1 and 3.2 of this review. The non-autonomous dissipative Hamiltonians H 2.5PN (x a , p a , t) and H 3 .5PN (x a , p a , t) are written as explicitly depending on time because they depend on the gravitational field variables (see Sect. 6.5 for more details). The dependence of the 4PN Hamiltonian H 4PN on x a and p a is both pointwise and functional (and this is why we have used square brackets for arguments of H 4PN ). We will display here the conservative Hamiltonians H N to H 4PN in the centreof-mass reference frame, relegating their generic, noncentre-of-mass forms, to Appendix C. In the ADM formalism the centre-of-mass reference frame is defined by the simple requirement p 1 + p 2 = 0. (6.2) Here we should point out that at the 3.5PN order for the first time recoil arises, hence the conservation of linear momentum is violated [see, e.g., Fitchett 1983 (derivation based on wave solutions of linearized field equations) and Junker and Schäfer 1992 (derivation based on wave solutions of non-linear field equations)]. This however has no influence on the energy through 6.5PN order, if P ≡ p 1 + p 2 = 0 holds initially, because up to 3PN order the Eq. (3.43) is valid and the change of the Hamiltonian H caused by nonconservation of P equals dH /dt = [(c 2 /H )P] 3PN · (dP/dt) 3.5PN = 0 through 6.5PN order. Let us define where the symmetric mass ratio 0 ≤ ν ≤ 1/4, with ν = 0 being the test-body case and ν = 1/4 for equal-mass binaries. It is convenient to introduce reduced (or rescaled) variables r and p (together with the rescaled time variablet), (6.5)

Conservative Hamiltonians through 4PN order
The conservative reduced 4PN-accurate two-point-mass Hamiltonian in the centre-ofmass frame readŝ The HamiltoniansĤ N throughĤ 3PN are local in time. They explicitly read where Pf T is a Hadamard partie finie with time scale T ≡ 2r 12 /c and where ... I i j denotes a third time derivative of the Newtonian quadrupole moment I i j of the binary system, (6.14) The Hadamard partie finie operation is defined as (Damour et al. 2014) Let us also note that in reduced variables the quadrupole moment I i j and its third time derivative ... I i j read Gr 2 4n i p j − 3(n · p)n i n j , (6.16) where · · · denotes a symmetric tracefree projection and where in ... I i j the time derivativesṙ,r, and ... r were eliminated by means of Newtonian equations of motion. From the reduced conservative Hamiltonians displayed above, where a factor of 1/ν is factorized out [through the definition (6.5) of the reduced Hamiltonian], the standard test-body dynamics is very easily obtained, simply by putting ν = 0. The conservative HamiltoniansĤ N throughĤ 4PN serve as basis of the EOB approach, where with the aid of a canonical transformation the two-body dynamics is put into test-body form of an effective particle moving in deformed Schwarzschild metric, with ν being the deformation parameter Damour 1999, 2000;Damour et al. 2000aDamour et al. , 2015. These Hamiltonians, both directly and through the EOB approach, constitute an important element in the construction of templates needed to detect gravitational waves emitted by coalescing compact binaries. Let us stress again that the complete 4PN Hamiltonian has been obtained only in 2014 (Damour et al. 2014), based on earlier calculations (Blanchet and Damour 1988;Bini and Damour 2013;Jaranowski and Schäfer 2013) and a work published later .

Nonlocal-in-time tail Hamiltonian at 4PN order
The nonlocal-in-time tail Hamiltonian at the 4PN level (derived and applied by Damour et al. 2014Damour et al. , 2015 is the most subtle part of the 4PN Hamiltonian. It certainly deserves some discussion. Let us remark that though the tail Hamiltonian derived in 2016 by Bernard et al. (2016) was identical with the one given in Damour et al. (2014), the derivation there of the equations of motion and the conserved energy was incorrectly done, as detailed by Damour et al. (2016), which was later confirmed by Bernard et al. (2017b).
The 4PN-level tail-related contribution to the action reads the tail Hamiltonian can also be written as (6.20) where Ω is the angular frequency along circular orbit ( p r = 0),

Another writing of the tail Hamiltonian is
Notice that the representation of Ω(t) as function of the still independent (dynamical equationṗ r = −∂ H 0 /∂r has not yet been used) canonical variables p ϕ (t) and r (t) (in Damour et al. 2014Damour et al. , 2016, a more concise representation for circular orbits has been applied, based on the orbital angular momentum as only variable where γ E = 0.577 . . . denotes Euler's constant. This representation has been used by Bernard et al. (2016), see Eq. (5.32) therein, for a straightforward comparison with the tail results presented by Damour et al. (2014).

