High-order half-integral conservative post-Newtonian coefficients in the redshift factor of black hole binaries
aa r X i v : . [ g r- q c ] S e p High-order half-integral conservative post-Newtonian coefficientsin the redshift factor of black hole binaries
Luc Blanchet, ∗ Guillaume Faye, † and Bernard F. Whiting
2, 1, ‡ G R ε C O , Institut d’Astrophysique de Paris — UMR 7095 du CNRS,Universit´e Pierre & Marie Curie, 98 bis boulevard Arago, 75014 Paris, France Institute for Fundamental Theory, Department of Physics,University of Florida, Gainesville, FL 32611, USA (Dated: July 24, 2018)
Abstract
The post-Newtonian approximation is still the most widely used approach to obtaining explicitsolutions in general relativity, especially for the relativistic two-body problem with arbitrary massratio. Within many of its applications, it is often required to use a regularization procedure.Though frequently misunderstood, the regularization is essential for waveform generation withoutreference to the internal structure of orbiting bodies. In recent years, direct comparison with theself-force approach, constructed specifically for highly relativistic particles in the extreme mass ratiolimit, has enabled preliminary confirmation of the foundations of both computational methods,including their very independent regularization procedures, with high numerical precision. In thispaper, we build upon earlier work to carry this comparison still further, by examining next-to-next-to-leading order contributions beyond the half integral 5.5PN conservative effect, which arise fromterms to cubic and higher orders in the metric and its multipole moments, thus extending scrutinyof the post-Newtonian methods to one of the highest orders yet achieved. We do this by explicitlyconstructing tail-of-tail terms at 6.5PN and 7.5PN order, computing the redshift factor for compactbinaries in the small mass ratio limit, and comparing directly with numerically and analyticallycomputed terms in the self-force approach, obtained using solutions for metric perturbations in theSchwarzschild space-time, and a combination of exact series representations possibly with moretypical PN expansions. While self-force results may be relativistic but with restricted mass ratio,our methods, valid primarily in the weak-field slowly-moving regime, are nevertheless in principleapplicable for arbitrary mass ratios.
PACS numbers: 04.25.Nx, 04.30.-w, 04.80.Nn, 97.60.Jd, 97.60.Lf ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]fl.edu . INTRODUCTION Over the last five years, comparison between post-Newtonian and gravitational self-forcecalculations has made rapid progress, in large part due to both high precision numericalcomputations from a self-force perspective [1–8] (either by directly linearizing the Einsteinfield equations or by using the Teukolsky equation [9–11] or the Regge-Wheeler and Zerilliequations [12, 13]), and extensive analytical computations within the post-Newtonian ap-proximation [3, 4, 14]. Much more recently, the possibility for this comparison has beendramatically extended. From the self-force side [15–18], this is due to the new application of(already more than 15 years old) techniques [19–21] with which to represent metric perturba-tion solutions for black hole space-times. On the post-Newtonian side, this has required thecomputation of previously unevaluated higher order terms including tail-of-tail effects [22]and, in particular, half-integral n PN terms that are nevertheless conservative. In this paper,we extend that most recent work. As will be seen, although the computations are indeedvery extensive, the results are quite simple to state and along with further motivation, theyare listed below, before we describe in detail the processes we have used in their derivation.
A. Motivation
The self-force problem concerns itself with computations for binary orbiting systemscomposed of compact bodies in which the mass ratio is extreme, such that a full numericalrelativity approach is unfeasible, due to the vastly different length scales associated withthe very different masses and physical sizes of the compact bodies. Foundations for thegravitational self-force (GSF) computations of compact binaries have developed over thelast two decades [23–27] (see Refs. [28–30] for reviews), following very early work by DeWitt and Brehme more than half a century ago [31]. For the conservative part of thedynamics, this has led to the recent possibility of high-order comparisons between self-force computations [1, 3, 4] on the one hand, and traditional post-Newtonian calculations(reviewed in Ref. [32]) on the other hand, with ever increasing precision.For compact binaries moving on exactly circular orbits, Detweiler [1] introduced a gaugeinvariant redshift factor, computed it numerically, and showed agreement with existing post-Newtonian (PN) analytical calculations [33] up to 2PN order. Then a systematic program ofcomparison was initiated in Refs. [3, 4] which showed that GSF and PN methods agree forthe 3PN term and specific logarithmic tail-induced contributions arising at 4PN and 5PNorders, and predicted numerically the values of high-order PN coefficients, notably the full4PN coefficient. The analytical 4PN coefficient was then obtained [16] using a combinationof analytical self-force (SF) computation and a partial derivation of the 4PN equations ofmotion in the Arnowitt-Deser-Misner (ADM) Hamiltonian formalism [34, 35], with verygood agreement with the numerical value computed in Ref. [4].Since that work, the accuracy of the numerical computation of the GSF has improveddrastically [15]. The PN coefficients of the redshift factor were obtained numerically to10.5PN order and for a subset of coefficients, also analytically, specifically those that areeither rational, or made of the product of π with a rational, or a simple sum of commonlyoccurring transcendentals [15]. An alternative self-force approach [17, 18] (based on the post-Minkowskian expansion of the Regge-Wheeler-Zerilli (RWZ) equation following Refs. [19–21]) has also obtained high order PN coefficients analytically, up to 8.5PN order.A feature of the post-Newtonian expansion at high order is the appearance of half-integral
2N coefficients (of type n PN where n is an odd integer) in the conservative dynamics ofbinary point particles, moving on exactly circular orbits. Using standard post-Newtonianmethods it was proved [22] that the dominant half-integral PN term occurs at the 5.5PNorder (confirming the finding of Ref. [15]) and originates from the non-linear “tail-of-tail”integrals [36]. Here we continue Paper I and compute, still using the traditional PN method(in principle applicable for any mass ratio), high-order half-integral PN terms at orders6.5PN and 7.5PN in the redshift factor, thus corresponding to the next-to-next-to-leadinghalf-integral contributions. B. Results
We have computed the redshift factor introduced in Ref. [1], for a particle moving on anexact circular orbit around a Schwarzschild black hole. The ensuing space-time is helicallysymmetric, with a helical Killing vector K α such that its value K α at the location of theparticle is proportional to the normalized four-velocity u α of the particle, u α = u T K α . (1.1)The redshift factor, denoted u T , is thus defined geometrically as the conserved quantityassociated with the helical Killing symmetry appropriate to conservative space-times withcircular orbits. However, adopting a coordinate system in which the helical Killing vectorreads K α ∂ α = ∂ t + Ω ∂ ϕ , where Ω is the orbital frequency of the circular motion, the redshiftfactor reduces to the t component d t/ d τ of the particle’s four-velocity (where d τ is theparticle’s proper time), and is thereby obtained as u T = (cid:20) − g αβ ( y ) v α v β c (cid:21) − / , (1.2)where g αβ ( y ) is the regularized metric evaluated at the particle’s location y α = ( ct, y i ),which we shall compute in detail in the present paper for insertion into the redshift fac-tor (1.2), and where v α = d y α / d t = ( c, v i ) is the coordinate velocity.In a first stage, our calculation is valid for a general extended matter source, in thevacuum region outside the source. Then, in a second stage, we use a matching argument tocontinue that solution inside the source, which is then specialized to a binary point particlesystem. Finally the metric is evaluated at the location of one of the particles, with thehelp of a self-field regularization, in principle dimensional regularization. Using the relativeframe of the center of mass and reducing the expressions to circular orbits, mindful of themodification of the relation between the orbital separation and the orbital frequency, wefinally obtain the redshift factor in the limit of a small mass ratio q = m /m (where m is the small particle and m is the black hole). In the test-mass limit the redshift factor isgiven by the Schwarzschild value, u T Schw = 1 √ − y , (1.3) Hereafter we refer to this paper as Paper I. y = ( Gm Ω /c ) / is the frequency-related parameter associated with the motion ofthe test-mass particle around the black hole. The self-force part to the redshift factor u T SF isthen defined as u T = u T Schw + q u T SF + O ( q ). We finally find that the half-integral conservativecontributions therein up to 2PN relative order are u T SF = − y − y − y + · · · − π y / + 810773675 π y / + 82561159467775 π y / + · · · , (1.4)where we have written only the relative 2PN terms relevant to our next-to-next-to-leadingorder calculation, i.e. the Newtonian, 1PN and 2PN terms for the dominant effects, and the5.5PN, 6.5PN and 7.5PN terms for the half-integral conservative corrections, with all theother terms, not computed in the present work, indicated by ellipsis. The result (1.4) isin full agreement with results derived by gravitational self-force methods, either numerical,semi-analytical or purely analytical [15, 17, 18].Let us emphasize again that the result (1.4) has been achieved from the traditionalpost-Newtonian approach. Contrary to various analytical and numerical self-force calcula-tions [15, 17, 18] the PN approach is completely general, i.e. it is not tuned to a particulartype of source as it is applicable to any extended post-Newtonian source with spatial com-pact support. It is remarkable that this general method can nevertheless be specialized tosuch degree that it is able to control terms up to the very high order 7.5PN.With the post-Newtonian coefficients in the redshift factor (1.4), one can straightfor-wardly obtain, by making use of the first law of black hole binary mechanics [14], the corre-sponding coefficients in the PN binding energy and angular momentum of the system [37]and the most important effective-one-body (EOB) potential [8, 38].In the remainder of this paper, we first discuss vacuum solutions in the exterior zone(Section II). Then we investigate tail-of-tail terms in the near zone (Section III), listing theterms which need to be evaluated, and introduce a gauge transformation to shorten thesubsequent calculation. In Section IV, we set up the PN iteration of tails of tails, thencompute the quadratic and cubic contributions in turn. We end with a brief discussionof our results (Section V), with Appendix A providing an alternative derivation of somekey results, and with Appendix B providing the source terms required for our tail-of-tailcalculations.
