Black hole perturbation theory and gravitational self-force
BBlack hole perturbation theory and gravitationalself-force
Adam Pound and Barry Wardell
Abstract
Much of the success of gravitational-wave astronomy rests on perturbationtheory. Historically, perturbative analysis of gravitational-wave sources has largelyfocused on post-Newtonian theory. However, strong-field perturbation theory is es-sential in many cases such as the quasinormal ringdown following the merger ofa binary system, tidally perturbed compact objects, and extreme-mass-ratio inspi-rals. In this review, motivated primarily by small-mass-ratio binaries but not limitedto them, we provide an overview of essential methods in (i) black hole perturbationtheory, (ii) orbital mechanics in Kerr spacetime, and (iii) gravitational self-force the-ory. Our treatment of black hole perturbation theory covers most common methods,including the Teukolsky and Regge-Wheeler-Zerilli equations, methods of metricreconstruction, and Lorenz-gauge formulations, casting them in a unified notation.Our treatment of orbital mechanics covers quasi-Keplerian and action-angle descrip-tions of bound geodesics and accelerated orbits, osculating geodesics, near-identityaveraging transformations, multiscale expansions, and orbital resonances. Our sum-mary of self-force theory’s foundations is brief, covering the main ideas and resultsof matched asymptotic expansions, local expansion methods, puncture schemes,and point particle descriptions. We conclude by combining the above methods ina multiscale expansion of the perturbative Einstein equations, leading to adiabaticand post-adiabatic evolution and waveform-generation schemes. Our presentationincludes some new results but is intended primarily as a reference for practitioners.
Adam PoundSchool of Mathematical Sciences and STAG Research Centre, University of Southampton,Southampton, United Kingdom, SO17 1BJ, e-mail:
Barry WardellSchool of Mathematics & Statistics, University College Dublin, Belfield, Dublin 4, Ireland, e-mail: [email protected] a r X i v : . [ g r- q c ] J a n ontents Black hole perturbation theory has a long and rich history, dating back at least as faras Regge and Wheeler’s study of odd-parity perturbations of Schwarzschild space-time in the late 1950s [167]. This was followed up in 1970 by Zerilli’s study ofeven-parity perturbations [213, 212]. Soon afterwards, Vishveshwara [197] identi-fied quasinormal modes in perturbations of Schwarzschild spacetime, Press [164]studied the associated quasinormal mode frequencies, and Chandrasekhar and De-tweiler [35] numerically computed the frequencies. Teukolsky’s success in derivingdecoupled and separable equations for perturbations of Kerr spacetime [187, 189]paved the way for similar progress in the Kerr case.The idea of a self-force has an even longer history, having been studied by Diracin 1938 in his relativistic generalization of the Abraham–Lorentz self-force to thecontext of an electric charge undergoing acceleration in flat spacetime [51]. In the1960s this was extended by DeWitt and Brehme to the curved spacetime case [50].The gravitational self-force acting on a point mass was studied in the late 1990sby Mino, Sasaki and Tanaka [130] and by Quinn and Wald [166], leading to theMiSaTaQuWa equation that is named after the authors of those first papers. Subse-quent work has put gravitational self-force theory on a very strong theoretical foot-ing [80, 151, 150] and has extended the formalism to second order in perturbationtheory [171, 44, 154, 79].The last 20 years have seen increasingly intense focus on the study of gravita-tional self-force in perturbations of black hole spacetimes. This has been motivatedto a large extent by the European Space Agency’s LISA mission, which is sched-uled for launch in the 2030s [114] and which will observe gravitational waves inthe millihertz frequency band. One of the key sources for LISA will be extreme-mass-ratio inspirals (EMRIs), binary systems consisting of a compact solar-massobject orbiting a massive black hole. The presence of a small parameter (the massratio, which is expected to be in the region of 10 − ) makes black hole perturbationtheory an ideal tool for the development of theoretical models of the gravitationalwaveforms from EMRIs. Over the several year timescale that the LISA missionis expected to run, the smaller body in an EMRI will execute ∼ –10 intricateorbits in the strong-field regime around the central black hole, acting as a preciseprobe and enabling high-precision measurements of the black hole’s parameters,tests of its Kerr nature, and tests of general relativity. Radiation reaction will causethe orbit to significantly evolve and possibly plunge into the black hole in that time,meaning that self-force effects will be important to include in waveform models. In-deed, in order to extract the maximum information from the observation of EMRIsby LISA it has been established that it will be necessary to incorporate informationat second order in perturbation theory by computing the second order gravitationalself-force [88, 100, 30]. Aside from EMRIs, gravitational self-force is also poten-tially highly accurate for intermediate-mass-ratio inspirals (IMRIs) [122], in whichthe mass ratio may be as large as ∼ − . This makes black hole perturbation theoryand self-force also relevant for the current generation of ground-based gravitationalwave detectors including LIGO [113], Virgo [196] and Kagra [102]. There are already numerous reviews of these topics in the literature. The clas-sic text by Chandrasekhar [34] provides a comprehensive introduction to black holephysics, linear black hole perturbation theory, and geodesic motion in black holespacetimes. Ref. [177] reviews linear black hole perturbation theory with an empha-sis on analytical post-Newtonian expansions of the perturbation equations. Ref. [19]provides a thorough introduction to quasinormal modes of black holes. Ref. [11]offers a broad introduction to self-force calculations for non-experts, including asurvey of concrete physical results through 2018. Refs. [148, 156] cover the foun-dations of self-force theory, and Ref. [85] provides a complementary view of thefoundations from a fully nonlinear perspective. Finally, Refs. [6, 205] provide de-tailed introductions to methods of computing the self-force.Our aim is to complement rather than reiterate these existing reviews. We keepour description of self-force theory brief, only summarizing the key ideas and meth-ods, and we forgo a survey of physical results. Instead, we focus on detailing themain perturbative methods required to model waveforms from small-mass-ratio bi-naries, leading ultimately to a multiscale expansion of the Einstein equations witha small-body source. At the same time, we keep much of the material sufficientlygeneral to apply to other scenarios of interest.Our review is divided into three distinct parts. Sections 2 and 3 briefly introducerelevant background material on perturbation theory in general relativity and theKerr spacetime. Sections 4, 5, and 6 review three disjoint topics: black hole pertur-bation theory; geodesics and accelerated orbits in Kerr spacetime; and the founda-tions of the “local problem” in self-force theory. These three sections are writtento be largely independent of one another, and they can be read in any order. Fi-nally, in Section 7 we bring together all three topics in a description of black holeperturbation theory with a (skeletal) small-body source, focusing on the multiscaleformulation and the waveform-generation scheme that comes along with it.