Dynamical invariants of two-body conservative dynamics
The observables of two-body systems that can be measured from infinity by, say, gravitational-wave observations, are describable in terms of dynamical invariants, i.e., functions which do not depend on the choice of phase-space coordinates. Dynamical invariants are easily obtained within a Hamiltonian framework of integrable systems. We start from the reduced conservative HamiltonianĤ (r, p) in the centre-of-mass frame (we are thus considering here a local-in-time Hamiltonian; for the local reduction of a nonlocal-in-time 4PN-level Hamiltonian see Sect. 6.3.2 below) and we employ reduced variables (r, p). The invariance ofĤ (r, p) under time translations and spatial rotations leads to the conserved quantities where E is the total energy and J is the total orbital angular momentum of the binary system in the centre-of-mass frame. We further restrict considerations to the plane of the relative trajectory endowed with polar coordinates (r , φ) and we use Hamilton-Jacobi approach to obtain the motion. To do this we separate the variablest ≡ t/(G M) and φ in the reduced planar actionŜ ≡ S/(GμM), which takes the form Here j ≡ |j| and the effective radial potential R(r , E, j) is obtained by solving the equation E =Ĥ (r, p) with respect to p r ≡ n · p, after making use of the relation p 2 = (n · p) 2 + (n × p) 2 = p 2 r + j 2 r 2 . (6.28) The Hamilton-Jacobi theory shows that the observables of the two-body dynamics can be deduced from the (reduced) radial action integral where the integration is defined from minimal to maximal radial distance. The dimensionless parameter k ≡ ΔΦ/(2π) (with ΔΦ ≡ Φ − 2π ) measuring the fractional periastron advance per orbit and the periastron-to-periastron period P are obtained by differentiating the radial action integral: It is useful to express the Hamiltonian as a function of the Delaunay (reduced) action variables (see, e.g., Goldstein 1981) defined by The angle variables conjugate to n, j, and m are, respectively: the mean anomaly, the argument of the periastron, and the longitude of the ascending node. In the quantum language, N / is the principal quantum number, J / the total angular-momentum quantum number, and J z / the magnetic quantum number. They are adiabatic invariants of the dynamics and they are, according to the Bohr-Sommerfeld rules of the old quantum theory, (approximately) quantized in integers. Knowing the Delaunay HamiltonianĤ (n, j, m) one computes the angular frequencies of the (generic) rosette motion of the binary system by differentiatingĤ with respect to the action variables. Namely, Here, ω radial is the angular frequency of the radial motion, i.e., the angular frequency of the return to the periastron, while ω periastron is the average angular frequency with which the major axis advances in space.

3PN-accurate results
The dynamical invariants of two-body dynamics were computed by Damour and Schäfer (1988) at the 2PN level and then generalized to the 3PN level of accuracy by Damour et al. (2000b). We are displaying here 3PN-accurate formulae. The periastron advance parameter k reads The 3PN-accurate formula for the orbital period reads These expressions have direct applications to binary pulsars (Damour and Schäfer 1988). Explicit analytic orbit solutions of the conservative dynamics through 3PN order are given by Memmesheimer et al. (2005). The 4PN periastron advance was first derived by Damour et al. (2015Damour et al. ( , 2016, with confirmation provided in a later rederivation (Bernard et al. 2017b; also see Le Tiec and Blanchet 2017). All conservative two-body Hamiltonians respect rotational symmetry, therefore the Delaunay variable m does not enter these Hamiltonians. The 3PN-accurate Delaunay Hamiltonian reads (Damour et al. 2000b) (6.37)  Damour et al. (2015) has shown that modulo some nonlocal-in-time shift of the phasespace coordinates, one can reduce a nonlocal dynamics defined by the Hamiltonian H [r, p; s] ≡Ĥ N (r, p) +Ĥ II 4PN [r, p; s] to an ordinary (i.e., local in time) one. We will sketch here this reduction procedure, which employs the Delaunay form of the Newtonian equations of motion.

Results at 4PN order
It is enough to consider the planar case. In that case the action-angle variables are (L, ; G, g), using the standard notation of Brouwer and Clemence (1961) (with L ≡ n and G ≡ j). The variable L is conjugate to the "mean anomaly" , while G is conjugate to the argument of the periastron g = ω. The variables L and G are related to the usual Keplerian variables a (semimajor axis) and e (eccentricity) via (6.42) We use here rescaled variables: in particular, a denotes the rescaled semimajor axis a ≡ a phys /(G M). We also use the rescaled time variablet ≡ t phys /(G M) appropriate for the rescaled Newtonian Hamiltonian The explicit expressions of the Cartesian coordinates (x, y) of a Newtonian motion in terms of action-angle variables are given by x(L, ; G, g) = cos g x 0 − sin g y 0 , y(L, ; G, g) = sin g x 0 + cos g y 0 , (6.44) For completeness, we also recall the expressions involving the "true anomaly" f (polar angle from the periastron) and the radius vector r : 1 + e cos f , (6.50) (6.51) The above expressions allows one to evaluate the expansions of x, y, and therefrom the components of the quadrupole tensor I i j , as power series in e and Fourier series in . Let us then consider the expression which enters the nonlocal-in-time piece (6.40) of the Hamiltonian. In order to evaluate the order-reduced value of F(t, τ ) one needs to use the equations of motion, both for computing the third time derivatives of I i j , and for expressing the phase-space variables at time t + τ in terms of the phase-space variables at time t. One employs the zeroth-order equations of motion following from the Newtonian Hamiltonian (6.43), where Ω(L) ≡ L −3 is the (t-time) rescaled Newtonian (anomalistic) orbital frequency Ω = G MΩ phys (it satisfies the rescaled third Kepler's law: Ω = a −3/2 ). The fact that g, L, and G are constant and that varies linearly with time, makes it easy to compute ... I i j (t + τ ) in terms of the values of ( , g, L, G) at time t. It suffices to use (denoting by a prime the values at time t ≡ t + τ ) C ± n 1 n 2 n 3 e n 1 cos(n 2 ± n 3 Ωτ ), (6.57) where n 1 , n 2 , n 3 are natural integers. (Because of rotational invariance, and of the result g = g, there is no dependence of F on g.) All the terms in the expansion (6.57) containing a nonzero value of n 2 will, after integrating overτ with the measure dτ /|τ | as indicated in Eq. (6.40), generate a corresponding contribution toĤ II 4PN which varies with proportionally to cos(n 2 ). One employs now the standard Delaunay technique: any term of the type A(L) cos(n ) in a first-order perturbation ε H 1 (L, ) ≡Ĥ II 4PN (L, ) of the leading-order Hamiltonian H 0 (L) ≡ H N (L) can be eliminated by a canonical transformation with generating function of the type εg(L, ) ≡ ε B(L) sin(n ). Indeed, whereF denotes the -average of F( ,τ ) [which is simply obtained by dropping all the terms with n 2 = 0 in the expansion (6.57)]. This procedure yields an averaged HamiltonianĤ II 4PN which depends only on L, G (and s) and which is given as an expansion in powers of e (because of the averaging this expansion contains only even powers of e). Damour et al. (2015)  (6.61) Equation (6.60) is the basic expression for the transition of the tail-related part of the 4PN dynamics to the EOB approach (Damour et al. 2015).
For another approach to the occurrence and treatment of the ( , )-structure in nonlocal-in-time Hamiltonians the reader is referred to Damour et al. (2016) (therein, is called λ).

