accounting for the energy which is radiated by the quadrupole waves (there is agreement with the ..... is defined in terms of these tensors. A particu...

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arXiv:gr-qc/9710037v1 7 Oct 1997

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arXiv:gr-qc/9710037v1 7 Oct 1997

Quadrupole-quadrupole gravitational waves

Luc Blanchet D´epartement d’Astrophysique Relativiste et de Cosmologie (UPR 176 du CNRS), Observatoire de Paris, 92195 Meudon Cedex, France

Short title: Quadrupole-quadrupole gravitational waves PACS number(s): 04.25.Nx, 04.30.Db Submitted to: Classical Quantum Grav. Date: 7 February 2008

1

Abstract.

This paper investigates the non-linear self-interaction of quadrupole

gravitational waves generated by an isolated system.

The vacuum Einstein field

equations are integrated in the region exterior to the system by means of a postMinkowskian algorithm. Specializing in the quadrupole-quadrupole interaction (at the quadratic non-linear order), we recover the known results concerning the nonlocal modification of the ADM mass-energy of the system accounting for the emission of quadrupole waves, and the non-local memory effect due to the re-radiation by the stress-energy distribution of linear waves.

Then we compute all the local

(instantaneous) terms which are associated in the quadrupole-quadrupole metric with the latter non-local effects. Expanding the metric at large distances from the system, we obtain the corresponding radiation-field observables, including all non-local and transient contributions. This permits notably the completion of the observable quadrupole moment at the 5/2 post-Newtonian order.

2

1. Introduction

In general relativity, the multipole moments of any finite distribution of energy and momentum interact with each other in vacuum, through the non-linearities of the field equations. In particular, the multipole moments which describe the gravitational waves emitted by an isolated system do not evolve independently, but rather couple together (including with themselves), giving rise to non-linear physical effects. The simplest multipole interaction which contributes to the radiation field is that between the (mass-type) quadrupole moment Mij and the mass monopole M . The latter moment is the constant mass-energy of the source as measured at spatial infinity (ADM mass). Associated with the multipole interaction Mij × M is the non-linear

effect of tails. This effect is due to the backscatter of linear waves (described by

Mij ) onto the space-time curvature generated by the mass-energy M . The tails can be computed within the theory of gravitational perturbations of the Schwarzschild background (see e.g. [1–3]). A consequence of the existence of tails is the non-locality in time, as the tails are in the form of integrals depending on the history of the source from −∞ in the past to the retarded time t − r/c [4–7]. It is known that the tails appear both in the radiation field and in the radiation reaction forces at the 3/2 post-

Newtonian order (1.5PN, or order c−3 when c → ∞) relatively to the quadrupole radiation [8,9].

Next in complexity is the interaction of the quadrupole moment with itself, or quadrupole self-interaction Mij × Mkl . Two closely related non-local (or hereditary)

effects are known with this particular interaction.

As shown by Bonnor and

collaborators [10,11,4,6], the total mass M gets modified by a non-local integral accounting for the energy which is radiated by the quadrupole waves (there is agreement with the Einstein quadrupole formula). On the other hand the radiation field involves a non-local contribution whose physical origin is the re-emission of waves by the linear waves [12–16,9]. This contribution can be easily computed by using as the source of waves in the right side of the Einstein field equations the effective stress-energy tensor of gravitational waves (averaged over several wavelengths). As shown by Christodoulou [14] and Thorne [15] this implies a permanent change in the wave amplitude from before to afterward a burst of gravitational waves, which can be interpreted as the contribution of gravitons in the known formulas for the linear 3

memory [17,18]. The latter non-linear memory integral appears at the 2.5PN order in the radiation field (1PN order relatively to the tail integral). Now the metric which corresponds to the quadrupole-quadrupole interaction involves also, besides the non-local contributions, many terms which by contrast depend on the multipole moments at the sole retarded instant t − r/c. In the following we shall often qualify these local terms as instantaneous (following the terminology of [8]). The instantaneous terms in the radiation field (to order 1/r) are transient in the sense that they return to zero after the passage of a burst of gravitational waves. On physical grounds, it can be argued that the instantaneous terms do not play a very important role. However, these terms do exist and form an integral part of the field generated by very relativistic sources like inspiraling compact binaries (see e.g. [19]). In fact, the complete quadrupole-quadrupole metric including all the hereditary and instantaneous contributions will be needed in the construction of very accurate theoretical wave forms to be used by the future detectors LIGO and VIRGO. The present paper is devoted to the computation of the quadrupole-quadrupole metric, using the so-called multipolar-post-Minkowskian method proposed by Blanchet and Damour [20,21] after previous work by Bonnor [10] (his double-series approximation method) and Thorne [22] (how to start the post-Minkowskian iteration using STF multipole moments). The instantaneous quadrupole-quadrupole terms have never been computed within the multipolar-post-Minkowskian method. However, they have been computed by Hunter and Rotenberg [6] within the double-series method, in the case where the source is axi-symmetric. The non-linear interaction between quadrupoles has also received attention more recently within the double-series method [23]. The main result of this paper (having in mind the application to astrophysical sources to be detected by VIRGO and LIGO) is the completion of the observable quadrupole moment of a general isolated source at the 2.5PN order. To reach this result we also take into account previous results concerning the tails at the 1.5PN order [9], and the multipole moments given by explicit integrals over the source to 2.5PN order [24,25]. (Note that the 2.5PN approximation in the observable quadrupole moment gives no contribution to the phase evolution of inspiraling compact binaries [25], however it will be required when we compute the 2.5PN wave form.) In a following paper [26], that we shall refer to as paper II, we shall investigate 4

the monopole-monopole-quadrupole interaction (at the cubic non-linear order) which enters the radiation field at the 3PN approximation. The present paper and paper II are part of the program of computing the field generated by inspiraling compact binaries with 3PN and even 3.5PN accuracy (see [27–30] for why such a very high accuracy is necessary). The plan of the paper is as follows. In Section 2 we summarize from [20,21] the method for computing the field non-linearities.

In Section 3 we investigate

(following [8]) the general structure of the quadratic non-linearities. Section 4 deals with the explicit computation of the quadrupole-quadrupole metric. The results are presented in the form of tables of numerical coefficients. Finally, in Section 5, we expand the metric at infinity and obtain the observable moments in the radiation field (essentially the quadrupole). The technical formulas for integrating the wave equation are relegated to Appendix A, and some needed results concerning the dipolequadrupole interaction are derived in Appendix B. Henceforth we pose c = 1, except when we discuss the post-Newtonian order at the end of Section 5.

2. Non-linearities in the external field We summarize the method set up in [20] for the computation of the quadratic and higher non-linearities in the field generated by an isolated system. The computation is performed in the exterior (vacuum) weak-field region of the system, where the components of the gravitational field hµν are numerically small as compared to one. √ Here hµν denotes the metric deviation hµν = −gg µν − η µν , with g µν the inverse and

g the determinant of the metric gµν , and η µν the Minkowski metric diag(−1, 1, 1, 1). The field hµν admits in the exterior weak-field region a post-Minkowskian expansion, 2 µν n µν hµν = Ghµν 1 + G h2 + ... + G hn + ... ,

(2.1)

where G is the Newtonian gravitational constant, which plays here the role of bookkeeping parameter in the non-linearity expansion. The first term in (2.1) satisfies the vacuum Einstein equations linearized around the Minkowski metric. In harmonic (or De Donder) coordinates this means two equations, 5

⊓ ⊔hµν 1 =0 ,

(2.2a)

∂ν hµν 1 =0 .

(2.2b)

In the first equation ⊓ ⊔ denotes the flat space-time wave (d’Alembertian) operator.

The second equation (divergenceless of the field) is the harmonic gauge condition. The equations (2.2), supplemented by the condition of retarded potentials, are solved by means of a multipolar expansion. It is known that only two sets of multipole moments,

the mass-type moments ML and current-type moments SL (both depending on the retarded time t − r), are sufficient to parametrize the general multipole expansion (see e.g. [22]). In terms of these moments the general solution reads [22]

h00 1 =−4 h0i 1

=4

X (−)ℓ ℓ≥0

ℓ!

X (−)ℓ ℓ≥1

+4

ℓ!

∂L r −1 ML (t − r) ,

(2.3a)

h i (1) −1 ∂L−1 r MiL−1 (t − r)

X (−)ℓ ℓ εiab ∂aL−1 r −1 SbL−1 (t − r) , (ℓ + 1)!