II. SOLVING THE EINSTEIN EQUATIONS IN THE EXTERIOR ZONE
In the present paper we shall continue and extend the method of Paper I. Namely wecompute a series of non-linear tail effects in the exterior vacuum region around a generalisolated source. We show that a crucial piece in the expansion of these non-linear tails canbe extended using a matching argument from the near zone of the source to the inner regionof the source, while the other pieces will not contribute to the half integral post-Newtonianorders in which we are interested. This crucial piece is then specialized to the case of pointparticle binaries and evaluated at the very location of one of the particles. Finally thecorresponding metric is inserted into the redshift factor of that particle and the small mass-ratio limit is computed in order to obtain the self-force prediction which is meaningfullycompared to direct analytical or numerical self-force calculations. The sign of the Newtonian term in Eq. (5.18) of Paper I should be changed and read − y . i.e. decomposed into multipolar spherical har-monics and iterated in a non-linear or post-Minkowskian way. Using harmonic coordinates,the equation that we have to solve at each post-Minkowskian order is a (flat) d’Alembertianequation for the components of the gothic metric deviation, whose right hand-side is knownfrom previous iterations. Furthermore, if we project out that equation on a basis of multi-polar spherical harmonics with multipole index ℓ , we end up with solving a generic equationof the type (cid:3) u L ( x , t ) = ˆ n L S ( r, t − r/c ) . (2.1)Here (cid:3) ≡ η µν ∂ µ ∂ ν is the flat space-time d’Alembertian operator, r = | x | is the coordinatedistance from the field point to the origin located inside the matter source, and ˆ n L is asymmetric-trace-free (STF) product of ℓ unit vectors n i = x i /r , which is equivalent to theusual basis of spherical harmonics. The solution of Eq. (2.1) for a source term S whichtends to zero sufficiently rapidly when r → u L ( x , t ) = c Z t − r/c −∞ d s ˆ ∂ L (cid:26) r (cid:20) R (cid:16) t − s − r/c , s (cid:17) − R (cid:16) t − s + r/c , s (cid:17)(cid:21)(cid:27) , (2.2)where R ( ρ, s ) denotes some intermediate function defined in terms of the source by R ( ρ, s ) = ρ ℓ Z ρ d λ ( ρ − λ ) ℓ ℓ ! (cid:18) λ (cid:19) ℓ − S ( λ, s ) . (2.3)For the present work, since we shall perform a matching of this solution to the inner field ofa post-Newtonian source, we shall need the expansion of the solution (2.2) in the near zone, i.e. when r → r → u L ( x , t ) = ˆ ∂ L ( G ( t − r/c ) − G ( t + r/c ) r ) + (cid:3) − (cid:2) ˆ n L S ( r, t − r/c ) (cid:3) . (2.4)The second term in that formula represents a particular solution of the equation (2.1), in theform of an expansion when r →
0, and given by the so-called operator of the instantaneouspotentials defined by (cid:3) − (cid:2) ˆ n L S ( r, t − r/c ) (cid:3) = + ∞ X i =0 (cid:18) ∂c∂t (cid:19) i ∆ − − i (cid:2) ˆ n L S ( r, t − r/c ) (cid:3) . (2.5)Note that such operator acts directly (term by term) on the formal expansion of the sourcewhen r →
0, given by the usual Tayor expansion of the retardation t − r/c , and does not For STF tensors we use the same notation as in Paper I: L = i · · · i ℓ denotes a multi-index composedof ℓ spatial indices ranging from 1 to 3; similarly L − i · · · i ℓ − ; ∂ L = ∂ i · · · ∂ i ℓ is the product of ℓ partial derivatives ∂ i ≡ ∂/∂x i ; x L = x i · · · x i ℓ is the product of ℓ spatial positions x i ; n L = n i · · · n i ℓ isthe product of ℓ unit vectors n i = x i /r ; the STF projection is indicated with a hat, i.e. ˆ x L ≡ STF[ x L ],ˆ n L ≡ STF[ n L ], ˆ ∂ L ≡ STF[ ∂ L ] (for instance ˆ ∂ ij = ∂ ij − δ ij ∆), or sometimes with angular bracketssurrounding the indices, e.g. x h i ℓ ∂ L − i ≡ STF[ x i ℓ ∂ L − ]; in the case of summed-up multi-indices L , we donot write the ℓ summations from 1 to 3 over the dummy indices. integral post-Newtonian approximations, and thus can be safely ignored whenlooking at the half-integral approximations. We shall check in Appendix B below that theproof of Paper I is still applicable to the extended calculation performed here. Thus all theeffects we are looking for come from the first term in Eq. (2.4), which is a homogeneoussolution of the wave equation of the type retarded minus advanced and is parametrized bythe function G ( u ) = c Z u −∞ d s R (cid:18) u − s , s (cid:19) . (2.6)Note that the retarded-minus-advanced solution is regular when r → G given the generic form of the source term S we need. As in Paper I we apply Eq. (2.6),together with Eq. (2.3), to source terms made of the requisite tails, that is, non-local in time(hereditary) terms having the form S ( r, t − r/c ) = r B − k Z + ∞ d x Q m ( x ) F ( t − rx/c ) , (2.7)where F denotes some time derivative of a multipole moment, k and m are integers, and Q m ( x ) is the Legendre function of the second kind, with branch cut from −∞ to 1, explicitlygiven in terms of the usual Legendre polynomial P m ( x ) by Q m ( x ) = 12 P m ( x ) ln (cid:18) x + 1 x − (cid:19) − m X j =1 j P m − j ( x ) P j − ( x ) . (2.8)Besides the hereditary source terms (2.7) we need also to include the case of instantaneous(non-tail) terms of the type S ( r, t − r/c ) = r B − k F ( t − r/c ), but these are immediately deducedfrom the hereditary case (2.7) by replacing formally Q m ( x ) by the truncated delta-functiondefined by δ + ( x −
1) = Y ( x − δ ( x − Y and δ are the usual Heaviside and deltafunctions. Hence we can handle all the terms given for completeness in Appendix B.Note that we systematically include inside the source term (2.7) a regularization factor r B , where B is a complex parameter destined to tend to zero at the end of the calculation.The presence of this factor ensures, when the real part ℜ ( B ) is large enough, that the sourceterm tends sufficiently rapidly toward zero when r →
0, so the applicability conditions of theintegration formulas (2.2) and (2.4) are fulfilled (see Refs. [39, 41]). From the initial domainof the complex plane where ℜ ( B ) is large enough, we extend the validity of the formulas byanalytic continuation to any complex B -values except isolated poles at integer values of B .Plugging the source term (2.7) into Eq. (2.3), and then substituting (2.3) into Eq. (2.6),we obtained in Paper I a more tractable expression of the function G that parametrizes theterm of interest to us in Eq. (2.4), namely G ( u ) = c B + ℓ − k +3 C k,ℓ,m ( B ) Z + ∞ d τ τ B F ( k − ℓ − ( u − τ ) . (2.9)Always implicit in expressions such as Eq. (2.9) is that we perform the Laurent expansion ofthe result when B → i.e. the coefficient6f the zero-th power of B . Depending on the relative values of k and ℓ (namely the power of1 /r and the multipole order of the term in question), the function F in Eq. (2.9) will appeareither multi time-differentiated or multi time-integrated, which we indicate in both cases bythe superscript ( p ) where p = k − ℓ − B -dependent coefficient C k,ℓ,m in Eq. (2.9) reads C k,ℓ,m ( B ) = 2 ℓ ℓ ! Γ( B − k + ℓ + 3)Γ( B + 1) Z + ∞ d y Q m (1 + y ) Z d z z B − k − ℓ +1 (1 − z ) ℓ (2 + yz ) B − k + ℓ +3 , (2.10)where Γ is the usual Eulerian function; see Paper I for more details. An alternative form ofEq. (2.10), also derived in Paper I, is C k,ℓ,m ( B ) = Γ( B − k − ℓ + 2)2Γ( B + 1) ℓ X i =0 ( ℓ + i )! i !( ℓ − i )! Γ( B − k + ℓ + 3)Γ( B − k + i + 3) Z + ∞ d y (cid:16) y (cid:17) i Q m (1 + y )(2 + y ) B − k +2 . (2.11)In order to control the tails present in the function G , and which are responsible forthe half-integral post-Newtonian approximations (Paper I), we need to control the poleparts when B → ∝ /B appear which will always be the case in thepresent paper, the tail part of the function G is given by G tail ( u ) = c ℓ − k +3 α ( − k,ℓ,m Z + ∞ d τ ln τ F ( k − ℓ − ( u − τ ) , (2.12)where α ( − k,ℓ,m denotes the residue ( i.e. coefficient of 1 /B ) in the Laurent expansion of thecoefficient C k,ℓ,m ( B ) when B →
0. The residue can be obtained either by carefully expandingEqs. (2.10) or (2.11) when B → e.g. ln( τ /P ), becauseany constant P will yield an intantaneous (non-tail) term that is safely ignored here. III. TAILS OF TAILS IN THE NEAR-ZONEA. Expressions in harmonic coordinates
