Conservative Dynamics of Compact Binaries

8 Conservative Dynamics of Compact Binaries

8.1 Concept of innermost circular orbit

Having in hand the conserved energy $E(x )$ for circular orbits given by Eq. (232), or even more accurate by (233), we define the innermost circular orbit (ICO) as the minimum, when it exists, of the energy function $E(x)$ – see e.g., Ref. [51 *]. Notice that the ICO is not defined as a point of dynamical general-relativistic unstability. Hence, we prefer to call this point the ICO rather than, strictly speaking, an innermost stable circular orbit or ISCO. A study of the dynamical stability of circular binary orbits in the post-Newtonian approximation is reported in Section 8.2.

The previous definition of the ICO is motivated by the comparison with some results of numerical relativity. Indeed we shall confront the prediction of the standard (Taylor-based) post-Newtonian approximation with numerical computations of the energy of binary black holes under the assumptions of conformal flatness for the spatial metric and of exactly circular orbits [228 *, 232 *, 133 *, 121 *]. The latter restriction is implemented by requiring the existence of an “helical” Killing vector (HKV), which is time-like inside the light cylinder associated with the circular motion, and space-like outside. The HKV will be defined in Eq. (273 *) below. In the numerical approaches of Refs. [228 *, 232 *, 133 *, 121 *] there are no gravitational waves, the field is periodic in time, and the gravitational potentials tend to zero at spatial infinity within a restricted model equivalent to solving five out of the ten Einstein field equations (the so-called Isenberg–Wilson–Mathews approximation; see Ref. [228 *] for a discussion). Considering an evolutionary sequence of equilibrium configurations the circular-orbit energy $E (Ω)$ and the ICO of binary black holes are obtained numerically (see also Refs. [92 , 229 , 301 ] for related calculations of binary neutron stars and strange quark stars).

Since the numerical calculations [232 *, 133 *] have been performed in the case of two corotating black holes, which are spinning essentially with the orbital angular velocity, we must for the comparison include within our post-Newtonian formalism the effects of spins appropriate to two Kerr black holes rotating at the orbital rate. The total relativistic masses of the two Kerr black holes (with $a = 1,2$ labelling the black holes) are given by⁵⁴

We assume the validity of the Christodoulou mass formula for Kerr black holes [127 , 129 ]; i.e., we neglect the influence of the companion. Here $Sa$ is the spin, related to the usual Kerr parameter by $Sa = maaa$ , and $μa ≡ mirar$ is the irreducible mass, not to be confused with the reduced mass of the binary system, and given by $√ --- 4π μa = Aa$ ( $Aa$ is the hole’s surface area). The angular velocity of the black hole, defined by the angular velocity of the outgoing photons that remain for ever at the location of the light-like horizon, is

We shall give in Eq. (284) below a more general formulation of the “internal structure” of the black holes. Combining Eqs. (243 *) – (244 *) we obtain $ma$ and $Sa$ as functions of $μa$ and $ωa$ ,

In the limit of slow rotation we get

where $3 Ia = 4μa$ is the moment of inertia of the black hole. We see that the total mass-energy $ma$ involves the irreducible mass augmented by the usual kinetic energy of the spin.

We now need the relation between the rotation frequency $ωa$ of each of the corotating black holes and the orbital frequency $Ω$ of the binary system. Indeed $Ω$ is the basic variable describing each equilibrium configuration calculated numerically in Refs. [232 *, 133 *], with the irreducible masses held constant along the numerical evolutionary sequences. Here we report the result of an investigation of the condition for corotation based on the first law of mechanics for spinning black holes [55 *], which concluded that the corotation condition at 2PN order reads

where $x$ denotes the post-Newtonian parameter (230 *) and $ν$ the symmetric mass ratio (215 *). The condition (247 *) is issued from the general relation which will be given in Eq. (285 *). Interestingly, notice that $ω1 = ω2$ up to the rather high 2PN order. In the Newtonian limit $x → 0$ or the test-particle limit $ν → 0$ we simply have $ωa = Ω$ , in agreement with physical intuition.

To take into account the spin effects our first task is to replace all the masses entering the energy function (232) by their equivalent expressions in terms of $ωa$ and the irreducible masses $μa$ , and then to replace $ωa$ in terms of $Ω$ according to the corotation prescription (247 *).⁵⁵ It is clear that the leading contribution is that of the spin kinetic energy given in Eq. (246b), and it comes from the replacement of the rest mass-energy $m = m1 + m2$ . From Eq. (246b) this effect is of order $Ω2$ in the case of corotating binaries, which means by comparison with Eq. (232) that it is equivalent to an “orbital” effect at the 2PN order (i.e., $∝ x2$ ). Higher-order corrections in Eq. (246b), which behave at least like $4 Ω$ , will correspond to the orbital 5PN order at least and are negligible for the present purpose. In addition there will be a subdominant contribution, of the order of $Ω8∕3$ equivalent to 3PN order, which comes from the replacement of the masses into the Newtonian part, proportional to $x ∝ Ω2∕3$ , of the energy $E$ ; see Eq. (232). With the 3PN accuracy we do not need to replace the masses that enter into the post-Newtonian corrections in $E$ , so in these terms the masses can be considered to be the irreducible ones.