The innermost stable circular orbit
The innermost stable circular orbit (ISCO) of a test-body orbiting in the Schwarzschild metric is located at R = 6MG/c 2 , in Schwarzschild coordinates. Within a Hamiltonian formalism the calculation of the ISCO for systems made of bodies of comparable masses is rather straightforward. It is relevant to start from discussion of dynamics of a two-body system along circular orbits. The centre-of-mass conservative HamiltonianĤ (r, p) can be reduced to circular orbits by setting p r = n · p = 0 and p 2 = j 2 /r 2 , thenĤ =Ĥ (r , j). Moreover, ∂Ĥ (r , j)/∂r = 0 along circular orbits, what gives the link between r and j, r = r ( j). Finally the energyÊ circ along circular orbits can be expressed as a function of j only, E circ ( j) ≡Ĥ (r ( j), j). The link between the (reduced) centre-of-mass energyÊ circ and the (reduced) angular momentum j is explicitly known up to the 4PN order. It reads (Bini and Damour 2013;Damour et al. 2014 In the test-mass limit ν → 0 (describing motion of a test particle on a circular orbit in the Schwarzschild spacetime) the linkÊ circ (x; ν) is exactly known, The location x ISCO = 1/6 of the ISCO in the test-mass limit corresponds to the minimum of the functionÊ circ (x; 0), i.e.
Therefore the most straightforward way of locating the ISCO for ν > 0 relies on looking for the minimum of the functionÊ circ (x; ν), i.e., for a given value of ν, the location of the ISCO is obtained by (usually numerically) solving the equation dÊ circ (x; ν)/(dx) = 0 (Blanchet 2002). Equivalently the location of the ISCO can be defined as a solution of the set of simultaneous equations ∂Ĥ (r , j)/∂r = 0 and ∂ 2Ĥ (r , j)/∂r 2 = 0. Both approaches are equivalent only for the exact Hamiltonian  H (r , j), see however Sect. IV A 2 in Buonanno et al. ( , 2006 for subtleties related to equivalence of both approaches when using post-Newtonian-accurate Hamiltonians. With the aid of the latter method, Schäfer and Wex (1993a)   To overcome the problem of the slow convergence of PN expansions several new methods of determination of the ISCO for comparable-mass binaries were devised by Damour et al. (2000a). They use different "resummation" techniques and are based on the consideration of gauge-invariant functions. One of the methods, called the " j-method" by Damour et al. (2000a), employs the invariant function linking the angular momentum and the angular frequency along circular orbits and uses Padé approximants. The ISCO is defined in this method as the minimum, for the fixed value of ν, of the function j 2 (x; ν), where j is the reduced angular momentum [introduced in Eq. (6.26)]. The function j 2 (x; ν) is known in the test-mass limit, , (6.69) and its minimum coincides with the exact "location" x ISCO = 1/6 of the test-mass ISCO. The form of this function suggests to use Padé approximants instead of direct Taylor expansions. It also suggests to require that all used approximants have a pole for some x pole , which is related with the test-mass "light-ring" orbit occurring for x lr = 1/3 in the sense that x pole (ν) → 1/3 when ν → 0. The 4PN-accurate function j 2 (x; ν) has the symbolic structure (1/x)(1 + x + · · · + x 4 + x 4 ln x). In the j-method the Taylor expansion at the 1PN level with symbolic form 1 + x is replaced by Padé approximant of type (0,1), at the 2PN level 1+x +x 2 is replaced by (1,1) approximant, at the 3PN level 1 + x + x 2 + x 3 is replaced by (2,1) approximant, and finally at the 4PN level 1 + x + x 2 + x 3 + x 4 is replaced by (3,1) Padé approximant [the explicit form of the (0,1), (1,1), and (2,1) approximants can be found in Eqs. (4.16) of Damour et al. 2000a]. At all PN levels the test-mass result is recovered exactly and Jaranowski and Schäfer (2013)