A straightforward extension of the analysis of Paper I (see Sec. II there) shows that inorder to control the half-integral post-Newtonian coefficients up to next-to-next-to-leadingorder, namely 2PN beyond the leading-order 5.5PN coefficient obtained in Paper I, weneed to compute the tails of tails associated with the mass-type quadrupole, octupole andhexadecapole moments, and with the current-type quadrupole and octupole moments. In thenotation of Paper I, this means that we have to take into account the multipole interactions M × M × I ij (that one was sufficient for Paper I), M × M × I ijk and M × M × I ijkl for massmoments, as well as M × M × J ij and M × M × J ijk for current moments. As will be discussedin Sec. IV, those interactions represent only the “seeds” for a subsequent post-Newtonianiteration, formally involving higher non-linear multipole interactions.7or all the “seed” multipole interactions we only need the functions G parametrizing theregular retarded-minus-advanced homogeneous solutions in Eq. (2.4). They are obtainedfrom applying Eqs. (2.9)–(2.11) to each one of the source terms corresponding to thesemultipole interactions. The computation is straightforward, and for completeness we presentin Appendix B the complete expressions of the required source terms, extending Eqs. (3.4)–(3.5) of Paper I. Typically all the coefficients in Eqs. (B3)–(B12) of Appendix B contributeto the final results. To ease the notation we use the following shorthand for an elementarymonopolar retarded-minus-advanced homogeneous wave, (cid:8) G ( t ) (cid:9) ≡ G ( t − r/c ) − G ( t + r/c ) r . (3.1)Corresponding multipolar retarded-minus-advanced waves are obtained by applying STFpartial space multi-derivative operators ˆ ∂ L (with multipolarity ℓ ). The near-zone expansionwhen r → ∂ L (cid:8) G ( t ) (cid:9) = − + ∞ X k =0 ˆ x L r k (2 k )!!(2 k + 2 ℓ + 1)!! G (2 k +2 ℓ +1) ( t ) c k +2 ℓ +1 . (3.2)Extending Eqs. (5.1) of Paper I, we present the multipolar tail-of-tail interactions cor-responding to the first term of Eq. (2.4), for each of the components of the gothic metricdeviation h µν ≡ √− gg µν − η µν in harmonic gauge, such that ∂ ν h µν = 0. All these contribu-tions are built from the source terms given in Eqs. (B3)–(B12). • Mass quadrupole moment:( h ) M × M × I ij = 11621 G M c Z + ∞ d τ ln τ ∂ ab (cid:8) I (3) ab ( t − τ ) (cid:9) , (3.3a)( h i ) M × M × I ij = 4105 G M c Z + ∞ d τ ln τ ˆ ∂ iab (cid:8) I (2) ab ( t − τ ) (cid:9) − G M c Z + ∞ d τ ln τ ∂ a (cid:8) I (4) ia ( t − τ ) (cid:9) , (3.3b)( h ij ) M × M × I ij = − G M c Z + ∞ d τ ln τ δ ij ∂ ab (cid:8) I (3) ab ( t − τ ) (cid:9) + 10435 G M c Z + ∞ d τ ln τ ˆ ∂ a ( i (cid:8) I (3) j ) a ( t − τ ) (cid:9) + 7615 G M c Z + ∞ d τ ln τ (cid:8) I (5) ij ( t − τ ) (cid:9) . (3.3c) • Mass octupole:( h ) M × M × I ijk = − G M c Z + ∞ d τ ln τ ∂ abc (cid:8) I (3) abc ( t − τ ) (cid:9) , (3.4a)( h i ) M × M × I ijk = − G M c Z + ∞ d τ ln τ ˆ ∂ iabc (cid:8) I (2) abc ( t − τ ) (cid:9) + 256245 G M c Z + ∞ d τ ln τ ∂ ab (cid:8) I (4) iab ( t − τ ) (cid:9) , (3.4b)8 h ij ) M × M × I ijk = 835 G M c Z + ∞ d τ ln τ δ ij ∂ abc (cid:8) I (3) abc ( t − τ ) (cid:9) − G M c Z + ∞ d τ ln τ ˆ ∂ ab ( i (cid:8) I (3) j ) ab ( t − τ ) (cid:9) − G M c Z + ∞ d τ ln τ ∂ a (cid:8) I (5) ija ( t − τ ) (cid:9) . (3.4c) • Mass hexadecapole:( h ) M × M × I ijkl = 189810395 G M c Z + ∞ d τ ln τ ∂ abcd (cid:8) I (3) abcd ( t − τ ) (cid:9) , (3.5a)( h i ) M × M × I ijkl = 11155 G M c Z + ∞ d τ ln τ ˆ ∂ iabcd (cid:8) I (2) abcd ( t − τ ) (cid:9) − G M c Z + ∞ d τ ln τ ∂ abc (cid:8) I (4) iabc ( t − τ ) (cid:9) , (3.5b)( h ij ) M × M × I ijkl = − G M c Z + ∞ d τ ln τ δ ij ∂ abcd (cid:8) I (3) abcd ( t − τ ) (cid:9) + 32495 G M c Z + ∞ d τ ln τ ˆ ∂ abc ( i (cid:8) I (3) j ) abc ( t − τ ) (cid:9) + 169945 G M c Z + ∞ d τ ln τ ∂ ab (cid:8) I (5) ijab ( t − τ ) (cid:9) . (3.5c) • Current quadrupole: ( h ) M × M × J ij = 0 , (3.6a)( h i ) M × M × J ij = 296105 G M c Z + ∞ d τ ln τ ε iab ∂ bc (cid:8) J (3) ac ( t − τ ) (cid:9) , (3.6b)( h ij ) M × M × J ij = − G M c Z + ∞ d τ ln τ ε ab ( i ˆ ∂ j ) bc (cid:8) J (2) ac ( t − τ ) (cid:9) − G M c Z + ∞ d τ ln τ ε ab ( i ∂ b (cid:8) J (4) j ) a ( t − τ ) (cid:9) . (3.6c) • Current octupole:( h ) M × M × J ijk = 0 , (3.7a)( h i ) M × M × J ijk = − G M c Z + ∞ d τ ln τ ε iab ∂ bcd (cid:8) J (3) acd ( t − τ ) (cid:9) , (3.7b)( h ij ) M × M × J ijk = 235 G M c Z + ∞ d τ ln τ ε ab ( i ˆ ∂ j ) bcd (cid:8) J (2) acd ( t − τ ) (cid:9) + 922735 G M c Z + ∞ d τ ln τ ε ab ( i ∂ bc (cid:8) J (4) j ) ac ( t − τ ) (cid:9) . (3.7c) Underlined indices mean that they should be excluded from the symmetrization T ( ij ) = ( T ij + T ji ). . Application of a gauge transformation As noticed in Paper I the tail-of-tail term M × M × I ij given by Eq. (3.3) is to be iteratedat higher non-linear order as there are some post-Newtonian terms which contribute at thesame level coming from higher non-linear iterations. However it was found that the detailsof that non-linear iteration depend on the adopted coordinate system. In Paper I twocomputations of the 5.5PN coefficient were made. One in the standard harmonic coordinatesystem, based on the previous expressions (3.3), and one in an alternative coordinate systemin which the 5.5PN terms in the 0 i and ij components of the metric are “transferred” to the00 component at that order. This alternative coordinate system has the great advantagethat it considerably simplifies the subsequent non-linear iteration. Actually, it was found inPaper I that at 5.5PN order in this coordinate system, there is no need to perform the non-linear iteration. Such a coordinate system is analogous to the Burke & Thorne coordinatesystem [42, 43] (see also [44]), in which the complete radiation reaction force at the 2.5PNorder is linear, with non-linear contributions arising only at higher post-Newtonian orders.In the present paper we shall systematically work in the alternative non-harmonic co-ordinate system so designed that it minimizes (but, at such high 7.5PN order, does notsuppress) the need for controlling non-linear contributions. Even in that optimized gaugewe shall find that the non-linear contributions are numerous and require two iterations. Wedid not attempt to perform these non-linear iterations in harmonic coordinates. Since theredshift factor we compute in fine is gauge invariant we are allowed to use whatever coor-dinate system we like. Thus we proceed with introducing appropriate gauge transformationvectors η µ to be applied to each of the multipolar pieces presented in Sec. III A. The com-plete gauge transformation is of course the sum of each of the separate multipolar pieces. Atleading 5.5PN order the mass quadrupole piece agrees with Eqs. (5.11) of Paper I, exceptthat here we do not yet focus our attention on the conservative part of the dynamics; a splitbetween conservative and dissipative parts will be made at a later stage, see Eqs. (4.43).Note also that the above gauge vectors generalize those of Paper I not only because theyinvolve more multipole interactions but also because they include all post-Newtonian terms, i.e. complete series expansions such as Eq. (3.2). • Mass quadrupole:( η ) M × M × I ij = 7715 G M c Z + ∞ d τ ln τ ∂ ab (cid:8) I (2) ab ( t − τ ) (cid:9) , (3.8a)( η i ) M × M × I ij = − G M c Z + ∞ d τ ln τ ˆ ∂ iab (cid:8) I (1) ab ( t − τ ) (cid:9) + 385 G M c Z + ∞ d τ ln τ ∂ a (cid:8) I (3) ia ( t − τ ) (cid:9) . (3.8b) • Mass octupole:( η ) M × M × I ijk = − G M c Z + ∞ d τ ln τ ∂ abc (cid:8) I (2) abc ( t − τ ) (cid:9) , (3.9a)( η i ) M × M × I ijk = 133 G M c Z + ∞ d τ ln τ ˆ ∂ iabc (cid:8) I (1) abc ( t − τ ) (cid:9) − G M c Z + ∞ d τ ln τ ∂ ab (cid:8) I (3) iab ( t − τ ) (cid:9) . (3.9b)10 Mass hexadecapole:( η ) M × M × I ijkl = 29504 G M c Z + ∞ d τ ln τ ∂ abcd (cid:8) I (2) abcd ( t − τ ) (cid:9) , (3.10a)( η i ) M × M × I ijkl = − G M c Z + ∞ d τ ln τ ˆ ∂ iabcd (cid:8) I (1) abcd ( t − τ ) (cid:9) + 169810 G M c Z + ∞ d τ ln τ ∂ abc (cid:8) I (3) iabc ( t − τ ) (cid:9) . (3.10b) • Current quadrupole:( η ) M × M × J ij = 0 , (3.11a)( η i ) M × M × J ij = − G M c Z + ∞ d τ ln τ ε iab ∂ bc (cid:8) J (2) ac ( t − τ ) (cid:9) . (3.11b) • Current octupole:( η ) M × M × J ijk = 0 , (3.12a)( η i ) M × M × J ijk = 461210 G M c Z + ∞ d τ ln τ ε iab ∂ bcd (cid:8) J (2) acd ( t − τ ) (cid:9) . (3.12b)Applying the latter linear gauge transformations we obtain new expressions for the gothicmetric coefficients, say h ′ µν . Our convention is that (for each multipole component) h ′ µν = h µν − ∂ µ η ν − ∂ ν η µ + η µν ∂ ρ η ρ . (3.13)The nice property of the metric in the new gauge is that the number ℓ of STF spatialderivatives ˆ ∂ L for each multipole is maximal, and equal to ℓ = m + s for mass moments and ℓ = m + s − m is the multipolarity of the multipole momentin question ( i.e. I M or J M ) and s is the number of spatial indices in the gothic metric ( i.e. s = 0 , , µν = 00 , i, ij ). From Eq. (3.2) we see that maximizingthe number of STF derivatives means pushing to the maximum the leading PN order, andtherefore minimizing the need of non-linear iterations at a given PN level. • Mass quadrupole:( h ′ ) M × M × I ij = 85635 G M c Z + ∞ d τ ln τ ∂ ab (cid:8) I (3) ab ( t − τ ) (cid:9) , (3.14a)( h ′ i ) M × M × I ij = − G M c Z + ∞ d τ ln τ ˆ ∂ iab (cid:8) I (2) ab ( t − τ ) (cid:9) , (3.14b)( h ′ ij ) M × M × I ij = 2143 G M c Z + ∞ d τ ln τ ˆ ∂ ijab (cid:8) I (1) ab ( t − τ ) (cid:9) . (3.14c) • Mass octupole:( h ′ ) M × M × I ijk = − G M c Z + ∞ d τ ln τ ∂ abc (cid:8) I (3) abc ( t − τ ) (cid:9) , (3.15a)11 h ′ i ) M × M × I ijk = 13027 G M c Z + ∞ d τ ln τ ˆ ∂ iabc (cid:8) I (2) abc ( t − τ ) (cid:9) , (3.15b)( h ′ ij ) M × M × I ijk = − G M c Z + ∞ d τ ln τ ˆ ∂ ijabc (cid:8) I (1) abc ( t − τ ) (cid:9) . (3.15c) • Mass hexadecapole:( h ′ ) M × M × I ijkl = 15714158 G M c Z + ∞ d τ ln τ ∂ abcd (cid:8) I (3) abcd ( t − τ ) (cid:9) , (3.16a)( h ′ i ) M × M × I ijkl = − G M c Z + ∞ d τ ln τ ˆ ∂ iabcd (cid:8) I (2) abcd ( t − τ ) (cid:9) , (3.16b)( h ′ ij ) M × M × I ijkl = 15711260 G M c Z + ∞ d τ ln τ ˆ ∂ ijabcd (cid:8) I (1) abcd ( t − τ ) (cid:9) . (3.16c) • Current quadrupole:( h ′ ) M × M × J ij = 0 , (3.17a)( h ′ i ) M × M × J ij = − G M c Z + ∞ d τ ln τ ε iab ∂ bc (cid:8) J (3) ac ( t − τ ) (cid:9) , (3.17b)( h ′ ij ) M × M × J ij = 171263 G M c Z + ∞ d τ ln τ ε ab ( i ˆ ∂ j ) bc (cid:8) J (2) ac ( t − τ ) (cid:9) . (3.17c) • Current octupole:( h ′ ) M × M × J ijk = 0 , (3.18a)( h ′ i ) M × M × J ijk = 6542 G M c Z + ∞ d τ ln τ ε iab ∂ bcd (cid:8) J (3) acd ( t − τ ) (cid:9) , (3.18b)( h ′ ij ) M × M × J ijk = − G M c Z + ∞ d τ ln τ ε ab ( i ˆ ∂ j ) bcd (cid:8) J (2) acd ( t − τ ) (cid:9) . (3.18c)Notice that h ′ ii (1) = 0 for all these pieces, which is a nice feature of the new gauge, shared infact with the harmonic gauge. Recall that expressions (3.14)–(3.18) are regular inside thesource and will be valid as they stand at the location of the particles in a binary system. IV. POST-NEWTONIAN ITERATION OF TAILS OF TAILSA. Setting up the iteration