Our second task is to include the specific relativistic effects due to the spins, namely the spin-orbit (SO) interaction and the spin-spin (SS) one. In the case of spins $S 1$ and $S 2$ aligned parallel to the orbital angular momentum (and right-handed with respect to the sense of motion) the SO energy reads

We shall review in Section 11 the most up-to-date results for the spin-orbit energy and related quantities; here we are simply employing the leading-order formula obtained in Refs. [27 *, 28 *, 275 *, 271 *] and given by the first term in Eq. (415). We immediately infer from this formula that in the case of corotating black holes the SO effect is equivalent to a 3PN orbital effect and thus must be retained with the present accuracy. With this approximation, the masses in Eq. (248 *) can be replaced by the irreducible ones. As for the SS interaction (still in the case of spins aligned with the orbital angular momentum) it is given by

The SS effect can be neglected here because it is of order 5PN for corotating systems. Summing up all the spin contributions to 3PN order we find that the supplementary energy due to the corotating spins is [51 *, 55 *]⁵⁶

The total mass $μ = μ1 + μ2$ , the symmetric mass ratio $η = μ1μ2∕μ2$ , and the dimensionless invariant post-Newtonian parameter $x = (μ Ω)2∕3 μ$ are now expressed in terms of the irreducible masses $μ a$ , rather than the masses $ma$ . The complete 3PN energy of the corotating binary is finally given by the sum of Eqs. (232) and (250 *), in which all the masses are now understood as being the irreducible ones, which must be assumed to stay constant when the binary evolves for the comparison with the numerical calculation.

Figure 1: The binding energy $EICO$ versus $ΩICO$ in the equal-mass case ( $ν = 1∕4$ ). Left panel: Comparison with the numerical relativity result of Gourgoulhon, Grandclément et al. [228 *, 232 *] valid in the corotating case (marked by a star). Points indicated by $nPN$ are computed from the minimum of Eq. (232), and correspond to irrotational binaries. Points denoted by $corot nPN$ come from the minimum of the sum of Eqs. (232) and (250 *), and describe corotational binaries. Note the very good convergence of the standard (Taylor-expanded) PN series. Right panel: Numerical relativity results of Cook, Pfeiffer et al. [133 *, 121 *] for quasi-equilibrium (QE) configurations and various boundary conditions for the lapse function, in the non-spinning (NS), leading-order non spinning (LN) and corotating (CO) cases. The point from [228 *, 232 *] (HKV-GGB) is also reported as in the left panel, together with IVP, the initial value approach with effective potential [132 *, 342 *], as well as standard PN predictions from the left panel and non-standard (EOB) ones. The agreement between the QE computation and the standard non-resummed 3PN point is excellent especially in the irrotational NS case.

The left panel of Figure 1 * shows the results for $EICO$ in the case of irrotational and corotational binaries. Since $ΔEcorot$ , given by Eq. (250 *), is at least of order 2PN, the result for $1PNcorot$ is the same as for 1PN in the irrotational case; then, obviously, $2PNcorot$ takes into account only the leading 2PN corotation effect, i.e., the spin kinetic energy given by Eq. (246b), while $3PNcorot$ involves also, in particular, the corotational SO coupling at the 3PN order. In addition we present the numerical point obtained by numerical relativity under the assumptions of conformal flatness and of helical symmetry [228 *, 232 *]. As we can see the 3PN points, and even the 2PN ones, are in good agreement with the numerical value. The fact that the 2PN and 3PN values are so close to each other is a good sign of the convergence of the expansion. In fact one might say that the role of the 3PN approximation is merely to “confirm” the value already given by the 2PN one (but of course, had we not computed the 3PN term, we would not be able to trust very much the 2PN value). As expected, the best agreement we obtain is for the 3PN approximation and in the case of corotation, i.e., the point $3PNcorot$ . However, the 1PN approximation is clearly not precise enough, but this is not surprising in the highly relativistic regime of the ICO. The right panel of Figure 1 * shows other very interesting comparisons with numerical relativity computations [133 *, 121 *], done not only for the case of corotational binaries but also in the irrotational (non-spinning) case. Witness in particular the almost perfect agreement between the standard 3PN point (PN standard, shown with a green triangle) and the numerical quasi-equilibrium point (QE, red triangle) in the case of irrotational non-spinning (NS) binaries.

However, we recall that the numerical works [228 *, 232 *, 133 *, 121 *] assume that the spatial metric is conformally flat, which is incompatible with the post-Newtonian approximation starting from the 2PN order (see [196 ] for a discussion). Nevertheless, the agreement found in Figure 1 * constitutes an appreciable improvement of the previous situation, because the first estimations of the ICO in post-Newtonian theory [274 ] and numerical relativity [132 , 342 , 29 ] disagreed with each other, and do not match with the present 3PN results.

