Matching to a Post-Newtonian Source

4 Matching to a Post-Newtonian Source

By Theorem 2 we control the most general class of solutions of the vacuum equations outside the source, in the form of non-linear functionals of the source multipole moments. For instance, these solutions include the Schwarzschild and Kerr solutions for black holes, as well as all their perturbations. By Theorem 4 we learned how to construct the radiative moments at infinity, which constitute the observables of the radiation field at large distances from the source, and we obtained in Section 3.3 explicit relationships between radiative and source moments. We now want to understand how a specific choice of matter stress-energy tensor $αβ T$ , i.e., a specific choice of some physical model describing the material source, selects a particular physical exterior solution among our general class, and therefore a given set of multipole moments for the source.

4.1 The matching equation

We shall provide the answer to that problem in the case of a post-Newtonian source for which the post-Newtonian parameter $𝜖 ∼ 1∕c$ defined by Eq. (1 *) is small. The fundamental fact that permits the connection of the exterior field to the inner field of the source is the existence of a “matching” region, in which both the multipole expansion and the post-Newtonian expansion are valid. This region is nothing but the exterior part of the near zone, such that $r > a$ (exterior) and $r ≪ λ$ (near zone); it always exists around post-Newtonian sources whose radius is much less than the emitted wavelength, $aλ ∼ 𝜖 ≪ 1$ . In our formalism the multipole expansion is defined by the multipolar-post-Minkowskian (MPM) solution; see Section 2. Matching together the post-Newtonian and MPM solutions in this overlapping region is an application of the method of matched asymptotic expansions, which has frequently been applied in the present context, both for radiation-reaction [114 *, 113 *, 7 , 58 *, 43 *] and wave-generation [59 *, 155 , 44 *, 49 *] problems.

Let us denote by $ℳ (h )$ the multipole expansion of $h$ (for simplicity, we suppress the space-time indices). By $ℳ (h)$ we really mean the MPM exterior metric that we have constructed in Sections 2.2 and 2.3:

This solution is formally defined for any radius $r > 0$ . Of course, the true solution $h$ agrees with its own multipole expansion in the exterior of the source, i.e.

By contrast, inside the source, $h$ and $ℳ (h)$ disagree with each other because $h$ is a fully-fledged solution of the field equations within the matter source, while $ℳ (h)$ is a vacuum solution becoming singular at $r = 0$ . Now let us denote by $h-$ the post-Newtonian expansion of $h$ . We have already anticipated the general structure of this expansion which is given in Eq. (54 *). In the matching region, where both the multipolar and post-Newtonian expansions are valid, we write the numerical equality

This “numerical” equality is viewed here in a sense of formal expansions, as we do not control the convergence of the series. In fact, we should be aware that such an equality, though quite natural and even physically obvious, is probably not really justified within the approximation scheme (mathematically speaking), and we simply take it here as part of our fundamental assumptions.

We now transform Eq. (102 *) into a matching equation, by replacing in the left-hand side $ℳ (h)$ by its near-zone re-expansion $ℳ--(h)$ , and in the right-hand side $h-$ by its multipole expansion $ℳ (h-)$ . The structure of the near-zone expansion ( $r → 0$ ) of the exterior multipolar field has been found in Theorem 3, see Eq. (53 *). We denote the corresponding infinite series $------ ℳ (h)$ with the same overbar as for the post-Newtonian expansion because it is really an expansion when $r∕c → 0$ , equivalent to an expansion when $c → ∞$ . Concerning the multipole expansion of the post-Newtonian metric, $-- ℳ (h)$ , we simply postulate for the moment its existence, but we shall show later how to construct it explicitly. Therefore, the matching equation is the statement that

