Newtonian-like Equations of Motion

7 Newtonian-like Equations of Motion

7.1 The 3PN acceleration and energy for particles

We present the acceleration of one of the particles, say the particle 1, at the 3PN order, as well as the 3PN energy of the binary, which is conserved in the absence of radiation reaction. To get this result we used essentially a “direct” post-Newtonian method (issued from Ref. [76 *]), which consists of reducing the 3PN metric of an extended regular source, worked out in Eqs. (144), to the case where the matter tensor is made of delta functions, and then curing the self-field divergences by means of the Hadamard regularization technique. The equations of motion are simply the 3PN geodesic equations explicitly provided in Eqs. (150 *) – (152); the metric therein is the regularized metric generated by the system of particles itself. Hadamard’s regularization permits to compute all the terms but one, and the Hadamard ambiguity parameter $λ$ is obtained from dimensional regularization; see Section 6.3. We also add the 3.5PN terms in harmonic coordinates which are known from Refs. [258 *, 259 *, 260 *, 336 *, 278 *, 322 *, 254 *]. These correspond to radiation reaction effects at relative 1PN order (see Section 5.4 for discussion on radiation reaction up to 1.5PN order).

Though the successive post-Newtonian approximations are really a consequence of general relativity, the final equations of motion must be interpreted in a Newtonian-like fashion. That is, once a convenient general-relativistic (Cartesian) coordinate system is chosen, we should express the results in terms of the coordinate positions, velocities, and accelerations of the bodies, and view the trajectories of the particles as taking place in the absolute Euclidean space of Newton. But because the equations of motion are actually relativistic, they must:

Stay manifestly invariant – at least in harmonic coordinates – when we perform a global post-Newtonian-expanded Lorentz transformation;
Possess the correct “perturbative” limit, given by the geodesics of the (post-Newtonian-expanded) Schwarzschild metric, when one of the masses tends to zero;
Be conservative, i.e., to admit a Lagrangian or Hamiltonian formulation, when the gravitational radiation reaction is turned off.

We denote by $r12 = |y1(t) − y2(t)|$ the harmonic-coordinate distance between the two particles, with $y = (yi) 1 1$ and $y = (yi) 2 2$ , by $n = (y − y )∕r 12 1 2 12$ the corresponding unit direction, and by $v1 = dy1 ∕dt$ and $a1 = dv1∕dt$ the coordinate velocity and acceleration of the particle 1 (and idem for 2). Sometimes we pose $v12 = v1 − v2$ for the relative velocity. The usual Euclidean scalar product of vectors is denoted with parentheses, e.g., $(n12v1) = n12 ⋅ v1$ and $(v1v2) = v1 ⋅ v2$ . The equations of the body 2 are obtained by exchanging all the particle labels $1 ↔ 2$ (remembering that $n12$ and $v12$ change sign in this operation):

The 2.5PN and 3.5PN terms are associated with gravitational radiation reaction.⁴⁹ The 3PN harmonic-coordinates equations of motion depend on two arbitrary length scales $r′ 1$ and $r′ 2$ associated with the logarithms present at the 3PN order. It has been proved in Ref. [71 *] that $′ r1$ and $′ r2$ are merely linked with the choice of coordinates – we can refer to $r′1$ and $r′2$ as “gauge constants”. In our approach [69 *, 71 ], the harmonic coordinate system is not uniquely fixed by the coordinate condition $∂μh αμ = 0$ . In fact there are infinitely many “locally-defined” harmonic coordinate systems. For general smooth matter sources, as in the general formalism of Part A, we expect the existence and uniqueness of a global harmonic coordinate system. But here we have some point-particles, with delta-function singularities, and in this case we do not have the notion of a global coordinate system. We can always change the harmonic coordinates by means of the gauge vector $ηα = δxα$ , satisfying $Δ ηα = 0$ except at the location of the two particles (we assume that the transformation is at the 3PN level, so we can consider simply a flat-space Laplace equation). More precisely, we can show that the logarithms appearing in Eq. (203), together with the constants $r′1$ and $r′2$ therein, can be removed by the coordinate transformation associated with the 3PN gauge vector (with $r1 = |x − y1(t)|$ and $r2 = |x − y2(t)|$ ; and $∂α = ηαμ∂μ$ ):

Therefore, the arbitrariness in the choice of the constants $r′1$ and $r′2$ is innocuous on the physical point of view, because the physical results must be gauge invariant. Indeed we shall verify that $r′ 1$ and $r′ 2$ cancel out in our final results.