8.2 Dynamical stability of circular orbits

In this section, following Ref. [79 *], we shall investigate the problem of the stability, against dynamical perturbations, of circular orbits at the 3PN order. We propose to use two different methods, one based on a linear perturbation at the level of the center-of-mass equations of motion (219 *) – (220) in (standard) harmonic coordinates, the other one consisting of perturbing the Hamiltonian equations in ADM coordinates for the center-of-mass Hamiltonian (223). We shall find a criterion for the stability of circular orbits and shall present it in an invariant way – the same in different coordinate systems. We shall check that our two methods agree on the result.

We deal first with the perturbation of the equations of motion, following Kidder, Will & Wiseman [275 *] (see their Section III.A). We introduce polar coordinates $(r,φ )$ in the orbital plane and pose $u ≡ ˙r$ and $Ω ≡ φ˙$ . Then Eq. (219 *) yields the system of equations

where $𝒜$ and $ℬ$ are given by Eqs. (220) as functions of $r$ , $u$ and $Ω$ (through $v2 = u2 + r2Ω2$ ).

In the case of an orbit that is circular apart from the adiabatic inspiral at the 2.5PN order (we neglect the radiation-reaction damping effects), we have $˙r0 = ˙u0 = Ω˙0 = 0$ hence $u0 = 0$ . In this section we shall indicate quantities associated with the circular orbit, which constitutes the zero-th approximation in our perturbation scheme, using the subscript $0$ . Hence Eq. (251b) gives the angular velocity $Ω0$ of the circular orbit as

Solving iteratively this relation at the 3PN order using the equations of motion (219 *) – (220), we obtain $Ω0$ as a function of the circular-orbit radius $r0$ in standard harmonic coordinates; the result agrees with Eq. (228).⁵⁷

We now investigate the linear perturbation around the circular orbit defined by the constants $r0$ , $u0 = 0$ and $Ω0$ . We pose

where $δr$ , $δu$ and $δΩ$ denote the linear perturbations of the circular orbit. Then a system of linear equations readily follows:

where the coefficients, which solely depend on the unperturbed circular orbit (hence the added subscript $0$ ), read as [275 *]

In obtaining these equations we use the fact that $𝒜$ is a function of the square $u2$ through $2 2 2 2 v = u + r Ω$ , so that $∂𝒜 ∕∂u$ is proportional to $u$ and thus vanishes in the unperturbed configuration (because $u = δu$ ). On the other hand, since the radiation reaction is neglected, $ℬ$ is also proportional to $u$ [see Eq. (220b)], so only $∂ℬ ∕∂u$ can contribute at the zero-th perturbative order. Now by examining the fate of perturbations that are proportional to some $eiσt$ , we arrive at the condition for the frequency $σ$ of the perturbation to be real, and hence for stable circular orbits to exist, as being [275 *]

Substituting into this $𝒜$ and $ℬ$ at the 3PN order we then arrive at the orbital-stability criterion

where we recall that $r 0$ is the radius of the orbit in harmonic coordinates.

Our second method is to use the Hamiltonian equations associated with the 3PN center-of-mass Hamiltonian in ADM coordinates $HADM$ given by Eq. (223). We introduce the polar coordinates $(R, Ψ )$ in the orbital plane – we assume that the orbital plane is equatorial, given by $Θ = π2$ in the spherical coordinate system $(R,Θ, Ψ )$ – and make the substitution

This yields a reduced Hamiltonian that is a function of $R$ , $PR$ and $PΨ$ , and describes the motion in polar coordinates in the orbital plane; henceforth we denote it by $ℋ = ℋ [R, PR, PΨ ] ≡ HADM ∕ μ$ . The Hamiltonian equations then read

Evidently the constant $PΨ$ is nothing but the conserved angular-momentum integral. For circular orbits we have $R = R0$ (a constant) and $PR = 0$ , so

which gives the angular momentum $0 PΨ$ of the circular orbit as a function of $R0$ , and

which yields the angular frequency of the circular orbit $Ω0$ , which is evidently the same numerical quantity as in Eq. (252 *), but is here expressed in terms of the separation $R0$ in ADM coordinates. The last equation, which is equivalent to $R = const = R0$ , is

It is automatically verified because $ℋ$ is a quadratic function of $P R$ and hence $∂ℋ ∕∂P R$ is zero for circular orbits.

We consider now a perturbation of the circular orbit defined by

The Hamiltonian equations (259), worked out at the linearized order, read as

where the coefficients, which depend on the unperturbed orbit, are given by

By looking to solutions proportional to some $iσt e$ one obtains some real frequencies, and therefore one finds stable circular orbits, if and only if

Using explicitly the Hamiltonian (223) we readily obtain

This result does not look the same as our previous result (257), but this is simply due to the fact that it depends on the ADM radial separation $R0$ instead of the harmonic one $r0$ . Fortunately we have derived in Section 7.2 the material needed to connect $R0$ to $r0$ with the 3PN accuracy. Indeed, with Eqs. (210 *) we have the relation valid for general orbits in an arbitrary frame between the separation vectors in both coordinate systems. Specializing that relation to circular orbits we find

Note that the difference between $R0$ and $r0$ starts only at 2PN order. That relation easily permits to perfectly reconcile both expressions (257) and (267).