Dissipative Hamiltonians
To discuss dissipative Hamiltonians it is convenient to use the toy model from Sect. 3.2 with the Routhian R(q, p; ξ,ξ ) and its corresponding Hamiltonian H (q, p; ξ, π) = πξ − R. The Hamilton equations of motion for the (q, p) variables reaḋ These autonomous equations in the (q, p) variables contain the full conservative and dissipative content of the (q, p) dynamics. The time-symmetric part of R yields the conservative equations of motion and the time-antisymmetric part of the dissipative ones. The conservative equations of motion agree with the Fokker-type ones showing the same boundary conditions for the (ξ,ξ) variables. When going from the (ξ,ξ) variables to the field variables h TT andḣ TT , those time-symmetric boundary conditions mean as much incoming as outgoing radiation.
To describe astrophysical systems one should use the physical boundary conditions of no incoming radiation and past stationarity. Clearly, radiative dissipation happens now and the time-symmetric part of the whole dynamics makes the conservative part. In linear theories the conservative part just results from the symmetric Green function G s , whereas the dissipative one from the antisymmetric Green function G a , which is a homogeneous solution of the wave equation. The both together combine to the retarded Green function G ret = G s + G a , with G s = (1/2)(G ret + G adv ) and G a = (1/2)(G ret − G adv ), where G adv denotes the advanced Green function. In non-linear theories time-symmetric effects can also result from homogeneous solutions, e.g., the tail contributions. For a binary system, the leading-order direct and tail radiation reaction enter the Routhian in the form Gm 1 m 2 r 12 n i 12 n j 12 , (6.73) where h TT rr i j (t) decomposes into a direct radiation-reaction term and a tail one , The last term on the right side results in a Routhian, which reproduces the corresponding tail effects in Blanchet (1993) and Galley et al. (2016). (6.75) and the dissipative (time-antisymmetric) one equals where use has been made of the relations The leading-order 2.5PN dissipative binary orbital dynamics is described by the non-autonomous Hamiltonian (Schäfer 1995), Gm 1 m 2 r 12 n i 12 n j 12 , (6.79) where I i j is the Newtonian mass-quadrupole tensor, Only after the Hamilton equations of motion have been obtained the primed position and momentum variables coming from ... I i j are allowed to be identified with the unprimed position and momentum variables, also see Galley (2013). Generally, the treatment of dissipation with Hamiltonians or Lagrangians necessarily needs doubling of variables (Bateman 1931). In quantum mechanics, that treatment was introduced by Schwinger (1961) and Keldysh (1965). In the EFT approach as well a doubling of variables is needed if one wants to treat dissipative systems in a full-fledged manner on the action level (see, e.g., Galley and Leibovich 2012;Galley et al. 2016). However, one should keep in mind that in quantum mechanics damping can also be treated without doubling of variables by making use of the fact that the Feynman Green function G F , the analogue of the retarded Green function of classical physics, decomposes into real and imaginary parts, G F = G s + (i/2)G (1) , where both G s from above and G (1) , Hadamard's elementary function, are symmetric Green functions, G (1) solving homogeneous wave equation as G a does. The imaginary part in e.g. the Eq. (8.7.57) in the book by Brown (1992) yields nothing but the dipole radiation loss formula and this without any doubling of variables (also see Sect. 9-4 in Feynman and Hibbs 1965).
Applications of the 2.5PN Hamiltonian can be found in, e.g., Kokkotas and Schäfer (1995), Ruffert et al. (1996), Buonanno and Damour (1999), and Gopakumar and , where in Gopakumar and Schäfer (2008) a transformation to the Burke-Thorne gauge (coordinate conditions) is performed. More information on the 2.5PN dissipation can be found in Damour (1987a). The 3.5PN Hamiltonian for many point-mass systems is known too, it is displayed in Appendix E (Jaranowski and Schäfer 1997;Königsdörffer et al. 2003). Regarding gravitational spin interaction, see the next section, also for this case radiation reaction Hamiltonians have been derived through leading order spin-orbit and spin-spin couplings (Steinhoff and Wang 2010;Wang et al. 2011). Recent related developments within the EFT formalism are found in Maia et al. (2017a, b).
Let us mention that the already cited article Galley et al. (2016) contains two interesting results improving upon and correcting an earlier article by Foffa and Sturani (2013b): on the one hand it confirms the conservative part of the tail action, particularly the additional rational constant 41/30 which corresponds to the famous 5/6 in the Lamb shift (see, e.g., Brown 2000), and on the other side it correctly delivers the dissipative part of the tail interaction. It is worth noting that in the both articles the involved calculations were performed in harmonic coordinates.

Generalized ADM formalism for spinning objects
In this section we review the recent generalization of ADM formalism describing dynamics of systems made of spinning point masses or, more precisely, pole-dipole particles. We start from reviewing the generalization which is of fully reduced form (i.e., without unresolved constraints, spin supplementary and coordinate conditions) and which is valid to linear order in spin variables [our presentation of linear-in-spins dynamics closely follows that of Steinhoff and Schäfer (2009a)].