The overarching framework for our review is perturbation theory in general relativ-ity. In self-force calculations, this is typically applied to the specific case of a smallobject in the spacetime of a Kerr black hole, and in much of the review we specializeto that scenario. But to allow for generality in some sections, we first consider themore generic case of smooth perturbations of an arbitrary vacuum spacetime. Weassume the metric can be expanded in powers of a small parameter ε , g exact µν = g µν + ε h ( ) µν + ε h ( ) µν + O ( ε ) , (1)where g µν is a vacuum metric, and that the stress-energy can be similarly expandedas T µν = ε T ( ) µν + ε T ( ) µν + O ( ε ) . (2) For later convenience, we define the total metric perturbation h µν = ∑ n > ε n h ( n ) µν .We also warn the reader that we will later treat ε as a formal counting parameterthat can be set equal to 1.To expand the Einstein equations G µν [ g + h ] = π T µν in powers of ε , we firstnote that the Einstein tensor of a metric g µν + h µν can be expanded in powers of theexact perturbation h µν : G µν [ g + h ] = G µν [ g ] + G ( ) µν [ h ] + G ( ) µν [ h , h ] + O ( | h | ) . Thequantities G ( n ) µν are easily obtained from the exact Riemann tensor (see, e.g., Ch. 7.5of Ref. [200]). For a vacuum background, the first two terms are G ( ) µν [ h ] = (cid:16) g µ α g ν β − g µν g αβ (cid:17) R ( ) αβ , (3) G ( ) µν [ h , h ] = (cid:16) g µ α g ν β − g µν g αβ (cid:17) R ( ) αβ − (cid:16) h µν g αβ − g µν h αβ (cid:17) R ( ) αβ , (4)where the linear and quadratic terms in the Ricci tensor are R ( ) αβ [ h ] = − (cid:0) (cid:3) h αβ + R α µ β ν h µν − h µ ( α ; µ β ) (cid:1) , (5) R ( ) αβ [ h , h ] = h µν ; α h µν ; β + h µ β ; ν (cid:0) h µα ; ν − h να ; µ (cid:1) − ¯ h µν ; ν (cid:0) h µ ( α ; β ) − h αβ ; µ (cid:1) − h µν (cid:0) h µ ( α ; β ) ν − h αβ ; µν − h µν ; αβ (cid:1) . (6)Here we have defined the trace-reversed perturbation ¯ h µν : = h µν − g µν g αβ h αβ andthe d’Alembertian (cid:3) : = g µν ∇ µ ∇ ν . A semicolon and ∇ both denote the covariantderivative compatible with g µν .So, substituting the expansions (1) and (2) into the Einstein equations and equat-ing powers of ε , we obtain G ( ) µν [ h ( ) ] = π T ( ) µν , (7) G ( ) µν [ h ( ) ] = π T ( ) µν − G ( ) µν [ h ( ) , h ( ) ] . (8)This perturbative expansion comes with the freedom to perform gauge transfor-mations h ( ) µν → h ( ) µν + £ ξ ( ) g µν , (9) h ( ) µν → h ( ) µν + £ ξ ( ) g µν + £ ξ ( ) g µν + £ ξ ( ) h ( ) µν , (10)where £ ξ is a Lie derivative, and ξ α ( n ) are freely chosen vector fields. In self-forcetheory, this freedom is commonly used to impose the Lorenz gauge condition, ∇ α ¯ h αβ = , (11)in which case the linearized Einstein tensor simplifies to G ( ) µν [ h ] = − (cid:16) (cid:3) ¯ h µν + R µ α ν β ¯ h αβ (cid:17) . (12) A perturbed metric will come hand in hand with a perturbed equation of motionfor objects in the spacetime: D z µ d τ = f µ ( ) + ε f µ ( ) + ε f µ ( ) + O ( ε ) . (13)Here z µ ( τ ) is a perturbed worldline, τ is its proper time as measured in the back-ground g µν , D z µ d τ = dz ν d τ ∇ ν dz µ d τ : = a µ is its covariant acceleration with respect to g µν ,and f µ ( n ) is the n th-order covariant force (per unit mass) driving the acceleration. Inour review, we will consider the general case including a zeroth-order force, but wewill focus primarily on cases with f µ ( ) =
0. The forces f µ ( n ) will arise from (partsof) the metric perturbations h ( n ) µν as well as from coupling of g µν to the matter thatcreates those perturbations.Here we have limited the treatment to first- and second-order perturbations,which are expected to be necessary and sufficient for modelling small-mass-ratiobinaries. In some sections we will restrict the context to first, linearized order. In most of our review, we take the background spacetime to be that of an isolated,stationary black hole. In this section we provide an overview of the properties ofthese spacetimes.