by which we really mean an infinite set of functional identities, valid $3 ∀(x, t) ∈ ℝ ∗ × ℝ$ , between the coefficients of the series in both sides of the equation. Note that such a meaning is somewhat different from that of a numerical equality like Eq. (102 *), which is valid only when $x$ belongs to some limited spatial domain. The matching equation (103 *) tells us that the formal near-zone expansion of the multipole decomposition is identical, term by term, to the multipole expansion of the post-Newtonian solution. However, the former expansion is nothing but the formal far-zone expansion, when $r → ∞$ , of each of the post-Newtonian coefficients. Most importantly, it is possible to write down, within the present formalism, the general structure of these identical expansions as a consequence of Eq. (53 *):

where the functions $∑ n FL,m,p = n≥1 G FL,m,p,n$ . The latter expansion can be interpreted either as the singular re-expansion of the multipole decomposition when $r → 0$ – i.e., the first equality in Eq. (104 *) –, or the singular re-expansion of the post-Newtonian series when $r → + ∞$ – the second equality.

We recognize the beauty of singular perturbation theory, where two asymptotic expansions, taken formally outside their respective domains of validity, are matched together. Of course, the method works because there exists, physically, an overlapping region in which the two approximation series are expected to be numerically close to the exact solution. As we shall detail in Sections 4.2 and 5.2, the matching equation (103 *), supplemented by the condition of no-incoming radiation [say in the form of Eq. (29 *)], permits determining all the unknowns of the problem: On the one hand, the external multipolar decomposition $ℳ (h)$ , i.e., the explicit expressions of the multipole moments therein (see Sections 4.2 and 4.4); on the other hand, the terms in the inner post-Newtonian expansion $-- h$ that are associated with radiation-reaction effects, i.e., those terms which depend on the boundary conditions of the radiative field at infinity, and which correspond in the present case to a post-Newtonian source which is isolated from other sources in the Universe; see Section 5.2.

4.2 General expression of the multipole expansion

Theorem 5. Under the hypothesis of matching, Eq. (103 *), the multipole expansion of the solution of the Einstein field equation outside a post-Newtonian source reads

$[ ] 4G +∑∞ (− )ℓ { 1 αβ } ℳ (hαβ) = ℱ 𝒫B=0 □ −r1et^rB ℳ (Λαβ) − --4 ----∂L -ℋ L (t − r∕c) , (105 ) c ℓ=0 ℓ! r$

where the “multipole moments” are given by

Here, $τ-αβ$ denotes the post-Newtonian expansion of the stress-energy pseudo-tensor in harmonic coordinates as defined by Eq. (23 *).

Proof (see Refs. [44 *, 49 *]): First notice where the physical restriction of considering a post-Newtonian source enters this theorem: The multipole moments (106 *) depend on the post-Newtonian expansion $-- τ αβ$ of the pseudo-tensor, rather than on $τ αβ$ itself. Consider $Δ αβ$ , namely the difference between $hαβ$ , which is a solution of the field equations everywhere inside and outside the source, and the first term in Eq. (105 *), namely the finite part of the retarded integral of the multipole expansion $αβ ℳ (Λ )$ :

From now on we shall generally abbreviate the symbols concerning the finite-part operation at $B = 0$ by a mere $ℱ 𝒫$ . According to Eq. (30 *), $hαβ$ is given by the retarded integral of the pseudo-tensor $ταβ$ . So,

In the second term the finite part plays a crucial role because the multipole expansion $αβ ℳ (Λ )$ is singular at $r = 0$ . By contrast, the first term in Eq. (108 *), as it stands, is well-defined because we are considering only some smooth field distributions: $ταβ ∈ C ∞ (ℝ4)$ . There is no need to include a finite part $ℱ 𝒫$ in the first term, but a contrario there is no harm to add one in front of it, because for convergent integrals the finite part simply gives back the value of the integral. The advantage of adding artificially the $ℱ 𝒫$ in the first term is that we can re-write Eq. (108 *) into the more interesting form

in which we have also used the fact that $ℳ (Λαβ) = 16π4G-ℳ (τα β) c$ because $Tαβ$ has a compact support. The interesting point about Eq. (109 *) is that $αβ Δ$ appears now to be the (finite part of a) retarded integral of a source with spatially compact support. This follows from the fact that the pseudo-tensor agrees numerically with its own multipole expansion when $r > a$ [by the same equation as Eq. (102 *)]. Therefore, $ℳ (Δ αβ)$ can be obtained from the known formula for the multipole expansion of the retarded solution of a wave equation with compact-support source. This formula, given in Appendix B of Ref. [59 ], yields the second term in Eq. (105 *),