Dynamics linear in spins
In this section Latin indices from the middle of the alphabet i, j, k, . . . are running through {1, 2, 3}. We utilize three different reference frames here, denoted by different indices. Greek indices refer to the coordinate frame (x μ ) and have the values μ = 0, 1, 2, 3. Lower case Latin indices from the beginning of the alphabet refer to the local Lorentz frame with its associated tetrad fields e . The basics of the tetrad formalism in GR can be found in, e.g., Sect. 12.5 of Weinberg (1972).
In GR, the coupling of a spinning object to a gravitational field, in terms of a Lagrangian density, reads The linear momentum variable is p μ and the spin tensor is denoted by S ab . The object's affine time variable is τ and δ (4) (x ν − z ν (τ )) is the 4-dimensional Dirac delta function (from now on we will abbreviate it to δ (4) ). The angle variables are represented by some Lorentz matrix satisfying Λ Aa Λ Bb η AB = η ab or Λ Aa Λ Bb η ab = η AB , where η AB = diag(−1, 1, 1, 1) = η ab , which must be respected upon infinitesimal Lorentz transformations (see Hanson and Regge 1974), so δθ ab ≡ Λ a C dΛ Cb = −δθ ba . The Ricci rotation coefficients ω ab μ are given by ω μαβ = e aα e bβ ω ab μ = −Γ βαμ + e c α,μ e cβ , with Γ (4) βαμ = 1 2 (g βα,μ +g βμ,α −g αμ,β ) as the 4-dimensional Christoffel symbols of the first kind with g μν = e aμ e bν η ab the 4-dimensional metric. As in Hanson and Regge (1974), the matrix Λ Ca can be subjected to right (or left) Lorentz transformations, which correspond to transformations of the local Lorentz reference frame (or the body-fixed frame, respectively). In the action (7.1) only a minimal coupling between spin variables and gravitational field is employed; for more general (than minimal) couplings, the reader is referred to Bailey and Israel (1975).
The matter constraints are given by, also in terms of a Lagrangian density, where m is the constant mass of the object, p 2 ≡ p μ p μ , and λ a 1 , λ 2[i] , λ 3 are the Lagrange multipliers. The constraint is called the spin supplementary condition (SSC), it states that in the rest frame the spin tensor contains the 3-dimensional spin S (i)( j) only (i.e., the mass-dipole part S (0)(i) vanishes). 7 The conjugate constraint Λ [i]a p a = 0 ensures that Λ Ca is a pure 3-dimensional rotation matrix in the rest frame (no Lorentz boosts), see Hanson and Regge (1974). Finally, the gravitational part is given by the usual Einstein-Hilbert Lagrangian density where g is the determinant of the 4-dimensional metric and R (4) is the 4-dimensional Ricci scalar. Using a second-order form of the gravitational action, i.e., not varying the connection independently, ensures that the torsion tensor vanishes, see, e.g., Nelson and Teitelboim (1978). The complete Lagrangian density is the sum We assume space-asymptotic flatness as a boundary condition of the spacetime. The total action is given in a second-order form, where the Ricci rotation coefficients are not independent field degrees of freedom and where no torsion of spacetime shows up. It reads and must be varied with respect to the tetrad field e aμ , the Lagrange multipliers λ a 1 , λ 2[i] , λ 3 , position z μ and linear momentum p μ of the object, as well as with respect to angle-type variables Λ Ca and spin tensor S ab associated with the object.
Variation of the action δW = 0 leads to the equations of motion for the matter variables (here d and D denote ordinary and covariant total derivatives, respectively) (7.8) as well as to the usual Einstein equations with the stress-energy tensor (cf. Tulczyjew 1957 andSect. 12.5 in Weinberg 1972 8 ) where R μρab is the 4-dimensional Riemann tensor in mixed indices, ||α denotes the 4-dimensional covariant derivative. Here it was already used that preservation of the constraints in time requires λ a 1 to be proportional to p a and λ 2[i] to be zero, so that λ a 1 and λ 2[i] drop out of the matter equations of motion and the stress-energy tensor. The Lagrange multiplier λ 3 = λ 3 (τ ) represents the reparametrization invariance of the action (notice λ 3 = √ −u 2 /m). Further, an antisymmetric part of the stress-energy tensor vanishes, (7.10) and T μν ||ν = 0 holds by virtue of the matter equations of motion. Obviously, the spin length s as defined by 2s 2 ≡ S ab S ab is conserved. A fully reduced action is obtained by the elimination of all constraints and gauge degrees of freedom. However, after that the action has still to be transformed into canonical form by certain variable transformations. To perform this reduction we employ 3+1 splitting of spacetime by spacelike hypersurfaces t = const. The timelike unit covector orthogonal to these hypersurfaces reads n μ = (−N , 0, 0, 0) or n μ = (1, −N i )/N . The three matter constraints can then be solved in terms of p i , S i j , and (7.13) We take L C = 0 from now on. A split of the Ricci rotation coefficients results in ω ki j = −Γ jik + e a i,k e aj , (7.14) where ;i denotes the 3-dimensional covariant derivative, Γ jik the 3-dimensional Christoffel symbols, and the extrinsic curvature K i j is given by It is convenient to employ here the time gauge (see Schwinger 1963a andalso Dirac 1962;Kibble 1963;Nelson and Teitelboim 1978), e μ (0) = n μ . (7.18) Then lapse and shift turn into Lagrange multipliers in the matter action, like in the ADM formalism for nonspinning matter points. The condition (7.18) leads to the following relations: j , (7.20) which effectively reduce the tetrad e aμ to a triad e (i) j . The matter part of the Lagrangian density, after making use of the covariant SSC (7.3), turns into where (td) denotes an irrelevant total divergence. After fixing the yet arbitrary parameter τ by choosing τ = z 0 = ct, where t is the time coordinate, the terms attributed to the kinetic matter part are given by where δ ≡ δ(x i − z i (t)) and A i j is defined by The matter parts of the gravitational constraints result from where the densities H matter and H matter i are computed from Eqs. (2.11)-(2.12) and (7.9). After employing the covariant SSC one gets (Steinhoff et al. 2008c) Further, some terms attributed to the kinetic part of the gravitational field appear as , which turn the kinetic matter part L M K into canonical form. The variable transformations read , (7.31) whereÂ i j is given by The NW variables have the important propertiesŜ (i)( j)Ŝ(i)( j) = 2s 2 = const and Λ (i) [k]Λ [k]( j) = δ i j , which implies that δθ (i)( j) ≡Λ (i) [k] dΛ [k]( j) is antisymmetric. The redefinitions of position, spin tensor, and angle-type variables are actually quite natural generalizations of their Minkowski space versions to curved spacetime, cf. Hanson and Regge (1974) and Fleming (1965). However, there is no difference between linear momentum p i and canonical momentum P i in the Minkowski case. In these NW variables, one has j,0 δ. The final step goes with the ADM action functional of the gravitational field (Arnowitt et al. 1962;De Witt 1967;Regge and Teitelboim 1974), but in tetrad form as derived by Deser and Isham (1976). The canonical momentum conjugate to e (k) j is given byπ where the momentum π i j is given by Legendre transformation leads tô In asymptotically flat spacetimes the quantity E i is given by [cf. Eq. (2.6)] The total energy then reads The constraint part of the gravitational Lagrangian density takes the form where R is the 3-dimensional Ricci scalar. Due to the symmetry of π i j , not all components ofπ (k) j are independent variables (i.e., the Legendre map is not invertible), leading to the additional constraint ([...] denotes anti-symmetrization) This constraint will be eliminated by going to the spatial symmetric gauge (for the frame e (i) j ) e (i) j = e i j = e ji , e (i) j = e i j = e ji . (7.46) Then the triad is fixed as the matrix square-root of the 3-dimensional metric, e i j e jk = γ ik , or, in matrix notation, This expression may be evaluated perturbatively, cf. Steinhoff et al. (2008c). One also has B kl i j δ kl = 0. Furthermore, with the new canonical field momentum The gravitational constraints arising from the variations δ N and δ N i read, They are eliminated by imposing the gauge conditions 3γ i j, j − γ j j,i = 0, π ii can = 0, (7.54) which allow for the decompositions where h TT i j and π i jTT can are transverse and traceless quantities, and longitudinal partπ i j can is related to a vector potential V i can bỹ Let us note that in the construction of V i can the operator Δ −1 is employed [see the text below Eq. (2.15)].
The gravitational constraints can now be solved for Ψ andπ i j can , leaving h TT i j and π i jTT can as the final degrees of freedom of the gravitational field. Notice that our gauge condition π ii can = 0 deviates from the original ADM one π ii = 0 by spin corrections (which enter at 5PN order). The final fully reduced action reads, The dynamics is completely described by the ADM energy E, which is the total Hamiltonian (E = H ) once it is expressed in terms of the canonical variables. This Hamiltonian can be written as the volume integral (7.58) The equal-time Poisson bracket relations take the standard form, TTi j mn is the TT-projection operator, see, e.g., Steinhoff et al. (2008c). Though the commutation relations (7.59) and (7.60) are sufficient for the variables on which the Hamiltonian (7.58) depends on, for completeness we add the non-trivial ones needed when a Hamiltonian, besidesŜ (i)( j) , also depends on the 3-dimensional rotation matrix Λ [i]( j) ("angle" variables). They read (7.61) The angular velocity tensorΩ (i)( j) , the Legendre dual toŜ (i)( j) , i.e.Ω (i)( [k]( j) , and the time derivative of the spin tensor thus readṡ The Hamiltonian H of Eq. (7.58) generates the time evolution in the reduced mat-ter+field phase space. Generalization and application to many-body systems is quite straightforward, see Steinhoff et al. (2008c). The total linear (P tot i ) and angular (J tot i j ) momenta take the forms (particle labels are denoted by a), (7.64) and are obtained from the reduced action in the standard Noether manner.