(2.3b)

ℓ≥1

hij 1

=−4

X (−)ℓ ℓ≥2

ℓ!

h i (2) −1 ∂L−2 r MijL−2 (t − r)

i h X (−)ℓ ℓ (1) −1 ∂aL−2 r εab(i Sj)bL−2 (t − r) . −8 (ℓ + 1)!

(2.3c)

ℓ≥2

Here the superscript (n) denotes n time derivatives. The index L is a shorthand for a multi-index composed of ℓ indices, L = i1 i2 ...iℓ (similarly, L − 1 = i1 i2 ...iℓ−1 ,

aL − 2 = ai1 ...iℓ−2 , and so on), and ∂L denotes a product of ℓ space derivatives, ∂L = ∂i1 ∂i2 ...∂iℓ . The multipole moments ML and SL are symmetric and tracefree

(STF) with respect to all their indices (see [31] for a recapitulation of our notation and conventions). A priori the multipole moments ML (t) and SL (t) are arbitrary functions of time. They describe potentially the physics of a general isolated source as seen in its exterior field [22,20]. The only physical restriction is that the mass monopole M (total massenergy or ADM mass), the mass dipole Mi (position of the center of mass times the 6

mass), and the current dipole Si (total angular momentum) are constant. Technically speaking, this is a consequence of the harmonic gauge condition (2.2b) (see e.g. [22]). In this paper we shall set the mass dipole Mi to zero by shifting the origin of coordinates to the center of mass. In order to describe an isolated system, we must implement a condition of no-incoming radiation, ensuring that the radiation field is entirely generated by the system. We assume that the field is stationary in the remote past, i.e. that the moments ML (t) and SL (t) are constant before some finite instant in the past, say when t ≤ −T (hence Mi cannot be a linear function of time and is necessarily constant or zero). This assumption may seem to be somewhat restrictive,

but we can check a posteriori that the formulas derived in this paper and paper II admit a well-defined limit when −T → −∞ in more general physical situations,

such as the formation of the system by initial gravitational scattering. The multipole moments ML (t) and SL (t), subject to the previous restrictions, play the role of “seed”

moments for the construction of the exterior field (2.1). In particular we shall express the results of this paper in terms of products of Mij with itself. We shall not use the expressions of the multipole moments ML and SL as explicit integrals over the source. However, these expressions are known in the post-Newtonian approximation [24,25], and should be used in applications. The coefficient of G2 in (2.1) is the quadratically non-linear metric, whose precise definition we recall. The field equations for this coefficient read, still using harmonic coordinates,

µν ⊓ ⊔hµν 2 = N2 ,

∂ν hµν 2 = 0 .

(2.4a) (2.4b)

The d’Alembertian equation involves a quadratic source N2µν = N µν (h1 , h1 ) generated by the linearized gravitational field (2.3), where

1 1 N µν (h, h) = − hρσ ∂ρ ∂σ hµν + ∂ µ hρσ ∂ ν hρσ − ∂ µ h∂ ν h 2 4 (µ ρ ν)σ µρ σ ν − 2∂ hρσ ∂ h + ∂σ h (∂ hρ + ∂ρ hνσ ) 1 1 1 µν λ ρσ ρ σ ρλ − ∂λ hρσ ∂ h + ∂ρ h∂ h + ∂ρ hσλ ∂ h . +η 4 8 2 7

(2.5)

From (2.4b) we deduce ∂ν N2µν = 0 .

(2.6)

Then the quadratic metric hµν 2 , solving (2.4) and the condition of stationarity in the past, is obtained as the sum of two distinct contributions, µν µν hµν 2 = u2 + v2 .

(2.7)

µν Basically the first contribution uµν 2 is the retarded integral of the source N2 .

However, in our case the source is in the form of a multipole expansion (valid only in the exterior of the system and singular at r = 0), so we cannot apply directly the usual retarded integral operator, whose range of integration intersects the system at retarded time. A way out of this problem, proposed in [20], consists of multiplying the actual source term N2µν by a factor (r/r0 )B , where B is a complex number and r0 denotes a certain constant having the dimension of a length. When the real part of B is large enough all the power-like (except for logarithms) singularities of the multipole expansion at the spatial origin of the coordinates r = 0 are cancelled. [Actually we consider separately each multipolar pieces, with given multipolarities ℓ, so that the maximal power of the singularities is finite, and B can indeed be chosen in such a way.] Applying the retarded integral on each multipolar pieces of the product (r/r0 )B N2µν results in a function of B whose definition can be analytically continued to a neighbourhood of B = 0, at which value it admits a Laurent expansion. The finite part at B = 0 (in short FPB=0 ) of the latter expansion is our looked-for solution, as µν µν it satisfies the correct wave equation (⊓ ⊔uµν in the form of 2 = N2 ), and is like N2

a multipole expansion. Note that the latter process represents simply a convenient

mean to find a solution of the wave equation whose source is in the form of a multipole expansion. Other processes could be used as well, but this one is particularly powerful as it yields many explicit formulas to be used in practical computations (see Appendix A below and Appendix A of paper II). Hence the first contribution in (2.7) reads uµν ⊔−1 2 = FPB=0 ⊓ R

"

where ⊓ ⊔−1 R denotes the usual retarded integral 8

r r0

B

N2µν

#

,

(2.8)

(⊓ ⊔−1 R f )(x, t)

1 =− 4π

ZZZ

d3 x′ f (x′ , t − |x − x′ |) . ′ |x − x |

(2.9)

When dealing with the metric at quadratic order, it can be proved that the Bdependent retarded integral in (2.8) is actually finite when B → 0 (the finite part

is not followed by any pole). So uµν 2 is simply given by the value at B = 0 of the retarded integral (see [8] and Section 4). But this is due to the special structure of the quadratic source N2µν , and does not remain true at cubic and higher non-linear approximations (see for instance paper II). The first contribution uµν 2 solves (2.4a), but not the harmonic gauge condition µ µν (2.4b). The divergence of uµν 2 , say w2 = ∂ν u2 , is a priori different from zero. Using

(2.6) we find w2µ

" # B r n i µi B = FPB=0 ⊓ ⊔−1 N . R r0 r 2

(2.10)

The explicit factor B comes from the differentiation of r B in (2.8) (we denote ni = ∂i r = xi /r). Owing to this factor the finite part in (2.10) is in fact a residue at B = 0, or coefficient of B −1 in the Laurent expansion. [The source term in (2.10) has a structure which is different from N2µν , and unlike in (2.8) the integral admits in general a (simple) pole at B = 0.] The second contribution v2µν in (2.7) is then defined in such a way as to compensate exactly the (a priori) non-zero divergence w2µ µ of uµν 2 , while being a homogeneous solution of (2.4a). This is possible because w2 is

a particular retarded solution of ⊓ ⊔w2µ = 0 (in the exterior region). As such, it admits

a unique multipolar decomposition in terms of four sets of STF tensors AL , BL , CL , DL , namely

w20 =

X ℓ≥0

w2i =

X ℓ≥0

+

∂L r −1 AL (t − r) , ∂iL r −1 BL (t − r)

X ℓ≥1

(2.11a)

∂L−1 r −1 CiL−1 (t − r) + εiab ∂aL−1 r −1 DbL−1 (t − r) . (2.11b)

These tensors can be computed straigthforwardly from the known expression (2.10), and the contribution v2µν is defined in terms of these tensors. A particular definition 9

was proposed in the equations (4.13) of [20], where it was denoted q2µν . Here we shall define this second contribution slightly differently, and accordingly we use the different notation v2µν . The various components of v2µν are given by

R R RR v200 = − r −1 A + ∂a r −1 − Aa + Ca − 3Ba , R R (1) − εiab ∂a r −1 Db v20i = r −1 − Ci + 3Bi X − ∂L−1 r −1 AiL−1 ,

(2.12a)

(2.12b)

ℓ≥2

v2ij

= − δij r

−1

B+

X ℓ≥2

2δij ∂L−1 r −1 BL−1 − 6∂L−2(i r −1 Bj)L−2

h i (1) (2) + ∂L−2 r −1 (AijL−2 + 3BijL−2 − CijL−2 ) −1 . − 2∂aL−2 r εab(i Dj)bL−2

(2.12c)

Like in (2.11) all the tensors are evaluated at the retarded time t − r. We note that

the formulas (2.12) are non-instantaneous, as they depend on the moments ML and SL at any time less than t − r through the first and second time anti-derivatives RR R t−r R R R t−r Ca = −∞ Ca (t′ )dt′ of A, Aa , Ca , denoted e.g. by A = −∞ A(t′ )dt′ and (see [20] for discussions). [To quadratic order the tensors AL , ..., DL are given by

some instantaneous functionals of the moments ML and SL .] The main property of v2µν is ∂ν v2µν = −w2µ , which is easily checked on the expressions (2.12). Furthermore

⊓ ⊔v2µν = 0, so the quadratic metric (2.7) is, indeed, a solution of both the wave equation (2.4a) and the gauge condition (2.4b). Note that the spatial trace v2ii = δ ij v2ij is

especially simple,

v2ii = −3r −1 B .