As mentioned above, we found in Paper I that in harmonic coordinates the computationof the 5.5PN coefficient requires the control of one non-linear PN iteration, but that nonon-linear iteration is needed in the alternative non-harmonic gauge. To extend the resultup to 7.5PN order, our rationale here is to systematically use the simpler non-harmonicgauge in which the metric components are given by Eqs. (3.14)–(3.18).In the iteration process we shall have to couple the tail-of-tail pieces (3.14)–(3.18) withthe lower order 1PN metric. Since the choice of non-harmonic gauge we have made above12ffects only the higher order tail-of-tail parts of the metric, we can take for the 1PN metricthe standard form in harmonic coordinates, given by h = − c V − c (cid:16) ˆ W + 4 V (cid:17) + O (cid:18) c (cid:19) , (4.1a) h i = − c V i + O (cid:18) c (cid:19) , (4.1b) h ij = − c (cid:18) ˆ W ij − δ ij ˆ W (cid:19) + O (cid:18) c (cid:19) , (4.1c)where we follow our usual notation for appropriate metric potentials V , V i , ˆ W ij and ˆ W =ˆ W kk , defined in Sec. 5.3 of Ref. [32] for general post-Newtonian sources. We then specializethese potentials to point-particle binary sources. Denoting the masses by m A ( A = 1 , y iA ( t ) and v iA ( t ) = d y iA ( t ) / d t , the distances to the field pointby r A = | x − y A | , and the separation by r = | y − y | , we have V = U + 1 c ∂ t U + O (cid:18) c (cid:19) , (4.2a) V i = U i + O (cid:18) c (cid:19) , (4.2b)ˆ W ij = U ij − δ ij U kk − G m h ∂ ij ln r + δ ij r i − G m h ∂ ij ln r + δ ij r i − G m m ∂ g∂y ( i ∂y j )2 + O (cid:18) c (cid:19) . (4.2c)Here U , U i and U ij refer to the compact-support parts of the potentials that are given(consistently with the approximation) explicitly by U = G ˜ µ r + G ˜ µ r , (4.3a) U = G ˜ µ r + G ˜ µ r , (4.3b) U i = Gm r v i + Gm r v i , (4.3c) U ij = Gm r v i v j + Gm r v i v j . (4.3d)The potential U is 1PN-accurate and we have introduced the effective time-dependent massesat 1PN order (which are pure functions of time),˜ µ = m (cid:20) − Gm c r + 32 v c (cid:21) , (4.4)and ˜ µ obtained by exchanging the particle labels. Note that the potential U so defined isthe “super-potential” of U , in the sense that∆ U = U . (4.5)13ater, we shall systematically make use of the notion of high-order super-potentials. Finallythe non-linear interaction term in (4.2c) is expressed by means of the well-known function [45] g = ln( r + r + r ) , (4.6)which is the super-potential of 1 / ( r r ), i.e. ∆ g = r r in the sense of distributions. Laterwe shall introduce the super-potential of g itself. In Eq. (4.2c) the function g is differentiatedwith respect to the two source points y iA as indicated.The most important problem we face is the mass quadrupole case, which we shall needto iterate two times. We need to control the covariant metric components g , g i and g ij up to order 7.5PN, which means c − , c − and c − included, i.e. up to remainders O ( c − ), O ( c − ) and O ( c − ) respectively. We first write the metric components in the newgauge obtained in Eqs. (3.14)–(3.18) up to the required order, with the help of Eq. (3.2).For convenience we simply denote e.g. δh µν (1) = ( h ′ µν ) M × M × I ij , forgetting about the primeindicating the new gauge and also about the type of multipole interaction. However we callthis piece δh µν (1) because we shall eventually obtain iterated and twice-iterated contributions δh µν (2) and δh µν (3) . • Mass quadupole: δh = − G M c ˆ x ab Z + ∞ d τ ln τ h I (8) ab ( t − τ )+ r c I (10) ab ( t − τ ) + r c I (12) ab ( t − τ ) i + O (cid:18) c (cid:19) , (4.7a) δh i (1) = 17122205 G M c ˆ x iab Z + ∞ d τ ln τ h I (9) ab ( t − τ ) + r c I (11) ab ( t − τ ) i + O (cid:18) c (cid:19) , (4.7b) δh ij (1) = − G M c ˆ x ijab Z + ∞ d τ ln τ I (10) ab ( t − τ ) + O (cid:18) c (cid:19) . (4.7c) • Mass octupole: δh = 2083969 G M c ˆ x abc Z + ∞ d τ ln τ h I (10) abc ( t − τ ) + r c I (12) abc ( t − τ ) i + O (cid:18) c (cid:19) , (4.8a) δh i (1) = − G M c ˆ x iabc Z + ∞ d τ ln τ I (11) abc ( t − τ ) + O (cid:18) c (cid:19) , (4.8b) δh ij (1) = O (cid:18) c (cid:19) . (4.8c) • Mass hexadecapole: δh = − G M c ˆ x abcd Z + ∞ d τ ln τ I (12) abcd ( t − τ ) + O (cid:18) c (cid:19) , (4.9a) δh i (1) = O (cid:18) c (cid:19) , (4.9b) δh ij (1) = O (cid:18) c (cid:19) . (4.9c)14 Current quadrupole: δh = 0 , (4.10a) δh i (1) = 68484725 G M c ε iab ˆ x bc Z + ∞ d τ ln τ h J (8) ac ( t − τ ) + r c J (10) ac ( t − τ ) i + O (cid:18) c (cid:19) , (4.10b) δh ij (1) = − G M c ε ab ( i ˆ x j ) bc Z + ∞ d τ ln τ J (9) ac ( t − τ ) + O (cid:18) c (cid:19) . (4.10c) • Current octupole: δh = 0 , (4.11a) δh i (1) = − G M c ε iab ˆ x bcd Z + ∞ d τ ln τ J (10) acd ( t − τ ) + O (cid:18) c (cid:19) , (4.11b) δh ij (1) = O (cid:18) c (cid:19) . (4.11c) B. Quadratic iteration
At quadratic non-linear order we have to solve the equation (cid:3) h µν (2) = N µν (2) , (4.12)where the source term N µν (2) is made of quadratic products of derivatives of h µν (1) , symbolicallywritten as ∼ ∂h (1) ∂h (1) and ∼ h (1) ∂ h (1) . Here h µν (1) is composed by the 1PN metric (4.1)augmented at high orders by all the previous tail-of-tail pieces. Note that Eq. (4.12) is validin the new gauge but with the assumption that at the next non-linear order the harmonicgauge condition is satisfied, i.e. ∂ ν h µν (2) = 0, and later, at still higher order, we shall assumethe same, ∂ ν h µν (3) = 0 (such choices are simply a matter of convenience). The quadratic termswe need for the present iteration, consistent with the order 7.5PN, are N + N ii (2) = − h ∂ h − h i (1) ∂ i h − h ij (1) ∂ ij h − ∂ i h ∂ i h −
12 ( ∂ h ) + 2 ∂ h i (1) ∂ h i (1) + 4 ∂ h ij (1) ∂ i h j (1) + 2 ∂ i h j (1) ∂ j h i (1) + ∂ i h jk (1) ∂ i h jk (1) , (4.13a) N i (2) = 34 ∂ h ∂ i h + ∂ j h ∂ i h j (1) − ∂ j h ∂ j h i (1) , (4.13b) N ij (2) = 14 ∂ i h ∂ j h − δ ij ∂ k h ∂ k h . (4.13c)Since we are ultimately interested in the covariant metric components g µν we are consideringthe combination 00 + ii of gothic metric components which appears dominantly into g .We now replace in (4.13) the gothic metric by its explicit form which reduces up to1PN order to Eqs. (4.1) and involves all the tail-of-tail pieces δh µν (1) . Obviously the iteratedquadratic tail-of-tail pieces will come from the cross products between the 1PN metric (4.1)15nd the linear tails of tails (4.7)–(4.11). When considering such cross products, we shall haveto integrate typical terms whose general structure is ˆ x L φ , where φ is any of the potentialsappearing in Eqs. (4.1), and where ˆ x L denotes a STF product of spatial vectors comingfrom Eqs. (4.7)–(4.11). Notice that the hereditary integrals therein are simply functions oftime [see e.g. (4.24)], and essentially play a spectator role in the process, with the notableexception of that in the dominant term for which we have to consider a retardation at therelative 1PN order. In addition, because of the 1PN retardation in δh µν (1) , we shall have tointegrate the slightly more complicated source term r ˆ x L φ [see e.g. Eq. (4.7a)]. Thus, theequations we have to solve are ∆Ψ L = ˆ x L φ , (4.14a)∆Φ L = r ˆ x L φ . (4.14b)To solve them we adopt the method of “super-potentials”. Namely we introduce, given thepotential φ , the hierarchy of its super-potentials denoted φ k +2 , for any positive integer k ,where φ = φ and ∆ φ k +2 = φ k . (4.15)We thus have ∆ k φ k = φ . The explicit formulas for the solutions of Eqs. (4.14) (the firstone being needed for ℓ = 4, and the second one only for ℓ = 0 ,