Spin-squared dynamics
For the construction of the spin-squared terms we resort to the well-known stressenergy tensor for pole-dipole particles but augmented for quadrupolar terms. The stress-energy tensor density then reads (Steinhoff et al. 2008b) √ −g T μν = dτ t μν δ (4) + (t μνα δ (4) ) ||α + (t μναβ δ (4) ) ||αβ . (7.65) The quantities t μν... = t νμ... only depend on the four-velocity u μ ≡ dz μ /dτ , where z μ (τ ) is the parametrization of the worldline in terms of its proper time τ , and on the spin and quadrupole tensors. Notice that, in general, the quadrupole expressions include not only the mass-quadrupole moment, but also the current-quadrupole moment and the stress-quadrupole moment (see, e.g., Steinhoff and Puetzfeld 2010). For the pole-dipole particle t μναβ is zero. In contrast to the stress-energy tensor of pole-dipole particles, the Riemann tensor shows up at the quadrupolar level. However, the source terms of the constraints, at the approximation considered here, do not include the Riemann tensor.
Regarding rotating black holes, the mass-quadrupole tensor Q i j 1 of object 1 is given by Steinhoff et al. (2008b) (also see, e.g., Thorne 1980;Damour 2001) (7.67) where S 1 = (S 1(i) ) is the three-dimensional Euclidean spin vector related to a spin tensorŜ 1i j with the help of a dreibein e i( j) byŜ 1i j = e i(k) e j(l) klm S 1(m) . The quantity S 2 1 is conserved in time, The source terms of the constraints in the static case (independent from the linear momenta P i of the objects, what means taking P i = 0, but p i = 0) read The c 1 is some constant that must be fixed by additional considerations, like matching to the Kerr metric. The noncovariant terms are due to the transition from threedimensional covariant linear momentum p i to canonical linear momentum P i given by [cf. Eq. (4.24) in Steinhoff et al. 2008c or Eq. (7.32) above] Thus the source terms are indeed covariant when the point-mass and linear-in-spin terms depending on the (noncovariant) canonical linear momentum are added, cf. Eqs. (7.26) and (7.27). The simple structure of the Q i j 1 term in Eq. (7.69) is just the structure of minimal coupling of the Minkowski space mass-quadrupole term to gravity. As shown by Steinhoff et al. (2008b), the most general ansatz for the spin-squared coupling including the three-dimensional Ricci tensor reduces to the shown term. Here we may argue that the correct limit to flat space on the one side and the occurrence of a multiplicative second delta-function through the Ricci tensor from the spinning "point" particle on the other side makes the ansatz unique. A deeper analysis of the structure of nonlinear-in-spin couplings can be found in, e.g., Levi and Steinhoff (2015).