When retaining the “even” relativistic corrections at the 1PN, 2PN and 3PN orders, and neglecting the “odd” radiation reaction terms at the 2.5PN and 3.5PN orders, we find that the equations of motion admit a conserved energy (and a Lagrangian, as we shall see); that energy can be straightforwardly obtained by guess-work starting from Eq. (203), with the result

To the terms given above, we must add the same terms but corresponding to the relabelling $1 ↔ 2$ . Actually, this energy is not conserved because of the radiation reaction. Thus its time derivative, as computed by means of the 3PN equations of motion themselves (i.e., by order-reducing all the accelerations), is purely equal to the 2.5PN effect,

The resulting energy balance equation can be better expressed by transfering to the left-hand side certain 2.5PN terms so that we recognize in the right-hand side the familiar form of a total energy flux. Posing

we find agreement with the standard Einstein quadrupole formula (4 *):

where the Newtonian trace-free quadrupole moment reads $Q = m (yiyj− 1δijy2) + 1 ↔ 2 ij 1 1 1 3 1$ . We refer to [258 *, 259 *] for the discussion of the energy balance equation up to the next 3.5PN order. See also Eq. (158) for the energy balance equation at relative 1.5PN order for general fluid systems.

7.2 Lagrangian and Hamiltonian formulations

The conservative part of the equations of motion in harmonic coordinates (203) is derivable from a generalized Lagrangian, depending not only on the positions and velocities of the bodies, but also on their accelerations: $a1 = dv1 ∕dt$ and $a2 = dv2∕dt$ . As shown in Ref. [147 ], the accelerations in the harmonic-coordinates Lagrangian occur already from the 2PN order. This fact is in accordance with a general result [308 ] that N-body equations of motion cannot be derived from an ordinary Lagrangian beyond the 1PN level, provided that the gauge conditions preserve the manifest Lorentz invariance. Note that we can always arrange for the dependence of the Lagrangian upon the accelerations to be linear, at the price of adding some so-called “multi-zero” terms to the Lagrangian, which do not modify the equations of motion (see, e.g., Ref. [169 ]). At the 3PN level, we find that the Lagrangian also depends on accelerations. It is notable that these accelerations are sufficient – there is no need to include derivatives of accelerations. Note also that the Lagrangian is not unique because we can always add to it a total time derivative $dF ∕dt$ , where $F$ is any function depending on the positions and velocities, without changing the dynamics. We find [174 *]

Witness the accelerations occurring at the 2PN and 3PN orders; see also the gauge-dependent logarithms of $′ r12∕r1$ and $′ r12∕r2$ . We refer to [174 *] for the explicit expressions of the ten conserved quantities corresponding to the integrals of energy [also given in Eq. (205)], linear and angular momenta, and center-of-mass position. Notice that while it is strictly forbidden to replace the accelerations by the equations of motion in the Lagrangian, this can and should be done in the final expressions of the conserved integrals derived from that Lagrangian.

Now we want to exhibit a transformation of the particles’ dynamical variables – or contact transformation, as it is called in the jargon – which transforms the 3PN harmonic-coordinates Lagrangian (209) into a new Lagrangian, valid in some ADM or ADM-like coordinate system, and such that the associated Hamiltonian coincides with the 3PN Hamiltonian that has been obtained by Jaranowski & Schäfer [261 *, 262 *]. In ADM coordinates the Lagrangian will be ordinary, depending only on the positions and velocities of the bodies. Let this contact transformation be $Y1 (t) = y1(t) + δy1(t)$ and $1 ↔ 2$ , where $Y1$ and $y1$ denote the trajectories in ADM and harmonic coordinates, respectively. For this transformation to be able to remove all the accelerations in the initial Lagrangian $Lharm$ up to the 3PN order, we determine [174 *] it to be necessarily of the form

