Post-Newtonian theory and the two-body problem
aa r X i v : . [ g r- q c ] M a r Post-Newtonian theory and the two-bodyproblem
Luc Blanchet
Abstract
Reliable predictions of general relativity theory are extracted using ap-proximation methods. Among these, the powerful post-Newtonian approximationprovides us with our best insights into the problems of motion and gravitationalradiation of systems of compact objects. This approximation has reached an im-pressive mature status, because of important progress regarding its theoretical foun-dations, and the successful construction of templates of gravitational waves emittedby inspiralling compact binaries. The post-Newtonian predictions are routinely usedfor searching and analyzing the very weak signals of gravitational waves in currentgenerations of detectors. High-accuracy comparisons with the results of numericalsimulations for the merger and ring-down of binary black holes are going on. Inthis article we give an overview on the general formulation of the post-Newtonianapproximation and present up-to-date results for the templates of compact binaryinspiral.
Although relativists admire the mathematical coherence — and therefore beauty— of Einstein’s general relativity, this theory is not easy to manage when draw-ing firm predictions for the outcome of laboratory experiments and astronomicalobservations. Indeed only few exact solutions of the Einstein field equations areknown, and one is obliged in most cases to resort to approximation methods. It isbeyond question that approximation methods in general relativity do work, and insome cases with some incredible precision. Many of the great successes of generalrelativity were in fact obtained using approximation methods. However because ofthe complexity of the field equations, such methods become awfully intricate at
Luc Blanchet G R e C O , Institut d’Astrophysique de Paris — C.N.R.S. & Universit´e Pierre et Marie Curie, 98 bis boulevard Arago, 75014 Paris, France. e-mail: [email protected] high approximation orders. On the other hand it is difficult to set up a formalismin which the approximation method would be perfectly well-defined and based onclear premises. Sometimes it is impossible to relate the approximation method tothe exact framework of the theory. In this case the only thing one can do is to relyon the approximation method as the only representation of real phenomena, and todiscover “empirically” that the approximation works well.The most important approximation scheme in general relativity is the post-Newtonian expansion, which can be viewed as an expansion when the speed oflight c tends to infinity, and is physically valid under the assumptions of weak gravi-tational field inside the source and of slow internal motion. The post-Newtonian ap-proximation makes sense only in the near-zone of an isolated matter source, definedas r ≪ l , where l ≡ c T is the wavelength of the emitted gravitational radiation,with T being a characteristic time scale of variation of the source. The approxima-tion has been formalized in the early days of general relativity by Einstein [1], DeSitter [2, 3], Lorentz & Droste [4]. It was subsequently developed notably by Ein-stein, Infeld & Hoffmann [5], Fock [6, 7], Plebanski & Bazanski [8], Chandrasekharand collaborators [9, 10, 11], Ehlers and his school [12, 13, 14], Papapetrou andcoworkers [15, 16].Several long-standing problems with the post-Newtonian approximation for gen-eral isolated slowly-moving systems have hindered progress untill recently. At highpost-Newtonian ordners some divergent Poisson-type integrals appear, casting adoubt on the physical soundness of the approximation. Linked to that, the domainof validity of the post-Newtonian approximation is limited to the near-zone of thesource, making it a priori difficult to incorporate into the scheme the condition ofno-incoming radiation, to be imposed at past null infinity from an isolated source.In addition, from a mathematical point of view, we do not know what the “reliabil-ity” of the post-Newtonian series is, i.e. if it comes from the Taylor expansion of afamily of exact solutions.The post-Newtonian approximation gives wonderful answers to the problems ofmotion and gravitational radiation, two of general relativity’s corner stones. Threecrucial applications are:1. The motion of N point-like objects at the first post-Newtonian level [5] i.e. is taken into account to describe the Solar System dynamics (motion of the cen-ters of mass of planets);2. The gravitational radiation-reaction force, which appears in the equations of mo-tion at the 2.5PN order [17, 18, 19, 20], has been experimentally verified by theobservation of the secular acceleration of the orbital motion of the Hulse-Taylorbinary pulsar PSR 1913+16 [21, 22, 23];3. The analysis of gravitational waves emitted by inspiralling compact binaries —two neutron stars or black holes driven into coalescence by emission of gravi- As usual, we refer to n PN as the order equivalent to terms ∼ ( v / c ) n in the equations of motion be-yond the Newtonian acceleration, and in the asymptotic waveform beyond the Einstein quadrupoleformula, where v denotes the binary’s orbital velocity and c is the speed of light.ost-Newtonian theory and the two-body problem 3 tational radiation — necessitate the prior knowledge of the equations of motionand radiation field up to high post-Newtonian order.Strategies to detect and analyze the very weak signals from inspiralling compactbinaries involve matched filtering of a set of accurate theoretical template wave-forms against the output of the detectors. Measurement-accuracy analyses haveshown that, in order to get sufficiently accurate theoretical templates, one must atleast include conservative post-Newtonian effects up to the 3PN level, and radiation-reaction effects up to 5.5PN order, i.e. point-particles , characterized solely by their masses m or m (and possiblytheir spins). Indeed, most of the non-gravitational effects usually plaguing the dy-namics of binary star systems, such as the effects of a magnetic field, of an inter-stellar medium, the influence of the internal structure of the compact bodies, aredominated by purely gravitational effects. Inspiralling compact binaries are veryclean systems which can essentially be described in pure general relativity.Although point-particles are ill-defined in the exact theory, they are admissiblein post-Newtonian approximations. Furthermore the model of point particles canbe pushed to high post-Newtonian order, where an a priori more realistic modelinvolving the internal structure of compact bodies would fail through becoming in-tractable. However there is an important worry: a process of regularization to dealwith the infinite self-field of point particles is crucially needed. The regularizationshould be carefully defined to be implemented at high orders. It should hopefully befollowed by a renormalization.The orbit of inspiralling compact binaries can be considered to be circular, apartfrom the gradual inspiral, with an excellent approximation. Indeed, gravitational ra-diation reaction forces tend to circularize rapidly the orbital motion. At each instantduring the gradual inspiral, the eccentricity e of the orbit is related to the instanta-neous frequency w ≡ p / P by [30] e ≃ const w − / (for e ≪ . (1)For instance one can check that the eccentricity of a system like the binary pulsarPSR 1913+16 will be e ≃ − when the gravitational waves become visible bythe detectors, i.e. when the signal frequency after its long chirp reaches f ≡ w / p ≃
10 Hz. Only systems formed very late, near their final coalescence, could have anon-negligible residual eccentricity.The intrinsic rotations or spins of the compact bodies could play an importantrole, yielding some relativistic spin-orbit and spin-spin couplings, both in the bi-nary’s equations of motion and radiation field. The spin of a rotating body is of theorder of S ∼ m a v spin , where m and a denote the mass and typical size of the body,and where v spin represents the velocity of the body’s surface. In the case of compactbodies we have a ∼ G m / c , and for maximal rotation v spin ∼ c ; for such objects themagnitude of the spin is roughly S ∼ G m / c . It is thus customary to introduce adimensionless spin parameter, generally denoted by c , defined by Luc Blanchet S = G m c c . (2)We have c ≤ c . . − .
74 for neutron stars (dependingon the equation of state of nuclear matter inside the neutron star). For binary pulsarssuch as PSR 1913+16, we have c . − . Considering models of evolution of ob-served binary pulsar systems when they become close to the coalescence we expectthat the spins will make a negligible contribution to the accumulated phase in thiscase. However, astrophysical observations suggest that black holes can have non-negligible spins, due to spin up driven by accretion from a companion during someearlier phase of the binary evolution. For a few black holes surrounded by matter,observations indicate a significant intrinsic angular momentum and the spin mayeven be close to its maximal value. However, very little is known about the black-hole spin magnitudes in binary systems. For black holes rotating near the maximalvalue the templates of gravitational waves need to take into account the effects ofspins, both for a successful detection and an accurate parameter estimation.We devote Part A of this article, including Sections 2 to 7, to a general overviewof the formulation of the post-Newtonian approximation for isolated sources. Weemphasize that the approximation can be carried out up to any post-Newtonianorder, without the aforesaid problem of divergences. The main technique used isthe matching of asymptotic expansions which permits to obtain a complete post-Newtonian solution incorporing the correct boundary conditions at infinity. ThenPart B, i.e. Sections 8 to 13, will deal with the application to systems composed ofcompact objects. The subtle issues linked with the self-field regularization of point-particles are discussed in some details. The results for the two-body equations ofmotion and radiation field at the state-of-the-art 3PN level are presented in the caseof circular orbits appropriate to inspiralling compact binaries. We put the accenton the description of spinning particles and particularly on the spin-orbit couplingeffect on the binary’s internal energy and gravitational-wave flux.
A Post-Newtonian formalism2 Einstein field equations
General relativity is based on two independent tenets, the first one concerned withthe dynamics of the gravitational field, the second one dealing with the coupling ofall the matter fields with the gravitational field. Accordingly the action of generalrelativity is made of two terms, S = c p G Z d x √− g R + S matter (cid:2) Y , g ab (cid:3) . (3)The first term represents the kinetic Einstein-Hilbert action for gravity and tells thatthe gravitational field g ab propagates like a pure spin-2 field. Here R is the Ricci ost-Newtonian theory and the two-body problem 5 scalar and g = det ( g ab ) is the determinant of the metric. The second term expressesthe fact that all matter fields (collectively denoted by Y ) are minimally coupled tothe metric g ab which defines the physical lengths and times as measured in locallaboratory experiments. The field equations are obtained by varying the action withrespect to the metric (such that d g ab = | x m | → ¥ ) and form a system of tendifferential equations of second order, G ab [ g , ¶ g , ¶ g ] = p Gc T ab [ g ] . (4)The Einstein tensor G ab ≡ R ab − R g ab is generated by the matter stress-energytensor T ab ≡ ( / √− g )( d S matter / d g ab ) . Four equations give, via the contractedBianchi identity, the conservation equation for the matter system as (cid:209) m G am ≡ = ⇒ (cid:209) m T am = , (5)which must be solved conjointly with the Einstein field equations for the gravita-tional field. The matter equation (5) reads also ¶ m (cid:0) √− g T ma (cid:1) = √− g T mn ¶ a g mn . (6)Let us introduce an asymptotically Minkowskian coordinate system such that thegravitational-wave amplitude, defined by h ab ≡ √− g g ab − h ab , is divergence-less, i.e. satisfies the de Donder or harmonic gauge condition ¶ m h am = . (7)With this coordinate choice the Einstein field equations can be recast into thed’Alembertian equation (cid:3) h ab = p Gc t ab . (8)Here (cid:3) = h mn ¶ m ¶ n is the usual ( flat -spacetime) d’Alembertian operator. The sourceterm t ab can rightly be interpreted as the “effective” stress-energy distribution ofthe matter and gravitational fields in harmonic coordinates; note that t ab is not atensor and we shall often call it a pseudo-tensor. It is conserved in the sense that ¶ m t am = . (9)This is equivalent to the condition of harmonic coordinates (7) and to the covariantconservation (5) of the matter tensor. The pseudo-tensor is made of the contributionof matter fields described by T ab , and of the gravitational contribution L ab whichis a complicated functional of the gravitational field variable h mn and its first andsecond derivatives. Thus, Here, g ab denotes the contravariant metric, inverse of the covariant metric g ab , and h ab repre-sents an auxiliary Minkowski metric. We assume that our spatial coordinates are Cartesian so that ( h ab ) = ( h ab ) = diag ( − , , , ) . Luc Blanchet t ab = | g | T ab + c p G L ab [ h , ¶ h , ¶ h ] . (10)The point is that L ab is at least quadratic in the field strength h mn , so the fieldequations (8) are naturally amenable to a perturbative non-linear treatment. Thegeneral expression is L ab = − h mn ¶ m ¶ n h ab + ¶ m h an ¶ n h bm + g ab g mn ¶ r h ms ¶ s h nr − g am g ns ¶ r h bs ¶ m h nr − g bm g ns ¶ r h as ¶ m h nr + g mn g rs ¶ r h am ¶ s h bn + (cid:0) g am g bn − g ab g mn (cid:1)(cid:16) g rs g ep − g se g rp (cid:17) ¶ m h rp ¶ n h se . (11)To select a physically sensible solution of the field equations in the case of abounded matter system, we must impose a boundary condition