Approximate Hamiltonians for spinning binaries
All the approximate Hamiltonians presented in this subsection have been derived or rederived in recent papers by one of the authors and his collaborators employing canonical formalisms presented in Sects. 7.1 and 7.2 (Damour et al. 2008c;Steinhoff et al. 2008c, b). They are two-point-particle Hamiltonians, which can be used to approximately model binaries made of spinning black holes. For the rest of this section, canonical variables (which are arguments of displayed Hamiltonians) are not hatted any further. We use a, b = 1, 2 as the bodies labels, and for a = b we define r ab n ab ≡ x a − x b with n 2 ab = 1. The Hamiltonian of leading-order (LO) spin-orbit coupling reads (let us note that in the following p a will denote the canonical linear momenta) (7.72) and the one of leading-order spin(1)-spin(2) coupling is given by (7.73) The more complicated Hamiltonian is the one with spin-squared terms because it relates to the rotational deformation of spinning black holes. To leading order, say for spin(1), it reads H LO S 2 1 = Gm 2 2c 2 m 1 r 3 12 [3(S 1 · n 12 )(S 1 · n 12 ) − (S 1 · S 1 )] . (7.74) The LO spin-orbit and spin(a)-spin(b) centre-of-mass vectors take the form The LO spin Hamiltonians have been applied to studies of binary pulsar and solar system dynamics, including satellites on orbits around the Earth (see, e.g., Barker and O'Connell 1979;Schäfer 2004). Another application to the coalescence of spinning binary black holes via the effective-one-body approach is given in Damour (2001). The LO spin dynamics was analysed for black holes and other extended objects in external fields by D'Eath (1975a) and Thorne and Hartle (1985), and for binary black holes in the slow-motion limit by D'Eath (1975b). In Barausse et al. ( , 2012) the spinning test-particle dynamics in the Kerr metric has been explored at LO within Hamiltonian formalism based on Dirac brackets. In the article Kidder (1995) the LO spin-orbit and spin1-spin2 dynamics for compact binaries is treated in full detail, even including their influence on the gravitational waves and the related gravitational damping, particularly the quasi-circular inspiraling and the recoil of the linear momentum from the LO spin coupling was obtained.

12
[(S 1 · S 2 ) − 2(S 1 · n 12 )(S 2 · n 12 )]. (7.77) The calculation of the LO and NLO S 2 1 -Hamiltonians needs employing the source terms (7.69)-(7.70). In the case of polar-dipolar-quadrupolar particles which are to model spinning black holes, Q i j 1 is the quadrupole tensor of the black hole 1 resulting from its rotational deformation and the value of the constant c 1 is fixed by matching to the test-body Hamiltonian in a Kerr background: c 1 = −1/2. Additionally one has to use the Poincaré algebra for unique fixation of all coefficients in momentumdependent part of the Hamiltonian. The NLO S 2 1 -Hamiltonian was presented for the first time by Steinhoff et al. (2008b). 9 It reads The spin precession equations corresponding to the Hamiltonians H NLO S 1 S 2 and H NLO S 2 1 have been calculated also by Porto and Rothstein (2008a, b), 10 respectively, where the first paper [Porto and Rothstein 2008b has benefited from Steinhoff et al. (2008a)] when forgotten terms from spin-induced velocity corrections in the LO spin-orbit coupling could be identified (so-called subleading corrections), see Eq. (57) in Porto and Rothstein (2008b). The NLO spin-orbit and spin(a)-spin(b) centre-of-mass vectors take the form Slightly earlier a fully dynamical calculation of that dynamics was made by Porto and Rothstein (2008a). This result turned out to be incomplete due to incorrect treatement of a specific Feynman diagram. 10 The final spin precession equation of the paper [Porto and Rothstein 2008a deviates from the corresponding one in Steinhoff et al. (2008c)]. A detailed inspection has shown that the last term in Eq. (60) of Porto and Rothstein (2008a) has wrong sign (Steinhoff and Schäfer 2009b). Using the correct sign, after redefinition of the spin variable, agreement with the Hamiltonian of Steinhoff et al. (2008c) is achieved. (7.81) We can sum up all centre-of-mass vectors displayed in this subsection in the following equation: (7.82) where G N up to G 4PN represent the pure orbital contributions, which do not depend on spin variables [the explicit formulae for them one can find in Jaranowski and Schäfer (2015)]. The last term in Eq. (7.82) can be obtained from the second last one by means of the exchange (1 ↔ 2) of the bodies' labels.
The currently known conservative two-point-particle Hamiltonians, modeling binaries made of spinning black holes, can be summarized as follows:  Hartung et al. (2013), their explicit forms can be found in Appendix D. Levi and Steinhoff (2016a)  ; we do not display them explicitly, as their derivation is not yet fully confirmed. All the Hamiltonians with labels containing linear momenta p 1 or p 2 and those quartic in the spins were derived by Hergt and Schäfer (2008a, b) with the aid of approximate ADMTT coordinates of the Kerr metric and application of the Poincaré algebra. 11 Their generalizations to general extended objects were achieved by Levi and Steinhoff (2015), where also for the first time the Hamiltonians H S 4 Hamiltonians cubic and quartic in the spins and displayed in Eq. (7.83) are explicitly given in Appendix D. Notice that not all Hamiltonians from Eq. (7.83) are necessarily given in the ADM gauge, because any use of the equations of motion in their derivation has changed gauge. E.g., for spinless particles the highest conservative Hamiltonian in ADM gauge is H 2PN .
(iv) The conservation law for the specific entropy s along the flow lines, The Newtonian gravitational field equation, where Δ is the Laplacian. The gravitational potential hereof reads Within the Hamilton framework the equations of motion are obtained from the relation ∂ t A(x, t) = {A(x, t), H }, valid for any function A(x, t) living in phase space, i.e. built out of the fundamental variables * , π i , and s, with the Hamiltonian given by H = H [ * , π i , s], where π i is the linear momentum density of the fluid (Holm 1985). The brackets {·, ·} are called Lie-Poisson brackets. They may be defined by where F is a functional of * , s, and π i , L ξ denotes the Lie derivative along the vector field ξ i , and δ F/δ(· · · ) are the Fréchet derivatives of the functional F [see, e.g., Appendix C of Blanchet et al. (1990) and references therein]. Explicitly, the equations in (i), (ii), and (iv) take the following Hamiltonian form [the equations in (iii) and (v) remain unchanged]: (i) The mass conservation equation (ii) The equations of motion (iv) The entropy conservation law The following kinematical Lie-Poisson bracket relations between the fundamental variables are fulfilled: (A.13) and zero otherwise. More explicitly the Hamiltonian of the fluid takes the form, For point masses, the momentum and mass densities are given by where the internal and self-energy terms have been dropped (after performing a proper regularization, see Sect. 4.2 in our review). Let us remark that for fluids a canonical formalism with standard Poisson brackets can be obtained with the transition to Lagrangian coordinates The variables b A and p B are canonically conjugate to each other, i.e.
The mass density in Lagrangian coordinates, say μ(b A , t), is defined by * d 3 x = μ d 3 b and relates to the usual mass density as