1) areΨ L = ∆ − (cid:0) ˆ x L φ (cid:1) = ℓ X k =0 ( − k ℓ !( ℓ − k )! x h L − K ∂ K i φ k +2 , (4.16a)Φ L = ∆ − (cid:0) r ˆ x L φ (cid:1) = ℓ X k =0 ( − k ℓ !( ℓ − k )! x h L − K ∂ K i h r φ k +2 + 2( k + 1)(2 k + 1) φ k +4 − k + 1) x i ∂ i φ k +4 i . (4.16b)These solutions are unique in the following sense. Suppose that φ admits an asymptoticexpansion when r → ∞ (with t fixed) on the set of basis functions r λ − n , labeled by n ∈ N and where the maximal power is λ ∈ R \ N ( i.e. , is not an integer). Then, for instance, thesolution Ψ L given by Eq. (4.16a) is the unique solution of Eq. (4.14a), valid in the senseof distribution theory [46], that admits an asymptotic expansion when r → ∞ on the basisfunctions r λ + ℓ +2 − n . Similarly Φ L is the unique solution in the sense of distributions whichadmits an asymptotic expansion on the basis r λ + ℓ +4 − n . The formulas (4.16) can be easilyproved by induction. They can also be iterated if necessary; for instance we find by iterating i times the first one that∆ − i Ψ L = ∆ − i − (cid:0) ˆ x L φ (cid:1) = ℓ X k =0 ( − k ( k + i )! ℓ ! k ! i !( ℓ − k )! x h L − K ∂ K i φ k +2 i +2 . (4.17)Let us give an example of the applicability of those formulas. The 1PN compact-supportpotential U was defined by Eq. (4.3a), where we recall that the effective masses ˜ µ A are merefunctions of time. Now the hierarchy of super-potentials of U is given by U k = 1(2 k )! h G ˜ µ r k − + G ˜ µ r k − i . (4.18)16or k = 1 we recover the potential U already met in Eq. (4.3b). Of course similar expressionsapply for the other potentials U i and U ij in Eqs. (4.3). Taking only the leading-order crossterm in the expression of the non-linear source (4.13a) we find that we have to solve (cid:3) Ψ = x i ∂ j V F ij ( t ) . (4.19)Here V is the retarded potential (4.2a) and F ij is a certain function of time, which we shalldefine below to be the hereditary integral (4.24). Note that Eq. (4.19) is to be solved includ-ing the first-order retardation at 1PN order, which is simply done by using the symmetricpropagator (cid:3) − = ∆ − + c ∂ t ∆ − + O ( c − ). Using then our elementary solution (4.16a) weget, up to 1PN relative order,Ψ = (cid:3) − h x i ∂ j V F ij ( t ) i = ( x i ∂ j U − ∂ ij U ) F ij (4.20)+ 1 c (cid:20) x i (cid:16) ∂ j ∂ t U F ij + 2 ∂ j ∂ t U F (1) ij + ∂ j U F (2) ij (cid:17) − ∂ ij ∂ t U F ij − ∂ ij ∂ t U F (1) ij − ∂ ij U F (2) ij (cid:21) + O (cid:18) c (cid:19) . Below we shall need not only the super-potentials of a compact-support potential like U ,but also those of more complicated potentials such as ˆ W ij defined by Eq. (4.2c). Its firstorder super-potential reads (to Newtonian order)ˆ W ij = U ij − δ ij U kk − G m (cid:20) ∂ ij (cid:18) r (cid:16) ln r − (cid:17)(cid:19) + δ ij ln r (cid:21) − G m (cid:20) ∂ ij (cid:18) r (cid:16) ln r − (cid:17)(cid:19) + δ ij ln r (cid:21) − G m m ∂ f∂y ( i ∂y j )2 , (4.21a)where we have used the super-potential of U ij as well as the one of the function g of Eq. (4.6),namely g = f / f = 13 r · r h g − i + 16 ( r r + r r − r r ) , (4.22)where r A = x − y A , which satisfies ∆ f = 2 g in the sense of distributions (see e.g. Ref. [47]).A full hierarchy of higher super-potentials for the function g could be defined similarly.Note that the super-potentials of the non-compact potential U are obtained thanks to thesuper-potentials of g (at Newtonian order say, i.e. assimilating ˜ µ A to m A ): U = G m r + G m r + 2 G m m r r , (4.23a) (cid:0) U ) = G m ln r + G m ln r + 2 G m m g , (4.23b) (cid:0) U ) = G m r (cid:16) ln r − (cid:17) + G m r (cid:16) ln r − (cid:17) + G m m f . (4.23c)We now define as a convenient short-hand the following hereditary function of time ap-propriate for the mass quadrupole moment, F ij ( t ) = − G M Z + ∞ d τ ln τ I (8) ij ( t − τ ) , (4.24)17nd obtain the full expressions of δh µν (2) up to the requested PN order as δh + δh ii (2) = 16 c ( x i ∂ j U − ∂ ij U ) F ij + 1 c (cid:20) x i (cid:16) ∂ i ∂ t U F ij + 2 ∂ i ∂ t U F (1) ij + ∂ i U F (2) ij (cid:17) + 16 (cid:16) − ∂ ij ∂ t U F ij − ∂ ij ∂ t U F (1) ij − ∂ ij U F (2) ij (cid:17) + 85 (cid:0) r x i ∂ j U − x i ∂ j U − x ik ∂ kj U − r ∂ ij U − ∂ ij U + 16 x k ∂ ijk U ) F (2) ij + 87 (cid:16) ˆ x ijk ∂ k U − x h ij ∂ k i ∂ k U + 24 x h i ∂ jk i ∂ k U −
48 ˆ ∂ ijk ∂ k U (cid:17) F (2) ij + 4 (cid:0) ˆ x ij U − x h i ∂ j i U + 8 ∂ ij U (cid:1) F (2) ij + 4 (cid:0) ˆ x ij ∂ t U − x h i ∂ j i ∂ t U + 8 ∂ ij ∂ t U (cid:1) F ij + 16 (cid:0) ˆ x i U j − ∂ i U j (cid:1) F (1) ij − (cid:0) ˆ x ijk ∂ k ∂ t U − x h ij ∂ k i ∂ k ∂ t U + 24 x h i ∂ jk i ∂ k ∂ t U −
48 ˆ ∂ ijk ∂ k ∂ t U (cid:17) F (1) ij + 527 (cid:0) ˆ x ijkl ∂ kl U − x h ijk ∂ l i ∂ kl U + 48 x h ij ∂ kl i ∂ kl U − x h i ∂ jkl i ∂ kl U + 384 ˆ ∂ ijkl ∂ kl U (cid:17) F (2) ij + 4 (cid:0) ˆ x ij ∂ t U − x h i ∂ j i ∂ t U + 8 ∂ ij ∂ t U (cid:1) F (1) ij + 8021 (cid:16) ˆ x ij ∂ k U k − x h i ∂ j i ∂ k U k + 8 ˆ ∂ ij ∂ k U k (cid:17) F (1) ij + 16021 (cid:16) ˆ x ik ∂ k U j − x h i ∂ k i ∂ k U j + 8 ˆ ∂ ik ∂ k U j (cid:17) F (1) ij − (cid:16) ˆ x ik ∂ j U k − x h i ∂ k i ∂ j U k + 8 ˆ ∂ ik ∂ j U k (cid:17) F (1) ij + 8 (cid:16) ˆ W ij + x i ∂ j (cid:0) ˆ W + 4 U (cid:1) − ∂ ij (cid:0) ˆ W + 4 U (cid:1) (cid:17) F ij (cid:21) + O (cid:18) c (cid:19) , (4.25a) δh i (2) = 1 c (cid:20) − x j ∂ t U − ∂ j ∂ t U ) F ij − x jk ∂ i U − x j ∂ ik U + 8 ∂ ijk U ) F (1) jk + 83 (cid:16) ˆ x jk ∂ k U − x h j ∂ k i ∂ k U + 8 ˆ ∂ jk ∂ k U (cid:17) F (1) ij − (cid:16) ˆ x ij ∂ k U − x h i ∂ j i ∂ k U + 8 ˆ ∂ ij ∂ k U (cid:17) F (1) jk − (cid:0) x j (cid:0) ∂ i U k − ∂ k U i (cid:1) − (cid:0) ∂ ij U k − ∂ jk U i (cid:1)(cid:1) F jk (cid:21) + O (cid:18) c (cid:19) , (4.25b)18 h ij (2) = 1 c (cid:20) − (cid:0) x k ∂ ( i U − ∂ k ( i U (cid:1) F j ) k + 2 δ ij ( x k ∂ l U − ∂ kl U ) F kl (cid:21) + O (cid:18) c (cid:19) . (4.25c)We must also do the same for the other multipole interactions, but these arise at higher PNorder and the iteration is much simpler. We need only to consider the mass octupole andcurrent quadrupole moments, for which we define G ijk ( t ) = 2083969 G M Z + ∞ d τ ln τ I (10) ijk ( t − τ ) , (4.26a) H ij ( t ) = 68484725 G M Z + ∞ d τ ln τ J (8) ij ( t − τ ) . (4.26b)For the mass octupole moment we obtain δh + δh ii (2) = 24 c (cid:20) x ij ∂ k U − x i ∂ jk U + 8 ∂ ijk U (cid:21) G ijk + O (cid:18) c (cid:19) , (4.27)while the other components are negligible. For the current quadrupole we have δh + δh ii (2) = 8 c (cid:20)(cid:16) ˆ x bc ∂ i ∂ t U − x h b ∂ c i ∂ i ∂ t U + 8 ˆ ∂ bc ∂ i ∂ t U (cid:17) ε iab H ac − (cid:16) ˆ x jbc ∂ ij U − x h jb ∂ c i ∂ ij U + 24 x h j ∂ bc i ∂ ij U −
48 ˆ ∂ jbc ∂ ij U (cid:17) ε iab H (1) ac + 2 (cid:0) x c ∂ i U j − ∂ ci U j (cid:1) ε ija H ac − (cid:0) x b ∂ i U j − ∂ bi U j (cid:1) ε iab H aj (cid:21) + O (cid:18) c (cid:19) , (4.28a) δh i (2) = 4 c (cid:20) − x c ∂ j U − ∂ cj U ) ε ija H ac + ( x b ∂ j U − ∂ bj U ) (cid:0) ε iab H aj − ε jab H ai (cid:1)(cid:21) + O (cid:18) c (cid:19) , (4.28b)while the ij components are negligible. C. Cubic iteration
At the next-to-next-to-leading 7.5PN order ( i.e. (cid:3) h µν (3) = M µν (3) + N µν (3) . (4.29)The cubic source term is the sum of two contributions: M µν (3) which is a direct product ofthree linear terms h µν (1) and can be symbolically written as ∼ h (1) ∂h (1) ∂h (1) , and N µν (3) which19s a product between a linear term h µν (1) and a quadratic one h µν (2) , symbolically written as ∼ ∂h (1) ∂h (2) . At cubic order only the dominant contribution in the combination 00 + ii ofthe components of the source terms will be needed.Considering first the M µν (3) piece we find that the dominant contribution therein is M + M ii (3) = − h ∂ i h ∂ i h . (4.30)We replace the linear metric h by its explicit expression made of the sum of Eqs. (4.1a)and (4.7a) (in which only the leading term of order c − is to be included), and find againthat the integration can be explicitly performed thanks to the method of super-potentials.However we need to compute the super-potentials of slightly more complicated potentialswith non-compact support. A first series is (with U Newtonian) (cid:0)