Finally let us give to $ˆC0$ an invariant meaning by expressing it with the help of the orbital frequency $Ω 0$ of the circular orbit, or, more conveniently, of the frequency-related parameter $x ≡ (Gm Ω ∕c3)2∕3 0 0$ – cf. Eq. (230 *). This allows us to write the criterion for stability as $C0 > 0$ , where $G2m2-ˆ C0 = x30 C0$ admits the gauge-invariant form

This form is more interesting than the coordinate-dependent expressions (257) or (267), not only because of its invariant form, but also because as we see the 1PN term yields exactly the Schwarzschild result that the innermost stable circular orbit or ISCO of a test particle (i.e., in the limit $ν → 0$ ) is located at $x = 1 ISCO 6$ . Thus we find that, at the 1PN order, but for any mass ratio $ν$ ,

One could have expected that some deviations of the order of $ν$ already occur at the 1PN order, but it turns out that only from the 2PN order does one find the occurrence of some non-Schwarzschildean corrections proportional to $ν$ . At the 2PN order we obtain

For equal masses this gives $x2PISNCO ≃ 0.187$ . Notice also that the effect of the finite mass corrections is to increase the frequency of the ISCO with respect to the Schwarzschild result, i.e., to make it more inward:⁵⁸

Finally, at the 3PN order, for equal masses $1 ν = 4$ , we find that according to our criterion all the circular orbits are stable. More generally, we find that at the 3PN order all orbits are stable when the mass ratio $ν$ is larger than some critical value $νc ≃ 0.183$ .

The stability criterion (269 *) has been compared in great details to various other stability criteria by Favata [191 *] and shown to perform very well, and has also been generalized to spinning black hole binaries in Ref. [190 ]. Note that this criterion is based on the physical requirement that a stable perturbation should have a real frequency. It gives an innermost stable circular orbit, when it exists, which differs from the innermost circular orbit or ICO defined in Section 8.1; see Ref. [378 ] for a discussion on the difference between an ISCO and the ICO in the PN context. Note also that the criterion (269 *) is based on systematic post-Newtonian expansions, without resorting for instance to Padé approximants. Nevertheless, it performs better than other criteria based on various resummation techniques, as discussed in Ref. [191 ].

8.3 The first law of binary point-particle mechanics

In this section we shall review a very interesting relation for binary systems of point particles modelling black hole binaries and moving on circular orbits, known as the “first law of point-particle mechanics”. This law was obtained using post-Newtonian methods in Ref. [289 *], but is actually a particular case of a more general law, valid for systems of black holes and extended fluid balls, derived by Friedman, Uryū & Shibata [208 *].

Before tackling the problem it is necessary to make more precise the notion of circular orbits. These are obtained from the conservative part of the dynamics, neglecting the dissipative radiation-reaction force responsible for the gravitational-wave inspiral. In post-Newtonian theory this means neglecting the radiation-reaction force at 2.5PN and 3.5PN orders, i.e., considering only the conservative dynamics at the even-parity 1PN, 2PN and 3PN orders. We have seen in Sections 5.2 and 5.4 that this clean separation between conservative even-parity and dissipative odd-parity terms breaks at 4PN order, because of a contribution originating from gravitational-wave tails in the radiation-reaction force. We expect that at any higher order 4PN, 4.5PN, 5PN, etc. there will be a mixture of conservative and dissipative effects; here we assume that at any higher order we can neglect the radiation-reaction dissipation effects.

Consider a system of two compact objects moving on circular orbits. We examine first the case of non-spinning objects. With exactly circular orbits the geometry admits a helical Killing vector (HKV) field $K α$ , satisfying the Killing equation $∇ αK β + ∇ βK α = 0$ . Imposing the existence of the HKV is the rigorous way to implement the notion of circular orbits. A Killing vector is only defined up to an overall constant factor. The helical Killing vector $α K$ extends out to a large distance where the geometry is essentially flat. There,

in any natural coordinate system which respects the helical symmetry [370 ]. We let this equality define the overall constant factor, thereby specifying the Killing vector field uniquely. In Eq. (273 *) $Ω$ denotes the angular frequency of the binary’s circular motion.

An observer moving with one of the particles (say the particle 1), while orbiting around the other particle, would detect no change in the local geometry. Thus the four-velocity $uα1$ of that particle is tangent to the Killing vector $K α$ evaluated at the location of the particle, which we denote by $K α1$ . A physical quantity is then defined as the constant of proportionality $uT 1$ between these two vectors, namely

The four-velocity of the particle is normalized by $μ ν (gμν)1u 1u1 = − 1$ , where $(gμν)1$ denotes the metric at the location of the particle. For a self-gravitating compact binary system, the metric at point 1 is generated by the two particles and has to be regularized according to one of the self-field regularizations discussed in Section 6. It will in fact be sometimes more convenient to work with the inverse of $uT1$ , denoted $z ≡ 1∕uT 1 1$ . From Eq. (274 *) we get

where $(u1K1 ) = (gμν)1uμK ν 1 1$ denotes the usual space-time dot product. Thus we can regard $z1$ as the Killing energy of the particle that is associated with the HKV field $α K$ . The quantity $z1$ represents also the redshift of light rays emitted from the particle and received on the helical symmetry axis perpendicular to the orbital plane at large distances from it [176 *]. In the following we shall refer to $z 1$ as the redshift observable.