(2.12d)

The choice of definition (2.12) adopted here, which differs from the choice adopted in [20], is for convenience in future work. Of course we are free to adopt one definition or another because such a choice is equivalent to a choice of gauge. However, the definition (2.12) is slightly preferable to the definition proposed in [20] when we want to express the multipole moments ML and SL as integrals over the source. Thus we take the present opportunity to redefine the construction of the exterior metric using 10

this new definition. [Actually it can be checked that all intermediate and final results of this paper and paper II are independent of the choice of definition.] To the third (cubic) and higher non-linear iterations the construction of the external metric proceeds exactly along the same line, namely µν µν hµν n = un + vn ,

(2.13)

where the first term uµν n is the finite part of the retarded integral of the source to B µν the nth post-Minkowskian order, uµν ⊔−1 n = FPB=0 ⊓ R [(r/r0 ) Nn (h1 , ..., hn−1 )], and

where the second term vnµν is defined from the divergence wnµ = ∂ν uµν n by the same

formulas (2.11)-(2.12). (See [20] for the proof that the construction of the metric can be implemented to any post-Minkowskian order.)

3. Structure of the quadratically non-linear field In this section we investigate the structure of the quadratic metric hµν 2 defined by (2.7)-(2.12). For simplicity we omit most of the numerical coefficients and indices, so as to focus our attention on the basic structure of the metric. A precise computation of the numerical coefficients is dealt with in Section 4. The structure of the linearized metric (2.3) is that of a sum of retarded multipolar waves, consisting of p spatial derivatives (say) acting on monopolar waves r −1 X(t−r), h1 ≈

X

∂P [r −1 X(t − r)] .

(3.1)

Our notation ≈ refers to the structure of the expression. By expanding the derivatives ∂P (which act both on the pre-factor r −1 and on the retardation t − r) we get h1 ≈

X j≥1

n ˆ Q r −j Z(t − r) ,

(3.2)

where the powers of 1/r are j ≥ 1, and where we have expressed the angular

dependence of each term using STF products of unit vectors n ˆ Q = n

In practice, one may compute (3.2) from (3.1) by 11

decomposing ∂P on the basis of STF spatial derivatives ∂ˆQ and using (A.15) in Appendix A. After insertion of (3.2) into the quadratic source term N2µν one finds

N2 ≈

X

k≥2

n ˆ L r −k F (t − r) ,

(3.3)

where the powers of 1/r start with k = 2 (as is clear from the fact that N2 is quadratic in h1 which is of order 1/r). The functions F are composed of sums of quadratic products of derivatives of the functions Z in (3.2). The main problem is to compute u2 defined by (2.8). In view of the structure (3.3), it is a priori required to compute the finite part FPB=0 of the retarded integral of any term n ˆ L r −k F (t − r) with multipolarity ℓ and radial dependence with k ≥ 2. All the

required formulas are listed in Appendix A (which summarizes results obtained mostly in previous works [20,8,9]). Notably, we know from Appendix A that the (finite part of the) retarded integral in the case k = 2 is irreducibly non-local or non-instantaneous. See (A.3)–(A.7) in Appendix A. When the power k satisfies 3 ≤ k ≤ ℓ + 2, where

ℓ is the multipolarity (this excludes the monopolar case ℓ = 0), we know that the corresponding retarded integral is instantaneous, and given by (A.11)–(A.12). Finally, when k ≥ ℓ + 3, we have again a non-local expression, given by (A.13)–(A.14), except in special combinations like (A.16) for which the non-local integrals cancel out.

The computation of u2 = FP⊓ ⊔−1 R N2 can be implemented by an algebraic computer

program, following the successive steps (3.1)–(3.3) and applying on each terms composing (3.3) the formulas (A.5), (A.11) and (A.13)–(A.14). This is probably the most efficient way to obtain u2 . However, in doing so we would discover that the only non-local integrals left in u2 come from the source terms having a radial dependence with k = 2, in other words all the non-local integrals coming from terms having k ≥ ℓ + 3 actually cancel out. Practically speaking the source terms having k ≥ ℓ + 3 turn out to combine into combinations such as (A.16) yielding purely instantaneous

contributions. This fact (proved in [8]) is special to the quadratic non-linearities, and does not stay true at higher orders, e.g. at the cubic order as seen in paper II. The proof of the latter assertion uses specifically the quadratic structure of the source N2 . Instead of inserting in (2.5) the linearized metric in expanded form (3.2) and then working out all derivatives to arrive at (3.3), we keep the structure of the source as it basically is, i.e. a sum of quadratic products of multipolar waves, 12

N2 ≈

X

h i h i −1 −1 ∂P r X(t − r) ∂Q r Y (t − r) ,

(3.4)

involving spatial multi-derivatives with P = i1 ...ip and Q = j1 ...jq and some functions of time X and Y . Then we perform on each term of (3.4) a sequence of operations by parts [i.e. ∂i A∂j B = ∂i (A∂j B) − A∂i ∂j B], by which the spatial derivatives acting on the wave in the left (say) are shifted in front and to the right. This leads to N2 ≈

X

o n ∂P r −1 X(t − r)∂R [r −1 Y (t − r)] .

(3.5)

Only at this stage does one expand the space derivatives ∂R (inside the curly brackets), while leaving the derivatives ∂P in front un-expanded. The result reads N2 ≈

X

2≤k≤ℓ+2

∂P n ˆ L r −k H(t − r) ,

(3.6)

where the functions H are sums of products of X and time-derivatives of Y , and where we have projected the angular dependence of the terms inside the brackets on STF tensors n ˆ L . The point is that the radial dependence of the terms inside the brackets is related to the multipolarity ℓ by 2 ≤ k ≤ ℓ + 2. This can easily be seen

from (3.5), as the expansion of the multipolar wave ∂R [r −1 Y ] is composed of a sum of terms ∂ˆL [r −1 Y ′ ] which have 1 ≤ k ≤ ℓ + 1 [see (A.1)], yielding 2 ≤ k ≤ ℓ + 2

after multiplication by the factor r −1 on the left. The next operation is to single out the terms with pure radial dependence k = 2. All these terms can be obtained by applying ∂P on the terms inside the brackets having k = 2. In this way one generates besides all the terms k = 2 many other terms with k ≥ 3, but the latter terms can be

recombined into terms of the same form as in (3.6) (and thus having 3 ≤ k ≤ ℓ + 2). See [8] for the proof. Thus (3.6) can be rewritten as N2 ≈ r −2 Q(n, t − r) +

X

3≤k≤ℓ+2

∂P n ˆ L r −k F (t − r) ,

(3.7)

where Q(n, t−r) denotes the coefficient of r −2 in the (finite) expansion of the quadratic source when r → ∞ with t − r = const. The definition of Q(n, t − r) is N2µν

1 = 2 Qµν (n, t − r) + O r 13

1 r3

.

(3.8)

Next, in anticipation of applying the finite part of the retarded integral, we multiply (3.7) by a factor r B , and introduce r B inside the brackets using again a series of operations by parts. In this way we get many new terms, but which all involve at least one factor B coming from the differentiation of r B during the latter operations. Thus

r B N2 ≈ r B−2 Q(n, t − r) +

X

3≤k≤ℓ+2

∂P n ˆ L r B−k F (t − r) + O(B) .