but in which the moments do not yet match the result (106 *); instead,²⁹

The reason is that we have not yet applied the assumption of a post-Newtonian source. Such sources are entirely covered by their own near zone (i.e., $a ≪ λ$ ), and, in addition, for them the integral (111 *) has a compact support limited to the domain of the source. In consequence, we can replace the integrand in Eq. (111 *) by its post-Newtonian expansion, valid over all the near zone:

Strangely enough, we do not get the expected result because of the presence of the second term in Eq. (112 *). Actually, this term is a bit curious, because the object $-------- ℳ (τα β)$ it contains is only known in the form of the formal series whose structure is given by the first equality in Eq. (104 *) (indeed $τ$ and $h$ have the same type of structure). Happily – because we would not know what to do with this term in applications – we are now going to prove that the second term in Eq. (112 *) is in fact identically zero. The proof is based on the properties of the analytic continuation as applied to the formal structure (104 *) of $-----αβ- ℳ (τ )$ . Each term of this series yields a contribution to Eq. (112 *) that takes the form, after performing the angular integration, of the integral $∫ ℱ 𝒫B=0 +0 ∞ drrB+b(lnr )p$ , and multiplied by some function of time. We want to prove that the radial integral $∫+ ∞ drrB+b(ln r)p 0$ is zero by analytic continuation ( $∀B ∈ ℂ$ ). First we can get rid of the logarithms by considering some repeated differentiations with respect to $B$ ; thus we need only to consider the simpler integral $∫ +∞ 0 drrB+b$ . We split the integral into a “near-zone” integral $∫ℛ drrB+b 0$ and a “far-zone” one $∫+ ∞ drrB+b ℛ$ , where $ℛ$ is some constant radius. When $ℜ (B )$ is a large enough positive number, the value of the near-zone integral is $B+b+1 ℛ ∕(B + b + 1)$ , while when $ℜ (B)$ is a large negative number, the far-zone integral reads the opposite, $− ℛB+b+1 ∕(B + b + 1)$ . Both obtained values represent the unique analytic continuations of the near-zone and far-zone integrals for any $B ∈ ℂ$ except $− b − 1$ . The complete integral $∫ +∞ drrB+b 0$ is equal to the sum of these analytic continuations, and is therefore identically zero ( $∀B ∈ ℂ$ , including the value $− b − 1$ ). At last we have completed the proof of Theorem 5:

The latter proof makes it clear how crucial the analytic-continuation finite part $ℱ 𝒫$ is, which we recall is the same as in our iteration of the exterior post-Minkowskian field [see Eq. (45 *)]. Without a finite part, the multipole moment (113 *) would be strongly divergent, because the pseudo-tensor $-α β τ$ has a non-compact support owing to the contribution of the gravitational field, and the multipolar factor $xL$ behaves like $rℓ$ when $r → + ∞$ . The latter divergence has plagued the field of post-Newtonian expansions of gravitational radiation for many years. In applications such as in Part B of this article, we must carefully follow the rules for handling the $ℱ 𝒫$ operator.

The two terms in the right-hand side of Eq. (105 *) depend separately on the length scale $r0$ that we have introduced into the definition of the finite part, through the analytic-continuation factor $^rB = (r∕r0)B$ introduced in Eq. (42 *). However, the sum of these two terms, i.e., the exterior multipolar field $ℳ (h)$ itself, is independent of $r0$ . To see this, the simplest way is to differentiate formally $ℳ (h)$ with respect to $r0$ ; the differentiations of the two terms of Eq. (105 *) cancel each other. The independence of the field upon $r 0$ is quite useful in applications, since in general many intermediate calculations do depend on $r0$ , and only in the final stage does the cancellation of the $r0$ ’s occur. For instance, we have already seen in Eqs. (93 *) – (94 *) that the source quadrupole moment $Iij$ depends on $r0$ starting from the 3PN level, but that this $r0$ is compensated by another $r0$ coming from the non-linear “tails of tails” at the 3PN order.

4.3 Equivalence with the Will–Wiseman formalism

Will & Wiseman [424 *] (see also Refs. [422 , 335 ]), extending previous work of Epstein & Wagoner [185 ] and Thorne [403 ], have obtained a different-looking multipole decomposition, with different definitions for the multipole moments of a post-Newtonian source. They find, instead of our multipole decomposition given by Eq. (105 *),

There is no $ℱ 𝒫$ operation in the first term, but instead the retarded integral is truncated, as indicated by the subscript $ℛ$ , to extend only in the “far zone”: i.e., $|x′| > ℛ$ in the notation of Eq. (31 *), where $ℛ$ is a constant radius enclosing the source ( $ℛ > a$ ). The near-zone part of the retarded integral is thereby removed, and there is no problem with the singularity of the multipole expansion $αβ ℳ (Λ )$ at the origin. The multipole moments $𝒲L$ are then given, in contrast with our result (106 *), by an integral extending over the “near zone” only:

Since the integrand is compact-supported there is no problem with the bound at infinity and the integral is well-defined (no need of a $ℱ 𝒫$ ).

Let us show that the two different formalisms are equivalent. We compute the difference between our moment $ℋL$ defined by Eq. (106 *), and the moment $𝒲L$ given by Eq. (115 *). For the comparison we split $ℋ L$ into far-zone and near-zone pieces corresponding to the radius $ℛ$ . Since the finite part $ℱ 𝒫$ present in $ℋL$ deals only with the bound at infinity, it can be removed from the near-zone piece, which is then seen to reproduce $𝒲L$ exactly. So the difference between the two moments is simply given by the far-zone piece:

We transform next this expression. Successively we write $ταβ = ℳ (ταβ)$ because we are outside the source, and $-------- ℳ (ταβ) = ℳ (τ αβ)$ thanks to the matching equation (103 *). At this stage, we recall from our reasoning right after Eq. (112 *) that the finite part of an integral over the whole space $3 ℝ$ of a quantity having the same structure as $-----αβ- ℳ (τ )$ is identically zero by analytic continuation. The main ingredient of the present proof is made possible by this fact, as it allows us to transform the far-zone integration $|x| > ℛ$ in Eq. (116 *) into a near-zone one $|x| < ℛ$ , at the price of changing the overall sign in front of the integral. So,

Finally, it is straightforward to check that the right-hand side of this equation, when summed up over all multipolarities $ℓ$ , accounts exactly for the near-zone part that was removed from the retarded integral of $ℳ (Λαβ )$ in the first term in Eq. (114 *), so that the “complete” retarded integral as given by the first term in our own definition (105 *) is exactly reconstituted. In conclusion, the formalism of Ref. [424 *] is equivalent to the one of Refs. [44 *, 49 ].

4.4 The source multipole moments

In principle, the bridge between the exterior gravitational field generated by the post-Newtonian source and its inner field is provided by Theorem 5; however, we still have to make the connection with the explicit construction of the general multipolar and post-Minkowskian metric in Section 2. Namely, we must find the expressions of the six STF source multipole moments $IL$ , $JL, ...,ZL$ parametrizing the linearized metric (35 *) – (37) at the basis of that construction.³⁰

To do this we first find the equivalent of the multipole expansion given in Theorem 5, which was parametrized by non-trace-free multipole functions $ℋLαβ$ , in terms of new multipole functions $ℱαL β$ that are STF in all their indices $L$ . The result is

where the STF multipole functions (witness the multipolar factor $ˆxL ≡ STF [xL]$ ) read

Notice the presence of an extra integration variable $z$ , ranging from $− 1$ to $1$ . The $z$ -integration involves the weighting function

which approaches the Dirac delta-function (hence its name) in the limit of large multipolarities, $lim ℓ→+ ∞ δℓ(z) = δ(z)$ , and is normalized in such a way that