If we choose a coordinate system such that Eq. (273 *) is satisfied everywhere, then in particular $Kt1 = 1$ , thus $uT1$ simply agrees with $ut1$ , the $t$ -component of the four-velocity of the particle. The Killing vector on the particle is then $K α = uα∕ut 1 1 1$ , and simply reduces to the particle’s ordinary coordinate velocity: $α α K 1 = v 1∕c$ where $α α v1 = dy1∕dt$ and $α y1(t) = [ct,y1(t)]$ denotes the particle’s trajectory in that coordinate system. The redshift observable we are thus considering is

It is important to note that for circular orbits this quantity does not depend upon the choice of coordinates; in a perturbative approach in which the perturbative parameter is the particles’ mass ratio $ν ≪ 1$ , it does not depend upon the choice of perturbative gauge with respect to the background metric. We shall be interested in the invariant scalar function $z (Ω) 1$ , where $Ω$ is the angular frequency of the circular orbit introduced when imposing Eq. (273 *).

We have obtained in Section 7.4 the expressions of the post-Newtonian binding energy $E$ and angular momentum $J$ for point-particle binaries on circular orbits. We shall now show that there are some differential and algebraic relations linking $E$ and $J$ to the redshift observables $z1$ and $z2$ associated with the two individual particles. Here we prefer to introduce instead of $E$ the total relativistic (ADM) mass of the binary system

where $m$ is the sum of the two post-Newtonian individual masses $m1$ and $m2$ – those which have been used up to now, for instance in Eq. (203). Note that in the spinning case such post-Newtonian masses acquire some spin contributions given, e.g., by Eqs. (243 *) – (246).

For point particles without spins, the ADM mass $M$ , angular momentum $J$ , and redshifts $za$ , are functions of three independent variables, namely the orbital frequency $Ω$ that is imposed by the existence of the HKV, and the individual masses $ma$ . For spinning point particles, we have also the two spins $Sa$ which are necessarily aligned with the orbital angular momentum. We first recover that the ADM quantities obey the “thermodynamical” relation already met in Eq. (235 *),

Such relation is commonly used in post-Newtonian theory (see e.g., [160 , 51 ]). It states that the gravitational-wave energy and angular momentum fluxes are strictly proportional for circular orbits, with $Ω$ being the coefficient of proportionality. This relation is used in computations of the binary evolution based on a sequence of quasi-equilibrium configurations [228 , 232 , 133 , 121 ], as discussed in Section 8.1.

The first law will be a thermodynamical generalization of Eq. (278 *), describing the changes in the ADM quantities not only when the orbital frequency $Ω$ varies with fixed masses, but also when the individual masses $m a$ of the particles vary with fixed orbital frequency. That is, one compares together different conservative dynamics with different masses but the same frequency. This situation is answered by the differential equations

Finally the three relations (278 *) – (279 *) can be summarized in the following result.

Theorem 11. The changes in the ADM mass and angular momentum of a binary system made of point particles in response to infinitesimal variations of the individual masses of the point particles, are related together by the first law of binary point-particle mechanics as [208 *, 289 *]

This law was proved in a very general way in Ref. [208 *] for systems of black holes and extended bodies under some arbitrary Killing symmetry. The particular form given in Eq. (280 *) is a specialization to the case of point particle binaries with helical Killing symmetry. It has been proved directly in this form in Ref. [289 *] up to high post-Newtonian order, namely 3PN order plus the logarithmic contributions occurring at 4PN and 5PN orders.

The first law of binary point-particle mechanics (280 *) is of course reminiscent of the celebrated first law of black hole mechanics $-κ δM − ωH δJ = 8πδA$ , which holds for any non-singular, asymptotically flat perturbation of a stationary and axisymmetric black hole of mass $M$ , intrinsic angular momentum (or spin) $J ≡ Ma$ , surface area $A$ , uniform surface gravity $κ$ , and angular frequency $ωH$ on the horizon [26 , 417 ]; see Ref. [289 *] for a discussion.

An interesting by-product of the first law (280 *) is the remarkably simple algebraic relation

which can be seen as a first integral of the differential relation (280 *). Note that the existence of such a simple algebraic relation between the local quantities $z1$ and $z2$ on one hand, and the globally defined quantities $M$ and $J$ on the other hand, is not trivial.