at infinity, namelythe famous no-incoming radiation condition, which ensures that the system is trulyisolated from other bodies in the Universe. In principle the no-incoming radiationcondition is to be formulated at past null infinity I − . Here, we can simplify theformulation by taking advantage of the presence of the Minkowski background h ab to define the no-incoming radiation condition with respect to Minkowskian past nullinfinity say I − h . Within approximate methods this is legitimate as we can view thegravitational field as propagating on the flat background h ab ; indeed h ab does existat any finite order of approximation.The no-incoming radiation condition should be such that it suppresses at I − h anyhomogeneous (regular in R ) solution of the d’Alembertian equation (cid:3) h hom = I − h . We have at our disposal the Kirchhoff formula whichexpresses h hom at some field point ( x ′ , t ′ ) in terms of its values and its derivatives ona sphere centered on x ′ with radius r ≡ | x ′ − x | and at retarded time t ≡ t ′ − r / c , h hom ( x ′ , t ′ ) = Z d W p (cid:20) ¶¶r ( r h hom ) + c ¶¶ t ( r h hom ) (cid:21) ( x , t ) , (12)where d W is the solid angle spanned by the unit direction N ≡ ( x − x ′ ) / r . Fromthis formula we deduce the no-incoming radiation condition as the following limitat I − h , i.e. when r ≡ | x | → + ¥ with t + r / c = const, lim I − h (cid:20) ¶¶ r ( rh ab ) + c ¶¶ t ( rh ab ) (cid:21) = . (13)Now if h ab satisfies (13), so does the pseudo-tensor t ab built on it, and then it isclear that the retarded integral of t ab does satisfy the same condition. Thus we inferthat the unique solution of the Einstein equation (8) reads In fact we obtain also the auxiliary condition that r ¶ m h ab should be bounded near I − h . Thiscomes from the fact that r differs from r and we have r = r − x ′ · n + O ( / r ) with n = x / r .ost-Newtonian theory and the two-body problem 7 h ab = p Gc (cid:3) − t ab , (14)where the retarded integral takes the standard form ( (cid:3) − t ab )( x , t ) ≡ − p Z d x ′ | x − x ′ | t ab (cid:0) x ′ , t − | x − x ′ | / c (cid:1) . (15)Notice that since t ab depends on h mn and its derivatives, the equation (14) is tobe viewed as an integro-differential equation equivalent to the Einstein equation (8)with no-incoming radiation condition. In this Section we proceed with the post-Newtonian iteration of the field equationsin harmonic coordinates in the near zone of an isolated matter distribution. We havein mind a general hydrodynamical fluid, whose stress-energy tensor is smooth, i.e.T ab ∈ C ¥ ( R ) . Thus the scheme a priori excludes the presence of singularities;these will be dealt with in later Sections.Let us remind that the post-Newtonian approximation in “standard” form ( e.g. [31, 13, 14]) is plagued with some apparently inherent difficulties, which crop up atsome high post-Newtonian order like 3PN. Up to the 2.5PN order the approxima-tion can be worked out without problems, and at the 3PN order the problems cangenerally be solved for each case at hand; but the problems worsen at higher or-ders. Historically these difficulties, even appearing at higher approximations, havecast a doubt on the actual soundness, from a theoretical point of view, of the post-Newtonian expansion. Practically speaking, they posed the question of the reliabil-ity of the approximation, when comparing the theory’s predictions with very preciseexperimental results. In this Section and the next one we assess the nature of thesedifficulties — are they purely technical or linked with some fundamental drawbackof the approximation scheme? — and eventually resolve them.We first distinguish the problem of divergences in the post-Newtonian expan-sion: in higher approximations some divergent Poisson-type integrals appear. Re-call that the post-Newtonian expansion replaces the resolution of an hyperbolic-like d’Alembertian equation by a perturbatively equivalent hierarchy of elliptic-likePoisson equations. Rapidly it is found during the post-Newtonian iteration that theright-hand-sides of the Poisson equations acquire a non-compact support (it is dis-tributed all over space), and that the standard Poisson integral diverges because ofthe bound of the integral at spatial infinity, i.e. r ≡ | x | → + ¥ , with t = const.The divergencies are linked to the fact that the post-Newtonian expansion is actu-ally a singular perturbation, in the sense that the coefficients of the successive pow-ers of 1 / c are not uniformly valid in space, typically blowing up at spatial infinitylike some positive powers of r . We know for instance that the post-Newtonian ex-pansion cannot be “asymptotically flat” starting at the 2PN or 3PN level, depending Luc Blanchet on the adopted coordinate system [32]. The result is that the standard Poisson inte-grals are in general badly-behaving at infinity. Trying to solve the post-Newtonianequations by means of the Poisson integral does not a priori make sense. This doesnot mean that there are no solutions to the problem, but simply that the Poisson in-tegral does not constitute the good solution of the Poisson equation in the contextof post-Newtonian expansions. So the difficulty is purely of a technical nature, andwill be solved once we succeed in finding the appropriate solution to the Poissonequation.We shall now prove (following [33]) that the post-Newtonian expansion can be indefinitely iterated without divergences. Let us denote by means of an overlinethe formal (infinite) post-Newtonian expansion of the field inside the source’s near-zone, which is of the form h ab ( x , t , c ) = + ¥ (cid:229) n = c n h n ab ( x , t , ln c ) . (16)The n -th post-Newtonian coefficient is naturally the factor of the n -th power of 1 / c ;however, we know [36] that the post-Newtonian expansion also involves some log-arithms of c , included for convenience here into the definition of the coefficients h n .For the stress-energy pseudo-tensor (10) we have the same type of expansion, t ab ( x , t , c ) = + ¥ (cid:229) n = − c n t n ab ( x , t , ln c ) . (17)The expansion starts with a term of order c corresponding to the rest mass-energy( t ab has the dimension of an energy density). Here we shall always understand theinfinite sums such as (16)–(17) in the sense of formal series, i.e. merely as an orderedcollection of coefficients. Because of our consideration of regular extended matterdistributions the post-Newtonian coefficients are smooth functions of space-time.Inserting the post-Newtonian ansatz into the Einstein field equation (8) andequating together the powers of 1 / c results is an infinite set of Poisson-type equa-tions ( ∀ n ≥ D h n ab = p G t n − ab + ¶ t h n − ab . (18)The second term comes from the split of the d’Alembertian into a Laplacian and asecond time derivative: (cid:3) = D − c ¶ t . This term is zero when n = i.e. fix some post-Newtonian order n , assume that we succeededin constructing the sequence of previous coefficients p h for p ≤ n −
1, and from thisshow how to infer the next-order coefficient n h .To cure the problem of divergencies we introduce a generalized solution of thePoisson equation with non-compact support source, in the form of an appropriate finite part of the usual Poisson integral obtained by regularization of the bound at An alternative solution to the problem of divergencies, proposed in [34, 35], is based on aninitial-value formalism, which avoids the appearance of divergencies because of the finiteness ofthe integration region.ost-Newtonian theory and the two-body problem 9 infinity by means of a specific process of analytic continuation. For any source termlike n t , we multiply it by the “regularization” factor | e x | B ≡ (cid:12)(cid:12)(cid:12)(cid:12) x r (cid:12)(cid:12)(cid:12)(cid:12) B , (19)where B ∈ C is a complex number and r denotes an arbitrary length scale. Only thendo we apply the Poisson integral, which therefore defines a certain function of B .The well-definedness of that integral heavily relies on the behavior of the integrandat the bound at infinity. There is no problem with the vicinity of the origin insidethe source because of the smoothness of the pseudo-tensor. Then one can prove [33]that the latter function of B generates a (unique) analytic continuation down to aneighborhood of the value of interest B =
0, except at B = B → B , of that expansion. This defines ourgeneralized Poisson integral: D − (cid:2) t n ab (cid:3) ( x , t ) ≡ − p FP B = Z d x ′ | x − x ′ | | e x ′ | B t n ab ( x ′ , t ) . (20)The integral extends over all three-dimensional space but with the latter finite-partregularization at infinity denoted FP B = . The main properties of our generalizedPoisson operator is that it does solve the Poisson equation, namely D (cid:16) D − (cid:2) t n ab (cid:3)(cid:17) = t n ab , (21)and that the so defined solution D − n t owns the same properties as its source n t , i.e. the smoothness and the same type of behavior at infinity.The most general solution of the Poisson equation (18) will be obtained by appli-cation of the previous generalized Poisson operator to the right-hand-side of (18),and augmented by the most general homogeneous solution of the Poisson equation.Thus, we can write h n ab = p G D − (cid:2) t n − ab (cid:3) + ¶ t D − (cid:2) h n − ab (cid:3) + + ¥ (cid:229) ℓ = B n ab L ( t ) ˆ x L . (22)The last term represents the general solution of the Laplace equation which is regularat the origin r ≡ | x | =
0. It can be written, using the symmetric-trace-free (STF)language, as a multipolar series of terms of the type ˆ x L , and multiplied by some Here L = i · · · i ℓ denotes a multi-index composed of ℓ multipolar spatial indices i , · · · , i ℓ (rangingfrom 1 to 3); x L ≡ x i · · · x i ℓ is the product of ℓ spatial vectors x i ; ¶ L = ¶ i · · · ¶ i ℓ is the product of ℓ partial derivatives ¶ i = ¶ / ¶ x i ; in the case of summed-up (dummy) multi-indices L , we do notwrite the ℓ summations from 1 to 3 over their indices; the STF projection is indicated with a hat, i.e. ˆ x L ≡ STF [ x L ] and similarly ˆ ¶ L ≡ STF [ ¶ L ] , or sometimes using brackets surrounding the indices, x < L > ≡ ˆ x L .0 Luc Blanchet STF-tensorial functions of time n B L ( t ) . These functions will be associated with theradiation reaction of the field onto the source; they will depend on which boundaryconditions are to be imposed on the gravitational field at infinity from the source.It is now trivial to iterate the process. We substitute for n − h in the right-hand-side of (22) the same expression but with n replaced by n −
2, and similarly comedown until we stop at either one of the coefficients h = h =
0. At this point n h is expressed in terms of the “previous” p t ’s and p B L ’s with p ≤ n −
2. To finalizethe process we introduce what we call the operator of the “instantaneous” potentials (cid:3) − . Our notation is chosen to contrast with the standard operators of the retardedand advanced potentials (cid:3) − and (cid:3) − , see (15). However, beware of the fact thatunlike (cid:3) − , A the operator (cid:3) − will be defined only when acting on a post-Newtonianseries such as t . Indeed, we pose (cid:3) − (cid:2) t ab (cid:3) ≡ + ¥ (cid:229) k = (cid:18) ¶ c ¶ t (cid:19) k D − k − (cid:2) t ab (cid:3) , (23)where D − k − is the k -th iteration of the operator (20). It is readily checked that inthis way we have a solution of the source-free d’Alembertian equation, (cid:3) (cid:16) (cid:3) − (cid:2) t ab (cid:3)(cid:17) = t ab . (24)On the other hand, the homogeneous solution in (22) will yield by iteration an ho-mogeneous solution of the d’Alembertian equation which is necessarily regular atthe origin. Hence it should be of the anti-symmetric type, i.e. be made of the dif-ference between a retarded multipolar wave and the corresponding advanced wave.We shall therefore introduce a new definition for some STF-tensorial functions A L ( t ) parametrizing those advanced-minus-retarded free waves. It will not be difficult torelate the post-Newtonian expansion of A L ( t ) to the functions n B L ( t ) which wereintroduced in (22). Finally the most general post-Newtonian solution, iterated adinfinitum and without any divergences, is obtained into the form h ab = p Gc (cid:3) − (cid:2) t ab (cid:3) − Gc + ¥ (cid:229) ℓ = ( − ) ℓ ℓ ! ˆ ¶ L ( A ab L ( t − r / c ) − A ab L ( t + r / c ) r ) . (25)We shall refer to the A L ( t ) ’s as the radiation-reaction functions. If we stay at thelevel of the post-Newtonian iteration which is confined into the near zone we can-not do more than (25); there is no means to compute the radiation-reaction func-tions A L ( t ) . We are here touching the second problem faced by the standard post-Newtonian approximation. ost-Newtonian theory and the two-body problem 11 As we now understand this problem is that of the limitation to the near zone. Indeedthe post-Newtonian expansion assumes that all retardations r / c are small, so it canbe viewed as a formal near-zone expansion when r →