where $F$ is a freely adjustable function of the positions and velocities, made of 2PN and 3PN terms, and where $X 1$ represents a special correction term, that is purely of order 3PN. The point is that once the function $F$ is specified there is a unique determination of the correction term $X1$ for the contact transformation to work (see Ref. [174 *] for the details). Thus, the freedom we have is entirely encoded into the function $F$ , and the work then consists in showing that there exists a unique choice of $F$ for which our Lagrangian $Lharm$ is physically equivalent, via the contact transformation (210 *), to the ADM Hamiltonian of Refs. [261 *, 262 *]. An interesting point is that not only the transformation must remove all the accelerations in $harm L$ , but it should also cancel out all the logarithms $ln(r12∕r′1)$ and $ln(r12∕r′2)$ , because there are no logarithms in ADM coordinates. The result we find, which can be checked to be in full agreement with the expression of the gauge vector in Eq. (204 *), is that $F$ involves the logarithmic terms

together with many other non-logarithmic terms (indicated by dots) that are entirely specified by the isometry of the harmonic and ADM descriptions of the motion. For this particular choice of $F$ the ADM Lagrangian reads

Inserting into this equation all our explicit expressions we find

The notation is the same as in Eq. (209), except that we use upper-case letters to denote the ADM-coordinates positions and velocities; thus, for instance $N12 = (Y1 − Y2 )∕R12$ and $(N12V1 ) = N12 ⋅ V1$ . The Hamiltonian is simply deduced from the latter Lagrangian by applying the usual Legendre transformation. Posing $P1 = ∂LADM ∕∂V1$ and $1 ↔ 2$ , we get [261 *, 262 *, 263 , 162 *, 174 *]

Arguably, the results given by the ADM-Hamiltonian formalism (for the problem at hand) look simpler than their harmonic-coordinate counterparts. Indeed, the ADM Lagrangian is ordinary – no accelerations – and there are no logarithms nor associated gauge constants $r′1$ and $′ r2$ .⁵⁰ Of course, one is free to describe the binary motion in whatever coordinates one likes, and the two formalisms, harmonic (209) and ADM (213) – (214), describe rigorously the same physics. On the other hand, the higher complexity of the harmonic-coordinates Lagrangian (209) enables one to perform more tests of the computations, notably by inquiring about the future of the constants $′ r1$ and $′ r2$ , that we know must disappear from physical quantities such as the center-of-mass energy and the total gravitational-wave flux.

7.3 Equations of motion in the center-of-mass frame

In this section we translate the origin of coordinates to the binary’s center-of-mass by imposing the vanishing of the binary’s mass dipole moment: $Ii = 0$ in the notation of Part A. Actually the dipole moment is computed as the center-of-mass conserved integral associated with the boost symmetry of the 3PN equations of motion [174 , 79 *]. This condition results in the 3PN-accurate relationship between the individual positions in the center-of-mass frame $y1$ and $y2$ , and the relative position $x ≡ y1 − y2$ and velocity $v ≡ v1 − v2 = dx ∕dt$ (formerly denoted $y12$ and $v12$ ). We shall also use the orbital separation $r ≡ |x|$ , together with $n = x ∕r$ and $˙r ≡ n ⋅ v$ . Mass parameters are: The total mass $m = m1 + m2$ (to be distinguished from the ADM mass denoted by $M$ in Part A); the relative mass difference $Δ = (m1 − m2 )∕m$ ; the reduced mass $μ = m1m2 ∕m$ ; and the very useful symmetric mass ratio

The usefulness of this ratio lies in its interesting range of variation: $0 < ν ≤ 1∕4$ , with $ν = 1∕4$ in the case of equal masses, and $ν → 0$ in the test-mass limit for one of the bodies. Thus $ν$ is numerically rather small and may be viewed as a small expansion parameter. We also pose $X1 = m1 ∕m$ and $X2 = m2 ∕m$ so that $Δ = X1 − X2$ and $ν = X1X2$ .

For reference we give the 3PN-accurate expressions of the individual positions in the center-of-mass frame in terms of relative variables. They are in the form

where all post-Newtonian corrections, beyond Newtonian order, are proportional to the mass ratio $ν$ and the mass difference $Δ$ . The two dimensionless coefficients $𝒫$ and $𝒬$ read

Up to 2.5PN order there is agreement with the circular-orbit limit of Eqs. (6.4) in Ref. [45 ]. Notice the 2.5PN radiation-reaction term entering the coefficient $𝒬$ ; such 2.5PN term is explicitly displayed for circular orbits in Eqs. (224) below. In Eqs. (217) the logarithms at the 3PN order appear only in the coefficient $𝒫$ . They contain a particular combination $′′ r0$ of the two gauge-constants $′ r1$ and $′ r2$ defined by

and which happens to be different from a similar combination $r′ 0$ we shall find in the equations of relative motion, see Eq. (221 *).