The Schwarzschild spacetime is a static, spherically symmetric solution of the vac-uum Einstein equations representing a non-rotating black hole with mass M . It hasa line element given by ds = − f ( r ) dt + f ( r ) − dr + r (cid:0) d θ + sin θ d φ (cid:1) , (14)where f ( r ) : = − Mr . The Schwarzschild spacetime may be generalized to allowthe black hole to have a charge per unit mass, Q , resulting in the Reissner-Nordstr¨omsolution of the Einstein-Maxwell equations, with line element ds = − (cid:18) − Mr + Q r (cid:19) dt + (cid:18) − Mr + Q r (cid:19) − dr + r (cid:0) d θ + sin θ d φ (cid:1) . (15)The spacetime of a spinning black hole is given by the Kerr metric with angularmomentum per unit mass a . In Boyer-Lindquist coordinates, its line-element is ds = − (cid:20) − Mr Σ (cid:21) dt − aMr sin θΣ dt d φ + Σ∆ dr + Σ d θ + (cid:20) ∆ + Mr ( r + a ) Σ (cid:21) sin θ d φ , (16)where Σ : = r + a cos θ and ∆ : = r − Mr + a = ( r − r + )( r − r − ) with r ± : = M ± √ M − a the locations of the inner and outer horizons. As was the casewith Schwarzschild spacetime, the Kerr spacetime may be generalized to allow theblack hole to have a charge per unit mass, Q , giving the Kerr-Newman solution ofthe Einstein-Maxwell equations. In Boyer-Lindquist coordinates, the Kerr-Newmanmetric is: ds = − (cid:20) − Mr − Q Σ (cid:21) dt − ( Mr − Q ) a sin θΣ dt d φ + Σ∆ + Q dr + Σ d θ + (cid:20) ∆ + Q + ( Mr − Q )( a + r ) Σ (cid:21) sin θ d φ . (17)In astrophysical scenarios, a charged black hole will quickly be neutralized. Forthat reason, in later sections we will restrict our attention to the Kerr spacetime. Wewill also later use Q to denote the Carter constant, associated with the Kerr metric’sthird, hidden symmetry discussed below. However, we include the charged blackhole metrics here for completeness. The black hole spacetimes above are all of Petrov type D and thus have two non-degenerate principal null directions. This gives us a natural way to define a complexnull tetrad by having two of the tetrad legs aligned with the principal null directions.Choosing l α : = e α ( ) to align with the outward null direction and n α : = e α ( ) to alignwith the inward null direction, there is still residual freedom in the choice of scalingof each tetrad leg, and also in the relative orientation of the remaining two tetradlegs, m α : = e α ( ) and ¯ m α : = e α ( ) . The two most common choices in Kerr spacetimeare Carter’s canonical tetrad [33] , l α = √ ∆ Σ (cid:104) r + a , ∆ , , a (cid:105) , n α = √ ∆ Σ (cid:104) r + a , − ∆ , , a (cid:105) , m α = √ Σ (cid:104) ia sin θ , , , i sin θ (cid:105) , ¯ m α = √ Σ (cid:104) − ia sin θ , , , − i sin θ (cid:105) , (18) Carter’s original tetrad had interchanged l µ ↔ n µ and m µ ↔ ¯ m µ . Carter also worked in differentcoordinates ( ˜ t = t − a φ , r , q = a cos θ , ˜ φ = φ / a ) which more fully reflect the inherent symmetriesof Kerr. We deviate from that here and keep with the convention of having l α point outwards andworking in the more common Boyer-Lindquist coordinates. and the Kinnersley tetrad [107], which is related to Carter’s canonical tetrad bya simple rescaling: l α = (cid:113) ∆ Σ l α K , n α = (cid:113) Σ∆ n α K , m α = ¯ ζ √ Σ m α K and ¯ m α = ζ √ Σ ¯ m α K ,where ζ : = r − ia cos θ (19)is an important quantity that will will encounter again later (note that Σ = ζ ¯ ζ ). TheCarter tetrad transforms as l ↔ − n , m ↔ ¯ m under { t , φ } → {− t , − φ } . Althoughthe Kinnersley tetrad formed a crucial part of Teukolsky’s separability result forperturbations of the Weyl tensor [187] it has two unfortunate features that makeit less than ideal for elucidating the symmetric structure of Kerr spacetime: (i) itviolates the { t , φ } → {− t , − φ } symmetry; and (ii) it destroys a symmetry in { r , θ } .Carter’s canonical tetrad does not suffer from either of these deficiencies and isslightly preferable from that point of view. Note, however, that all of the results thatfollow can be derived using either tetrad. Much of the success in studying Kerr spacetime has arisen from the inherent sym-metries it possesses. Two of these are associated with the existence of two Killingvectors, ξ α and η α , which satisfy Killing’s equation, ξ ( α ; β ) = = η ( α ; β ) . (20)In Kerr spacetime these are related to the timelike and axial symmetries, ξ α = δ α t , η α = δ α ˜ φ = a ( δ αφ + a δ α t ) . (21)The spacetime also admits a conformal Killing-Yano tensor, f αβ = ( ζ + ¯ ζ ) n [ α l β ] − ( ζ − ¯ ζ ) ¯ m [ α m β ] , (22)which satisfies f α ( β ; γ ) = g βγ ξ α − g α ( β ξ γ ) . (23)Here, we have introduced the Killing spinor coefficient, ζ , which we previouslyencountered as a coordinate expression in Sec. 3.2. Its appearance here can be con-sidered more fundamental, and does not depend on any particular coordinate choice.The divergence of this conformal Killing-Yano tensor is a Killing vector, ξ α = f αβ ; β , (24)and its Hodge dual, Note that the Killing vector δ αφ = a η α − a δ α t is often used in place of η α when working inBoyer-Lindquist coordinates.0 (cid:63) f αβ = ε αβ γδ f γδ = i ( ζ − ¯ ζ ) n [ α l β ] − i ( ζ + ¯ ζ ) ¯ m [ α m β ] , (25)is a Killing-Yano tensor satisfying (cid:63) f α ( β ; γ ) = . (26)The products of these Killing-Yano tensors generate two conformal Killing tensors, K αβ = f αγ f β γ = ( ζ + ¯ ζ ) n ( α l β ) − ( ζ − ¯ ζ ) ¯ m ( α m β ) , (27) (cid:63) K αβ = f αγ (cid:63) f β γ = i ( ζ − ¯ ζ )( n ( α l β ) + ¯ m ( α m β ) ) , (28)which satisfy K ( αβ ; γ ) = g ( αβ K γ ) , (cid:63) K ( αβ ; γ ) = g ( αβ (cid:63) K γ ) , (29)where K α = ( K βα ; β + K β β ; α ) and (cid:63) K α = ( (cid:63) K βα ; β + (cid:63) K β β ; α ) . They also generatea Killing tensor, (cid:63)(cid:63) K αβ = (cid:63) f αγ (cid:63) f β γ = − ( ζ − ¯ ζ ) n ( α l β ) + ( ζ + ¯ ζ ) ¯ m ( α m β ) , (30)satisfying (cid:63)(cid:63) K ( αβ ; γ ) = . (31)This Killing tensor gives a relationship between the two Killing vectors, η α = − (cid:63)(cid:63) K αβ ξ β . (32) We now consider perturbations of the isolated black hole spacetimes. We de-scribe, in a unified notation, how to calculate metric perturbations in the mostcommonly used gauges: radiation gauges, Regge-Wheeler-Zerilli gauges, and theLorenz gauge. Our focus is particularly on reconstruction methods, in which mostor all of the metric perturbation is reconstructed from decoupled scalar variables.Since the left-hand sides of the perturbative Einstein equations (7) and (8) arethe same at every order, we specialize to the first-order case. We refer the readerto Refs. [31, 29] for general discussions of second-order perturbation theory inSchwarzschild and Kerr spacetimes. Teukolsky [187] showed that the equations governing perturbations of rotating blackhole spacetimes can be recast into a form where they are given by decoupled equa-tions. These equations further have the remarkable property of being separable, re-ducing the problem to the solution of a set of uncoupled ordinary differential equa-tions. In the case of metric perturbations, Teukolsky’s results yield solutions for thespin-weight ± ± Our exposition makes use of the formalism of Geroch, Held and Penrose (GHP)[76], which is a simplified and more explicitly covariant version of the Newman-Penrose (NP) [132] formalism originally used by Teukolsky. Here we provide aconcise review of the key features of the formalism needed to understand metricperturbations of black hole spacetimes; see Refs. [165, 4, 141] for more thoroughtreatments.The GHP formalism prioritises the concepts of spin- and boost-weights; withinthe formalism, everything has a well-defined type { p , q } , which is related to its spin-weight s = ( p − q ) / b = ( p + q ) /