(3.9)

Applying the retarded integral on both sides of (3.9), commuting ⊓ ⊔−1 R with ∂P , and taking the finite part, we are left with (the finite part of) retarded integrals of three

types of terms: (i) the first term in (3.9) which has radial dependence r −2 , (ii) the terms in the brackets of (3.9) having radial dependence such that 3 ≤ k ≤ ℓ + 2, and (iii) the terms O(B) which have the structure (3.3) with any radial dependence

k but carry at least one factor B. In Case (ii), the retarded integrals are given by the instantaneous expressions (A.11). In Case (iii), the retarded integrals are given by (A.18) when the power of B is one, and are zero for higher powers. So in Case (iii) the retarded integrals are also instantaneous. Therefore we can state the following result [8]: the only non-local integrals in u2 = FP⊓ ⊔−1 R N2 come from Case (i), namely

from the source terms whose radial dependence is r −2 , and which are denoted by r −2 Q(n, t − r) in (3.8). These non-local integrals are given by (A.3)–(A.7). This

result holds true only in the case of the quadratic non-linearity. In cubic and higher non-linearities, some hereditary integrals are generated by source terms with radial dependence such that k ≥ ℓ + 3. Incidentally, note that the decomposition (3.9) of the source shows that the BB dependent retarded integral ⊓ ⊔−1 R [r N2 ] is finite at B = 0, i.e. does not involve any

pole when B → 0. Thus the finite part at B = 0 is simply equal to the value of

B ⊓ ⊔−1 R [r N2 ] at B = 0. This can be checked from the formulas (A.3), (A.11) and (A.18)

in Appendix A, which are all finite at B = 0. Here again this is a peculiarity of the quadratic approximation.

We are now in the position to write down the structure of the first contribution u2 . From (3.9) and (A.3), (A.11) and (A.18) we have 14

u2 ≈ t 2 +

X

k≥1

n ˆ L r −k G(t − r) ,

(3.10)

where the functions G(t − r) depend instantaneously on the multipole moments ML

and SL (i.e. at the retarded time t − r only), and where the first term is non-local and given by

−2 µν tµν ⊔−1 Q (n, t − r)] . 2 =⊓ R [r

(3.11)

On the other hand the second contribution v2 , defined by (2.11)-(2.12), involves some time anti-derivatives of quadratic products of moments. We denote these time antiderivatives by s2 . Thus the quadratic-order metric can be written as h2 ≈ t2 + s2 +

X

k≥1

n ˆ L r −k P (t − r) ,

(3.12)

where t2 is the non-local integral (3.11), where s2 are some anti-derivatives [given by (4.12) below in the case of the quadrupole-quadrupole interaction], and where we have many instantaneous terms. See (4.13) and Table 2 below for the complete expression of h2 in the case of the quadrupole-quadrupole interaction.

4. The quadrupole-quadrupole metric We specialize the previous investigation in the case of the interaction between two quadrupole moments Mab and Mcd . Thus we keep in the linearized metric (2.3) only the terms corresponding to Mab ,

−1 h00 Mab , 1 = −2∂ab r (1) −1 , r M h0i = 2∂ a 1 ai (2)

−1 hij Mij . 1 = −2r

(4.1a) (4.1b) (4.1c)

(Henceforth we use the same notation for the metric constructed out of the quadrupole moment as for the complete metric involving all multipolar contributions.) 15

Inserting (4.1) into the quadratic source (2.5), we can work out explicitly all the terms composing the source either in the all-expanded form (3.3) or in the more elaborate form (3.7). Notably, we find that the terms with radial dependence r −2 take the classic form of the stress-energy tensor of a massless field, Qµν (n, t − r) = k µ k ν Π(n, t − r) ,

(4.2)

where k µ denotes the Minkowskian null vector k µ = (1, ni), and where Π is given by (3)

(3)

(3)

(3)

(3)

(3) Π = nabcd Mab Mcd − 4nab Mac Mbc + 2Mab Mab .

(4.3)

The quantity Π is proportional to the power (per unit of steradian) carried by the linearized waves. In the general case, Qµν involves besides the quadrupole-quadrupole terms all the interacting terms between multipole moments with ℓ ≥ 2 and the mass monopole M . We introduce the STF multipole decomposition of Π, Π(n, u) =

X

nL ΠL (u) .

(4.4)

ℓ≥0

From (4.3), the only non-zero multipolar coefficients are 4 (3) (3) M M , 5 ab ab 24 (3) (3) Πij = − Ma

(4.5a) (4.5b) (4.5c)

(Π0 denotes the coefficient with multipolarity ℓ = 0 or spherical average.) With this definition, and with the help of (A.7) in Appendix A, we can write the non-local integral t2 as tµν 2

=

⊓ ⊔−1 R

Z +∞ Z dΩ′ k ′µ k ′ν kµ kν ds Π = − Π(n′ , t − s) . ′ r2 4π s − rn.n r

(4.6)

Some equivalent expressions follow from (A.3)–(A.5). For instance, we can write tµν 2 in the explicit form

tµν 2

1 = 2

Z

+∞ r

h i (3) (3) µν (3) (3) µν (3) (3) ds Mij Mkl ∂ijkl {6} − 4Mai Maj ∂ij {4} + 2Mab Mab ∂ µν {2} ,

(4.7a)

16

where the moments in the integrand are evaluated at the time t − s, where the multi-

µν derivative operators mean for instance ∂ij = ∂ µ ∂ ν ∂i ∂j with ∂ µ = (−∂/∂s, ∂i), and

where we use the special notation

{p} =

(s − r)p ln(s − r) − (s + r)p ln(s + r) p!r

(4.7b)

[see (A.4) in Appendix A]. Having the term tµν 2 , we undertake the computation of all the instantaneous terms Mab × Mcd in uµν 2 [second terms in (3.10)]. The computation is straigthforward but

tedious. As said above, when doing practical computations (notably by computer), the best method is the somewhat brute force method consisting of obtaining the source in the all-expanded form (3.3), and applying the (finite part of the) retarded integral on each term of (3.3) with k ≥ 3 using the formulas (A.11) and (A.13). This is simpler than working out the source in the more elaborate form (3.9), and using

the manifestly instantaneous formulas (A.11) and (A.18). The brute force method has also the advantage that one can check that all the non-local integrals but those coming from the r −2 term cancel out (as well as the associated logarithms of r). The term u2 takes the form

u00 2

=

t00 2

6−k 6 X 1 X k (6−k−m) (m) am n ˆ abcd Mab Mcd + k r m=0 k=1

+ u0i 2

=

t0i 2

(m) (6−k−m) ˆ ab Mac Mbc bkm n

+

(6−k−m) (m) Mab ckm Mab

6 6−k X 1 X k (6−k−m) (m) + d n ˆ iabcd Mab Mcd r k m=0 m

,

(4.8a)

k=1

(m)

(6−k−m) + ekm n ˆ iab Mac Mbc

+ uij 2

=

tij 2

(6−k−m) (m) k gm n ˆ abc Mia Mbc

(6−k−m)

k + fm ni Mab

+

(m)

Mab

(6−k−m) (m) hkm na Mib Mab

6 6−k X 1 X k (6−k−m) (m) + pm n ˆ ijabcd Mab Mcd k r m=0

, (4.8b)

k=1

(m)

k (6−k−m) + qm n ˆ ijab Mac Mbc (6−k−m)

+ skm n ˆ ij Mab

(6−k−m)

+ ukm δij Mab

17

(6−k−m)

k + rm δij n ˆ abcd Mab

(m)

(m)

(6−k−m) Mab + tkm δij n ˆ ab Mac Mbc (m)

(6−k−m)

k Mab + vm n ˆ abc(i Mj)a

(m)

Mbc

(m)

Mcd

(6−k−m)

k + wm n ˆ a(i Mj)b

(m)

(6−k−m)

Mab + xkm nab Mij

(m)

Mab (m) (6−k−m) (m) (6−k−m) k k Mj)a , Mj)b + zm Ma(i + ym n ˆ ab Ma(i

(4.8c)

k where all moments are evaluated at time t − r and where akm , bkm , ..., zm are purely

numerical coefficients. Using the algebraic computer program Mathematica [32] we have obtained the numerical coefficients listed in Table 1. Next we follow the second part of the construction of the metric, and compute the divergence w2µ = ∂ν uµν 2 [see (2.10)-(2.12)]. To this end we need the divergence µν of the integral tµν 2 , which is easily evaluated by noticing that ∂ν t2 can be written,

analogously to (2.10), as some residue at B = 0 of a retarded integral (because of the explicit factor B), and furthermore that the radial dependence of the integrand is merely r −3 (because it comes from the differentiation of the source term r −2 ). From (A.18), we know that when k = 3 the residue is non-zero only when the multipolarity is ℓ = 0. This yields immediately ∂ν tµν 2

" # Z B µ r k dΩ µ 1 −1 = FPB=0 ⊓ ⊔R B Π =− k Π(n, t − r) . 3 r0 r r 4π

(4.9)

The µ = 0 component of (4.9) is proportional to the angular average of Π, already computed in (4.5a). One can check that the µ = i component is zero. Thus,

4 −1 (3) (3) ∂ν t0ν Mab Mab , 2 =− r 5 ∂ν tiν 2 =0 .

(4.10a) (4.10b)

Knowing (4.10) we can obtain w2µ by direct differentiation of the expressions (4.8), using the coefficients in Table 1.

Again the computation is quite lengthy, but

it provides us with an important check. Indeed, from (2.11) one must find that the divergence w2µ is a solution of the source-free d’Alembertian equation, namely ⊓ ⊔w2µ = 0. This test is very stringent, as a single erroneous coefficient in Table 1 would

signify almost certainly its failure. Thus we determine all the tensors AL , ..., DL in (2.11). The non-zero ones are given by

A=−

2 (6) 176 (5) (1) 22 (4) (2) 8 (3) (3) Mab Mab − Mab Mab − Mab Mab − Mab Mab , (4.11a) 75 225 45 15 18

24 (4) (3) (1) (4.11b) Ma

v2µν . In particular, we find some time anti-derivatives which define sµν 2 [see (3.12)] as t−r

s00 2

4 = r −1 5

s0i 2

Z t−r 4 (2) −1 (3) duMce Mde (u) , = − εiab ∂a r εbcd 5 −∞

Z

−∞

(3)

(3)

duMab Mab (u) ,

sij 2 = 0 .

(4.12a) (4.12b) (4.12c)

The physical interpretation of these anti-derivatives is clear. Indeed the linearized metric (2.3) depends in particular on the mass monopole M and current dipole Si of the source (both are constant). Physically M and Si represent the total massenergy and angular momentum of the system before the emission of gravitational radiation. Now the integral s00 2 given by (4.12a) represents a small modification, due to the emission of radiation, of the initial mass M . This is clear from the comparison of (4.12a) and (2.3a), showing that there is exact agreement with the energy loss by radiation as given by the standard Einstein quadrupole formula. This result is originally due to Bonnor [10], and Bonnor and Rotenberg [4]. Similarly the integral s0i 2 given by (4.12b) represents a modification of the total angular momentum in agreement with the quadrupole formula for the angular momentum loss. [There is no loss of total linear momentum at the level of the quadrupole-quadrupole interaction (one needs to consider also the mass octupole Mabc and/or the current quadrupole Sab ).] The quadratic metric hµν 2 can now be completed. We add up the two contributions µν uµν 2 [given by (4.8) and Table 1] and v2 [given by (4.11) and (2.12)]. The local terms

are written in the same form as in (4.8). Thus,

hµν 2

=

tµν 2

+

sµν 2

6 6−k X 1 X + same expressions as in (4.8abc) but r k m=0 k=1

19

with coefficients

′k a′k m , ..., zm

,

(4.13)

µν where the non-local integrals tµν 2 and s2 are given by (4.7) and (4.12), and where all ′k the coefficients a′k m , ..., zm are listed in Table 2.

5. The quadrupole-quadrupole metric in the far zone We investigate the behaviour of the quadrupole-quadrupole field hµν 2 in the far zone, near future null infinity (i.e. at large distances when we recede from the source at the speed of light). The degrees of freedom of the radiation field, at leading order in the inverse of the distance, are contained in the transverse and tracefree (TT) TT projection of the spatial components ij of the metric (say gij ). These are the so-called

observable (or radiative) multipole moments, which are “measured” in an experiment TT located far away from the system. The TT projection gij to first order in 1/r reads

TT gij

∞ X 4 2ℓ 1 1 = Pijab nL−2 UijL−2 − naL−2 εab(i Vj)bL−2 + O r ℓ! ℓ+1 r2 ℓ=2

(5.1)

(with G = c = 1), where UL and VL denote the mass-type and current-type observable moments (both are functions of t − r), and where the TT projection operator is 1 Pijab (n) = (δia − ni na )(δjb − nj nb ) − (δij − ni nj )(δab − na nb ) . 2

(5.2)

The ℓ-dependent coefficients in (5.1) are chosen so that UL and VL agree at the linearized order with the ℓth time derivatives of the moments ML and SL [compare with (2.3c)]. Let us consider first the non-local integral tµν 2 [see (4.6)-(4.7)]. As we know from (A.8), the asymptotic expansion when r → ∞, t − r = const of the retarded integral

of a source with radial dependence r −2 is composed of terms 1/r n and ln r/r n . As such, tµν 2 behaves like ln r/r when r → ∞, t − r = const. The logarithm is due

to the deviation of the flat cones t − r = const in harmonic coordinates from the 20

true space-time null cones. The metric in harmonic coordinates is not of the normal Bondi-type at future null infinity [33,34]. Removal of the logarithm is done using radiative coordinates, so defined that the associated flat cones agree, asymptotically when r → +∞, with the true null cones (see e.g. [21]). This method, adopted in [9],

permits to compute the observable moments in the non-local term tµν 2 . Here we follow another method, found by Thorne [15] and Wiseman and Will [16], which consists of applying first the TT projection operator (5.2) on tµν 2 . Because the TT projection kills any (linear) gauge term in the 1/r part of the metric, this method shortcuts the need of a transformation to radiative coordinates. However, one must be cautious in taking the limit r → ∞, t − r = const using (4.6). It is not allowed, for instance, to

work out a leading 1/r term from the second expression in (4.6) because this term

would involve a divergent integral (in accordance with the fact that the leading term is actually ln r/r). But, as pointed out in [15,16], the divergent parts of the integral cancel out after application of the TT projection, and at the end one recovers the correct result. Thus, we compute TT (tij 2 )

= −Pijab (n)

Z

+∞

ds

r

Z

dΩ′ n′a n′b Π(n′ , t − s) . 4π s − rn.n′

(5.3)

[Note that the TT projection as defined in (5.2) is purely algebraic. Strictly speaking it agrees with the true TT projection only when acting on the leading 1/r term.] With the multipole decomposition (4.4) we have TT (tij 2 )

= −Pijab (n)

XZ ℓ≥0

t−r

du ΠL (u)

−∞

Z

dΩ′ n′a n′b n′L . 4π t − u − rn.n′

(5.4)

We decompose the product of unit vectors n′a n′b n′L on the basis of STF tensors [31], we drop the terms having zero TT-projection, and we express the remaining terms using the Legendre function of the second kind Qℓ [see (A.6c)]. Restoring the traces on the STF tensors, and droping further terms having zero TT-projection, we obtain

TT (tij 2 )

Z t−r X 1 ℓ(ℓ − 1) = − Pijab du ΠabL−2 (u) nL−2 r (2ℓ + 3)(2ℓ + 1)(2ℓ − 1) −∞ ℓ≥2 t−u t−u t−u − 2(2ℓ + 1)Qℓ + (2ℓ + 3)Qℓ−2 . × (2ℓ − 1)Qℓ+2 r r r (5.5) 21

The limit at future null infinity can now be applied. Indeed it suffices to insert the expansion of Qℓ (x) when x → 1+ as given by (A.8). As expected the limit is finite,

because the terms ln(x − 1)/2 in the expansions of the Qℓ ’s cancel out. This yields [13,15,16,9]

TT (tij 2 )

X 2 nL−2 = − Pijab r (ℓ + 1)(ℓ + 2) ℓ≥2

Z

t−r

du ΠabL−2 (u) + O −∞

ln r r2

.

(5.6)

In the case of the quadrupole-quadrupole interaction we find [using (4.5)]

TT (tij 2 )

1 = Pijab r

Z

t−r −∞

4 (3) (3) ln r 1 (3) (3) du Mcc − ncd M

(5.7)

(where the brackets <> denote the STF projection). The non-local integral (5.7) represents the quadrupole-quadrupole contribution to the non-linear memory [14-16,9]. We now turn our attention to the instantaneous part of the metric. All the terms have been obtained in Section 4. We need only to apply the TT projection on the 1/r part of hij 2 as given by (4.13) and the coefficients listed in Table 2. After several transformations using STF techniques we obtain

(hij 2

−

TT tij 2 )

2 (5) 1 10 (4) (1) 4 (3) (2) = Pijab − Mc

(where the overbar on e means that the index e is to be excluded from the STF projection). With (5.7) and (5.8) we deduce the observable moments by comparison with (5.1). The quadrupole-quadrupole interaction contributes only to the observable mass quadrupole moment Uij , mass 24 -pole moment Uijkl , and current octupole moment Vijk . We find 22

2 δUij = − 7

δUijkl

δVijk

2 = 5

Z

t−r

−∞

t−r

1 (5) 5 (4) (1) 2 (3) (2) (3) (3) Ma

21 (5) 63 (4) (1) 102 (3) (2) M

(3)

(3)

M

Note that the non-local integrals are present in Uij and Uijkl , but not in the current moment Vijk .