The 3PN and even 3.5PN center-of-mass equations of motion are obtained by replacing in the 3.5PN equations of motion (203) in a general frame, the positions and velocities by their center-of-mass expressions (216) – (217), applying as usual the order-reduction of all accelerations where necessary. We write the relative acceleration in the center-of-mass frame in the form

and find that the coefficients $𝒜$ and $ℬ$ are [79 *]

Up to the 2.5PN order the result agrees with Ref. [302 ]. The 3.5PN term is issued from Refs. [258 *, 259 *, 260 *, 336 *, 278 *, 322 *, 254 *]. At the 3PN order we have some gauge-dependent logarithms containing a constant $r′ 0$ which is the “logarithmic barycenter” of the two constants $r′ 1$ and $r′ 2$ :

The logarithms in Eqs. (220), together with the constant $′ r0$ therein, can be removed by applying the gauge transformation (204 *), while still staying within the class of harmonic coordinates. The resulting modification of the equations of motion will affect only the coefficients of the 3PN order in Eqs. (220); let us denote them by $𝒜3PN$ and $ℬ3PN$ . The new values of these coefficients, obtained after removal of the logarithms by the latter harmonic gauge transformation, will be denoted $MH 𝒜 3PN$ and $MH ℬ3PN$ . Here MH stands for the modified harmonic coordinate system, differing from the SH (standard harmonic) coordinate system containing logarithms at the 3PN order in the coefficients $𝒜3PN$ and $ℬ3PN$ . See Ref. [9 *] for a full description of the coordinate transformation between SH and MH coordinates for various quantities. We have [320 *, 9 *]

Again, the other terms in the equations of motion (219 *) – (220) are unchanged. These gauge-transformed coefficients in MH coordinates are useful because they do not yield the usual complications associated with logarithms. However, they must be handled with care in applications such as in Ref. [320 ], because one must ensure that all other quantities in the problem (energy, angular momentum, gravitational-wave fluxes, etc.) are defined in the same specific MH gauge avoiding logarithms. In the following we shall no longer use the MH coordinate system leading to Eqs. (222), except when constructing the generalized quasi-Keplerian representation of the 3PN motion in Section 10.2. Therefore all expressions we shall derive below, notably all those concerning the radiation field, are valid in the SH coordinate system in which the equations of motion are fully given by Eq. (203) or, in the center-of-mass frame, by Eqs. (219 *) – (220).

For future reference let also give the 3PN center-of-mass Hamiltonian in ADM coordinates derived in Refs. [261 , 262 , 162 ]. In the center-of-mass frame the conjugate variables are the relative separation $X = Y1 − Y2$ and the conjugate momentum (per unit reduced mass) $P$ such that $μP = P1 = − P2$ where $P1$ and $P2$ are defined in Section 7.2). Posing $N ≡ X ∕R$ with $R ≡ |X |$ , together with $P 2 ≡ P 2$ and $P ≡ N ⋅ P R$ , we have

7.4 Equations of motion and energy for quasi-circular orbits

Most inspiralling compact binaries will have been circularized by the time they become visible by the detectors LIGO and VIRGO; see Section 1.2. In the case of orbits that are circular – apart from the gradual radiation-reaction inspiral – the complicated equations of motion simplify drastically, since we have $5 r˙= (nv ) = 𝒪(1∕c )$ . For circular orbits, up to the 2.5PN order, the relation between center-of-mass variables and the relative ones reads

where we recall $X = m ∕m 1 1$ , $X = m ∕m 2 2$ and $Δ = X − X 1 2$ . See Eqs. (216) – (217) for more general formulas. To conveniently display the successive post-Newtonian corrections, we employ the post-Newtonian parameter

Notice that there are no corrections of order 1PN in Eqs. (224) for circular orbits; the dominant term is of order 2PN, i.e., is proportional to $2 4 γ = 𝒪 (1∕c )$ . See Ref. [79 *] for a systematic calculation of Eqs. (224) to higher order.