U ∂ i U (cid:1) = G m ∂ i ln r + G m ∂ i ln r − G m m (cid:18) ∂g∂y i + ∂g∂y i (cid:19) , (4.31a) (cid:0) U ∂ i U (cid:1) = G m ∂ i (cid:20) r (cid:18) ln r − (cid:19)(cid:21) + G m ∂ i (cid:20) r (cid:18) ln r − (cid:19)(cid:21) − G m m (cid:18) ∂f∂y i + ∂f∂y i (cid:19) , (4.31b)where f has been defined by Eq. (4.22). To define another series we introduce the super-potential of U ∆ U namely (at Newtonian order) K = ( U ∆ U ) = Gm r ( U ) + Gm r ( U ) , (4.32)with ( U ) = Gm /r and ( U ) = Gm /r being the values of U at the locations of theparticles. Notice that in fact (at Newtonian order) K is related to the trace ˆ W = ˆ W kk ofEq. (4.2c) by K = ˆ W + U U kk . (4.33)Then we can write, with the super-potentials of K computed similarly to Eq. (4.18), (cid:0) ∂ i U ∂ i U (cid:1) = − K + U , (4.34a) (cid:0) ∂ i U ∂ i U (cid:1) = − K + G m r + G m r + G m m g , (4.34b) (cid:0) ∂ i U ∂ i U (cid:1) = − K + G m r (cid:18) ln r − (cid:19) + G m r (cid:18) ln r − (cid:19) + G m m f . (4.34c)With these results we obtain in fully closed form the solution corresponding to the directcubic source term (4.30) as δh + δh ii (3) = 18 c (cid:20) − x i (cid:0) U ∂ j U (cid:1) + 8 ∂ i (cid:0) U ∂ j U (cid:1) − x ij (cid:0) ∂ k U ∂ k U (cid:1) + 4 x i ∂ j (cid:0) ∂ k U ∂ k U (cid:1) − ∂ ij (cid:0) ∂ k U ∂ k U (cid:1) (cid:21) F ij + O (cid:18) c (cid:19) , (4.35)20ith, as we said, the other components 0 i and ij being negligible at this stage.Considering next the N µν (3) piece of the cubic source term (4.29) we find that only thefollowing contributions are needed: N + N ii (3) = − h ij (1) ∂ ij h − h ij (2) ∂ ij h − ∂ i h ∂ i h . (4.36)Again the method of super-potentials works for all the terms encountered. The neededsuper-potentials are some straightforward extensions or variants of the ones in Eqs. (4.31)and (4.34). Let us add that the super-potentials of r /r and r /r are also needed. Theseare obtained by appropriate exchanges between the field point x and the source points y A in Eq. (4.22). Posing f = − r · r h g − i + 16 ( r r + r r − r r ) , (4.37a) f = 13 r · r h g − i + 16 ( r r + r r − r r ) , (4.37b)we have ∆ f = r /r and ∆ f = r /r . Those solutions appear in the more complicatedsuper-potential (cid:0) U ∂ ij U (cid:1) = − G m (cid:20) ∂ ij (cid:18) r (cid:18) ln r − (cid:19)(cid:19) − δ ij ln r (cid:21) − G m (cid:20) ∂ ij (cid:18) r (cid:18) ln r − (cid:19)(cid:19) − δ ij ln r (cid:21) + G m m (cid:18) ∂ f ∂y ij + ∂ f ∂y ij (cid:19) . (4.38a)Finally we encounter a series of super-potentials with compact support generalizingEq. (4.32), of the type( φ ∆ U ) k +2 = 1(2 k )! h Gm ( φ ) r k − + Gm ( φ ) r k − i . (4.39)where ( φ ) A denotes the value of φ at the particle A . We are finally in a position to writedown the complete explicit form of the cubic solution of Eq. (4.36) as δh + δh ii (3) = 8 c (cid:20) x i (7 U ∂ j U − U ∂ j U ) − U ∂ ij U + 2 ∂ i U ∂ j U − x i (cid:0) U ∂ j U (cid:1) + 12 ∂ i (cid:0) U ∂ j U (cid:1) + 2 (cid:0) U ∂ ij U (cid:1) − x i (cid:0) ∂ j U ∆ U (cid:1) + x i ∂ j (cid:0) U ∆ U (cid:1) + 14 ∂ i (cid:0) ∂ j U ∆ U (cid:1) − ∂ ij (cid:0) U ∆ U (cid:1) + 14 (cid:0) ∂ ij U ∆ U (cid:1) − ∂ i (cid:0) ∂ j U ∆ U (cid:1) (cid:21) F ij + O (cid:18) c (cid:19) . (4.40) D. Miscellaneous
A few operations are still in order before obtaining the relevant metric and the result forthe redshift factor (1.2). Of course we have to sum up all the results, thereby obtaining the21ull (iterated and twice-iterated) tail-of-tail contributions in the gothic metric deviation, δh µν = δh µν (1) + δh µν (2) + δh µν (3) , (4.41)where δh µν (1) is itself the sum of Eqs. (4.7) to (4.11), δh µν (2) is the sum of (4.25), (4.27) and (4.28),and δh µν (3) is the sum of (4.35) and (4.40). The corresponding contributions in the usualcovariant metric, say δg µν , must then be deduced from (4.41). This is a straightforward stepand we get, up to the requested PN order, δg = − (cid:18) h + h ii (cid:19) (cid:0) δh + δh ii (cid:1) − h δh + h i δh i + 12 h ij δh ij − h ) δh + O (cid:18) c (cid:19) , (4.42a) δg i = (cid:18) h (cid:19) δh i + 12 h i δh + O (cid:18) c (cid:19) , (4.42b) δg ij = − δh ij + 12 (cid:0) − δh + δh kk (cid:1) δ ij − h δh δ ij + O (cid:18) c (cid:19) , (4.42c)where h µν is the 1PN gothic metric (4.1).Next we have to single out the conservative part of the metric, i.e. neglect the dissi-pative radiation reaction effects. As in Paper I we assume that the split between conser-vative and dissipative effects is equivalent to a split between “time-symmetric” and “time-antisymmetric” contributions in the following sense. We decompose each of the tail-of-tailintegrals like for instance F ij defined in Eq. (4.24), into F ij = F cons ij + F diss ij where F cons ij ( t ) = − G M Z + ∞ d τ ln τ h I (8) ij ( t − τ ) + I (8) ij ( t + τ ) i , (4.43a) F diss ij ( t ) = − G M Z + ∞ d τ ln τ h I (8) ij ( t − τ ) − I (8) ij ( t + τ ) i , (4.43b)and keep only the conservative part that is time-symmetric. This was justified in Paper I bythe fact that the equations of motion of compact binaries associated with the conservativepart of the metric defined in that way are indeed conservative, i.e. the acceleration is purelyradial for circular orbits.From the equations of motion reduced to circular orbits we obtain the relation between theseparation r between the particles and the orbital frequency Ω. This relation is importantwhen we reduce the expressions to the frame of the center of mass and then to circularorbits. We have checked that the results obtained in Eqs. (5.16)–(5.17) of Paper I aresufficient for the present purpose. However it is important that in all relations (such as theone between orbital separation and frequency) we take into account the lowest order 2PNcorrections, appropriate when performing a next-to-next-to-leading computation. For thesame reason it is also important, when we replace the complete covariant metric g µν in theredshift factor defined by Eq. (1.2), to include not only all the high order tail-of-tail pieces,but also the lower order covariant metric up to 2PN order, because of couplings between the2PN metric and the various iterated tail-of-tail pieces at next-to-next-to-leading order. Wedo not reproduce here the 2PN metric at the location of each particle since it is given in fullform by Eqs. (7.6) of Ref. [33]. 22 . DISCUSSION In this paper, using standard post-Newtonian methods (see e.g. [32]), we have computednext-to-next-to-leading contributions to Detweiler’s redshift variable [1] at odd powers inthe post-Newtonian expansion, by examining the conservative post-Newtonian dynamics ofcompact binaries moving on exactly circular orbits. Conservative PN effects at odd powers inthe PN expansion necessarily involve non-local in time or hereditary (tail) integrals extendingover the whole past history of the source [22]. They have been shown to appear first at the5.5PN order in the redshift factor for circular orbits [15]. In the standard PN approximationthey have been proved to originate from the so-called tails of tails associated with the massquadrupole moment of the source [22].Here we have extended our previous effort to 2PN order beyond the leading 5.5PN con-tribution, thus obtaining the 6.5PN and 7.5PN coefficients in the redshift factor (at linearorder in the mass ratio), which are perhaps the highest orders ever reached by traditionalPN methods. This work involved computing high-order tails of tails associated with highermass and current multipole moments. For this purpose, we have systematically workedin a preferred gauge for which the computation drastically simplifies, with respect to, say,the harmonic gauge. In addition we have employed a more efficient method to obtain theprecise coefficients of tail-of-tail integrals in the near zone of general matter sources. Fur-thermore, we could perform the non-linear iteration of tails of tails thanks to an integrationmethod based on the use of hierarchical “super-potentials”. Our analytical post-Newtoniancalculation gives results in full agreement with numerical and analytical self-force calcula-tions [15, 18].The present work is an addition to the body of works [1, 3, 4, 14, 22] that have demon-strated the beautiful consistency between analytical post-Newtonian methods, valid for anymatter source but limited to the weak-field slow-motion regime of the source, and gravita-tional self-force methods, which give an accurate description of extreme mass ratio compactbinaries even in the relativistic and strong-field regime. The agreement between PN andGSF approaches provides an indirect check that the dimensional regularization procedureinvoked in the PN calculation when it is applied to point particle binary sources, is in factequivalent to the very different procedure of subtraction of the singular field which is em-ployed in the GSF approach. Although the dimensional regularization has not been explicitlyused in the present paper, this check between very different regularization procedures wasa central motivation for our initial works [3, 4]. Our recent work [22] together with thepresent paper confirm that the machinery used in the traditional PN approach to computenon-linear effects and their associated hereditary-type integrals like tails, tails of tails andso on, is correct.In principle, the prospects of extending the present analysis to yet higher PN orders aregood. The main challenge would be to control higher non-linear multipole interactions. Inparticular, the computation of the coefficient at 8.5PN order would be feasible since we knowthe mass quadrupole moment to 3PN order. On the other hand, extension to higher-orderin the mass ratio would be possible only by controlling other multipole couplings such asthe double mass-quadrupole interaction coupled with mass monopoles.The success of the comparison performed in this paper has obviously important im-plications: post-Newtonian calculations of tails of tails at 3PN order beyond the (2.5PN)quadrupole term already play a role [32] in the generation of template waveforms for compa-rable mass compact binaries (made of neutron star or black holes) to be analyzed in ground23r space based detectors. By contrast, self-force computations are designed with the viewto generate waveforms for comparison with the extreme mass ratio inspiral signals expectedfrom future space based detectors.