0, valid only in the regionsurrounding the source that is of small extent with respect to the wavelength ofthe emitted radiation: r ≪ l . As we have seen, a consequence is that the post-Newtonian coefficients blow up at infinity, when r → + ¥ . It is thus not possible, a priori , to implement within the post-Newtonian scheme the physical informationthat the matter system is isolated from the rest of the Universe. The no-incomingradiation condition imposed at past null infinity I − h cannot be taken into account, a priori , within the scheme.The near-zone limitation can be circumvented to the lowest post-Newtonian or-ders by considering retarded integrals that are formally expanded when c → + ¥ asseries of “instantaneous” Poisson-like integrals [31]. This procedure works well upto the 2.5PN level and has been shown to correctly fix the dominant radiation reac-tion term at the 2.5PN order [13, 14]. Unfortunately such a procedure assumes fun-damentally that the gravitational field, after expansion of all retardations r / c → t , in keeping with the instan-taneous character of the Newtonian interaction. However, we know that the post-Newtonian field (as well as the source’s dynamics) will cease at some stage to begiven by a functional of the source parameters at a single time, because of the im-print of gravitational-wave tails in the near zone field, in the form of some modifi-cation of the radiation reaction force at the 1.5PN relative order [37, 38]. Since thereaction force is itself of order 2.5PN this means that the formal post-Newtonianexpansion of retarded Green functions is no longer valid starting at the 4PN order.The solution of the problem resides in the matching of the near-zone field to theexterior field, a solution of the vacuum equations outside the source which has beendeveloped in previous works using some post- Minkowskian and multipolar expan-sions [36, 39]. In the case of post-Newtonian sources, the near zone, i.e. r ≪ l ,covers entirely the source, because the source’s radius itself is such that a ≪ l .Thus the near zone overlaps with the exterior zone where the multipole expansion isvalid. Matching together the post-Newtonian and multipolar-post-Minkowskian so-lutions in this overlapping region is an application of the method of matched asymp-totic expansions, which has frequently been applied in the present context, both forradiation-reaction [40, 41, 37, 38] and wave-generation [42, 43, 44] problems.In the previous Section we obtained the most general solution (25) for the post-Newtonian expansion, as parametrized by the set of unknown radiation-reactionfunctions A L ( t ) . We shall now impose the matching condition M ( h ab ) ≡ M ( h ab ) , (26)telling that the multipole decomposition of the post-Newtonian expansion h of theinner field, agrees with the near-zone expansion of the multipole expansion M ( h ) of the external field. Here the calligraphic letter M stands for the multipole de- composition or far-zone expansion, while the overbar denotes the post-Newtonianor near-zone expansion. The matching equation results from the numerical equality h = M ( h ) , clearly verified in the exterior part of the near-zone, namely our over-lapping region a < r ≪ l . The left-hand-side is expanded when r → + ¥ yielding M ( h ) while the right-hand-side is expanded when r → M ( h ) . Thematching equation is thus physically justified only for post-Newtonian sources, forwhich the exterior near-zone exists. It is actually a functional identity; it identifies, term-by-term , two asymptotic expansions, each of them being formally taken out-side its own domain of validity. In the present context, the matching equation insiststhat the infinite far-zone expansion ( r → ¥ ) of the inner post-Newtonian field isidentical to the infinite near-zone expansion ( r →
0) of the exterior multipolar field.Let us now state that (26), plus the condition of no-incoming radiation, permits de-termining all the unknowns of the problem: i.e. , at once, the external multipolardecomposition M ( h ) and the radiation-reaction functions A L and hence the innerpost-Newtonian expansion h .When applied to a multipole expansion such as that of the pseudo-tensor, i.e. M ( t ab ) , we have to define a special type of generalized inverse d’Alembertianoperator, built on the standard retarded integral (15), viz (cid:3) − (cid:2) M ( t ab ) (cid:3) ( x , t ) ≡ − p FP B = Z d x ′ | x − x ′ | | e x ′ | B M ( t ab )( x ′ , t − | x − x ′ | / c ) . (27)Like (15) this integral extends over the whole three-dimensional space, but a regu-larization factor | e x ′ | B given by (19) has been “artificially” introduced for applicationof the finite part operation FP B = . The reason for introducing such regularization isto cure the divergencies of the integral when | x ′ | →
0; these are coming from thefact that the multipolar expansion is singular at the origin. We notice that this reg-ularization factor is the same as the one entering the generalized Poisson integral(20), however its role is different, as it takes care of the bound at | x ′ | = (cid:3) (cid:16) (cid:3) − (cid:2) M ( t ab ) (cid:3)(cid:17) = M ( t ab ) . (28)Therefore M ( h ab ) should be given by that solution plus a retarded homogeneoussolution of the d’Alembertian equation (imposing the no-incoming radiation condi-tion), i.e. be of the type M ( h ab ) = p Gc (cid:3) − (cid:2) M ( t ab ) (cid:3) − Gc + ¥ (cid:229) ℓ = ( − ) ℓ ℓ ! ˆ ¶ L ( F ab L ( t − r / c ) r ) . (29)Now the matching equation (26) will determine both the A L ’s in (25) and the F L ’s in(29). We summarize the results which have been obtained in [33].The functions F L ( t ) will play an important role in the following, because theyappear as the multipole moments of a general post-Newtonian source as seen fromits exterior near zone. Their closed-form expression obtained by matching reads ost-Newtonian theory and the two-body problem 13 F ab L ( t ) = FP B = Z d x | e x | B ˆ x L Z + − d z d ℓ ( z ) t ab ( x , t − z | x | / c ) . (30)Again the integral extends over all space but the bound at infinity (where the post-Newtonian expansion becomes singular) is regularized by means of the same finitepart. The z -integration involves a weighting function d ℓ ( z ) defined by d ℓ ( z ) = ( ℓ + ) !!2 ℓ + ℓ ! ( − z ) ℓ . (31)The integral of that function is normalized to one: R + − d z d ℓ ( z ) =
1. Furthermore itapproaches the Dirac function in the limit of large multipoles: lim ℓ → ¥ d ℓ ( z ) = d ( z ) .The multipole moments (30) are physically valid only for post-Newtonian sources.As such, they must be considered only in a perturbative post-Newtonian sense. Withthe result (30) the multipole expansion (29) is fully determined and will be exploitedin the next Section.Concerning the near-zone field (25) we find that the radiation-reaction functions A L are composed of the multipole moments F L which will also characterize “linear-order” radiation reaction effects starting at 2.5PN order, and of an extra contribution R L which will be due to non-linear effets in the radiation reaction and turn out toarise at 4PN order. Thus, A ab L = F ab L + R ab L , (32)where F L is given by (30) and where R L is defined from the multipole expansion ofthe pseudo-tensor as R ab L ( t ) = FP B = Z d x | e x | B ˆ x L Z + ¥ d z g ℓ ( z ) M ( t ab ) ( x , t − z | x | / c ) . (33)Here the regularization deals with the bound of the integral at | x | =
0. Since thevariable z extends up to infinity these functions truly depend on the whole past-history of the source. The weighting function therein is simply given by g ℓ ( z ) ≡− d ℓ ( z ) , this definition being motivated by the fact that the integral of that functionis normalized to one: R + ¥ d z g ℓ ( z ) = The specific contributions due to R L in thepost-Newtonian metric (25) are associated with tails of waves [37, 38]. The fact thatthe external multipolar expansion M ( t ) is the source term for the function R L , andtherefore will enter the expression of the near-zone metric (25), is a result of thematching condition (26) and reflects of course the no-incoming radiation conditionimposed at I − h .The post-Newtonian metric (25) is now fully determined. However let us nowderive an interesting alternative formulation of it [45]. To this end we introduce stillanother object which will be made of the expansion of the standard retarded integral(15) when c → ¥ , but acting on a post-Newtonian source term t , This integral is a priori divergent, however its value can be obtained by invoking complex analyticcontinuation in ℓ ∈ C .4 Luc Blanchet (cid:3) − (cid:2) t ab (cid:3) ( x , t ) ≡ − p + ¥ (cid:229) n = ( − ) n n ! (cid:18) ¶ c ¶ t (cid:19) n FP B = Z d x ′ | e x ′ | B | x − x ′ | n − t ab ( x ′ , t ) . (34)Each of the terms is regularized by means of the finite part to deal with the boundat infinity where the post-Newtonian expansion is singular. This regularization iscrucial and the object should carefully be distinguished from the “global” solution (cid:3) − [ t ] defined by (14) and in which the pseudo-tensor is not expanded in post-Newtonian fashion. We emphasize that (34) constitutes merely the definition of a(formal) post-Newtonian expansion, each term of which being built from the post-Newtonian expansion of the pseudo-tensor. Such a definition is of interest because itcorresponds to what one would intuitively think as the “natural” way of performingthe post-Newtonian iteration, i.e. by Taylor expanding the retardations as in [31].Moreover, each of the terms of the series (34) is mathematically well-defined thanksto the finite part, and can therefore be implemented in practical computations. Thepoint is that (34) solves, in a post-Newtonian sense, the wave equation, (cid:3) (cid:16) (cid:3) − (cid:2) t ab (cid:3)(cid:17) = t ab , (35)so constitutes a good prescription for a particular solution of the wave equation— as legitimate a prescription as (23). Therefore (23) and (34) should differ byan homogeneous solution of the wave equation which is necessarily of the anti-symmetric type. Detailed investigations yield (cid:3) − (cid:2) t ab (cid:3) = (cid:3) − (cid:2) t ab (cid:3) − p + ¥ (cid:229) ℓ = ( − ) ℓ ℓ ! ˆ ¶ L ( F ab L ( t − r / c ) − F ab L ( t + r / c ) r ) , (36)in which the homogeneous solution is parametrized precisely by the multipole-moment functions F L ( t ) . This formula is the basis of our writing of the new form ofthe post-Newtonian expansion. Indeed, by combining (25) and (36), we nicely get h ab = p Gc (cid:3) − (cid:2) t ab (cid:3) − Gc + ¥ (cid:229) ℓ = ( − ) ℓ ℓ ! ˆ ¶ L ( R ab L ( t − r / c ) − R ab L ( t + r / c ) r ) , (37)which is our final expression for the general solution of the post-Newtonian fieldin the near-zone of any isolated matter distribution. This expression is probably themost convenient and fruitful when doing practical applications.We recognize in the first term of (37) (notwithstanding the finite part therein)the old way of performing the post-Newtonian expansion as it was advocated byAnderson & DeCanio [31]. For computations limited to the 3.5PN order, i.e. upto the level of the 1PN correction to the radiation reaction force, such first termis sufficient. However, at the 4PN order there is a fundamental breakdown of thisscheme and it becomes necessary to take into account the second term in (37) whichcorresponds to non-linear radiation reaction effects associated with tails. ost-Newtonian theory and the two-body problem 15 Note that the post-Newtonian solution, in either form (25) or (37), has been ob-tained without imposing the condition of harmonic coordinates in an explicit way,see (7). We have simply matched together the post-Newtonian and multipolar ex-pansions, satisfying the “relaxed” Einstein field equations (8) in their respective do-mains, and found that the matching determines uniquely the solution. An importantcheck (carried out in [33, 45]) is therefore to verify that the harmonic coordinatecondition (7) is indeed satisfied as a consequence of the conservation of the pseudo-tensor (9), so that we really grasp a solution of the full Einstein field equations.
The multipole expansion of the field outside a general post-Newtonian source hasbeen obtained in the previous Section as M ( h ab ) = − Gc + ¥ (cid:229) ℓ = ( − ) ℓ ℓ ! ˆ ¶ L ( F ab L ( t − r / c ) r ) + u ab , (38)where the multipole moments are explicitly given by (30), and the second piecereflects the non-linearities of the Einstein field equations and reads u ab = (cid:3) − (cid:2) M ( L ab ) (cid:3) . (39)To write the latter expression we have used the fact that since the matter tensor T ab has a spatially compact support we have M ( T ab ) =
0. Thus u ab is indeed generatedby the non-linear gravitational source term (11). We notice that the divergence of thispiece, say w a ≡ ¶ m u am , is a retarded homogeneous solution of the wave equation, i.e. of the same type as the first term in (38). Now from w a we can construct asecondary object v ab which is also a retarded homogeneous solution of the waveequation, and furthermore whose divergence satisfies ¶ m v am = − w a , so that it willcancel the divergence of u ab (see [44] for details). With the above construction of v ab we are able to define the following combination, G h ab ( ) ≡ − Gc + ¥ (cid:229) ℓ = ( − ) ℓ ℓ ! ˆ ¶ L ( F ab L ( t − r / c ) r ) − v ab , (40)which will constitute the linearized approximation to the multipolar expansion M ( h ab ) outside the source. Then we have M ( h ab ) = G h ab ( ) + u ab + v ab . (41) An alternative formulation of the multipole expansion for a post-Newtonian source, with non-STF multipole moments, has been developed by Will and collaborators [46, 47, 48].6 Luc Blanchet
Having singled out such linearized part, it is clear that the sum of the second andthird terms should represent the non-linearities in the external field. If we indexthose non-linearities by Newton’s constant G , then we can prove indeed that u ab + v ab = O ( G ) . More precisely we can decompose u ab + v ab as a complete non-linearity or “post-Minkowskian” expansion of the type u ab + v ab = + ¥ (cid:229) m = G m h ab ( m ) . (42)One can effectively define a post-Minkowskian “algorithm” [36, 44] able to con-struct the non-linear series up to any post-Minkowskian order m . The post-Minkows-kian expansion represents the most general solution of the Einstein field equationsin harmonic coordinates valid in the vacuum region outside an isolated source.The above linearized approximation h ( ) solves the linearized vacuum Einsteinfield equations in harmonic coordinates and it can be decomposed into multipolemoments in a standard way [49]. Modulo an infinitesimal gauge transformation pre-serving the harmonic gauge, namely h ab ( ) = k ab ( ) + ¶ a j b ( ) + ¶ b j a ( ) − h ab ¶ m j m ( ) , (43)where the infinitesimal gauge vector j a ( ) satisfies (cid:3) j a ( ) =