The relative acceleration $a ≡ a1 − a2$ of two bodies moving on a circular orbit at the 3.5PN order is then given by

where $x ≡ y − y 1 2$ is the relative separation (in harmonic coordinates) and $Ω$ denotes the angular frequency of the circular motion. The second term in Eq. (226 *), opposite to the velocity $v ≡ v1 − v2$ , represents the radiation reaction force up to 3.5PN order, which comes from the reduction of the coefficients of $1∕c5$ and $1∕c7$ in Eqs. (220). The radiation-reaction force is responsible for the secular decrease of the separation $r$ and increase of the orbital frequency $Ω$ :

Concerning conservative effects, the main content of the 3PN equations (226 *) is the relation between the frequency $Ω$ and the orbital separation $r$ , which is given by the following generalized version of Kepler’s third law:

The length scale $r′0$ is given in terms of the two gauge-constants $r′1$ and $r′2$ by Eq. (221 *). As for the energy, it is inferred from the circular-orbit reduction of the general result (205). We have

This expression is that of a physical observable $E$ ; however, it depends on the choice of a coordinate system, as it involves the post-Newtonian parameter $γ$ defined from the harmonic-coordinate separation $r$ . But the numerical value of $E$ should not depend on the choice of a coordinate system, so $E$ must admit a frame-invariant expression, the same in all coordinate systems. To find it we re-express $E$ with the help of the following frequency-related parameter $x$ , instead of the post-Newtonian parameter $γ$ :⁵¹

We readily obtain from Eq. (228) the expression of $γ$ in terms of $x$ at 3PN order,

that we substitute back into Eq. (229), making all appropriate post-Newtonian re-expansions. As a result, we gladly discover that the logarithms together with their associated gauge constant $r′0$ have cancelled out. Therefore, our result is [160 *, 69 ]

For circular orbits one can check that there are no terms of order $x7∕2$ in Eq. (232), so this result is actually valid up to the 3.5PN order. We shall discuss in Section 11 how the effects of the spins of the two black holes affect the latter formula.

The formula (232) has been extended to include the logarithmic terms $∝ lnx$ at the 4PN and 5PN orders [67 *, 289 *], that are due to tail effects occurring in the near zone, see Sections 5.2 and 5.4. Adding also the Schwarzschild test-mass limit⁵² up to 5PN order, we get:

We can write also a similar expression for the angular momentum,

For circular orbits the energy $E$ and angular momentum $J$ are known to be linked together by the so-called “thermodynamic” relation

which is actually just one aspect of the “first law of binary black hole mechanics” that we shall discuss in more details in Section 8.3.

We have introduced in Eqs. (233) – (234) some non-logarithmic 4PN and 5PN coefficients $e4(ν)$ , $j4(ν)$ and $e5(ν)$ , $j5(ν)$ , which can however be proved to be polynomials in the symmetric mass ratio $ν$ .⁵³ Recent works on the 4PN approximation to the equations of motion by means of both EFT methods [204 ] and the traditional ADM-Hamiltonian approach [264 , 265 ], and complemented by an analytic computation of the gravitational self-force in the small mass ratio $ν$ limit [36 *], have yielded the next-order 4PN coefficient as ( $γE$ being Euler’s constant)

The numerical value $e4(0) ≃ 153.88$ was predicted before thanks to a comparison with numerical self-force calculations [289 *, 287 *].

7.5 The 2.5PN metric in the near zone

The near-zone metric is given by Eqs. (144) for general post-Newtonian matter sources. For point-particles binaries all the potentials $V$ , $Vi$ , $⋅⋅⋅$ parametrizing the metric must be computed and iterated for delta-function sources. Up to the 2.5PN order it is sufficient to cure the divergences due to singular sources by means of the Hadamard self-field regularization. Let us point out that the computation is greatly helped – and indeed is made possible at all – by the existence of the following solution $g$ of the elementary Poisson equation

which takes the very nice closed analytic form [202 ]

where $ra = |x − ya|$ and $r12 = |y1 − y2 |$ . Furthermore, to obtain the metric at the 2.5PN order, the solutions of even more difficult elementary Poisson equations are required. Namely we meet

with $a∂i$ denoting the partial derivatives with respect to the source points $i ya$ (and as usual $∂i$ being the partial derivative with respect to the field point $xi$ ). It is quite remarkable that the solutions of the latter equations are known in closed analytic form. By combining several earlier results from Refs. [120 , 324 , 377 ], one can write these solutions into the form [64 *, 76 *]

where $Δa$ are the Laplacians with respect to the two source points.