B Hamiltonian dynamics of ideal fluids in GR
The general-relativistic equations governing the dynamics of gravitating ideal fluids are as follows (see, e.g., Holm 1985;Blanchet et al. 1990).
(i) The equation for the conservation of mass, where denotes the proper rest-mass density and u μ the four-velocity field of the fluid (g μν u μ u ν = −1), * = √ −gu 0 is the coordinate mass density and v the velocity field of the fluid, v i = cu i /u 0 . (ii) The equations of motion, is the stress-energy tensor of the fluid with pressure p and specific enthalpy h. (v) The Einsteinian field equations for gravitational potential (or metric) functions g μν , The variables of the canonical formalism get chosen to be * = √ −gu 0 , s, They do fulfill the same (universal, free of spacetime metric) kinematical Lie-Poisson bracket relations as in the Newtonian theory (see Holm 1985 or also Blanchet et al. 1990), (B.10) Written as Hamiltonian equations of motion, i.e. ∂ t A(x, t) = {A(x, t), H }, the equations in (i), (ii), and (iv) take the following form [the equations in (iii) and (v) remain unchanged]: (i) The mass conservation equation (ii) The equations of motion (iv) The entropy conservation law where the Hamiltonian functional is given by H = H [ * , π i , s], see Holm (1985).
Point-mass systems fulfill h = p = s = 0, (B.14) (just as for dust) and the momentum and mass densities read Remarkably, the difference to the Newtonian theory solely results from the Hamiltonian, so the difference between GR and the Newtonian theory is essentially a dynamical and not a kinematical one. This statement refers to the matter only and not to the gravitational field. The latter is much more complicated in GR, dynamically and kinematically as well.
(E.20) adding a total time derivative, compatible with the 4PN-accurate Hamiltonian which can be obtained from the exact 1PM Hamiltonian (F.6). Let us also note that Damour (2016) has shown that, after a suitable canonical transformation, the rather complicated Hamiltonian (F.6) is equivalent (modulo the EOB energy map) to the much simpler Hamiltonian of a test particle moving in a (linearized) Schwarzschild metric. The binary centre-of-mass 2PM Hamiltonian has been derived most recently by Damour (2018) in an EOB-type form and also the gravitational spin-orbit coupling in binary systems has been achieved at 2PM order by Bini and Damour (2018) (for other 2PM results see, e.g., Bel et al. 1981;Westpfahl 1985).

G Skeleton Hamiltonian for binary black holes
The skeleton approach to GR developed by Faye et al. (2004), is a truncation of GR such that an analytic PN expansion exists to arbitrary orders which, at the same time, is explicitly calculable. The approach imposes the conformal flat condition for the spatial three-metric for all times (not only initially as for the Brill-Lindquist solution), together with a specific truncation of the field-momentum energy density. It exactly recovers the general relativity dynamical equations in the limits of test-body and 1PN dynamics. The usefulness of the skeleton approach in the construction of initial data needed for numerical solving binary black hole dynamics was studied by Bode et al. The solution of these equations is constructed under the condition that π j i is purely longitudinal, i.e., This condition is part of the definition of the skeleton model. In the Isenberg-Wilson-Mathews approach to general relativity only the conformal flat condition is employed. Through 2PN order, the Isenberg-Wilson-Mathews energy of a binary is given bŷ The difference betweenĤ IWM andĤ sk shows the effect of truncation in the fieldmomentum part ofĤ sk through 2PN order and the difference betweenĤ IWM and H ≤3PN reveals the effect of conformal flat truncation. In the test-body limit, ν = 0, the three Hamiltonians coincide.