0, we can decompose k ( ) = − c (cid:229) ℓ ≥ ( − ) ℓ ℓ ! ¶ L (cid:18) r I L (cid:19) , (44a) k i ( ) = c (cid:229) ℓ ≥ ( − ) ℓ ℓ ! (cid:26) ¶ L − (cid:18) r I ( ) iL − (cid:19) + ℓℓ + e iab ¶ aL − (cid:18) r J bL − (cid:19)(cid:27) , (44b) k i j ( ) = − c (cid:229) ℓ ≥ ( − ) ℓ ℓ ! (cid:26) ¶ L − (cid:18) r I ( ) i jL − (cid:19) + ℓℓ + ¶ aL − (cid:18) r e ab ( i J ( ) j ) bL − (cid:19)(cid:27) . (44c)This decomposition defines two types of multipole moments, both assumed to beSTF: the mass-type I L ( u ) and the current-type J L ( u ) . These moments can be arbi-trary functions of the retarded time u ≡ t − r / c , except that the monopole and dipolemoments (having ℓ ≤
1) satisfy standard conservation laws, namelyI ( ) = I ( ) i = J ( ) i = . (45)The gauge transformation vector admits a decomposion in similar fashion, j ( ) = c (cid:229) ℓ ≥ ( − ) ℓ ℓ ! ¶ L (cid:18) r W L (cid:19) , (46a) The supersript ( k ) refers to k time derivatives of the moments; e abc is the Levi-Civita antisymmet-ric symbol such that e =
1. From here on the spatial indices such as i , j , . . . will be raised andlowered with the Kronecker metric d i j . They will be located lower or upper depending on context.ost-Newtonian theory and the two-body problem 17 j i ( ) = − c (cid:229) ℓ ≥ ( − ) ℓ ℓ ! ¶ iL (cid:18) r X L (cid:19) (46b) − c (cid:229) ℓ ≥ ( − ) ℓ ℓ ! (cid:26) ¶ L − (cid:18) r Y iL − (cid:19) + ℓℓ + e iab ¶ aL − (cid:18) r Z bL − (cid:19)(cid:27) . (46c)The six sets of STF multipole moments I L , J L , W L , X L , Y L and Z L will collectivelybe called the multipole moments of the source. They contain the full physical in-formation about any isolated source as seen from its exterior near zone. Actuallyit should be clear that the main moments are I L and J L because the other momentsW L , · · · , Z L parametrize a linear gauge transformation and thus have no physicalimplications at the linearized order. However because the theory is covariant withrespect to non-linear diffeomorphisms and not merely with respect to linear gaugetransformations, the moments W L , · · · , Z L do play a physical role starting at thenon-linear level. We shall occasionaly refer to the moments W L , X L , Y L and Z L asthe gauge moments.To express in the best way the source multipole moments, we introduce the fol-lowing notation for combinations of components of the pseudo-tensor t ab , S ≡ t + t ii c , (47a) S i ≡ t i c , (47b) S i j ≡ t i j , (47c)where t ii ≡ d i j t i j . Here the overbar reminds us that we are exclusively dealing withpost-Newtonian-expanded expressions, i.e. formal series of the type (17). Then thegeneral expressions of the “main” source multipole moments I L and J L in the caseof the time-varying moments for which ℓ ≥
2, areI L ( u ) = FP B = Z d x | e x | B Z − d z (cid:26) d ℓ ˆ x L S − ( ℓ + ) c ( ℓ + )( ℓ + ) d ℓ + ˆ x iL S ( ) i + ( ℓ + ) c ( ℓ + )( ℓ + )( ℓ + ) d ℓ + ˆ x i jL S ( ) i j (cid:27) , (48a)J L ( u ) = FP B = Z d x | e x | B Z − d z e ab h i ℓ (cid:26) d ℓ ˆ x L − i a S b − ℓ + c ( ℓ + )( ℓ + ) d ℓ + ˆ x L − i ac S ( ) bc (cid:27) . (48b)The integrands are computed at the spatial point x and at time u + z | x | / c , where u = t − r / c is the retarded time at which are evaluated the moments. We recall that z is theargument of the function d ℓ defined in (31). Similarly we can write the expressionsof the gauge-type moments W L , · · · , Z L . Notice that the source multipole moments (48) have no invariant meaning; they are defined for the harmonic coordinate systemwe have chosen.Of what use are these results for the multipole moments I L and J L ? From (44)these moments parametrize the linearized metric h ( ) which is the “seed” of an in-finite post-Minkowskian algorithm symbolized by (42). For a specific application, i.e. for a specific choice of matter tensor like the one we shall describe in Section8, the expressions (48) have to be worked out up to a given post-Newtonian order.The moments should then be inserted into the post-Minkowskian series (42) for thecomputation of the non-linearities. The result will be in the form of a non-linearmultipole decomposition depending on the source moments I L , J L , · · · , Z L , say M ( h ab ) = + ¥ (cid:229) m = G m h ab ( m ) [ I L , J L , · · · ] . (49)In the next Section we shall expand this metric at (retarded) infinity from the sourcein order to obtain the observables of the gravitational radiation field. The asymptotic waveform at future null infinity from an isolated source is thetransverse-traceless (TT) projection of the metric deviation at the leading order 1 / R in the distance R = | X | to the source, in a radiative coordinate system X m = ( c T , X ) . The waveform can be uniquely decomposed [49] into radiative multipole compo-nents parametrized by mass-type moments U L and current-type ones V L . We shalldefine the radiative moments in such a way thay they agree with the ℓ -th time deriva-tives of the source moments I L and J L at the linear level, i.e. U L = I ( ℓ ) L + O ( G ) , (50a)V L = J ( ℓ ) L + O ( G ) . (50b)At the non-linear level the radiative moments will crucially differ from the sourcemoments; the relations between these two types of moments will be discussed in thenext Section. The radiative moments U L ( U ) and V L ( U ) are functions of the retardedtime U ≡ T − R / c in radiative coordinates.The asymptotic waveform at distance R and retarded time U is then given by h TT i j = Gc R P i jkl + ¥ (cid:229) ℓ = c ℓ ℓ ! (cid:26) N L − U klL − − ℓ c ( ℓ + ) N aL − e ab ( k V l ) bL − (cid:27) . (51) Radiative coordinates T and X , also called Bondi-type coordinates [50], are such that the met-ric coefficients admit an expansion when R → + ¥ with U ≡ T − R / c being constant, in simplepowers of 1 / R , without the logarithms of R plaguing the harmonic coordinate system. Here U isa null or asymptotically null characteristic. It is known that the “far-zone” logarithms in harmoniccoordinates can be removed order-by-order by going to radiative coordinates [51].ost-Newtonian theory and the two-body problem 19 We denote by N = X / R = ( N i ) the unit vector pointing from the source to the far-away detector. The TT projection operator reads P i jkl = P ik P jl − P i j P kl where P i j = d i j − N i N j is the projector orthogonal to the unit direction N . We project outthe asymptotic waveform (51) on polarization directions in a standard way. We de-note the two unit polarisation vectors by P and Q , which are orthogonal and trans-verse to the direction of propagation N (hence P i j = P i P j + Q i Q j ). Our conventionsand choice for P and Q will be specified in Section 12. Then the two “plus” and“cross” polarisation states of the waveform are h + ≡ P i P j − Q i Q j h TT i j , (52a) h × ≡ P i Q j + P j Q i h TT i j . (52b)Although the multipole decomposition (51) entirely describes the waveform,it is also important, especially having in mind the comparison between the post-Newtonian results and numerical relativity [52], to consider separately the modes ( ℓ, m ) of the waveform as defined with respect to a basis of spin-weighted sphericalharmonics. To this end we decompose h + and h × as (see e.g. [52, 53]) h + − i h × = + ¥ (cid:229) ℓ = ℓ (cid:229) m = − ℓ h ℓ m Y ℓ m − ( Q , F ) , (53)where the spin-weighted spherical harmonics of weight − ( Q , F ) defining the direction of propagation N and reads Y ℓ m − = r ℓ + p d ℓ m ( Q ) e i m F , (54a) d ℓ m ≡ k (cid:229) k = k D ℓ mk (cid:18) cos Q (cid:19) ℓ + m − k − (cid:18) sin Q (cid:19) k − m + , (54b) D ℓ mk ≡ ( − ) k k ! p ( ℓ + m ) ! ( ℓ − m ) ! ( ℓ + ) ! ( ℓ − ) ! ( k − m + ) ! ( ℓ + m − k ) ! ( ℓ − k − ) ! . (54c)Here k = max ( , m − ) and k = min ( ℓ + m , ℓ − ) . Using the orthonormality prop-erties of these harmonics we obtain the separate modes h ℓ m from the surface integral(with the overline denoting the complex conjugate) h ℓ m = Z d W h h + − i h × i Y ℓ m − ( Q , F ) . (55)On the other hand, we can also write h ℓ m directly in terms of the radiative multipolemoments U L and V L , with result h ℓ m = − G √ R c ℓ + (cid:20) U ℓ m − i c V ℓ m (cid:21) , (56) where U ℓ m and V ℓ m are the radiative mass and current moments in non-STF guise.These are given in terms of the STF moments byU ℓ m = ℓ ! s ( ℓ + )( ℓ + ) ℓ ( ℓ − ) a ℓ mL U L , (57a)V ℓ m = − ℓ ! s ℓ ( ℓ + ) ( ℓ + )( ℓ − ) a ℓ mL V L . (57b)Here a ℓ mL denotes the STF tensor connecting together the usual basis of sphericalharmonics Y ℓ m to the set of STF tensors ˆ N L ≡ STF ( N L ) , recalling that Y ℓ m and ˆ N L represent two basis of an irreducible representation of weight ℓ of the rotation group.They are related by ˆ N L ( Q , F ) = ℓ (cid:229) m = − ℓ a ℓ mL Y ℓ m ( Q , F ) , (58a) Y ℓ m ( Q , F ) = ( ℓ + ) !!4 p ℓ ! a ℓ mL ˆ N L ( Q , F ) , (58b)with the STF tensorial coefficient being a ℓ mL = Z d W ˆ N L Y ℓ m . (59)The decomposition in spherical harmonic modes is especially useful if some of theradiative moments are known to higher post-Newtonian order than others. In thiscase the comparison with the numerical calculation [52, 53] can be made for theseindividual modes with higher post-Newtonian accuracy. The basis of our computation is the general solution of the Einstein field equa-tions outside an isolated matter system computed iteratively in the form of a post-Minkowskian or non-linearity expansion (49) (see details in [36, 39]). Here we givesome results concerning the relation between the set of radiative moments { U L , V L } and the sets of source moments { I L , J L } and gauge moments { W L , · · · , Z L } . Com-plete results up to 3PN order are available and have recently been used to controlthe 3PN waveform of compact binaries [54].Armed with definitions for all those moments, we proceed in a modular way.We express the radiative moments { U L , V L } in terms of some convenient intermedi-ate constructs { M L , S L } called the canonical moments. Essentially these canonicalmoments take into account the effect of the gauge transformation present in (43).Therefore they differ from the source moments { I L , J L } only at non-linear order. We ost-Newtonian theory and the two-body problem 21 shall see that in terms of a post-Newtonian expansion the canonical and source mo-ments agree with each other up to 2PN order. The canonical moments are then con-nected to the actual source multipole moments { I L , J L } and { W L , · · · , Z L } . The pointof the above strategy is that the source moments (including gauge moments) admitclosed-form expressions as integrals over the stress-energy distribution of matterand gravitational fields in the source, as shown in (48).The mass quadrupole moment U i j (having ℓ =
2) is known up to the 3PN order[55]. At that order it is made of quadratic and cubic non-linearities, and we haveU i j ( U ) = M ( ) i j ( U ) + G M c Z U − ¥ d u (cid:20) ln (cid:18) U − u u (cid:19) + (cid:21) M ( ) i j ( u )+ Gc (cid:26) − Z U − ¥ d u M ( ) a h i ( u ) M ( ) j i a ( u )+
17 M ( ) a h i M j i a −
57 M ( ) a h i M ( ) j i a −
27 M ( ) a h i M ( ) j i a + e ab h i M ( ) j i a S b (cid:27) + (cid:18) G M c (cid:19) Z U − ¥ d u (cid:20) ln (cid:18) U − u u (cid:19) + (cid:18) U − u u (cid:19) + (cid:21) M ( ) i j ( u )+ O (cid:18) c (cid:19) . (60)Notice the quadratic tail integral at 1.5PN order, the cubic tail-of-tail integral at 3PNorder, and the non-linear memory integral at 2.5PN order [56, 57, 58, 39]. The tail iscomposed of the coupling between the mass quadrupole moment M i j and the massmonopole moment or total mass M; the tail-of-tail is a coupling between M i j andtwo monopoles M × M; the non-linear memory is a coupling M i j × M kl . All these“hereditary” integrals imply a dependence of the waveform on the complete historyof the source, from infinite past up to the current retarded time U ≡ T − R / c . Theconstant u in the tail integrals is defined by u ≡ r / c , where r is the arbitrarylength scale introduced in (19).Note that the dominant hereditary integral is the tail arising at 1.5PN order in allradiative moments. For general ℓ we have at that orderU L = M ( ℓ ) L + G M c Z U − ¥ d u (cid:20) ln (cid:18) U − u u (cid:19) + k ℓ (cid:21) M ( ℓ + ) L ( u ) + O (cid:18) c (cid:19) , (61a)V L = S ( ℓ ) L + G M c Z U − ¥ d u (cid:20) ln (cid:18) U − u u (cid:19) + p ℓ (cid:21) S ( ℓ + ) L ( u ) + O (cid:18) c (cid:19) , (61b)where the constants k ℓ and p ℓ are given by k ℓ = ℓ + ℓ + ℓ ( ℓ + )( ℓ + ) + ℓ − (cid:229) k = k , (62a) p ℓ = ℓ − ℓ ( ℓ + ) + ℓ − (cid:229) k = k . (62b) Now it can be proved that the retarded time U in radiative coordinates reads U = t − rc − G M c ln (cid:18) rr (cid:19) + O (cid:18) c (cid:19) , (63)where ( t , r ) are the harmonic coordinates. Inserting U into (61) we obtain the radia-tive moments expressed in terms of local source-rooted coordinates ( t , r ) , e.g. U L = M ( ℓ ) L ( t − r / c )+ G M c Z t − r / c − ¥ d u (cid:20) ln (cid:18) t − u − r / c r / c (cid:19) + k ℓ (cid:21) M ( ℓ + ) L ( u )+ O (cid:18) c (cid:19) . (64)This no longer depends on the constant u — i.e. the u gets replaced by the retar-dation time r / c . More generally it can be checked that u always disappears fromphysical results at the end. On the other hand we can be convinced that the constant k ℓ (and p ℓ as well) depends on the choice of source-rooted coordinates ( t , r ) . For in-stance if we change the harmonic coordinate system ( t , r ) to some “Schwarzschild-like” coordinates ( t ′ , r ′ ) such that t ′ = t and r ′ = r + G M / c , we get a new constant k ′ ℓ = k ℓ + /
2. Thus we have k = /
12 in harmonic coordinates [as shown in(60)], but k ′ = /
12 in Schwarzschild coordinates.We still have to relate the canonical moments { M L , S L } to the source multipolemoments. As we said the difference between these two types of moments comesfrom the gauge transformation (46) and arises only at the small 2.5PN order. Theconsequence is that we have to worry about this difference only for high post-Newtonian waveforms. For the mass quadrupole moment M i j , the requisite cor-rection is given byM i j = I i j + Gc h W ( ) I i j − W ( ) I ( ) i j i + O (cid:18) c (cid:19) , (65)where I i j denotes the source mass quadrupole, and where W is the monopole corre-sponding to the gauge moments W L ( i.e. the moment having ℓ = i = { I L , J L , W L , X L , Y L , Z L } as the fundamental variables describing the source. B Inspiralling compact binaries8 Stress-energy tensor of spinning particles