2. Only objects of the sametype can be added together, providing a useful consistency check on any equations.Multiplication of two quantities yields a resulting object with type given by the sumof the types of its constituents.We first introduce a null tetrad { e α ( a ) } = { l α , n α , m α , ¯ m α } with normalisation l α n α = − , m α ¯ m α = , (33)and with all other inner products vanishing. In terms of the tetrad vectors, the metricmay be written as g αβ = − l ( α n β ) + m ( α ¯ m β ) . (34)There are three discrete transformations that reflect the inherent symmetry in theGHP formalism, corresponding to simultaneous interchange of the tetrad vectors:1. (cid:48) : l α ↔ n α and m α ↔ ¯ m α , { p , q } → {− p , − q } ;2. ¯ : m α ↔ ¯ m α , { p , q } → { q , p } ;3. ∗ : l α → m α , n α → − ¯ m α , m α → − l α , ¯ m α → ¯ n α .We next introduce the spin coefficients , defined to be the 12 directional derivativesof the tetrad vectors. Of these, the 8 with well-defined GHP type are κ = − l µ m ν ∇ µ l ν , σ = − m µ m ν ∇ µ l ν , ρ = − ¯ m µ m ν ∇ µ l ν , τ = − n µ m ν ∇ µ l ν , (35)along with their primed variants, κ (cid:48) , σ (cid:48) , ρ (cid:48) and τ (cid:48) . These have GHP type given by κ : { , } , σ : { , − } , ρ : { , } , τ : { , − } . The remaining 4 spin coefficients are used to define the GHP derivative operators(that depend on the GHP type of the object on which they are acting), Þ : = ( l α ∇ α − p ε − q ¯ ε ) , Þ (cid:48) : = ( n α ∇ α + p ε (cid:48) + q ¯ ε (cid:48) ) , ð : = ( m α ∇ α − p β + q ¯ β (cid:48) ) , ð (cid:48) : = ( ¯ m α ∇ α + p β (cid:48) − q ¯ β ) , (36)where β = ( m µ ¯ m ν ∇ µ m ν − m µ n ν ∇ µ l ν ) , ε = ( l µ ¯ m ν ∇ µ m ν − l µ n ν ∇ µ l ν ) , (37)along with their primed variants, β (cid:48) and ε (cid:48) . These spin coefficients have no well-defined GHP type and never appear explicitly in covariant equations. The action ofa GHP derivative causes the type to change by an amount { p , q } → { p + r , q + s } where { r , s } for each of the operators is given by Þ : { , } , Þ (cid:48) : {− , − } , ð : { , − } , ð (cid:48) : {− , } . In this sense we interpret Þ and Þ (cid:48) as boost raising and lowering operators, respec-tively, while we interpret ð and ð (cid:48) as spin raising and lowering operators, respec-tively. The adjoints of the GHP operators are given by Þ † : = − ( Þ − ρ − ¯ ρ ) , Þ (cid:48) † : = − ( Þ (cid:48) − ρ (cid:48) − ¯ ρ (cid:48) ) , ð † : = − ( ð − τ − ¯ τ (cid:48) ) , ð (cid:48) † : = − ( ð (cid:48) − τ (cid:48) − ¯ τ ) , (38)or, alternatively, (cid:68) † = − ( Ψ ¯ Ψ ) / (cid:68) ( Ψ ¯ Ψ ) − / , (cid:68) ∈ { Þ , Þ (cid:48) , ð , ð (cid:48) } . (39)In vacuum spacetimes, the only non-zero components of the Riemann tensor aregiven by the tetrad components of the Weyl tensor, which can be represented by fivecomplex Weyl scalars, Ψ = C lmlm , Ψ = C lnlm , Ψ = C lm ¯ mn , Ψ = C ln ¯ mn , Ψ = C n ¯ mn ¯ m , (40)with types inherited from the tetrad vectors that appear in their definition, Ψ : { , } , Ψ : { , } , Ψ : { , } , Ψ : {− , } , Ψ : {− , } . Many of the results that follow will be specialised to type-D spacetimes with l µ and n µ aligned to the two principal null directions, in which case the Goldberg-Sachs theorem implies that 4 of the of the spin coefficients vanish, κ = κ (cid:48) = σ = σ (cid:48) = , (41)and also that most of the Weyl scalars vanish Ψ = Ψ = Ψ = Ψ = . (42)The GHP equations give relations between the Weyl scalars and the directionalderivatives of the spin coefficients. For type-D spacetimes