Next, we report the result of a generalization of the first law applicable to systems of point particles with spins (moving on circular orbits).⁵⁹ This result is valid through linear order in the spin of each particle, but holds also for the quadratic coupling between different spins (interaction spin terms $S1 × S2$ in the language of Section 11). To be consistent with the HKV symmetry, we must assume that the two spins $Sa$ are aligned or anti-aligned with the orbital angular momentum. We introduce the total (ADM-like) angular momentum $J$ which is related to the orbital angular momentum $L$ by $J = L + ∑ S a a$ for aligned or anti-aligned spins. The first law now becomes [55 *]

where $Ωa = |Ωa |$ denotes the precession frequency of the spins. This law has been derived in Ref. [55 *] from the canonical Hamiltonian formalism. The spin variables used here are the canonical spins $Sa$ , that are easily seen to obey, from the algebra satisfied by the canonical variables, the usual Newtonian-looking precession equations $dSa ∕dt = Ωa × Sa$ . These variables are identical to the “constant-in-magnitude” spins which will be defined and extensively used in Section 11. Similarly, to Eq. (281 *) we have also a first integral associated with the variational law (282 *):

Notice that the relation (282 *) has been derived for point particles and arbitrary aligned spins. We would like now to derive the analogous relation for binary black holes. The key difference is that black holes are extended finite-size objects while point particles have by definition no spatial extension. For point particle binaries the spins can have arbitrary magnitude and still be compatible with the HKV. In this case the law (282 *) would describe also (super-extremal) naked singularities. For black hole binaries the HKV constraints the rotational state of each black hole and the binary system must be corotating.

Let us derive, in a heuristic way, the analogue of the first law (282 *) for black holes by introducing some “constitutive relations” $m (μ ,S ,⋅⋅⋅) a a a$ specifying the energy content of the bodies, i.e., the relations linking their masses $ma$ to the spins $Sa$ and to some irreducible masses $μa$ . More precisely, we define for each spinning particle the analogue of an irreducible mass $irr μa ≡ m a$ via the variational relation $δma = caδμa + ωaδSa$ ,⁶⁰ in which the “response coefficient” $ca$ of the body and its proper rotation frequency $ωa$ are associated with the internal structure:

For instance, using the Christodoulou mass formula (243 *) for Kerr black holes, we obtain the rotation frequency $ωa$ given by Eq. (244 *). On the other hand, the response coefficient $ca$ differs from 1 only because of spin effects, and we can check that $ca = 1 + 𝒪(S2a)$ .

Within the latter heuristic model a condition for the corotation of black hole binaries has been proposed in Ref. [55 ] as

This condition determines the value of the proper frequency $ωa$ of each black hole appropriate to the corotation state. When expanded to 2PN order the condition (285 *) leads to Eq. (247 *) that we have already used in Section 8.1. With Eq. (285 *) imposed, the first law (282 *) simplifies considerably:

This is almost identical to the first law for non-spinning binaries given by Eq. (280 *); indeed it simply differs from it by the substitutions $ca → 1$ and $μa → ma$ . Since the irreducible mass $μa$ of a rotating black hole is the spin-independent part of its total mass $ma$ , this observation suggests that corotating binaries are very similar to non-spinning binaries, at least from the perspective of the first law. Finally we can easily reconcile the first law (286 *) for corotating systems with the known first law of binary black hole mechanics [208 ], namely

Indeed it suffices to make the formal identification in Eq. (286 *) of $caza$ with $4μa κa$ , where $κa$ denotes the constant surface gravity, and using the surface areas $Aa = 16 πμ2a$ instead of the irreducible masses of the black holes. This shows that the heuristic model based on the constitutive relations (284) is able to capture the physics of corotating black hole binary systems.

8.4 Post-Newtonian approximation versus gravitational self-force

The high-accuracy predictions from general relativity we have drawn up to now are well suited to describe the inspiralling phase of compact binaries, when the post-Newtonian parameter (1 *) is small independently of the mass ratio $q ≡ m1 ∕m2$ between the compact bodies. In this section we investigate how well does the post-Newtonian expansion compare with another very important approximation scheme in general relativity: The gravitational self-force approach, based on black-hole perturbation theory, which gives an accurate description of extreme mass ratio binaries having $q ≪ 1$ or equivalently $ν ≪ 1$ , even in the strong field regime. The gravitational self-force analysis [317 , 360 , 178 *, 231 ] (see [348 , 177 , 23 ] for reviews) is thus expected to provide templates for extreme mass ratio inspirals (EMRI) anticipated to be present in the bandwidth of space-based detectors.

Consider a system of two (non-spinning) compact objects with $q = m ∕m ≪ 1 1 2$ ; we shall call the smaller mass $m1$ the “particle”, and the larger mass $m2$ the “black hole”. The orbit of the particle around the black hole is supposed to be exactly circular as we neglect the radiation-reaction effects. With circular orbits and no dissipation, we are considering the conservative part of the dynamics, and the geometry admits the HKV field (273 *). Note that in self-force theory there is a clean split between the dissipative and conservative parts of the dynamics (see e.g., [22 ]). This split is particularly transparent for an exact circular orbit, since the radial component (along $r$ ) is the only non-vanishing component of the conservative self-force, while the dissipative part of the self-force are the components along $t$ and $φ$ .