Finally we add back in (5.9) the factor G and the powers of 1/c which are required in order to have the correct dimensionality. When this is done we find that δUij is of order 1/c5 in the post-Newtonian expansion (2.5PN order), while both δUijkl and δVijk are of order 1/c3 or 1.5PN. This permits writing the complete expressions of Uij to 2.5PN order, and Uijkl , Vijk to 1.5PN order. In the case of Uij the reasoning has been done in [25], which shows that besides the quadrupole-quadrupole terms Mab × Mcd there is an interaction between Mab and the (static) current dipole Sc also

at 2.5PN order, and there is the standard contribution of tails (computed in [9]) at 1.5PN order. [In principle there is also an interaction between the mass octupole Mabc and the mass dipole Md at 2.5PN order, but we have chosen Md = 0.] Thus, from (5.9a) and the equation (5.7) in [25], Z 2GM t−r/c 11 t − r/c − u (4) Uij (t − r/c) = + + Mij (u) du ln c3 2b 12 −∞ Z t−r/c 1 (5) 2 G (3) (3) duMa

The tail integral involves a constant 11/12 computed in Appendix B of [9]. (See also [9] for the definition of the constant b.) The coefficient in front of the term Mab × Sc is

computed in Appendix B below. In a similar way, we find that Uijkl and Vijk involve the same types of multipole interactions, but that the terms Mab × Mcd are of the

same 1.5PN order as the tail terms. Thus,

23

Z t−r/c G 59 t − r/c − u (6) Uijkl (t − r/c) = + 3 2M + Mijkl (u) du ln c 2b 30 −∞ Z t−r/c 21 (5) 2 (3) (3) duM

The constants 59/30 and 4/3 are obtained from Appendix C in [24]. The coefficient (4)

of the term S

Appendix 1. Formulas to compute the quadratic non-linearities This appendix presents a unified compendium of formulas, many of them issued from previous works [20,8,9], which permit the computation of the quadratic nonlinearities (involving any interaction between two multipole moments). The source of the quadratic non-linearities takes the structure (3.3), so we present the formulas for the finite part [as defined in (2.8)] of the retarded integral of any term n ˆ L r −k F (t − r)

with multipolarity ℓ and radial dependence r −k where k is an integer ≥ 2. [For practical computations it is convenient to use the source in expanded form (3.3) rather 24

than in the more precise form (3.9).] The problem was solved in [20] which obtained a basic formula for the B-dependent retarded integral

⊓ ⊔−1 R

B

n ˆL 2k−3 F (t − r) = rk (2r0 )B (B − k + 2)(B − k + 1) · · · (B − k − ℓ + 2) Z +∞ (s − r)B−k+ℓ+2 − (s + r)B−k+ℓ+2 ˆ × ds F (t − s)∂L . (A.1) r r

r r0

This formula is valid (by analytic continuation) for all values of B in the complex plane, except possibly at integer values of B where there is a simple pole. Note that to the STF product of unit vectors n ˆ L in the left side corresponds a STF product of spatial derivatives ∂ˆL in the right side [31]. Our first case of interest is that of a source having k = 2. In this case (A.1) becomes

⊓ ⊔−1 R

r r0

B

n ˆL 1 F (t − r) = 2 B r 2(2r0 ) B(B − 1) · · · (B − ℓ) Z +∞ B+ℓ B+ℓ (s − r) − (s + r) , × ds F (t − s)∂ˆL r r

(A.2)

of which we compute the finite part in the Laurent expansion when B → 0. We repeat

briefly the reasoning of [8]: the coefficient in front of the integral admits a simple pole at B = 0, but at the same time the integral vanishes at B = 0 thanks to the identity (A36) in [20] (see also (4.20a) in [8]). As a result the right side of (A.2) is finite at B = 0, in agreement with the fact that the retarded integral in its usual form (2.9) is convergent, with value

⊓ ⊔−1 R

Z n ˆL (−)ℓ +∞ ds F (t − s) F (t − r) = r2 2 r ℓ ℓ (s − r) ln(s − r) − (s + r) ln(s + r) × ∂ˆL , ℓ!r

(A.3)

where we have removed the reference to taking the finite part at B = 0. Note that the length scale r0 drops out in the result (this is thanks to (4.20a) in [8]). The formula (A.3) can be generalized to the case where the angular dependence is contained in any (non-tracefree) product kα1 ...kαℓ of ℓ Minkowskian null vectors kα = (−1, n), 25

⊓ ⊔−1 R

Z kα1 ...kαℓ (−)ℓ +∞ ds F (t − s) F (t − r) = r2 2 r (s − r)ℓ ln(s − r) − (s + r)ℓ ln(s + r) × ∂α1 ... ∂αℓ , ℓ!r

(A.4)

where the space-time derivatives in the right side are defined by ∂α = (∂/∂s, ∂i). A useful alternative form of (A.3), proved e.g. in Appendix A of [9], reads ⊓ ⊔−1 R

Z s n ˆ L +∞ n ˆL dsF (t − s)Qℓ , F (t − r) = − r2 r r r

(A.5)

where Qℓ (x) denotes the ℓth-order Legendre function of the second kind (with branch cut from −∞ to 1, so x > 1). The Legendre function is given by 1

dy x−y −1 X ℓ 1 x+1 1 = Pℓ (x)ln − Pℓ−j (x)Pj−1 (x) , 2 x−1 j

1 Qℓ (x) = 2

Z

Pℓ (y)

(A.6a) (A.6b)

j=1

where Pℓ is the Legendre polynomial (see e.g. [35]). Note that by combining (A.6a) and the expansion of 1/(x − n.n′ ) in terms of Legendre polynomials (see e.g. (A26)

in [20]), one has

n ˆ L Qℓ (x) =

Z

dΩ′ n ˆ ′L , 4π x − n.n′

(A.6c)

where the angular integration dΩ′ is associated with the unit direction n′i . Combining (A.5) and (A.6c) we obtain another alternative formula, ⊓ ⊔−1 R

Z +∞ Z 1 dΩ′ F (n′ , t − s) F (n, t − r) = − ds . r2 4π s − rn.n′ r

(A.7)

This formula is valid for any function F (n, u) [not only for a function having a definite multipolarity ℓ like in (A.5)]. It can also be recovered from the retarded integral in its usual form (2.9). The leading terms in the expansion when r → ∞ (with t − r = const) follow from

the expansion of the Legendre function when x → 1+ . The expression (A.6b) yields 26

1 Qℓ (x) = − ln 2

x−1 2

−

ℓ X 1 j=1

j

+ O[(x − 1) ln(x − 1)] ,

(A.8a)

and with (A.5) this implies [9]

⊓ ⊔−1 R

X Z ℓ n ˆL 2 n ˆ L +∞ λ dλF (t − r − λ) ln + F (t − r) = r2 2r 0 2r j j=1 ln r . +O r2

(A.8b)

This formula gives the leading term ln r/r and the sub-dominant term 1/r. If we try to find the leading term using (A.7) instead of (A.5) [i.e. by changing the variable s = r + λ in (A.7) and expanding the integrand when r → ∞, t − r = const], we get

formally a 1/r term but in factor of a divergent integral, as expected since the leading

term is actually ln r/r. In the case of a source term corresponding to k ≥ 3, the relevant formula is

obtained by performing k − 2 integrations by parts of (A.1). We get in this way

⊓ ⊔−1 R

r r0

B

B k−3 X n ˆL F (i) (t − r) r α (B)ˆ n F (t − r) = i L rk r0 r k−i−2 i=0 B r n ˆ L (k−2) −1 + β(B)⊓ ⊔R F (t − r) , r0 r2

(A.9)

where the second term is a retarded integral of the type studied before, and where the coefficients are