they are given by Þ ρ = ρ , Þ τ = ρ ( τ − ¯ τ (cid:48) ) , ð τ = τ , ð ρ = τ ( ρ − ¯ ρ ) , Þ (cid:48) ρ = ρ ¯ ρ (cid:48) − τ ¯ τ − Ψ + ð (cid:48) τ , (43)along with the Bianchi identity, Þ Ψ = ρΨ , ð Ψ = τΨ , (44)and the conjugate, prime, and prime conjugate of these equations. Similarly thecommutator of any pair of directional derivatives can be written in terms of a linearcombination of spin coefficients multiplying single directional derivatives. Againfor type-D, they are given by [ Þ , Þ (cid:48) ] = ( ¯ τ − τ (cid:48) ) ð + ( τ − ¯ τ (cid:48) ) ð (cid:48) − p ( Ψ − ττ (cid:48) ) − q ( ¯ Ψ − ¯ τ ¯ τ (cid:48) ) , (45a) [ Þ , ð ] = ¯ ρ ð − ¯ τ (cid:48) Þ + q ¯ ρ ¯ τ (cid:48) , (45b) [ ð , ð (cid:48) ] = ( ¯ ρ (cid:48) − ρ (cid:48) ) Þ + ( ρ − ¯ ρ ) Þ (cid:48) + p ( Ψ + ρρ (cid:48) ) − q ( ¯ ρ ¯ ρ (cid:48) + ¯ Ψ ) , (45c)along with the conjugate, prime, and prime conjugate of these.If we further restrict to spacetimes that admit a Killing tensor, (cid:63)(cid:63) K αβ , the associatedsymmetries lead to additional identities relating the spin coefficients, ρ ¯ ρ = ρ (cid:48) ¯ ρ (cid:48) = − τ ¯ τ (cid:48) = − τ (cid:48) ¯ τ = ¯ M / M / Ψ / ¯ Ψ / = ¯ ζζ , (46)for some complex function M that is annihilated by Þ . Here, we have used the factthat the Killing spinor coefficient is related to Ψ by ζ = − M / Ψ − / . (47) In the case of Kerr spacetime, M is the mass of the spacetime as one might anticipate.4 These identities can be used along with the GHP equations to obtain a complemen-tary set of identities, Þ τ (cid:48) = ρτ (cid:48) = ð (cid:48) ρ , (48a) Þ (cid:48) ρ = ρρ (cid:48) + τ (cid:48) ( τ − ¯ τ (cid:48) ) − Ψ − ¯ ζ ζ ¯ Ψ , (48b) ð (cid:48) τ = ττ (cid:48) + ρ ( ρ (cid:48) − ¯ ρ (cid:48) ) + Ψ − ¯ ζ ζ ¯ Ψ , (48c)along with the conjugate, prime, and prime conjugate of these equations. A conse-quence of these additional relations is that there is an operator £ ξ = − ζ (cid:0) − ρ (cid:48) Þ + ρ Þ (cid:48) + τ (cid:48) ð − τ ð (cid:48) ) − p ζΨ − q ζ ¯ Ψ , (49)associated with the Killing vector ξ α = − ζ ( − ρ (cid:48) l α + ρ n α + τ (cid:48) m α − τ ¯ m α ) . (50)There is a second operator £ η = − ζ (cid:2) ( ζ − ¯ ζ ) ( ρ (cid:48) Þ − ρ Þ (cid:48) ) − ( ζ + ¯ ζ ) ( τ (cid:48) ð − τ ð (cid:48) ) (cid:3) + p η h + q η ¯ h (51)where η h = ζ ( ζ + ¯ ζ ) Ψ − ζ ¯ ζ ¯ Ψ + ρρ (cid:48) ζ ( ¯ ζ − ζ ) + ττ (cid:48) ζ ( ¯ ζ + ζ ) . (52)This is associated with the second Killing vector η α = − ζ (cid:2) ( ζ − ¯ ζ ) ( ρ (cid:48) l α − ρ n α ) − ( ζ + ¯ ζ ) ( τ (cid:48) m α − τ ¯ m α ) (cid:3) . (53)Both £ ξ and £ η commute with all of the GHP operators and annihilate all of the spincoefficients and Ψ . We now consider perturbations of vacuum type-D spacetimes. Teukolsky [187]showed that the perturbations to Ψ and Ψ (which we will denote by ψ and ψ ) aregauge invariant and satisfy decoupled and fully separable second order equations.These perturbations may be written in GHP form as ψ = C ( ) lmlm [ h ] = T h , ψ = C ( ) n ¯ mn ¯ m [ h ] = T h , (54)where the operators T I are given by T h = − (cid:104) ( ð − ¯ τ (cid:48) )( ð − ¯ τ (cid:48) ) h ll + ( Þ − ¯ ρ )( Þ − ¯ ρ ) h mm − (cid:0) ( Þ − ¯ ρ )( ð − τ (cid:48) ) + ( ð − ¯ τ (cid:48) )( Þ − ρ ) (cid:1) h ( lm ) (cid:105) , (55a) T h = − (cid:104) ( ð (cid:48) − ¯ τ )( ð (cid:48) − ¯ τ ) h nn + ( Þ (cid:48) − ¯ ρ (cid:48) )( Þ (cid:48) − ¯ ρ (cid:48) ) h ¯ m ¯ m − (cid:0) ( Þ (cid:48) − ¯ ρ (cid:48) )( ð (cid:48) − τ ) + ( ð (cid:48) − ¯ τ )( Þ (cid:48) − ρ (cid:48) ) (cid:1) h ( n ¯ m ) (cid:105) . (55b)We will later also need the adjoints of these, which are given by ( T †0 Ψ ) αβ = − (cid:104) l α l β ( ð − τ )( ð − τ ) + m α m β ( Þ − ρ )( Þ − ρ ) − l ( α m β ) (cid:0) ( ð − τ + ¯ τ (cid:48) )( Þ − ρ ) + ( Þ − ρ + ¯ ρ )( ð − τ ) (cid:1)(cid:105) Ψ , (56a) ( T †4 Ψ ) αβ = − (cid:104) n α n β ( ð (cid:48) − τ (cid:48) )( ð (cid:48) − τ (cid:48) ) + ¯ m α ¯ m β ( Þ (cid:48) − ρ (cid:48) )( Þ (cid:48) − ρ (cid:48) ) − n ( α ¯ m β ) (cid:0) ( ð (cid:48) − τ (cid:48) + ¯ τ )( Þ (cid:48) − ρ (cid:48) ) + ( Þ (cid:48) − ρ (cid:48) + ¯ ρ (cid:48) )( ð (cid:48) − τ (cid:48) ) (cid:1)(cid:105) Ψ . (56b)The scalars ψ and ψ satisfy the Teukolsky equations, O ψ = π S T , O (cid:48) ψ = π S T , (57)where O : = (cid:0) Þ − s ρ − ¯ ρ (cid:1)(cid:0) Þ (cid:48) − ρ (cid:48) (cid:1) − (cid:0) ð − s τ − ¯ τ (cid:48) (cid:1)(cid:0) ð (cid:48) − τ (cid:48) (cid:1) + (cid:2)(cid:0) s − (cid:1) − s (cid:3) Ψ (58)is the spin-weight s Teukolsky operator. The decoupling operators ( S T ) = ( ð − ¯ τ (cid:48) − τ ) (cid:2) ( Þ − ρ ) T ( lm ) − ( ð − ¯ τ (cid:48) ) T ll (cid:3) + ( Þ − ρ − ¯ ρ ) (cid:2) ( ð − τ (cid:48) ) T ( lm ) − ( Þ − ¯ ρ ) T mm (cid:3) , (59a) ( S T ) = ( ð (cid:48) − ¯ τ − τ (cid:48) ) (cid:2) ( Þ (cid:48) − ρ (cid:48) ) T ( n ¯ m ) − ( ð (cid:48) − ¯ τ ) T nn (cid:3) + ( Þ (cid:48) − ρ (cid:48) − ¯ ρ (cid:48) ) (cid:2) ( ð (cid:48) − τ ) T ( n ¯ m ) − ( Þ (cid:48) − ¯ ρ (cid:48) ) T ¯ m ¯ m (cid:3) , (59b)allow the sources for the Teukolsky equations to be constructed from the stress-energy tensor. We will later also need the adjoints of these, which are given by Note that O (cid:48) ψ = ζ − O ζ ψ and O ψ = ζ − O ζ ψ . Some authors (e.g. [199, 81]) define O to be the operator with s = +
2. Then, the operator for thenegative s fields is its adjoint O † .6 ( S †0 ) αβ = − l α l β ( ð − τ )( ð + τ ) − m α m β ( Þ − ρ )( Þ + ρ )+ l ( α m β ) (cid:2) ( Þ − ρ + ¯ ρ )( ð + τ ) + ( ð − τ + ¯ τ (cid:48) )( Þ + ρ )] Ψ , (60a) ( S †4 ) αβ = − n α n β ( ð (cid:48) − τ (cid:48) )( ð (cid:48) + τ (cid:48) ) − ¯ m α ¯ m β ( Þ (cid:48) − ρ (cid:48) )( Þ (cid:48) + ρ (cid:48) )+ n ( α ¯ m β ) (cid:2) ( Þ (cid:48) − ρ (cid:48) + ¯ ρ (cid:48) )( ð (cid:48) + τ (cid:48) ) + ( ð (cid:48) − τ (cid:48) + ¯ τ )( Þ (cid:48) + ρ (cid:48) )] Ψ . (60b)Introducing the index-free linearised Einstein operator ( E h ) αβ : = G ( ) αβ [ h ] , wesee that Teukolsky’s result for decoupling the equations are a consequence of theoperator identities S E = OT , S E = O (cid:48) T . (61)In vacuum Kerr-NUT spacetimes, the Teukolsky operator may be written in man-ifestly separable form by rewriting it in terms of the commuting operators [4] (cid:82) : = ζ ¯ ζ ( Þ − ρ − ¯ ρ )( Þ (cid:48) − b ρ (cid:48) ) + b − ( ζ + ¯ ζ ) £ ξ , (62)and (cid:83) : = ζ ¯ ζ ( ð − τ − ¯ τ (cid:48) )( ð (cid:48) − s τ (cid:48) ) + s − ( ζ − ¯ ζ ) £ ξ . (63)Then, Teukolsky operator is given by ζ ¯ ζ O = (cid:82) − (cid:83) . (64)The symmetry operators satisfy the commutation relations (cid:2) (cid:82) , (cid:83) (cid:3) = { p , } object. We will see later that when written as a coordinateexpression in Boyer-Lindquist coordinates in Kerr spacetime the operators (cid:82) and (cid:83) reduce to the radial Teukolsky and spin-weighted spheroidal operators (with acommon eigenvalue). In regions where they satisfy the homogeneous Teukolsky equations, the scalars ψ and ψ are not independent. Instead, they are related by the Teukolsky-Starobinskyidentities, which are given in GHP form by Þ ζ ψ = ð (cid:48) ζ ψ − M£ ξ ¯ ψ , (65a) Þ (cid:48) ζ ψ = ð ζ ψ + M£ ξ ¯ ψ , (65b)where we recall that M = − ζ ψ . From these, we can also derive eighth-orderTeukolsky-Starobinsky identities that do not mix the scalars Þ ¯ ζ Þ (cid:48) ζ ψ = ð (cid:48) ¯ ζ ð ζ ψ − M £ ξ ψ , (66a) Þ (cid:48) ¯ ζ Þ ζ ψ = ð ¯ ζ ð (cid:48) ζ ψ − M £ ξ ψ . (66b) Solutions of the Teukolsky equations can be related back to solutions for the metricperturbation h αβ by use of a Hertz potential [199, 36, 104, 115, 207]. In fact, thereare