Acknowledgments
We thank Sylvain Marsat for his independent proof of the basic formulas (4.16)–(4.17).This work was supported by NSF Grants PHY 1205906 and PHY 1314529 to UF. BFWacknowledges sabbatical support from the CNRS through the IAP, during the initial stagesof this work, as well as support of the French state funds managed by the ANR within theInvestissements d’Avenir programme under reference ANR-11-IDEX-0004-02. LB acknowl-edges hospitality and support from NSF and the Physics Department at UF, where part ofthis work was carried out during its final stages.
Appendix A: Alternative computation of pole part contributions
According to our discussion in Section II, in the coefficients given by Eq. (2.11), namely: C k,ℓ,m ( B ) = ℓ X i =0 γ k,ℓ,i ( B ) Z + ∞ d y y i Q m (1 + y )(2 + y ) B − k +2 , (A1a)where γ k,ℓ,i ( B ) = ( ℓ + i )! i !( ℓ − i )! Γ( B − k − ℓ + 2)2 i +1 Γ( B + 1) Γ( B − k + ℓ + 3)Γ( B − k + i + 3) , (A1b)we should compute the pole part when B →
0. Instead of expanding directly Eqs. (A1)when B → C k,ℓ,m ( B ) that merely differs from the original one by a finiteremainder O ( B ). This suggests an alternative method for computing the poles at B = 0,based on the fact that the integral I νm ≡ Z + ∞ d y y ν Q m (1 + y ) = 2 ν [Γ( ν + 1)] Γ( m − ν )Γ( m + ν + 2) , (A2)is known by analytic continuation for any ν ∈ C , except at isolated poles at integer values of ν (see e.g. [48]). The idea is to reshape the right-hand side of Eq. (A1a) in order to expressit in terms of integrals that possess the required form (A2). We proceed in two steps.(i) We perform the following transformation on the integrand of Eq. (A1a). For k >
2, wewrite the denominator in the original integrand as (2 + y ) k − / (2 + y ) B and expand (2 + y ) k − by means of the binomial theorem. This gives ( k > C k,ℓ,m ( B ) = ℓ X i =0 γ k,ℓ,i ( B ) i + k − X j = i i + k − j − (cid:18) k − j − i (cid:19) Z + ∞ d y y j Q m (1 + y )(2 + y ) B , (A3)where (cid:0) k − j − i (cid:1) is the usual binomial coefficient. For k = 1, we replace the factor y i by theequivalent form (2 + y ) P i − j =0 ( − i − j − y j + ( − i , which yields C ,ℓ,m ( B ) = ℓ X i =0 γ ,ℓ,i ( B ) i − X j =0 ( − i − j − Z + ∞ d y y j Q m (1 + y )(2 + y ) B + 12 Z + ∞ d y Q m (1 + y )(2 + y ) B +1 . (A4)24he last integral in this expression corresponds to the contribution to C k,ℓ,m ( B ) producedby the term ( − i . Its coefficient has been simplified by means of the following identity, ℓ X i =0 ( − i γ ,ℓ,i ( B ) = 12 , (A5)resulting from the Gauss theorem on hypergeometric functions. Remarkably, the last termin (A4) has no pole at B = 0 for m ∈ N , since the integral is well defined in the limit B → C ,ℓ,m ( B ) = ℓ X i =0 γ ,ℓ,i ( B ) i − X j =0 ( − i − j − Z + ∞ d y y j Q m (1 + y )(2 + y ) B + O ( B ) . (A6)With Eqs. (A3) and (A6) in hands, we see that all elementary integrands that may beassociated with poles are now of the type y j Q m (1 + y ) / (2 + y ) B (with j ∈ N ).(ii) It is immediately possible to check that the pre-factors in Eqs. (A3) and (A6) cannothave more than a simple pole, so that it is sufficient to control the integrals of y j Q m (1 + y ) / (2 + y ) B at order O ( B ), neglecting remainders O ( B ). If j < m the integral is convergentwhen B = 0 and its value is given by Eq. (A2). The problem is more difficult when j > m +1.In that case we introduce the asymptotic expansion of y j Q m (1 + y ) when y → + ∞ . It isobtained by expanding when y → + ∞ the monomials, say (1 + y ) − q − m − ( q ∈ N ), in thehypergeometric series defining Q m (1+ y ). After some technical manipulation involving againthe Gauss theorem, we get y j Q m (1 + y ) = j − m − X p =0 f j,m,p y p + O (cid:18) y (cid:19) , (A7a)where f j,m,p = ( − ) m ( − j − p − [( j − p − ( j − m − p − j + m − p )! . (A7b)Concretely, we shall resort to the following lemma, valid in the limit B → Lemma : Z + ∞ d y y j Q m (1 + y )(2 + y ) B = Z + ∞ d y y j − B Q m (1+ y )+ j − m − X p =0 ( − p +1 p + 1 f j,m,p + O ( B ) . (A8)This permits us to relate the remaining integrals in (A3) and (A6) to the simpler integralsthat admit the closed-form analytic expression (A2), with ν = j − B . The proof relies onthe observation that, in the limit where B → Z + ∞ d y (cid:18) y ) B − y B (cid:19)(cid:20) y j Q m (1 + y ) − j − m − X p =0 f j,m,p y p (cid:21) = O ( B ) . (A9)This follows from the fact that the second factor inside the integrand behaves like O (1 /y )when y → + ∞ [see Eq. (A7a)]; so the integral is well-defined in a neighborhood of B = 0and vanishes at that point. In addition, we can compute explicitly, in the sense of analyticcontinuation in B and in the limit B → Z + ∞ d y y p (cid:18) y ) B − y B (cid:19) = ( − p +1 p + 1 + O ( B ) . (A10)25he two facts (A9)–(A10) imply Eq. (A8).Finally, transforming the integrals that enter Eqs. (A3) and (A6) by means of ourlemma (A8), when combined with the expressions (A7b) for the coefficients f j,m,p and (A2)for the integral I j − Bm , we obtain, in the cases k > k = 1 respectively, C k,ℓ,m ( B ) = 2 k − ( k − − k − ℓ + B )Γ( ℓ + 3 − k + B )Γ(1 + B ) ×× ℓ + k − X i =0 c k,ℓ,i ( B ) i ! (cid:20) [Γ( i + 1 − B )] Γ( m − i + B )2 B Γ( m + i + 2 − B ) + ( − ) m + i e i,m (cid:21) + O ( B ) , (A11a) C ,ℓ,m ( B ) = Γ(1 − ℓ + B )Γ( ℓ + 2 + B )4Γ(1 + B ) ×× ℓ X i =1 d ℓ,i ( B ) (cid:20) [Γ( i − B )] Γ( m + 1 − i + B )2 B Γ( m + i + 1 − B ) + ( − ) m + i +1 e i − ,m (cid:21) + O ( B ) . (A11b)The coefficients therein read c k,ℓ,i ( B ) = min( ℓ,i ) X j =max(0 ,i +2 − k ) (cid:18) ij (cid:19) ( ℓ + j )!( ℓ − j )!( k + j − i − j + 3 − k + B ) , (A12a) d ℓ,i ( B ) = ℓ − i X j =0 ( − ) j ( ℓ + i + j )!( ℓ − i − j )!( i + j )! 1Γ( i + j + 2 + B ) , (A12b) e i,m = i − m − X j =0 [( i − j − ( j + 1)( i − j − m − m + i − j )! . (A12c)The Laurent expansion when B → Appendix B: Source terms for the tails of tails
The tail-of-tail terms associated with the various multipole moments I L or J L (symbolizedby K L say) obey a wave equation of the type (cid:3) h αβM × M × K L = Λ αβM × M × K L , (B1)where Λ M × M × K L is a cubic source term composed of non-linear interactions between twostatic mass monopoles M and the time-varying multipole K L . This source term has beenderived in Eqs. (2.14)–(2.16) of Ref. [36] for the tails of tails associated with the massquadrupole moment I ij , and this result was the basis of the computation of Paper I. In thisAppendix we provide similar expressions for the sources of the tails of tails associated withthe mass moments I ijk , I ijkl and current moments J ij , J ijk that are also required for thepresent computations. They have been obtained by means of the same algorithm as in thequadrupolar case, using the xAct package bundle for Mathematica [49]. As in Ref. [36] and26aper I we split the source terms into an instantaneous (local-in-time) part and a hereditary(past-dependent) one, say Λ αβM × M × K L = I αβM × M × K L + H αβM × M × K L . (B2) • Mass quadrupole moment: I M × M × I ij = M n ab r − (cid:26) − I ab − rI (1) ab − r I (2) ab − r I (3) ab + 108 r I (4) ab + 40 r I (5) ab (cid:27) , (B3a) I iM × M × I ij = M ˆ n iab r − (cid:26) I (1) ab + 4 rI (2) ab − r I (3) ab + 43 r I (4) ab − r I (5) ab (cid:27) + M n a r − (cid:26) − I (1) ai − rI (2) ai − r I (3) ai − r I (4) ai + 1245 r I (5) ai (cid:27) , (B3b) I ijM × M × I ij = M ˆ n ijab r − (cid:26) − I (2) ab − rI (3) ab − r I (4) ab − r I (5) ab (cid:27) + M δ ij n ab r − (cid:26) I (2) ab + 1767 rI (3) ab − r I (4) ab − r I (5) ab (cid:27) + M ˆ n a ( i r − (cid:26) − I (2) j ) a − rI (3) j ) a + 4007 r I (4) j ) a + 1047 r I (5) j ) a (cid:27) + M r − (cid:26) − I (2) ij − rI (3) ij − r I (4) ij − r I (5) ij (cid:27) . (B3c) H M × M × I ij = M n ab r − Z + ∞ d x (cid:26) Q I (4) ab + (cid:20) Q + 1685 Q (cid:21) rI (5) ab + 32 Q r I (6) ab (cid:27) , (B4a) H iM × M × I ij = M ˆ n iab r − Z + ∞ d x (cid:26) − Q I (4) ab + (cid:20) − Q + 83 Q (cid:21) rI (5) ab (cid:27) + M n a r − Z + ∞ d x (cid:26) Q I (4) ai + (cid:20) Q + 1125 Q (cid:21) rI (5) ai + 32 Q r I (6) ai (cid:27) , (B4b) H ijM × M × I ij = M ˆ n ijab r − Z + ∞ d x (cid:26) − Q I (4) ab + (cid:20) − Q − Q (cid:21) rI (5) ab (cid:27) + M δ ij n ab r − Z + ∞ d x (cid:26) − Q I (4) ab + (cid:20) − Q + 247 Q (cid:21) rI (5) ab (cid:27) + M ˆ n a ( i r − Z + ∞ d x (cid:26) Q I (4) j ) a + (cid:20) Q − Q (cid:21) rI (5) j ) a (cid:27) We pose G = c = 1 in this Appendix. M r − Z + ∞ d x (cid:26) Q I (4) ij + (cid:20) Q − Q (cid:21) rI (5) ij + 32 Q r I (6) ij (cid:27) . (B4c) • Mass octupole: I M × M × I ijk = M ˆ n abc r − (cid:26) − I abc − rI (1) abc − r I (2) abc − r I (3) abc + 763 r I (4) abc + 4849 r I (5) abc + 1129 r I (6) abc (cid:27) , (B5a) I iM × M × I ijk = M ˆ n iabc r − (cid:26) I (1) abc + 6 rI (2) abc − r I (3) abc − r I (4) abc − r I (5) abc − r I (6) abc (cid:27) + M ˆ n ab r − (cid:26) − I (1) abi − rI (2) abi − r I (3) abi − r I (4) abi + 56863 r I (5) abi + 57263 r I (6) abi (cid:27) , (B5b) I ijM × M × I ijk = M ˆ n ijabc r − (cid:26) − I (2) abc − rI (3) abc − r I (4) abc − r I (5) abc − r I (6) abc (cid:27) + M δ ij ˆ n abc r − (cid:26) I (2) abc + 24 rI (3) abc − r I (4) abc − r I (5) abc − r I (6) abc (cid:27) + M ˆ n ab ( i r − (cid:26) − I (2) j ) ab − rI (3) j ) ab + 383 r I (4) j ) ab + 2309 r I (5) j ) ab + 103 r I (6) j ) ab (cid:27) + M n a r − (cid:26) − I (2) aij − rI (3) aij − r I (4) aij − r I (5) aij + 247 r I (6) aij (cid:27) . (B5c) H M × M × I ijk = M ˆ n abc r − Z + ∞ d x (cid:26) I (5) abc Q + 83 (cid:20) Q + 4 Q (cid:21) rI (6) abc + 323 Q r I (7) abc (cid:27) , (B6a) H iM × M × I ijk = M ˆ n iabc r − Z + ∞ d x (cid:26) − Q I (5) abc − (cid:20) Q − Q (cid:21) rI (6) abc (cid:27) + M ˆ n ab r − Z + ∞ d x (cid:26) Q I (5) abi + 8105 (cid:20) Q + 93 Q (cid:21) rI (6) abi + 323 Q r I (7) abi (cid:27) , (B6b) H ijM × M × I ijk = M ˆ n ijabc r − Z + ∞ d x (cid:26) − Q I (5) abc − (cid:20) Q + 4 Q (cid:21) rI (6) abc (cid:27) + M δ ij ˆ n abc r − Z + ∞ d x (cid:26) − Q I (5) abc − (cid:20) Q − Q (cid:21) rI (6) abc (cid:27) M ˆ n ab ( i r − Z + ∞ d x (cid:26) Q I (5) j ) ab + 3263 (cid:20) Q − Q (cid:21) rI (6) j ) ab (cid:27) + M n a r − Z + ∞ d x (cid:26) Q I (5) aij + 32735 (cid:20) Q − Q (cid:21) rI (6) aij + 323 Q r I (7) aij (cid:27) . (B6c) • Mass hexadecapole: I M × M × I ijkl = M ˆ n abcd r − (cid:26) − I abcd − rI (1) abcd − r I (2) abcd − r I (3) abcd − r I (4) abcd + 84518 r I (5) abcd + 1336 r I (6) abcd + 299 r I (7) abcd (cid:27) , (B7a) I iM × M × I ijkl = M ˆ n iabcd r − (cid:26) I (1) abcd + 9 rI (2) abcd − r I (3) abcd − r I (4) abcd − r I (5) abcd − r I (6) abcd − r I (7) abcd (cid:27) + M ˆ n abc r − (cid:26) − I (1) abci − rI (2) abci − r I (3) abci − r I (4) abci − r I (5) abci + 64681 r I (6) abci + 7027 r I (7) abci (cid:27) , (B7b) I ijM × M × I ijkl = M ˆ n ijabcd r − (cid:26) − I (2) abcd − rI (3) abcd − r I (4) abcd − r I (5) abcd − r I (6) abcd − r I (7) abcd (cid:27) + M δ ij ˆ n abcd r − (cid:26) I (2) abcd + 34511 rI (3) abcd + 355198 r I (4) abcd − r I (5) abcd − r I (6) abcd − r I (7) abcd (cid:27) + M ˆ n abc ( i r − (cid:26) − I (2) j ) abc − rI (3) j ) abc − r I (4) j ) abc + 180899 r I (5) j ) abc + 45255 r I (6) j ) abc + 104165 r I (7) j ) abc (cid:27) + M ˆ n ab r − (cid:26) − I (2) abij − rI (3) abij − r I (4) abij − r I (5) abij + 110189 r I (6) abij + 314189 r I (7) abij (cid:27) . (B7c) H M × M × I ijkl = M ˆ n abcd r − Z + ∞ d x (cid:26) Q I (6) abcd + 227 (cid:20) Q + 35 Q (cid:21) rI (7) abcd + 83 Q r I (8) abcd (cid:27) , (B8a)29 iM × M × I ijkl = M ˆ n iabcd r − Z + ∞ d x (cid:26) − Q I (6) abcd − (cid:20) Q − Q (cid:21) rI (7) abcd (cid:27) + M ˆ n abc r − Z + ∞ d x (cid:26) Q I (6) abci + 8189 (cid:20) Q + 41 Q (cid:21) rI (7) abci + 83 Q r I (8) abci (cid:27) , (B8b) H ijM × M × I ijkl = M ˆ n ijabcd r − Z + ∞ d x (cid:26) − Q I (6) abcd − (cid:20) Q + 5 Q (cid:21) rI (7) abcd (cid:27) + M δ ij ˆ n abcd r − Z + ∞ d x (cid:26) − Q I (6) abcd − (cid:20) Q − Q (cid:21) rI (7) abcd (cid:27) + M ˆ n abc ( i r − Z + ∞ d x (cid:26) Q I (6) j ) abc + 16297 (cid:20) Q − Q (cid:21) rI (7) j ) abc (cid:27) + M ˆ n ab r − Z + ∞ d x (cid:26) Q I (6) abij + 16567 (cid:20) Q − Q (cid:21) rI (7) abij + 83 Q r I (8) abij (cid:27) . (B8c) • Current quadrupole: I M × M × J ij = 0 , (B9a) I iM × M × J ij = M ε iab ˆ n ac r − (cid:26) J bc + 88 rJ (1) bc + 80 r J (2) bc + 1523 r J (3) bc − r J (4) bc − r J (5) bc (cid:27) , (B9b) I ijM × M × J ij = M ε ab ( i ˆ n j ) ac r − (cid:26) J (1) bc + 643 rJ (2) bc − r J (3) bc − r J (4) bc − r J (5) bc (cid:27) + M ε ab ( i n a r − (cid:26) J (1) j ) b + 30415 rJ (2) j ) b + 36815 r J (3) j ) b + 83245 r J (4) j ) b − r J (5) j ) b (cid:27) . (B9c) H M × M × J ij = 0 , (B10a) H iM × M × J ij = M ε iab ˆ n ac r − Z + ∞ d x (cid:26) Q J (4) bc − (cid:20) Q + 3 Q (cid:21) rJ (5) bc − Q r J (6) bc (cid:27) , (B10b) H ijM × M × J ij = M ε ab ( i ˆ n j ) ac r − Z + ∞ d x (cid:26) − Q J (5) bc (cid:27) + M ε ab ( i n a r − Z + ∞ d x (cid:26) − Q J (5) j ) b − Q rJ (6) j ) b (cid:27) . (B10c)30 Current octupole: I M × M × J ijk = 0 , (B11a) I iM × M × J ijk = M ε iab ˆ n acd r − (cid:26) J bcd + 270 rJ (1) bcd + 188 r J (2) bcd + 98 r J (3) bcd − r J (4) bcd − r J (5) bcd − r J (6) bcd (cid:27) , (B11b) I ijM × M × J ijk = M ε ab ( i ˆ n j ) acd r − (cid:26) J (1) bcd + 54 rJ (2) bcd − r J (3) bcd − r J (4) bcd − r J (5) bcd − r J (6) bcd (cid:27) + M ε ab ( i ˆ n ac r − (cid:26) J (1) j ) bc + 3607 rJ (2) j ) bc + 2867 r J (3) j ) bc + 1667 r J (4) j ) bc − r J (5) j ) bc − r J (6) j ) bc (cid:27) . (B11c) H M × M × J ijk = 0 , (B12a) H iM × M × J ijk = M ε iab ˆ n acd r − Z + ∞ d x (cid:26) Q J (5) bcd − (cid:20) Q + 2 Q (cid:21) rJ (6) bcd − Q r J (7) bcd (cid:27) , (B12b) H ijM × M × J ijk = M ε ab ( i ˆ n j ) acd r − Z + ∞ d x (cid:26) − Q J (6) bcd (cid:27) + M ε ab ( i ˆ n ac r − Z + ∞ d x (cid:26) − Q J (6) j ) bc − Q rJ (7) j ) bc (cid:27) . 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