2i (B − k + 2 + i)..(B − k + 3) αi (B) = , (B − k + 2 − ℓ + i)..(B − k + 2 − ℓ)(B − k + 3 + ℓ + i)..(B − k + 3 + ℓ) (A.10a) k−2 2 B(B − 1)..(B − ℓ) . (A.10b) β(B) = (B − k + 2)..(B − k − ℓ + 2)(B + ℓ)..(B − k + ℓ + 3) The factors symbolized by dots decrease by steps of one unit from left to right. Before taking the finite part one must study the occurence of poles at B = 0 in the coefficients (A.10). Two cases must be distinguished. In the case 3 ≤ k ≤ ℓ + 2, none of the 27

denominators in (A.10) vanish at B = 0. This implies that the second term in (A.9) is zero at B = 0, owing to the explicit factor B in the numerator of β(B). Thus we obtain (in the case 3 ≤ k ≤ l + 2) a finite result when B → 0, which is local and

independent of r0 ,

⊓ ⊔−1 R

r r0

B

n ˆL F (t − r) rk

|B=0

=− ×

2k−3 (k − 3)!(ℓ + 2 − k)! n ˆL (ℓ + k − 2)!

k−3 X j=0

(ℓ + j)! F (k−3−j) (t − r) . 2j j!(ℓ − j)! r j+1

(A.11)

This formula can be checked by applying the d’Alembertian operator on both sides. When r → ∞, t − r = const, it gives

⊓ ⊔−1 R

B

r r0

n ˆL F (t − r) rk

|B=0

2k−3 (k − 3)!(ℓ + 2 − k)! F (k−3) (t − r) n ˆL (ℓ + k − 2)! r 1 +O . (A.12) r2

=−

Next, in the case k ≥ ℓ + 3, the denominators of both α(B) and β(B) vanish at

B = 0, so α(B) involves a (simple) pole, while β(B) is finite (because the pole is

compensated by the factor B in the numerator). In this case (k ≥ ℓ + 3) the finite part reads

FPB=0 ⊓ ⊔−1 R

r r0

k−3 X

B

n ˆL F (t − r) rk

F (i) (t − r) (−)k+ℓ 2k−2 (k − 3)! = α ˜in ˆ L k−i−2 + r (k + ℓ − 2)!(k − ℓ − 3)! i=0 ˆ L (k−2) r ˆ F (k−ℓ−3) (t − r) (−)ℓ −1 n +⊓ ⊔R ln ∂L F (t − r) . × 2 r0 r r2 (A.13) The coefficients are given, when 0 ≤ i ≤ k − ℓ − 4, by α ˜i =

(−2)i (k − 3)! (k + ℓ − 3 − i)!(k − ℓ − 4 − i)! , (k + ℓ − 2)!(k − ℓ − 3)! (k − 3 − i)! 28

(A.14a)

and, when k − ℓ − 3 ≤ i ≤ k − 3, by (k + ℓ − 3 − i)! 2i (−)k+ℓ (k − 3)! α ˜i = (k + ℓ − 2)!(k − ℓ − 3)! (k − 3 − i)!(ℓ − k + i + 3)! k−ℓ−3 k−3 k+ℓ−2 X 1 ℓ+3+i−k X 1 X 1 X 1 . × − − + j j j j j=1 j=1 j=k−2−i

(A.14b)

j=k+ℓ−2−i

[One can express (A.14) with the help of the Euler Γ-function and its logarithmic derivative ψ.] Note that (A.13) has a genuine dependence on the length scale r0 through the logarithm of r/r0 , which is in factor of a homogeneous solution, say ∂ˆL

G(t − r) r

ℓ

= (−) n ˆL

ℓ X j=0

(ℓ + j)! G(ℓ−j) (t − r) . 2j j!(ℓ − j)! r j+1

(A.15)

The last term in (A.13) involves a non-local integral known from (A.3)–(A.5) or (A.7). There exists a special combination of retarded integrals (A.13) which is purely local, finite at B = 0, logarithm-free and independent of r0 . This combination, particularly useful in practical computations, reads

⊓ ⊔−1 R

r r0

B

F (1) (t − r) F (t − r) n ˆ L 2(k − 2) + [(k − 1)(k − 2) − ℓ(ℓ + 1)] k+1 rk r |B=0 (k−ℓ−2) F (t − r) F (t − r) =n ˆ L k−1 + γ ∂ˆL , (A.16a) r r

where

γ=

i (−)k 2k−2 (k − 3)! h (k + ℓ − 1)(k − ℓ − 2) − (2k − 3)(k − 2) . (k + ℓ − 1)!(k − ℓ − 2)!

(A.16b)

The formula is valid when k ≥ ℓ+2; when k = ℓ+2 we recover (A.11). The second term

in (A.16a) is an homogeneous solution fixed by our particular way of integrating the wave equation. Thus the first term by itself is a solution of the required equation, but the homogeneous solution must absolutly be taken into account when doing practical computations with this method. To leading-order when r → ∞ with t − r = const, (A.13) reads 29

FPB=0 ⊓ ⊔−1 R

B n ˆL (−)k+ℓ 2k−3 (k − 3)! n ˆL r F (t − r) = r0 rk (k + ℓ − 2)!(k − ℓ − 3)! r k−ℓ−3 Z +∞ X 1 k+ℓ−2 X 1 λ (k−2) × dλF (t − r − λ) ln + + 2r0 j j 0 j=1 j=k−2 1 . (A.17) +O r2

As in the case k = 2 given by (A.8b), the leading 1/r term is non-local. However, in contrast with (A.8b), the expansion is free of logarithms of r (this is true to all orders in 1/r), and depends irreducibly on the length scale r0 . Finally we give the expression of the pole part when B → 0 of the retarded integral

in the case k ≥ ℓ + 3. The result, easily obtained from (A.9)-(A.10), is

⊓ ⊔−1 R

B (k−ℓ−3) n ˆL r (−)k 2k−3 (k − 3)! F (t − r) B . F (t − r) = ∂ˆL r0 rk (k + ℓ − 2)!(k − ℓ − 3)! r |B=0 (A.18)

As there are only simple poles, the result is zero when the power of B is strictly larger than one.

Appendix 2. The quadrupole-(current-)dipole interaction We start from the linearized metric (2.3) written for the mass quadrupole Mab and the (static) current dipole Sa , −1 h00 Mab ), 1 = − 2∂ab (r

(B.1a) (1)

−1 h0i )Sb + 2∂a (r −1 Mai ), 1 = − 2εiab ∂a (r

(B.1b)

−1 hij Mij , 1 = − 2r

(B.1c)

(2)

and replace it into (2.5), keeping only the terms Mab × Sc .

Once the source N2µν for this interaction is known in all-expanded form (3.3), we

can apply on each of the terms the formulas of Appendix A. However, as we are only interested in the two coefficients of the terms Mab × Sc in (5.10) and (5.12), it is

sufficient to obtain the 1/r part of the metric hµν 2 when r → ∞, t − r = const. Thus 30

we proceed like in Appendix C of [24], which computed the 1/r part of hµν 2 in the cases of quadrupole-monopole interactions. We recall the necessary formulas derived in [24],

Z h i (−)ℓ n ˆL ∞ B−1 ˆ −1 dλF (ℓ) (t − r − λ) r ∂L r F (t − r) = 2r 0 X ℓ 1 ln r λ + +O , (B.2a) × ln 2 2r j r j=1 h i B −1 ˆ −1 r ∂ (r ) ∂ r F (t − r) FPB=0 ⊓ ⊔−1 i L R (−)ℓ F (ℓ) (t − r) 1 = (ni n ˆ L − δi ) +O , (B.2b) 2(ℓ + 1) r r2 h i B −1 ˆ −1 FPB=0 ⊓ ⊔−1 r ∂ (r ) ∂ r F (t − r) ij L R (−)ℓ+1 4 = − δij δℓ0 + (nij + δij )ˆ nL 2(ℓ + 1)(ℓ + 2) 3 (ℓ+1) F (t − r) − 2[δi nj + δj ni ] + 2δi r 1 . (B.2c) +O r2 FPB=0 ⊓ ⊔−1 R

We have added in (B.2c) the term − 34 δij δℓ0 (where δij and δℓ0 are Kronecker symbols)

with respect to the formula (C3) given in Appendix C of [24]. Indeed this term is mistakenly missing in the formula (C3) of [24] (but it does not change any of the results derived in [24]). The formulas (B.2) yield straightforwardly the 1/r term in the first part uµν 2 of the metric [see (2.8)]. Then we compute the (1/r term of the) divergence w2µ = ∂ν uµν 2 and deduce from (2.11)-(2.12) the second part v2µν of the metric. We find that v2µν is non zero in the case Mab × Sc , contrarily to the cases M × ML and M × SL studied

in [24] for which the v2µν ’s are zero. The result of the computation is

4 −1 (4) h00 εabc nad Mbd Sc + O(r −2 ) , 2 =− r 3 5 4 (4) (4) 0i −1 − εabc na Mib Sc − εiab nacd Mcd Sb + O(r −2 ) , h2 =r 3 6 8 (4) (4) −1 1 εabc nijad Mbd Sc − εabc na(i Mj)b Sc hij 2 =r 3 3 31

(B.3a) (B.3b)

(4)

(4)

+ δij εabc nad Mbd Sc − 2εab(i nac Mj)c Sb 1 (4) + εab(i nj)acd Mcd Sb + O(r −2 ) . 3

(B.3c)

From (B.3c) we obtain the TT projection and then deduce the associated observable moments.