Figure 2: Different analytical approximation schemes and numerical techniques to study black hole binaries, depending on the mass ratio $q = m ∕m 1 2$ and the post-Newtonian parameter $2 2 2 2 𝜖 ∼ v ∕c ∼ Gm ∕(c r12)$ . Post-Newtonian theory and perturbative self-force analysis can be compared in the post-Newtonian regime ( $𝜖 ≪ 1$ thus $2 r12 ≫ Gm ∕c$ ) of an extreme mass ratio ( $m1 ≪ m2$ ) binary.

The problem of the comparison between the post-Newtonian and perturbative self-force analyses in their common domain of validity, that of the slow-motion and weak-field regime of an extreme mass ratio binary, is illustrated in Figure 2 *. This problem has been tackled by Detweiler [176 *], who computed numerically within the self-force (SF) approach the redshift observable $T u1$ associated with the particle, and compared it with the 2PN prediction extracted from existing post-Newtonian results [76 ]. This comparison proved to be successful, and was later systematically implemented and extended to higher post-Newtonian orders in Refs. [68 *, 67 *]. In this section we review the works [68 *, 67 *] which have demonstrated an excellent agreement between the analytical post-Newtonian result derived through 3PN order, with inclusion of specific logarithmic terms at 4PN and 5PN orders, and the exact numerical SF result.

For the PN-SF comparison, we require two physical quantities which are precisely defined in the context of each of the approximation schemes. The orbital frequency $Ω$ of the circular orbit as measured by a distant observer is one such quantity and has been introduced in Eq. (273 *); the second quantity is the redshift observable $uT1$ (or equivalently $z1 = 1∕uT1$ ) associated with the smaller mass $m1 ≪ m2$ and defined by Eqs. (274 *) or (275 *). The truly coordinate and perturbative-gauge independent properties of $Ω$ and the redshift observable $T u 1$ play a crucial role in this comparison. In the perturbative self-force approach we use Schwarzschild coordinates for the background, and we refer to “gauge invariance” as a property which holds within the restricted class of gauges for which (273 *) is a helical Killing vector. In all other respects, the gauge choice is arbitrary. In the post-Newtonian approach we work with harmonic coordinates and compute the explicit expression (276 *) of the redshift observable.

The main difficulty in the post-Newtonian calculation is the control to high PN order of the near-zone metric $(gμν)1$ entering the definition of the redshift observable (276 *), and which has to be regularized at the location of the particle by means of dimensional regularization (see Sections 6.3 – 6.4). Up to 2.5PN order the Hadamard regularization is sufficient and the regularized metric has been provided in Eqs. (242). Here we report the end result of the post-Newtonian computation of the redshift observable including all terms up to the 3PN order, and augmented by the logarithmic contributions up to the 5PN order (and also the known Schwarzschild limit) [68 *, 67 *, 289 *]:

We recall that $x$ denotes the post-Newtonian parameter (230 *), $ν$ is the mass ratio (215 *), and $Δ = (m1 − m2 )∕m$ . The redshift observable of the other particle is deduced by setting $Δ → − Δ$ .

In Eq. (288) we denote by $u4(ν)$ , $v4(ν)$ and $u5(ν)$ , $v5(ν )$ some unknown 4PN and 5PN coefficients, which are however polynomials of the symmetric mass ratio $ν$ . They can be entirely determined from the related coefficients $e4(ν)$ , $j4(ν)$ and $e5(ν)$ , $j5(ν)$ in the expressions of the energy and angular momentum in Eqs. (233) and (234). To this aim it suffices to apply the differential first law (280 *) up to 5PN order; see Ref. [289 ] for more details.

The post-Newtonian result (288) is valid for any mass ratio, and for comparison purpose with the SF calculation we now investigate the small mass ratio regime $q ≪ 1$ . We introduce a post-Newtonian parameter appropriate to the small mass limit of the “particle”,

We express the symmetric mass ratio in terms of the asymmetric one: $ν = q∕(1 + q)2$ , together with $Δ = (q − 1)∕(q + 1)$ . Then Eq. (288), expanded through first order in $q$ , which means including only the linear self-force level, reads

The Schwarzschildean result is known in closed form as

and for the self-force contribution one obtains⁶¹

The analytic coefficients were determined up to 2PN order in Ref. [176 *]; the 3PN term was computed in Ref. [68 *] making full use of dimensional regularization; the logarithmic contributions at the 4PN and 5PN orders were added in Refs. [67 *, 146 ].

The coefficients $α4$ and $α5$ represent some pure numbers at the 4PN and 5PN orders. By an analytic self-force calculation [36 ] the coefficient $α4$ has been obtained as

Using the first law (280 *), we know how to deduce from the PN coefficients in the redshift variable the corresponding PN coefficients in the energy function (233). Thus, the result reported in Eq. (236) for the 4PN term in the energy function for circular orbits has been deduced from Eq. (293 *) by application of the first law.