We report here the complete expression of the 2.5PN metric in harmonic coordinates valid at any field point in the near zone. Posing $g αβ = ηαβ + kαβ$ we have [76 *]

Here we pose $S = r1 + r2 + r12$ and $1 ↔ 2$ refers to the same quantity but with all particle labels exchanged. To higher order one needs the solution of elementary equations still more intricate than (239) and the 3PN metric valid in closed form all over the near zone is not currently known.

Let us also display the latter 2.5PN metric computed at the location of the particle 1 for instance, thanks to the Hadamard self-field regularization, i.e., in the sense of the Hadamard partie finie defined by Eq. (160 *). We get

When regularized at the location of the particles, the metric can be computed to higher order, for instance 3PN. We shall need it when we compute the so-called redshift observable in Sections (8.3) and (8.4); indeed, see Eq. (276 *).

	Abstract
1	Introduction
	1.1	Analytic approximations and wave generation formalism
	1.2	The quadrupole moment formalism
	1.3	Problem posed by compact binary systems
	1.4	Post-Newtonian equations of motion
	1.5	Post-Newtonian gravitational radiation
A	Post-Newtonian Sources
2	Non-linear Iteration of the Vacuum Field Equations
	2.1	Einstein’s field equations
	2.2	Linearized vacuum equations
	2.3	The multipolar post-Minkowskian solution
	2.4	Generality of the MPM solution
	2.5	Near-zone and far-zone structures
3	Asymptotic Gravitational Waveform
	3.1	The radiative multipole moments
	3.2	Gravitational-wave tails and tails-of-tails
	3.3	Radiative versus source moments
4	Matching to a Post-Newtonian Source
	4.1	The matching equation
	4.2	General expression of the multipole expansion
	4.3	Equivalence with the Will–Wiseman formalism
	4.4	The source multipole moments
5	Interior Field of a Post-Newtonian Source
	5.1	Post-Newtonian iteration in the near zone
	5.2	Post-Newtonian metric and radiation reaction effects
	5.3	The 3.5PN metric for general matter systems
	5.4	Radiation reaction potentials to 4PN order
B	Compact Binary Systems
6	Regularization of the Field of Point Particles
	6.1	Hadamard self-field regularization
	6.2	Hadamard regularization ambiguities
	6.3	Dimensional regularization of the equations of motion
	6.4	Dimensional regularization of the radiation field
7	Newtonian-like Equations of Motion
	7.1	The 3PN acceleration and energy for particles
	7.2	Lagrangian and Hamiltonian formulations
	7.3	Equations of motion in the center-of-mass frame
	7.4	Equations of motion and energy for quasi-circular orbits
	7.5	The 2.5PN metric in the near zone
8	Conservative Dynamics of Compact Binaries
	8.1	Concept of innermost circular orbit
	8.2	Dynamical stability of circular orbits
	8.3	The first law of binary point-particle mechanics
	8.4	Post-Newtonian approximation versus gravitational self-force
9	Gravitational Waves from Compact Binaries
	9.1	The binary’s multipole moments
	9.2	Gravitational wave energy flux
	9.3	Orbital phase evolution
	9.4	Polarization waveforms for data analysis
	9.5	Spherical harmonic modes for numerical relativity
10	Eccentric Compact Binaries
	10.1	Doubly periodic structure of the motion of eccentric binaries
	10.2	Quasi-Keplerian representation of the motion
	10.3	Averaged energy and angular momentum fluxes
11	Spinning Compact Binaries
	11.1	Lagrangian formalism for spinning point particles
	11.2	Equations of motion and precession for spin-orbit effects
	11.3	Spin-orbit effects in the gravitational wave flux and orbital phase
	Acknowledgments
	References
	Footnotes
	Updates
	Figures
	Tables