So far the post-Newtonian formalism has been developed for arbitrary matter dis-tributions. We want now to apply it to material systems made of compact objects ost-Newtonian theory and the two-body problem 23 (neutron stars or black holes) which can be described with great precision by pointmasses. We thus discuss the modelling of point-particles possibly carrying someintrinsic rotation or spin. This means finding the appropriate stress-energy tensorwhich will have to be inserted into the general post-Newtonian formulas such as theexpressions of the source moments (48).In the general case the stress-energy tensor will be the sum of a “monopo-lar” piece, which is a linear combination of monopole sources, i.e. made of Diracdelta-functions, plus the “dipolar” or spin piece, made of gradients of Dirac delta-functions. Hence we write T ab = T ab mono + T ab spin . (66)The monopole part takes the form of the stress-energy tensor for N particles (la-belled by A = , · · · , N ) without spin, reading in a four-dimensional picture T ab mono = c N (cid:229) A = Z + ¥ − ¥ d t A p ( a A u b ) A d ( ) ( x − y A ) p − ( g ) A . (67)Here d ( ) is the four-dimensional Dirac function. The world-line of particle A , de-noted y a A , is parametrized by the particle’s proper time t A . The four-velocity is givenby u a A = d y a A / d t A and is normalized to ( g mn ) A u m A u n A = − c , where ( g mn ) A denotesthe metric at the particle’s location. The four-vector p a A is the particle’s linear mo-mentum. For particles without spin we shall simply have p a A = m A u a A . However forspinning particles p a A will differ from that and include some contributions from thespins as given by (75) below.The dipolar part of the stress-energy tensor depends specifically on the spins,and reads in the classic formalism of spinning particles (due to Tulczyjew [59, 60],Trautman [61], Dixon [62], Bailey & Israel [63]), T ab spin = − N (cid:229) A = (cid:209) m (cid:20) Z + ¥ − ¥ d t A S m ( a A u b ) A d ( ) ( x − y A ) p − ( g ) A (cid:21) , (68)where (cid:209) m is the covariant derivative, and the anti-symmetric tensor S ab A representsthe spin angular momentum of particle A . In this formalism the momentum-likequantity p a A [entering (67)] is a time-like solution of the equationD S ab A d t A = (cid:16) p a A u b A − p b A u a A (cid:17) , (69)where D / d t A denotes the covariant proper time derivative. The equation of trans-lational motion of the spinning particle, equivalent to the covariant conservation (cid:209) m T am = p a A d t A = − S mn A u r A ( R a rmn ) A . (70) The Riemann tensor is evaluated at the particle’s position A . This equation can alsobe derived directly from an action principle [63].It is well-known that a choice must be made for a supplementary spin conditionin order to fix unphysical degrees of freedom associated with an arbitrariness in thedefinition of the spin tensor S ab . This arbitrariness can be interpreted, in the case ofextended bodies, as a freedom in the choice for the location of the center-of-massworldline of the body, with respect to which the angular momentum is defined (see e.g. [65]). An elegant spin condition is the covariant one, S am A p A m = , (71)which allows a natural definition of a spin four-(co)vector S A a such that S ab A = − p − ( g ) A e abmn p A m m A c S A n . (72)For the spin vector S A a itself, we can choose a four-vector which is purely spatialin the particle’s instantaneous rest frame, where u a A = ( , ) . Therefore, we deducethat in any frame S A m u m A = . (73)As a consequence of the covariant spin condition (71), we easily verify that the spinscalar is conserved along the trajectories, i.e.S mn A S A mn = const . (74)Furthermore, we can check, using (71) and also the law of motion (70), that the massdefined by m A c = − p m A p A m is also constant along the trajectories: m A = const. Fi-nally, the relation linking the four-momentum p a A and the four-velocity u a A is readilydeduced from the contraction of (69) with the four-momentum, which results in p a A ( pu ) A + m A c u a A = S am A S nr A u s A ( R msnr ) A , (75)where ( pu ) A ≡ p A m u m A . Contracting further this relation with the four-velocity onededuces the expression of ( pu ) A and inserting this back into (75) yields the desiredrelation between p a A and u a A .Focusing our attention on spin-orbit interactions, which are linear in the spins,we can neglect quadratic and higher spin corrections denoted O ( S ) ; drastic sim-plifications of the formalism occur in this case. Since the right-hand-side of (75) isquadratic in the spins, we find that the four-momentum is linked to the four-velocityby the simple proportionality relation p a A = m A u a A + O ( S ) . (76)Hence, the spin condition (71) becomes ost-Newtonian theory and the two-body problem 25 S am A u A m = O ( S ) . (77)Also, the equation of evolution for the spin, sometimes called the precessional equa-tion, follows immediately from the relationship (69) together with the law (76) asD S ab A d t A = O ( S ) ⇐⇒ D S A a d t A = O ( S ) . (78)Hence the spin vector S A a satisfies the equation of parallel transport, which meansthat it remains constant in a freely falling frame, as could have been expected be-forehand. Of course the norm of the spin vector is preserved along trajectories, S A m S m A = const . (79)In Section 13 we shall apply this formalism to the study of spin-orbit effects in theequations of motion and energy flux of compact binaries. The stress-energy tensor of point masses has been defined in the previous Section bymeans of Dirac functions, and involves metric coefficients evaluated at the locationsof the point particles, namely ( g ab ) A . However, it is clear that the metric g ab be-comes singular at the particles. Indeed this is already true at Newtonian order wherethe potential generated by N particles reads U = N (cid:229) B = Gm B r B , (80)where r B ≡ | x − y B | is the distance between the field point and the particle B . Thusthe values of U and hence of the metric coefficients [recall that g = − + U / c + O ( c − ) ], are ill-defined at the locations of the particles. What we need is a self-fieldregularization , i.e. a prescription for removing the infinite self-field of the pointmasses. Arguably the choice of a particular regularization constitutes a fully quali-fied element of our physical modelling of compact objects. At Newtonian order theregularization of the potential (80) should give the well-known result ( U ) A = (cid:229) B = A Gm B r AB , (81)where r AB ≡ | y A − y B | and the infinite self interaction term has simply been dis-carded from the summation. At high post-Newtonian orders the problem is not triv-ial and the self-field regularization must be properly defined.The post-Newtonian formalism reviewed in Sections 2–7 assumed from the starta continuous (smooth) matter distribution. Actually this formalism will be applica- ble to singular point-mass sources, described by the stress-energy tensor of Section8, provided that we suplement the scheme by the self-field regularization. Note thatthis regularization has nothing to do with the finite-part process FP B = extensivelyused in the case of extended matter sources. The latter finite part was an ingredientof the rigorous derivation of the general post-Newtonian solution [see (20)], whilethe self-field regularization is an assumption regarding a particular type of singularsource.Our aim is to compute up to 3PN order the metric coefficients at the locationof one of the particles: ( g ab ) A . At this stage different self-field regularizations arepossible. We first review Hadamard’s regularization [66, 67], which has proved tobe very efficient for doing practical computations, but suffers from the importantdrawback of yielding some “ambiguity parameters”, which cannot be determinedwithin the regularization, starting at the 3PN order.Iterating the Einstein field equations with point-like matter sources (delta func-tions with spatial supports localized on y A ) yields a generic form of functions rep-resenting the metric coefficients in successive post-Newtonian approximations. Thegeneric functions, say F ( x ) , are smooth except at the points y A , around which theyadmit singular Laurent expansions in powers and inverse powers of r A ≡ | x − y A | .When r A → P ∈ N ) F ( x ) = P (cid:229) p = p r pA f A p ( n A ) + o ( r PA ) . (82)The coefficients A f p of the various powers of r A depend on the unit direction n A ≡ ( x − y A ) / r A of approach to the singular point A . The powers p are relative integers,and are bounded from below by p ∈ Z . The Landau o -symbol for remainders takesits standard meaning. The A f p ’s depend also on the (coordinate) time t , throughtheir dependence on velocities v B ( t ) and relative positions y BC ( t ) ≡ y B ( t ) − y C ( t ) ;however the time t is purely “spectator” in the regularization process, and thus willnot be indicated. The coefficients A f p for which p < singular coefficients of F around A .The function F being given that way, we define the Hadamard partie finie as thefollowing value of F at the location of the particle A , ( F ) A = h f A i ≡ Z d W A p f A ( n A ) , (83)where d W A denotes the solid angle element centred on y A and sustained by n A . Thebrackets hi mean the angular average. The second notion of Hadamard partie finieconcerns the integral R d x F , which is generically divergent at the points y A . Itspartie finie (in short Pf) is defined byPf s ··· s N Z d x F = lim s → (cid:26) Z R \ S B A ( s ) d x F + p N (cid:229) A = D A ( s ) (cid:27) . (84) With this definition it is immediate to check that the previous Newtonian result (81) will hold.ost-Newtonian theory and the two-body problem 27
The first term integrates over the domain R \ S NA = B A ( s ) defined as R deprivedfrom the N spherical balls B A ( s ) ≡ { x ; r A ≤ s } of radius s and centred on the points y A . The second term is the opposite of the sum of divergent parts associated withthe first term around each of the particles in the limit where s →
0. We have D A ( s ) = − (cid:229) p = p s p + p + h f A p i + ln (cid:18) ss A (cid:19) h f A − i . (85)Since the divergent parts are cancelled (by definition) the Hadamard partie finie isobtained in the limit s →
0. Notice that as indicated in (84) the Hadamard partie-finieintegral is not fully specified: it depends on N strictly positive and a priori arbitraryconstants s , · · · , s N parametrizing the logarithms in (85).We have seen that the post-Newtonian scheme consists of breaking the hyper-bolic d’Alembertian operator (cid:3) into the elliptic Laplacian D and the retardationterm c − ¶ t considered to be small, and put in the right-hand-side of the equationwhere it can be iterated; see (18). We thus have to deal with the regularization ofPoisson integrals, or iterated Poisson integrals, of some generic function F . ThePoisson integral will be divergent and we apply the prescription (84). Thus, P ( x ′ ) = − p Pf s ··· s N Z d x | x − x ′ | F ( x ) . (86)This definition is valid for each field point x ′ different from the y A ’s, and we wantto investigate the singular limit when x ′ tends to one of the source points y A , soas to define the object ( P ) A . The definition (83) is not directly applicable becausethe expansion of the Poisson integral P ( x ′ ) when x ′ → y A will involve besides thenormal powers of r ′ A ≡ | x ′ − y A | some logarithms of r ′ A . The proper way to definethe Hadamard partie finie in this case is to include the ln r ′ A into the definition (83)as if it were a mere constant parameter. With this definition we arrive at [68] ( P ) A = − p Pf s ··· s N Z d x r A F ( x ) + (cid:20) ln (cid:18) r ′ A s A (cid:19) − (cid:21) h f A − i . (87)The first term is given by a partie-finie integral following the definition (84); thesecond involves the logarithm of r ′ A . The constants s , · · · , s N come from (86). Since r ′ A is actually tending to zero, ln r ′ A represents a formally infinite “constant”, whichwill ultimately parametrize the final Hadamard regularized 3PN equations of mo-tion. In the two-body case we shall find that the constants r ′ A are unphysical in thesense that they can be removed by a coordinate transformation [69]. Note that theapparent dependence of (87) on the constant s A is illusory. Indeed the dependenceon s A cancels out between the first and the second terms in the right-hand-side of(87), so the result depends only on r ′ A and the s B ’s for B = A . We thus have a simpler We consider only the local divergencies due to the singular points y A . The problem of divergen-cies of Poisson integrals at infinity is part of the general post-Newtonian formalism and has beentreated in Section 3.8 Luc Blanchet rewriting of (87) as ( P ) A = − p Pf s ··· r ′ A ··· s N Z d x r A F ( x ) − h f A − i . (88)Unfortunately, the constants s B for B = A remaining in the result (88) will be thesource of a genuine ambiguity. This ambiguity can in fact be traced back to the so-called “non-distributivity” of the Hadamard partie finie, a consequence of the pres-ence of the angular integration in (83), and implying that ( FG ) A = ( F ) A ( G ) A in gen-eral. The non-distributivity arises precisely at the 3PN order both in the equations ofmotion and radiation field of point-mass binaries. At that order we are loosing withHadamard’s regularization an elementary rule of ordinary calculus. Consequentlywe expect that some basic symmetries of general relativity such as diffeomorphisminvariance will be lost. However Hadamard’s regularization can still be efficientlyused to compute most of the terms in the equations of motion and radiation fieldat 3PN order; only a few ambiguous terms will show up which have then to bedetermined by another method.