On the self-force front the main problem is to control the numerical resolution of the computation of the redshift observable in order to distinguish more accurately the contributions of very high order PN terms. The comparison of the post-Newtonian expansion (292) with the numerical SF data has confirmed with high precision the determination of the 3PN coefficient [68 *, 67 *]: Witness Table 1 where the agreement with the analytical value involves 7 significant digits. Notice that such agreement provides an independent check of the dimensional regularization procedure invoked in the PN expansion scheme (see Sections 6.3 – 6.4). It is remarkable that such procedure is equivalent to the procedure of subtraction of the singular field in the SF approach [178 ].

Table 1: Numerically determined value of the 3PN coefficient for the SF part of the redshift observable defined by Eq. (292). The analytic post-Newtonian computation [68 *] is confirmed with many significant digits.

3PN coefficient	SF value
$121- 41-2 α3 ≡ − 3 + 32π = − 27.6879026 ⋅⋅⋅$	$− 27.6879034 ± 0.0000004$

Table 2: Numerically determined values of higher-order PN coefficients for the SF part of the redshift observable defined by Eq. (292). The uncertainty in the last digit (or two last digits) is indicated in parentheses. The 4PN numerical value agrees with the analytical expression (293 *).

PN coefficient	SF value
$α4$	$− 114.34747 (5)$
$α5$	$− 245.53 (1)$
$α6$	$− 695 (2 )$
$β6$	$+339.3 (5 )$
$α7$	$− 5837 (16 )$

Furthermore the PN-SF comparison has permitted to measure the coefficients $α4$ and $α5$ with at least 8 significant digits for the 4PN coefficient, and 5 significant digits for the 5PN one. In Table 2 we report the result of the analysis performed in Refs. [68 *, 67 *] by making maximum use of the analytical coefficients available at the time, i.e., all the coefficients up to 3PN order plus the logarithmic contributions at 4PN and 5PN orders. One uses a set of five basis functions corresponding to the unknown non-logarithmic 4PN and 5PN coefficients $α4$ , $α5$ in Eq. (292), and augmented by the 6PN and 7PN non-logarithmic coefficients $α6$ , $α 7$ plus a coefficient $β 6$ for the logarithm at 6PN. A contribution $β 7$ from a logarithm at 7PN order is likely to confound with the $α7$ coefficient. There is also the possibility of the contribution of a logarithmic squared at 7PN order, but such a small effect is not permitted in this fit.

Gladly we discover that the more recent analytical value of the 4PN coefficient, Eq. (293 *), matches the numerical value which was earlier measured in Ref. [67 *] (see Table 2). This highlights the predictive power of perturbative self-force calculations in determining numerically new post-Newtonian coefficients [176 , 68 , 67 ]. This ability is obviously due to the fact (illustrated in Figure 2 *) that perturbation theory is legitimate in the strong field regime of the coalescence of black hole binary systems, which is inaccessible to the post-Newtonian method. Of course, the limitation of the self-force approach is the small mass-ratio limit; in this respect it is taken over by the post-Newtonian approximation.

More recently, the accuracy of the numerical computation of the self-force, and the comparison with the post-Newtonian expansion, have been drastically improved by Shah, Friedman & Whiting [383 *]. The PN coefficients of the redshift observable were obtained to very high 10.5PN order both numerically and also analytically, for a subset of coefficients that are either rational or made of the product of $π$ with a rational. The analytical values of the coefficients up to 6PN order have also been obtained from an alternative self-force calculation [38 *, 37 *]. An interesting feature of the post-Newtonian expansion at high order is the appearance of half-integral PN coefficients (i.e., of the type $n 2$ PN where $n$ is an odd integer) in the conservative dynamics of binary point particles, moving on exactly circular orbits. This is interesting because any instantaneous (non-tail) term at any half-integral PN order will be zero for circular orbits, as can be shown by a simple dimensional argument [77 *]. Therefore half-integral coefficients can appear only due to truly hereditary (tail) integrals. Using standard post-Newtonian methods it has been proved in Refs. [77 , 78 ] that the dominant half-integral PN term in the redshift observable (292) occurs at the 5.5PN order (confirming the earlier finding of Ref. [383 *]) and originates from the non-linear “tail-of-tail” integrals investigated in Section 3.2. The results for the 5.5PN coefficient in Eq. (292), and also for the next-to-leading 6.5PN and 7.5PN ones, are

and fully agree between the PN and SF computations. We emphasize that the results (294 *) are achieved by the traditional PN approach, which is completely general (contrary to various analytical and numerical SF calculations [383 *, 38 , 37 , 268 ]), i.e., is not tuned to a particular type of source but is applicable to any extended PN source (see Part A). Note that Eqs. (294 *) represent the complete coefficients as there are no logarithms at these orders.

To conclude, the consistency of this “cross-cultural” comparison between the analytical post-Newtonian and the perturbative self-force approaches confirms the soundness of both approximations in describing the dynamics of compact binaries. Furthermore this interplay between PN and SF efforts (which is now rapidly growing [383 ]) is important for the synthesis of template waveforms of EMRIs to be analysed by space-based gravitational wave detectors, and has also an impact on efforts of numerical relativity in the case of comparable masses.

	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