Only the mass quadrupole and current octupole moments receive a

contribution, 1 (4) δUij = εab

32

(B.4a) (B.4b)

References [1] Peters P C 1966 Phys. Rev. 146 938 [2] Price R H 1972 Phys. Rev. D 5 2419 [3] Bardeen J M and Press W H 1973 J. Math. Phys. 14 7 [4] Bonnor W B and Rotenberg M A 1966 Proc. R. Soc. London A 289 247 [5] Couch W E, Torrence R J, Janis A I and Newman E T 1968 J. Math. Phys. 9 484 [6] Hunter A J and Rotenberg M A 1969 J. Phys. A 2 34 [7] Bonnor W B 1974 in Ondes et radiations gravitationnelles (CNRS, Paris) p. 73 [8] Blanchet L and Damour T 1988 Phys. Rev. D 37 1410 [9] Blanchet L and Damour T 1992 Phys. Rev. D 46 4304 [10] Bonnor W B 1959 Philos. Trans. R. Soc. London A 251 233 [11] Bonnor W B and Rotenberg M A 1961 Proc. R. Soc. London A 265 109 [12] Payne P N 1983 Phys. Rev. D 28 1894 [13] Blanchet, L 1990 Th`ese d’habilitation, Universit´e P. et M. Curie (1990) (unpublished) [14] Christodoulou D 1991 Phys. Rev. Lett. 67 1486 [15] Thorne K S 1992 Phys. Rev. D 45 520 [16] Wiseman A G and Will C M 1991 Phys. Rev. D 44 R2945 [17] Braginsky V B and Grishchuk L P 1985 Sov. Phys. JETP 62 427 [18] Braginsky V B and Thorne K S 1987 Nature 327 123 [19] Blanchet L 1997 in Relativistic gravitation and gravitational radiation Lasota J P and Marck J A eds. (Cambridge Univ. Press) p. 33 [20] Blanchet L and Damour T 1986 Philos. Trans. R. Soc. London A 320 379 [21] Blanchet L 1987 Proc. R. Soc. Lond. A 409 383 [22] Thorne K S 1980 Rev. Mod. Phys. 52 299 [23] Bonnor W B and Piper M S Implosion of quadrupole gravitational waves grqc/9610015 [24] Blanchet L 1995 Phys. Rev. D 51 2559 [25] Blanchet L 1996 Phys. Rev. D 54 1417 [26] Blanchet L Gravitational-wave tails of tails (referred to as paper II) 33

[27] Cutler C, Apostolatos T A, Bildsten L, Finn L S, Flanagan E E, Kennefick D, Markovic D M, Ori A, Poisson E, Sussman G J and Thorne K S 1993 Phys. Rev. Lett. 70 2984 [28] Cutler C, Finn L S, Poisson E and Sussmann G J 1993 Phys. Rev. D 47 1511 [29] Tagoshi H and Nakamura T 1994 Phys. Rev. D 49 4016 [30] Poisson E 1995 Phys. Rev. D 52 5719 [31] Our notation is the following: signature − + ++; greek indices =0,1,2,3; latin indices =1,2,3; g = det (gµν ); ηµν = η µν = flat metric = diag (1,1,1,1); r = |x| = (x21 + x22 + x23 )1/2 ; ni = ni = xi /r; ∂i = ∂/∂xi;

nL = nL = ni1 ni2 . . . niℓ and ∂L = ∂i1 ∂i2 . . . ∂iℓ , where L = i1 i2 . . . iℓ is

a multi-index with ℓ indices; nL−1 = ni1 . . . niℓ−1 , naL−1 = na nL−1 , etc. . . ; n ˆ L and ∂ˆL are the (symmetric) and trace-free (STF) parts of nL and ∂L , also denoted by n

34

Table 1: The numerical values of coefficients entering the expression of uµν 2 as defined in (4.8).

m 0 1 2 3 4 5 0 1 2 3 4 0 1 2 3 0 1 2 0 1 0

arXiv:gr-qc/9710037v1 7 Oct 1997

k

1

2

3

4 5 6

akm

bkm

ckm

dkm

ekm

k fm

k gm

hkm

pkm

k qm

k rm

skm

tkm

ukm

k vm

5/12

−4/7

1/50

−1/120

−13/108

23/210

59/108

−8/35

−1/120

1/33

−5/66

−5/756

103/378

−11/3150

97/330

−

29/24

−3/7

−2/45

−1/15

−23/108

29/105

37/27

34/35

−1/15

8/33

−61/264

−103/756

275/378

8/315

38/33

−

37/24

11/7

11/30

−37/120

43/54

−31/210

133/54

26/35

−37/120

37/33

−47/264

−187/378

110/189

221/630

52/33 −47/33

37/24

11/7

11/30

−37/120

43/54

−31/210

26/27

54/35

−37/120

37/33

−47/264

−187/378

110/189

221/630

29/24

−3/7

−2/45

−1/15

−23/108

29/105

121/108

−22/35

−1/15

8/33

−61/264

−103/756

275/378

8/315

−23/66

5/12

−4/7

1/50

−1/120

−13/108

23/210

8/27

−8/35

−1/120

1/33

−5/66

−5/756

103/378

−11/3150

−1/165

25/6

−12/7

0

−1/8

−13/18

23/210

59/18

−8/35

−7/40

10/33

−25/33

−5/252

103/126

0

97/33

83/12

−1/7

−2/15

−7/8

−5/9

1/6

53/18

2/5

−49/40

70/33

−205/132

−7/18

86/63

0

283/33

25/4

4/7

1/3

−3/2

1/3

4/35

17/6

16/35

−9/4

46/11

35/44

−6/7

−6/7

9/35

−29/11

83/12

−1/7

−2/15

−7/8

−5/9

1/6

89/18

−2/5

−49/40

70/33

−205/132

−7/18

86/63

0

−113/33

25/6

−12/7

0

−1/8

−13/18

23/210

16/9

−8/35

−7/40

10/33

−25/33

−5/252

103/126

0

−2/33

69/4

−18/7

−1/5

−7/8

−65/36

0

187/36

−6/5

−7/4

15/11

−75/22

−5/252

103/126

0

291/22

51/8

−20/7

−17/30

−3

−37/12

−19/70

−71/6

−38/35

−27/4

72/11

45/88

−19/28

−1/2

−9/70

45/11

51/8

−20/7

−17/30

−3

−37/12

−19/70

125/12

−24/35

−27/4

72/11

45/88

−19/28

−1/2

−9/70

−207/22

69/4

−18/7

−1/5

−7/8

−65/36

0

40/9

0

−7/4

15/11

−75/22

−5/252

103/126

0

−3/11

27

−39/7

−9/10

−9/4

−13/4

−23/70

−49/4

−102/35

−8

45/11

−153/44

5/63

37/63

−11/210

279/22

−291/8

−44/7

−53/30

−9/2

−13/2

−23/35

−17/4

−78/35

−35/2

117/11

333/88

−29/63

−16/9

−7/30

−225/22

27

−39/7

−9/10

−9/4

−13/4

−23/70

8

24/35

−8

45/11

−153/44

5/63

37/63

−11/210

−9/11

−63/4

−9

−21/10

−9/4

−13/4

−23/70

−49/4

−102/35

−75/4

90/11

−9/44

25/84

−29/42

−11/70

−18/11

−63/4

−9

−21/10

−9/4

−13/4

−23/70

8

24/35

−75/4

90/11

−9/44

25/84

−29/42

−11/70

−18/11

−63/4

−9

−21/10

0

0

0

0

0

−75/4

90/11

−9/44

25/84

−29/42

−11/70

−18/11

1