10 Dimensional regularization
Dimensional regularization is an extremely powerful regularization which is freeof ambiguities (at least up to the 3PN order). The main reason is that it is able topreserve the symmetries of classical general relativity; in fact dimensional regular-ization was invented [70, 71] as a means to preserve the gauge symmetry of pertur-bative quantum field theories. In the present context we shall show that dimensionalregularization permits to resolve the problem of ambiguities arising at the 3PN or-der in Hadamard’s regularization. We shall employ dimensional regularization notmerely as a trick to compute some particular integrals which would otherwise be di-vergent or ambiguous, but as a fundamental tool for solving in a consistent way theEinstein field equations with singular sources. We therefore assume that the correcttheory is general relativity in D = d + d ∈ C . In particular we shall analytically continue d down to thevalue of interest 3 and pose d = + e . (89)The Einstein field equations in d spatial dimensions take the same form as pre-sented in Section 2, with the exception that the explicit expression of the gravita-tional source term L ab now depends on d . We find that only the last term in (11)acquires a dependence on d ; namely the factor in g rs g ep − g se g rp should nowread d − . In addition the d -dimensional gravitational constant is related to the usualthree-dimensional Newton constant G by G ( d ) = G ℓ e , (90) ost-Newtonian theory and the two-body problem 29 where ℓ is a characteristic length associated with dimensional regularization.In the post-Newtonian iteration performed in d dimensions we shall meet theanalogue of the function F , which we denote by F ( d ) ( x ) where x ∈ R d . It turns outthat in the vicinity of the singular points y A , the function F ( d ) admits an expansionricher than in (82), and of the type F ( d ) ( x ) = P (cid:229) p = p q (cid:229) q = q r p + q e A f A ( e ) p , q ( n A ) + o ( r PA ) . (91)The coefficients A f ( e ) p , q ( n ) depend on the dimension through e ≡ d − ℓ . The powers of r A are now of the type p + q e where the two relativeintegers p , q ∈ Z have values limited as indicated. Because F ( d ) reduces to F when e = q (cid:229) q = q f A ( ) p , q = f A p . (92)To proceed with the iteration we need the Green function of the Laplace operatorin d dimensions. Its explicit form is u ( d ) A = K r − dA . (93)It satisfies D u ( d ) A = − pd ( d ) A , where d ( d ) A ≡ d ( d ) ( x − y A ) denotes the d -dimensionalDirac function. The constant K is given by K = G (cid:0) d − (cid:1) p d − , (94)where G is the usual Eulerian function. It reduces to one when d →
3. Note that thevolume W d − of the sphere with d − K by W d − = p ( d − ) K . (95)With these results the Poisson integral of F ( d ) , constituting the d -dimensional ana-logue of (86), reads P ( d ) ( x ′ ) = − K p Z d d x | x − x ′ | d − F ( d ) ( x ) . (96)In dimensional regularization the singular behavior of this integral is automaticallytaken care of by analytic continuation in d . Next we evaluate the integral at thesingular point x ′ = y A . In contrast with Hadamard’s regularization where the resultwas given by (87), in dimensional regularization this is quite easy, as we are allowedto simply replace x ′ by y A into the explicit integral form (96). So we simply have P ( d ) ( y A ) = − K p Z d d x r d − A F ( d ) ( x ) . (97)The main step of our strategy [72, 73] will now consist of computing the differ-ence between the d -dimensional Poisson potential (97) and its 3-dimensional coun-terpart which is defined from Hadamard’s regularization as (87). We shall then addthis difference (in the limit e = d − →
0) to the result obtained by Hadamard reg-ularization in order to get the corresponding dimensional regularization result. Thisstrategy is motivated by the fact that as already mentionned most of the terms do notpresent any problems and have already been correctly computed using Hadamard’sregularization. Denoting the difference between the two regularizations by means ofthe script letter D , we write D ( P ) A ≡ P ( d ) ( y A ) − ( P ) A . (98)We shall only compute the first two terms of the Laurent expansion of D ( P ) A when e →
0, which will be of the form D ( P ) A = a − e − + a + O ( e ) . This is the in-formation needed to determine the value of the ambiguity parameters. Notice thatthe difference D ( P ) A comes exclusively from the contribution of terms developingsome poles (cid:181) / e in the d -dimensional calculation. The ambiguity in Hadamard’sregularization at 3PN order is reflected by the appearance of poles in d dimensions.The point is that in order to obtain the difference D ( P ) A we do not need the expres-sion of F ( d ) for an arbitrary source point x but only in the vicinity of the singularpoints y A . Thus this difference depends only on the singular coefficients of the localexpansions of F ( d ) near the singularities. We find [73] D ( P ) A = − e ( + e ) q (cid:229) q = q "(cid:18) q + e h ln r ′ A − i(cid:19) h f A ( e ) − , q i + (cid:229) B = A (cid:18) q + + e ln s B (cid:19) + ¥ (cid:229) ℓ = ( − ) ℓ ℓ ! ¶ L (cid:18) r + e AB (cid:19) h n LB f B ( e ) − ℓ − , q i + O ( e ) . (99)We still use the bracket notation to denote the angular average but this time per-formed in d dimensions, i.e. h f A ( e ) p , q i ≡ Z d W d − ( n A ) W d − f A ( e ) p , q ( n A ) . (100)The above differences for all the Poisson and interated Poisson integrals compos-ing the equations of motion ( i.e. the accelerations of the point masses) are added tothe corresponding results of the Hadamard regularization in the variant of it calledthe “pure Hadamard-Schwartz” regularization (see [73] for more details). In thisway we find that the equations of motion in dimensional regularization are com-posed of a pole part (cid:181) / e which is purely 3PN, followed by a finite part when ost-Newtonian theory and the two-body problem 31 e →
0, plus the neglected terms O ( e ) . It has been shown (in the two-body case N =
2) that:1. The pole part (cid:181) / e of the accelerations can be renormalized into some shifts ofthe “bare” world-lines by y A → y A + xxx A , with xxx A containing the poles, so thatthe result expressed in terms of the “dressed” world-lines is finite when e → xxx A , if and only if the ambiguities in the Hadamard regularization are fully anduniquely determined ( i.e. take specific values).These results [73] provide an unambiguous determination of the equations of motionof compact binaries up to the 3PN order. A related strategy with similar completeresults has been applied to the problem of multipole moments and radiation field ofpoint-mass binaries [74]. This finally completed the derivation of the general rela-tivistic prediction for compact binary inspiral up to 3PN order (and even to 3.5PNorder). In later Sections we shall review some features of the 3.5PN gravitational-wave templates of inspiralling compact binaries.Why should the final results of the employed regularization scheme be unique, inagreement with our expectation that the problem is well-posed and should possessa unique physical answer? The results can be justified by invoking the “effacingprinciple” of general relativity [20] — namely that the internal structure of the com-pact bodies does not influence the equations of motion and emitted radiation untila very high post-Newtonian order. Only the masses m A of the bodies should drivethe motion and radiation, and not for instance their “compactness” Gm A / ( c a A ) . Amodel of point masses should therefore give the correct physical answer, which weexpect to be also valid for black holes, provided that the regularization scheme ismathematically consistent.
11 Energy and flux of compact binaries
The equations of motion of compact binary sources, up to the highest known post-Newtonian order which is 3.5PN, will serve in the definition of the gravitational-wave templates for two purposes:1. To compute the center-of-mass energy E appearing in the left-hand-side of theenergy balance equation to be used for deducing the orbital phase,d E d t = − F ; (101)2. To order-reduce the accelerations coming from the time derivatives of the sourcemultipole moments required to compute the gravitational-wave energy flux F inthe right-hand-side of the balance equation. We consider two compact objects moving under purely gravitational mutual in-teraction. In a first stage we assume that the bodies are non-spinning so the motiontakes place in a fixed plane, say the x-y plane. The relative position x = y − y ,velocity v = d x / d t , and acceleration a = d v / d t are given by x = r n , (102a) v = ˙ r n + r wl ll , (102b) a = ( ¨ r − r w ) n + ( r ˙ w + r w ) lll , (102c)The orbital frequency w is related in the usual way to the orbital phase f by w = ˙ f (time derivatives are denoted with a dot). Here the vector lll = ˆ z × n is perpendicularto the unit vector ˆ z along the z-direction orthogonal to the orbital plane, and to thebinary’s separation unit direction n ≡ x / r .Through 3PN order, it is possible to model the binary’s orbit as a quasi-circular orbit decaying by the effect of radiation reaction at the 2.5PN order. The restric-tion to quasi-circular orbits is both to simplify the presentation, and for physicalreasons because the orbit of inspiralling compact binaries detectable by current de-tectors should be circular (see the discussion in Section 1). The radiation-reactioneffect at 2.5PN order yields ˙ r = − r Gmr ng / + O (cid:18) c (cid:19) , (103a)˙ w = Gmr ng / + O (cid:18) c (cid:19) , (103b)where g is defined as the small [ i.e. g = O ( c − ) ] post-Newtonian parameter g ≡ Gmrc . (104)Substituting these results into (102), we obtain the expressions for the velocity andacceleration during the inspiral, v = r wl ll − r Gmr ng / n + O (cid:18) c (cid:19) , (105a) a = − w x − r Gmr ng / v + O (cid:18) c (cid:19) . (105b)Notice that while ˙ r = O ( c − ) , we have ¨ r = O ( c − ) which is of the order of the square of radiation-reaction effects and is thus zero with the present approximation. However the 3PN equations of motion are known in an arbitrary frame and for general orbits. Mass parameters are the total mass m ≡ m + m , the symmetric mass ratio n ≡ m m / m satisfying 0 < n ≤ /
4, and for later use the mass difference ratio D ≡ ( m − m ) / m .ost-Newtonian theory and the two-body problem 33 A central result of post-Newtonian calculations is the expression of the orbitalfrequency w in terms of the binary’s separation r up to 3PN order. This result hasbeen obtained independently by three groups. Two are working in harmonic co-ordinates: Blanchet & Faye [69, 75, 73] use a direct post-Newtonian iteration ofthe equations of motion, while Itoh & Futamase [76, 77, 78] apply a variant ofthe surface-integral approach (`a la Einstein-Infeld-Hoffmann [5]) valid for compactbodies without the need of a self-field regularization. The group of Jaranowski &Sch¨afer [79, 80, 72] employs Arnowitt-Deser-Misner coordinates within the Hamil-tonian formalism of general relativity. The 3PN orbital frequency in harmoniccoordinates is w = Gmr (cid:26) + (cid:16) − + n (cid:17) g + (cid:18) + n + n (cid:19) g (106) + (cid:18) − + (cid:20) − + p +
22 ln (cid:18) rr ′ (cid:19)(cid:21) n + n + n (cid:19) g + O (cid:18) c (cid:19)(cid:27) . Note the logarithm at 3PN order coming from a Hadamard self-field regularizationscheme, and depending on a constant r ′ defined by m ln r ′ ≡ m ln r ′ + m ln r ′ ,where r ′ A ≡ | x ′ − y A | are arbitrary “constants” discussed in Section 9. We shall seethat r ′ disappears from final results — it can be qualified as a gauge constant.To obtain the 3PN energy we need to go back to the equations of motion forgeneral non-circular orbits, and deduce the energy as the integral of the motion as-sociated with a Lagrangian formulation of (the conservative part of) these equations[75]. Once we have the energy for general orbits we can reduce it to quasi-circularorbits. We find E = − Gm n r (cid:26) + (cid:18) − + n (cid:19) g + (cid:18) − + n + n (cid:19) g (107) + (cid:18) − + (cid:20) − p +
223 ln (cid:18) rr ′ (cid:19)(cid:21) n + n + n (cid:19) g + O (cid:18) c (cid:19)(cid:27) . A convenient post-Newtonian parameter x = O ( c − ) is now used in place of g ; it isdefined from the orbital frequency as x = (cid:18) G m w c (cid:19) / . (108)The interest in this parameter stems from its invariant meaning in a large class of co-ordinate systems including the harmonic and ADM coordinates. By inverting (106)we find at 3PN order g = x (cid:26) + (cid:16) − n (cid:17) x + (cid:18) − n (cid:19) x (109) This approach is extensively reviewed in the contribution of Gerhard Sch¨afer in this volume.4 Luc Blanchet + (cid:18) + (cid:20) − − p −
223 ln (cid:18) rr ′ (cid:19)(cid:21) n + n + n (cid:19) x + O (cid:18) c (cid:19)(cid:27) . This is substituted back into (107) to get the 3PN energy in invariant form. We hap-pily observe that the logarithm and the gauge constant r ′ cancel out in the processand our final result is E = − m n c x (cid:26) + (cid:18) − − n (cid:19) x + (cid:18) − + n − n (cid:19) x + (cid:18) − + (cid:20) − p (cid:21) n − n − n (cid:19) x + O (cid:18) c (cid:19)(cid:27) . (110)The conserved energy E corresponds to the Newtonian, 1PN, 2PN and 3PN con-servative orders in the equations of motion; the damping part is associated withradiation reaction and arises at 2.5PN order. The radiation reaction at the dominant2.5PN level will correspond to the “Newtonian” gravitational-wave flux only. Hencethe flying-color 3PN flux F we are looking for cannot be computed from the 3PNequations of motion alone. Instead we have to apply all the machinery of the post-Newtonian wave generation formalism described in Sections 4–7. The final result at3.5PN order is [81, 82, 74] F = c G n x (cid:26) + (cid:18) − − n (cid:19) x + p x / + (cid:18) − + n + n (cid:19) x + (cid:18) − − n (cid:19) p x / + (cid:20) + p − C − ( x )+ (cid:18) − + p (cid:19) n − n − n (cid:21) x + (cid:18) − + n + n (cid:19) p x / + O (cid:18) c (cid:19)(cid:27) . (111)Here C = . · · · is the Euler constant. This result is fully consistent with black-hole perturbation theory: using it Sasaki & Tagoshi [83, 84, 85] obtain (111) inthe small mass-ratio limit n →