The next step is to impose the harmonic-gauge conditions (21 *) onto the multipole decomposition (118 *), and to decompose the multipole functions $ℱ αLβ(u)$ into STF irreducible pieces with respect to both $L$ and their spatial indices contained into $αβ = 00,0i,ij$ . This technical part of the calculation is identical to the one of the STF irreducible multipole moments of linearized gravity [154 ]. The formulas needed in this decomposition read

where the ten tensors $RL, (+)TL+1, ...,(− 2)UL −2,VL$ are STF, and are uniquely given in terms of the $αβ ℱ L$ ’s by some inverse formulas. Finally, the latter decompositions yield the following.

Theorem 6. The STF multipole moments $IL$ and $JL$ of a post-Newtonian source are given, formally up to any post-Newtonian order, by ( $ℓ ≥ 2$ )

These moments are the ones that are to be inserted into the linearized metric $αβ h(1)$ that represents the lowest approximation to the post-Minkowskian field $hαβ = ∑ Gnh αβ ext n≥1 (n)$ defined in Eq. (50 *).

In these formulas the notation is as follows: Some convenient source densities are defined from the post-Newtonian expansion (denoted by an overbar) of the pseudo-tensor $ταβ$ by

(where $τii ≡ δijτij$ ). As indicated in Eqs. (123) all these quantities are to be evaluated at the spatial point $x$ and at time $u + zr∕c$ .

For completeness, we give also the formulas for the four auxiliary source moments $WL, ...,ZL$ , which parametrize the gauge vector $φ α1$ as defined in Eqs. (37):

As discussed in Section 2, one can always find two intermediate “packages” of multipole moments, namely the canonical moments $ML$ and $SL$ , which are some non-linear functionals of the source moments (123) and (125), and such that the exterior field depends only on them, modulo a change of coordinates. However, the canonical moments $ML$ , $SL$ do not admit general closed-form expressions like (123) – (125).³¹

These source moments are physically valid for post-Newtonian sources and make sense only in the form of a post-Newtonian expansion, so in practice we need to know how to expand the $z$ -integrals as series when $c → +∞$ . Here is the appropriate formula:

Since the right-hand side involves only even powers of $1∕c$ , the same result holds equally well for the advanced variable $u + zr∕c$ or the retarded one $u − zr∕c$ . Of course, in the Newtonian limit, the moments $IL$ and $JL$ (and also $ML$ and $SL$ ) reduce to the standard Newtonian expressions. For instance, $I (u) = Q (u ) + 𝒪 (1∕c2) ij ij$ recovers the Newtonian quadrupole moment (3 *).³²

Needless to say, the formalism becomes prohibitively difficult to apply at very high post-Newtonian approximations. Some post-Newtonian order being given, we must first compute the relevant relativistic corrections to the pseudo stress-energy-tensor $τ-αβ$ ; this necessitates solving the field equations inside the matter, which we shall investigate in the next Section 5. Then $τ-αβ$ is to be inserted into the source moments (123) and (125), where the formula (126 *) permits expressing all the terms up to that post-Newtonian order by means of more tractable integrals extending over $3 ℝ$ . Given a specific model for the matter source we then have to find a way to compute all these spatial integrals; this is done in Section 9.1 for the case of point-mass binaries. Next, we must substitute the source multipole moments into the linearized metric (35 *) – (37), and iterate them until all the necessary multipole interactions taking place in the radiative moments $UL$ and $VL$ are under control. In fact, we have already worked out these multipole interactions for general sources in Section 3.3 up to the 3PN order in the full waveform, and 3.5PN order for the dominant $(2,2)$ mode. Only at this point does one have the physical radiation field at infinity, from which we can build the templates for the detection and analysis of gravitational waves. We advocate here that the complexity of the formalism simply reflects the complexity of the Einstein field equations. It is probably impossible to devise a different formalism, valid for general sources devoid of symmetries, that would be substantially simpler.

	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