0. The generalization of (111) to arbitrary eccentric(bound) orbits has also been worked out [86, 87].
12 Waveform of compact binaries
We specify our conventions for the orbital phase and polarization vectors definingthe polarization waveforms (52) in the case of a non-spinning compact binary mov-ing on a quasi-circular orbit. If the orbital plane is chosen to be the x-y plane as in ost-Newtonian theory and the two-body problem 35
Section 11, with the orbital phase f measuring the direction of the unit separationvector n = x / r , then n = ˆ x cos f + ˆ y sin f , (112)where ˆ x and ˆ y are the unit directions along x and y. Following [88, 89] we choosethe polarization vector P to lie along the x-axis and the observer to be in the y-zplane in the direction N = ˆ y sin i + ˆ z cos i , (113)where i is the orbit’s inclination angle. With this choice P lies along the intersectionof the orbital plane with the plane of the sky in the direction of the ascending node , i.e. that point at which the bodies cross the plane of the sky moving toward theobserver. Hence the orbital phase f is the angle between the ascending node and thedirection of body 1. The rotating orthonormal triad ( n , lll , ˆ z ) describing the motionof the binary and used in (102) is related to the fixed polarization triad ( N , P , Q ) by n = P cos f + (cid:0) Q cos i + N sin i (cid:1) sin f , (114a) lll = − P sin f + (cid:0) Q cos i + N sin i (cid:1) cos f , (114b)ˆ z = − Q sin i + N cos i . (114c)The 3.5PN expression of the orbital phase f as function of the orbital frequencyor equivalently the x -parameter is obtained from the energy balance equation (101)in which the binary’s conservative center-of-mass energy E and total gravitational-wave flux F have been obtained in (110)–(111). For circular orbits the orbital phaseis computed from f ≡ Z w d t = − Z w F d E d w d w . (115)Various methods (numerical or analytical) are possible for solving (115) given theexpressions (110) and (111). This yields different waveform families all valid atthe same 3.5PN order, but which may differ when extrapolated beyond the normaldomain of validity of the post-Newtonian expansion, i.e. in this case very near thecoalescence. Such differences must be taken into account when comparing the post-Newtonian waveforms to numerical results [52].It is convenient to perform a change of phase, from the actual orbital phase f tothe new phase variable y = f − G M w c ln (cid:18) ww (cid:19) , (116)where M is the binary’s total mass monopole moment and w = u exp [ − C ] is related to the constant u ≡ r / c entering the tail integrals in (60). The logarith-mic term in y corresponds physically to some spreading of the different frequencycomponents of the wave along the line of sight from the source to the detector, and The mass monopole M differs from m = m + m as it includes the contribution of the gravita-tional binding energy. At 1PN order it is given for circular orbits by M = m [ − n g ] + O ( c − ) .6 Luc Blanchet expresses the tail effect as a small delay in the arrival time of gravitational waves.This effect, although of formal 1.5PN order in (116), represents in fact a very smallmodulation of the orbital phase: compared to the dominant phase evolution whoseorder is that of the inverse of 2.5PN radiation reaction, this modulation is of order4PN and can thus be neglected with the present accuracy.The spherical harmonic modes of the polarization waveforms can now be ob-tained at 3PN order using the angular integration formula (55). We start from theexpressions of the wave polarizations h + and h × as functions of the inclination an-gle i and of the phase y . We use the known dependence of the spherical harmonics(54) on the azimuthal angle. Denoting h ≡ h + − i h × = h ( i , y ) we find that the latterangular integration becomes h ℓ m = ( − i ) m e − i m y Z p d y ′ Z p d i sin i h ( i , y ′ ) Y ℓ m − ( i , y ′ ) , (117)exhibiting the azimuthal factor e − i m y appropriate for each mode. Let us introducea normalized mode coefficient H ℓ m starting by definition with one at the Newtonianorder for the dominant mode having ( ℓ, m ) = ( , ) . This means posing h ℓ m = G m n xR c r p H ℓ m e − i m y . (118)All the modes have been given in [54] up to the 3PN order. The dominant mode ( , ) , which is primarily needed for the comparison between post-Newtonian cal-culations and numerical simulations, reads at 3PN order H = + (cid:18) − + n (cid:19) x + p x / (119) + (cid:18) − − n + n (cid:19) x + (cid:18) − p + pn − n (cid:19) x / + (cid:18) − C + p − ( x ) + p + (cid:20) − + p (cid:21) n − n + n (cid:19) x + O (cid:18) c (cid:19) .
13 Spin-orbit contributions in the energy and flux
To successfully detect the gravitational waves emitted by spinning, precessing bi-naries and to estimate the binary parameters, spin effects should be included in thetemplates. For maximally spinning compact bodies the spin-orbit coupling (linear inthe spins) appears dominantly at the 1.5PN order, while the spin-spin one (which isquadratic) appears at 2PN order. The spin effect on the free motion of a test particlewas obtained by Papapetrou [64] in the form of a coupling to curvature. Seminal ost-Newtonian theory and the two-body problem 37 later works by Barker & O’Connell [90, 91] obtained the leading order spin-orbitand spin-spin contributions in the post-Newtonian equations of motion. Based onthese works, the spin-orbit and spin-spin terms were obtained in the radiation field[92, 65, 93, 94], enabling the derivation of the orbital phase evolution (the crucialquantity that determines the templates). Finding the 1PN corrections to the leadingspin-orbit coupling in both the (translational) equations of motion and radiation fieldwas begun in [95, 96] and completed in [97, 98]. The result [97] for the equations ofmotion was confirmed by an alternative derivation based on the ADM-Hamiltonianformalism [99].In Section 8 we discussed a covariant formalism for spinning particles [59, 60,61, 62, 63]. We want now to find a convenient three-dimensional variable for thespin. Restricting ourselves to spin-orbit effects, i.e. neglecting O ( S ) , we can writethe components of the spin tensor S ab A as S iA = − c p − ( g ) A e i jk u Aj S Ak , (120a) S i jA = − c p − ( g ) A e i jk (cid:20) u A S Ak + u Ak v lA c S Al (cid:21) . (120b)We have used the momentum-velocity relation (76) and have taken into accountthe spin condition (73). A first possibility is to adopt as the basic spin variable the contravariant components of the spin covector S Ai in (120), which are obtained byraising the index by means of the contravariant spatial metric, viz S A = ( S iA ) with S iA ≡ ( g i j ) A S Aj , (121)where g i j is the inverse of the covariant spatial metric g i j ≡ g i j i.e. satisfies g ik g k j = d ij . The choice of spin variable (121) has been adopted in [95, 96].However to express final results (to be used in gravitational-wave templates) it isbetter to use a different set of spin variables characterized by having some conserved Euclidean lengths. Such spins with constant Euclidean magnitude will be denotedby S c A . They can be computed in a straightforward way at a given post-Newtonianorder in terms of the previous variables (121). For instance we find up to 1PN order, S c A = (cid:20) + ( U ) A c (cid:21) S A − c ( v A · S A ) v A + O (cid:18) c (cid:19) . (122)The (regularized) gravitational potential ( U ) A is defined by (81). The constant-magnitude spin variable S c A obeys a spin precession equation which is necessarilyof the form d S c A d t = WWW A × S c A . (123)Indeed this equation implies that | S c A | = const. The precession angular frequencyvector WWW A for two-body systems has been computed for the leading spin-orbit andspin-spin contributions [65] and for the 1PN correction to the spin-orbit [97, 98, 99]. For two bodies we conveniently use the following combinations (introduced in [65])of the two spins: S c ≡ S c1 + S c2 , (124a) SSS c ≡ m (cid:20) S c2 m − S c1 m (cid:21) . (124b)Furthermore, recalling the orthonormal triad { n , lll , ˆ z } used in Section 11, where ˆ z is the unit vector in the direction perpendicular to the orbital plane, we denote by S cz ≡ S c · ˆ z and S cz ≡ SSS c · ˆ z the projections along that perpendicular direction.The spin-orbit terms have been computed at 1PN order both in the equations ofmotion and in the radiation field. In the equations of motion they correct the orbitalfrequency and invariant conserved energy with terms at orders 1.5PN and 2.5PN.In the presence of spins the energy gets modified to E = E mono + E spin where themonopole part has been obtained in (110) and where the spin terms read E spin = − c G m n x (cid:26) x / (cid:20) S cz + DS cz (cid:21) + x / (cid:20)(cid:18) − n (cid:19) S cz + (cid:18) − n (cid:19) DS cz (cid:21) + O (cid:18) c (cid:19)(cid:27) . (125)We recall that D ≡ ( m − m ) / m ; see the footnote 13. This expression is valid forquasi-circular orbits, and we neglect the spin-spin terms. Similarly the gravitational-wave flux will be modified at the same 1.5PN and 2.5PN orders. Posing F = F mono + F spin where F mono is given by (111), we find F spin = c G m n x (cid:26) x / (cid:20) − S cz − DS cz (cid:21) + x / (cid:20)(cid:18) − + n (cid:19) S cz + (cid:18) − + n (cid:19) DS cz (cid:21) + O (cid:18) c (cid:19)(cid:27) . (126)Having in hand the spin contributions in E and F , we can deduce the evolutionof the orbital phase from the energy balance equation (101). In absence of pre-cession of the orbital plane, e.g. for spins aligned or anti-aligned with the orbitalangular momentum, the gravitational-wave phase will reduce to the “carrier” phase f GW ≡ f (keeping only the dominant harmonics), where f is the orbital phasewhich is obtained by integrating the orbital frequency. However, in the general caseof non-aligned spins, we must take into account the effect of precession of the or-bital plane induced by spin modulations. Then the gravitational-wave phase is givenby F GW = f GW + df GW , where the precessional correction df GW arises from thechanging orientation of the orbital plane, and can be computed by standard meth-ods using numerical integration [100]. Thus, the carrier phase f GW constitutes themain theoretical output to be provided for the gravitational-wave templates, and candirectly be obtained numerically from using the integration formula (115). ost-Newtonian theory and the two-body problem 39 References
1. A. Einstein. Sitzber. Preuss. Akad. Wiss. Berlin, 1916.2. W. De Sitter.
Mon. Not. Roy. Astron. Soc. , 76:699, 1916.3. W. De Sitter.
Mon. Not. Roy. Astron. Soc. , 77:155, 1916.4. H.A. Lorentz and J. Droste. Nijhoff, The Hague, 1937. Versl. K. Akad. Wet. Amsterdam ,392 and 649 (1917).5. A. Einstein, L. Infeld, and B. Hoffmann. The gravitational equations and the problem ofmotion. Ann. Math. , 39:65–100, 1938.6. V. Fock. On motion of finite masses in general relativity.
J. Phys. (Moscow) , 1(2):81–116,1939.7. V.A. Fock.
Theory of space, time and gravitation . Pergamon, London, 1959.8. J. Plebanski and S.L. Bazanski. The general fokker action principle and its application ingeneral relativity theory.
Acta Phys. Polonica , 18:307–345, 1959.9. S. Chandrasekhar. The post-newtonian equations of hydrodynamics in general relativity.
Astrophys. J. , 142:1488–1540, 1965.10. S. Chandrasekhar and Y. Nutku. The second post-newtonian equations of hydrodynamics ingeneral relativity.
Astrophys. J. , 158:55–79, 1969.11. S. Chandrasekhar and F.P. Esposito. The 5/2-post-newtonian equations of hydrodynamicsand radiation reaction in general relativity.
Astrophys. J. , 160:153–179, 1970.12. J. Ehlers.
Ann. N.Y. Acad. Sci. , 336:279, 1980.13. G.D. Kerlick.
Gen. Relativ. Gravit. , 12:467, 1980.14. G.D. Kerlick.
Gen. Relativ. Gravit. , 12:521, 1980.15. A. Papapetrou. Equations of motion in general relativity.
Proc. Phys. Soc. A , 64:57–75,1951.16. A. Papapetrou and B. Linet. Equation of motion including the reaction of gravitational radi-ation.
Gen. Relativ. Gravit. , 13:335, 1981.17. T. Damour and N. Deruelle. Radiation reaction and angular momentum loss in small anglegravitational scattering.
Phys. Lett. A , 87:81, 1981.18. T. Damour and N. Deruelle. Lagrangien g´en´eralis´e du syst`eme de deux masses ponctuelles,`a l’approximation post-post-newtonienne de la relativit´e g´en´erale.
C. R. Acad. Sc. Paris ,293:537, 1981.19. T. Damour. Gravitational radiation reaction in the binary pulsar and the quadrupole formulacontrovercy.
Phys. Rev. Lett. , 51:1019–1021, 1983.20. T. Damour. Gravitational radiation and the motion of compact bodies. In N. Deruelle andT. Piran, editors,
Gravitational Radiation , pages 59–144, Amsterdam, 1983. North-HollandCompany.21. J.H. Taylor, L.A. Fowler, and P.M. McCulloch. Measurements of general relativistic effectsin the binary pulsar psr1913+16.
Nature , 277:437–440, 1979.22. J.H. Taylor and J.M. Weisberg. A new test of general relativity: Gravitational radiation andthe binary pulsar psr 1913+16.
Astrophys. J. , 253:908–920, 1982.23. J.H. Taylor. Pulsar timing and relativistic gravity.
Class. Quant. Grav. , 10:167–174, 1993.24. C. Cutler, T.A. Apostolatos, L. Bildsten, L.S. Finn, E.E. Flanagan, D. Kennefick, D.M.Markovic, A. Ori, E. Poisson, G.J. Sussman, and K.S. Thorne. The last three minutes: Is-sues in gravitational-wave measurements of coalescing compact binaries.
Phys. Rev. Lett. ,70:2984–2987, 1993.25. C. Cutler, L.S. Finn, E. Poisson, and G.J. Sussman. Gravitational radiation from a particle incircular orbit around a black hole. ii. numerical results for the nonrotating case.
Phys. Rev.D , 47:1511–1518, 1993.26. L.S. Finn and D.F. Chernoff. Observing binary inspiral in gravitational radiation: One inter-ferometer.
Phys. Rev. D , 47:2198–2219, 1993.27. C. Cutler and E.E. Flanagan. Gravitational waves from merging compact binaries: Howaccurately can one extract the binary’s parameters from the inspiral waveform?
Phys. Rev.D , 49:2658–2697, 1994.0 Luc Blanchet28. H. Tagoshi and T. Nakamura. Gravitational waves from a point particle in circular orbitaround a black hole: Logarithmic terms in the post-newtonian expansion.
Phys. Rev. D ,49:4016–4022, 1994.29. E. Poisson. Gravitational radiation from a particle in circular orbit around a black-hole. vi.accuracy of the post-newtonian expansion.
Phys. Rev. D , 52:5719–5723, 1995. ErratumPhys. Rev. D , 7980, (1997).30. P.C. Peters. Gravitational radiation and the motion of two point masses. Phys. Rev. ,136:B1224–B1232, 1964.31. J.L. Anderson and T.C. DeCanio.
Gen. Relativ. Gravit. , 6:197, 1975.32. A.D. Rendall. On the definition of post-newtonian approximations.
Proc. R. Soc. Lond. A ,438:341, 1992.33. O. Poujade and L. Blanchet. Post-newtonian approximation for isolated systems calculatedby matched asymptotic expansions.
Phys. Rev. D , 65:124020, 2002.34. T. Futamase and B.F. Schutz.
Phys. Rev. D , 28:2363, 1983.35. T. Futamase.
Phys. Rev. D , 28:2373, 1983.36. L. Blanchet and T. Damour. Radiative gravitational fields in general relativity i. generalstructure of the field outside the source.
Phil. Trans. Roy. Soc. Lond. A , 320:379–430, 1986.37. L. Blanchet and T. Damour. Tail transported temporal correlations in the dynamics of agravitating system.
Phys. Rev. D , 37:1410, 1988.38. L. Blanchet. Time asymmetric structure of gravitational radiation.
Phys. Rev. D , 47:4392–4420, 1993.39. L. Blanchet and T. Damour. Hereditary effects in gravitational radiation.
Phys. Rev. D ,46:4304–4319, 1992.40. W.L. Burke and K.S. Thorne. Gravitational radiation damping. In M. Carmeli, S.I. Fickler,and L. Witten, editors,
Relativity , pages 209–228, New York and London, 1970. PlenumPress.41. W.L. Burke. Gravitational radiation damping of slowly moving systems calculated usingmatched asymptotic expansions.
J. Math. Phys. , 12:401, 1971.42. L. Blanchet and T. Damour. Post-newtonian generation of gravitational waves.
Annales Inst.H. Poincar´e Phys. Th´eor. , 50:377–408, 1989.43. T. Damour and B. R. Iyer. Postnewtonian generation of gravitational waves. 2. the spinmoments.
Annales Inst. H. Poincar´e, Phys. Th´eor. , 54:115–164, 1991.44. L. Blanchet. On the multipole expansion of the gravitational field.
Class. Quant. Grav. ,15:1971–1999, 1998.45. L. Blanchet, G. Faye, and Samaya Nissanke. Structure of the post-newtonian expansion ingeneral relativity.
Phys. Rev. D , 72:044024, 2005.46. C.M. Will and A.G. Wiseman. Gravitational radiation from compact binary systems: Gravi-tational waveforms and energy loss to second post-newtonian order.
Phys. Rev. D , 54:4813–4848, 1996.47. M.E. Pati and C.M. Will. Post-newtonian gravitational radiation and equations of motion viadirect integration of the relaxed einstein equations: Foundations.
Phys. Rev. D , 62:124015,2000.48. M.E. Pati and C.M. Will. Post-newtonian gravitational radiation and equations of motionvia direct integration of the relaxed einstein equations. ii. two-body equations of motion tosecond post-newtonian order, and radiation-reaction to 3.5 post-newtonian order.
Phys. Rev.D , 65:104008, 2002.49. K.S. Thorne. Multipole expansions of gravitational radiation.
Rev. Mod. Phys. , 52:299–339,1980.50. H. Bondi, M.G.J. van der Burg, and A.W.K. Metzner. Gravitational waves in general rela-tivity vii. waves from axi-symmetric isolated systems.
Proc. R. Soc. London, Ser. A , 269:21,1962.51. L. Blanchet. Radiative gravitational fields in general relativity. 2. asymptotic behaviour atfuture null infinity.
Proc. Roy. Soc. Lond. A , 409:383–399, 1987.52. A. Buonanno, G. B. Cook, and F. Pretorius.
Phys. Rev. D , 75:124018, 2007.ost-Newtonian theory and the two-body problem 4153. L.E. Kidder. Using full information when computing modes of post-newtonian waveformsfrom inspiralling compact binaries in circular orbits.
Phys. Rev. D , 77:044016, 2008.54. L. Blanchet, G. Faye, B. R. Iyer, and Siddhartha Sinha. The third post-newtonian gravita-tional wave polarisations and associated spherical harmonic modes for inspiralling compactbinaries in quasi-circular orbits.
Class. Quant. Grav. , 25:165003, 2008.55. L. Blanchet. Gravitational-wave tails of tails.
Class. Quant. Grav. , 15:113–141, 1998. Erra-tum
Class. Quant. Grav. , 22:3381, 2005.56. D. Christodoulou. Nonlinear nature of gravitation and gravitational-wave experiments.
Phys.Rev. Lett. , 67:1486–1489, 1991.57. K.S. Thorne. Gravitational-wave bursts with memory: The christodoulou effect.
Phys. Rev.D , 45:520, 1992.58. A.G. Wiseman and C.M. Will. Christodoulou’s nonlinear gravitational-wave memory: Eval-uation in the quadrupole approximation.
Phys. Rev. D , 44:R2945–R2949, 1991.59. W. Tulczyjew.
Bull. Acad. Polon. Sci. , 5:279, 1957.60. W. Tulczyjew.
Acta Phys. Polon. , 18:37, 1959.61. A. Trautman. Lectures on general relativity.
Gen. Relat. Grav. , 34:721, 2002. reprinted fromlectures delivered in 1958.62. W.G. Dixon. In J. Ehlers, editor,
Isolated systems in general relativity , page 156, Amsterdam,1979. North Holland.63. I. Bailey and W. Israel.
Ann. Phys. , 130:188, 1980.64. A. Papapetrou. Spinning test-particles in general relativity. i.
Proc. R. Soc. London A ,209:248, 1951.65. L.E. Kidder. Coalescing binary systems of compact objects to 5/2-post-newtonian order. v.spin effects.
Phys. Rev. D , 52:821–847, 1995.66. J. Hadamard.
Le probl`eme de Cauchy et les ´equations aux d´eriv´ees partielles lin´eaires hy-perboliques . Hermann, Paris, 1932.67. L. Schwartz.
Th´eorie des distributions . Hermann, Paris, 1978.68. L. Blanchet and G. Faye. Hadamard regularization.
J. Math. Phys. , 41:7675–7714, 2000.69. L. Blanchet and G. Faye. General relativistic dynamics of compact binaries at the thirdpost-newtonian order.
Phys. Rev. D , 63:062005, 2001.70. G. ’t Hooft and M. Veltman.
Nucl. Phys. , B44:139, 1972.71. C. G. Bollini and J. J. Giambiagi.
Phys. Lett. B , 40:566, 1972.72. T. Damour, P. Jaranowski, and G. Sch¨afer. Dimensional regularization of the gravitationalinteraction of point masses.
Phys. Lett. B , 513:147–155, 2001.73. L. Blanchet, T. Damour, and G. Esposito-Far`ese. Dimensional regularization of thethird post-newtonian dynamics of point particles in harmonic coordinates.
Phys. Rev. D ,69:124007, 2004.74. L. Blanchet, T. Damour, G. Esposito-Far`ese, and B. R. Iyer. Gravitational radiation frominspiralling compact binaries completed at the third post-newtonian order.
Phys. Rev. Lett. ,93:091101, 2004.75. V.C. de Andrade, L. Blanchet, and G. Faye. Third post-newtonian dynamics of compactbinaries: Noetherian conserved quantities and equivalence between the harmonic-coordinateand adm-hamiltonian formalisms.
Class. Quant. Grav. , 18:753–778, 2001.76. Y. Itoh, T. Futamase, and H. Asada. Equation of motion for relativistic compact binaries withthe strong field point particle limit: The second and half post-newtonian order.
Phys. Rev. D ,63:064038, 2001.77. Y. Itoh and T. Futamase.
Phys. Rev. D , 68:121501(R), 2003.78. Y. Itoh.
Phys. Rev. D , 69:064018, 2004.79. P. Jaranowski and G. Sch¨afer. Third post-newtonian higher order adm hamilton dynamicsfor two-body point-mass systems.
Phys. Rev. D , 57:7274–7291, 1998.80. P. Jaranowski and G. Sch¨afer. Binary black-hole problem at the third post-newtonian approx-imation in the orbital motion: Static part.
Phys. Rev. D , 60:124003–1–12403–7, 1999.81. L. Blanchet, B. R. Iyer, and B. Joguet. Gravitational waves from inspiralling compact bi-naries: Energy flux to third post-newtonian order.
Phys. Rev. D , 65:064005, 2002. Erratum
Phys. Rev. D , 71:129903(E), 2005.2 Luc Blanchet82. L. Blanchet, G. Faye, B. R. Iyer, and B. Joguet. Gravitational-wave inspiral of compactbinary systems to 7/2 post-newtonian order.
Phys. Rev. D , 65:061501(R), 2002. Erratum
Phys. Rev. D , 71:129902(E), 2005.83. M. Sasaki. Post-newtonian expansion of the ingoing-wave regge-wheeler function.
Prog.Theor. Phys. , 92:17–36, 1994.84. H. Tagoshi and M. Sasaki. Post-newtonian expansion of gravitational-waves from a particlein circular orbit around a schwarzschild black-hole.
Prog. Theor. Phys. , 92:745–771, 1994.85. T. Tanaka, H. Tagoshi, and M. Sasaki. Gravitational waves by a particle in circular orbitaround a schwarzschild black hole.
Prog. Theor. Phys. , 96:1087–1101, 1996.86. K.G. Arun, L. Blanchet, B. R. Iyer, and M. S. Qusailah. Tail effects in the 3pn gravitationalwave energy flux of compact binaries in quasi-elliptical orbits.
Phys. Rev. D , 77:064034,2008.87. K.G. Arun, L. Blanchet, B. R. Iyer, and M. S. Qusailah. Inspiralling compact binaries inquasi-elliptical orbits: The complete 3pn energy flux.
Phys. Rev. D , 77:064035, 2008.88. L. Blanchet, B. R. Iyer, C. M. Will, and A. G. Wiseman. Gravitational wave forms frominspiralling compact binaries to second-post-newtonian order.
Class. Quant. Grav. , 13:575–584, 1996.89. K.G. Arun, L. Blanchet, B. R. Iyer, and M. S. Qusailah. The 2.5pn gravitational wave polar-isations from inspiralling compact binaries in circular orbits.
Class. Quant. Grav. , 21:3771,2004. Erratum
Class. Quant. Grav. , 22:3115, 2005.90. B.M. Barker and R.F. O’Connell.
Phys. Rev. D , 12:329, 1975.91. B.M. Barker and R.F. O’Connell.
Gen. Relativ. Gravit. , 11:149, 1979.92. L.E. Kidder, C.M. Will, and A.G. Wiseman. Spin effects in the inspiral of coalescing compactbinaries.
Phys. Rev. D , 47:R4183–R4187, 1993.93. L. Gergely. Spin-spin effects in radiating compact binaries.
Phys. Rev. D , 61:024035, 1999.94. B. Mik´oczi, M. Vas´uth, and L. Gergely. Self-interaction spin effects in inspiralling compactbinaries.
Phys. Rev. D , 71:124043, 2005.95. B.J. Owen, H. Tagoshi, and A. Ohashi. Nonprecessional spin-orbit effects on gravitationalwaves from inspiraling compact binaries to second post-newtonian order.
Phys. Rev. D ,57:6168–6175, 1998.96. H. Tagoshi, A. Ohashi, and B.J. Owen. Gravitational field and equations of motion of spin-ning compact binaries to 2.5-post-newtonian order.
Phys. Rev. D , 63:044006, 2001.97. G. Faye, L. Blanchet, and A. Buonanno. Higher-order spin effects in the dynamics of com-pact binaries i. equations of motion.
Phys. Rev. D , 74:104033, 2006.98. L. Blanchet, A. Buonanno, and G. Faye. Higher-order spin effects in the dynamics of com-pact binaries ii. radiation field.
Phys. Rev. D , 74:104034, 2006. Erratum
Phys. Rev. D ,75:049903, 2007.99. T. Damour, Piotr Jaranowski, and Gerhard Sch¨afer. Hamiltonian of two spinning compactbodies with next-to-leading order gravitational spin-orbit coupling.
Phys. Rev. D , 77:064032,2008.100. T.A. Apostolatos, C. Cutler, G.J. Sussman, and K.S. Thorne.