two different Hertz potentials: ψ IRG , which produces a metric perturbation inthe ingoing radiation gauge; and ψ ORG , which produces a metric perturbation in theoutgoing radiation gauge.In the ingoing radiation gauge (IRG), the metric perturbation may be recon-structed by applying a second-order differential operator to a scalar Hertz potential ψ IRG of type {− , } (i.e. the same type as ψ ). In terms of this Hertz potential, theIRG metric perturbation is given explicitly by h IRG αβ = ℜ (cid:2) ( S †0 ψ IRG ) αβ ) (cid:3) . (67)where S †0 is the operator given in Eq. (60a). The IRG Hertz potential satisfies O ψ IRG = η IRG , where η IRG satisfies 2 ℜ ( T †0 η IRG ) αβ = π T αβ . In other words, ψ IRG is a solution of the equation satisfied by ζ ψ (equivalently, the adjoint of theequation satisfied by ψ ), but with a different source.The IRG Hertz potential manifestly satisfies the gauge conditions l α h αβ = h = ( E h IRG ) ll = = T ll . Computing the perturbedWeyl scalars from it, we find ψ = Þ ψ IRG (68a) ψ = ð (cid:48) ψ IRG − M ζ − £ ξ ψ IRG , + (cid:104) ζ − O ζ + ζ − £ ξ − ( τ (cid:48) τ − ρ (cid:48) ρ − Ψ ) (cid:105) η IRG . (68b)The IRG Hertz potential may therefore be obtained either by solving the sourced(adjoint) Teukolsky equation or by solving either of the fourth-order equationssourced by the perturbed Weyl scalars. The equations involving ψ and ψ are oftenreferred to as the “radial” and “angular” inversion equations, respectively. Acting onthe perturbed Weyl scalars with the Teukolsky operator and commuting operators,we find O ψ = ( Þ − ρ − ¯ ρ ) η IRG (69a) O (cid:48) ψ = ( ð (cid:48) − τ (cid:48) − ¯ τ ) η IRG − M ζ − £ ξ η IRG , + O (cid:48) (cid:104) ζ − O ζ + ζ − £ ξ − ( τ (cid:48) τ − ρ (cid:48) ρ − Ψ ) (cid:105) η IRG . (69b)Thus, in regions where the Hertz potential satisfies the homogenous equation O ψ IRG = h ORG αβ = ℜ (cid:2) ( S †4 ψ ORG ) αβ (cid:3) , (70)where the ORG Hertz potential, ψ ORG , is of type { , } (i.e. the same as ψ ). TheORG Hertz potential satisfies O (cid:48) ψ ORG = η ORG , where η ORG satisfies 2 ℜ ( T †4 η ORG ) αβ = π T αβ . In other words, ψ ORG is a solution of the equation satisfied by ζ ψ (equiv-alently, the adjoint of the equation satisfied by ψ ), but with a different source.The ORG Hertz potential manifestly satisfies the gauge conditions n α h αβ = h =
0, and it necessarily requires that ( E h IRG ) nn = = T nn . Computing theperturbed Weyl scalars from it, we find ψ = ð ψ ORG + M ζ − £ ξ ψ ORG + (cid:104) ζ O ζ − − ζ − £ ξ − ( τ (cid:48) τ − ρ (cid:48) ρ − Ψ ) (cid:105) η ORG (71a) ψ = Þ (cid:48) ψ ORG , (71b)The ORG Hertz potential may therefore be obtained either by solving the sourced(adjoint) Teukolsky equation or by solving either of the fourth-order equationssourced by the perturbed Weyl scalars. The equations involving ψ and ψ are oftenreferred to as the “angular” and “radial” inversion equations, respectively. Acting onthe perturbed Weyl scalars with the Teukolsky operator and commuting operators,we find O ψ = ( ð − τ − ¯ τ (cid:48) ) η ORG + M ζ − £ ξ η ORG + O (cid:104) ζ O ζ − − ζ − £ ξ − ( τ (cid:48) τ − ρ (cid:48) ρ − Ψ ) (cid:105) η ORG (72a) O (cid:48) ψ = ( Þ (cid:48) − ρ (cid:48) − ¯ ρ (cid:48) ) η ORG . (72b) Some authors [123] define a slightly different ORG Hertz potential related to the one here byˆ ψ ORG = ζ − ψ ORG and ( ˆ S †4 ) αβ = ( S †4 ζ ) αβ . Both conventions yield the same metric perturba-tion, ( ˆ S †4 ˆ ψ ORG ) αβ = ( S †4 ψ ORG ) αβ ) .9 Thus, in regions where the Hertz potential satisfies the homogenous equation O (cid:48) ψ ORG = ψ and ψ weobtain Teukolsky-Starobinsky identities relating them, Þ ψ IRG = ð (cid:48) ψ ORG + M ζ − £ ξ ψ ORG (73) Þ (cid:48) ψ ORG = ð ψ IRG − M ζ − £ ξ ψ IRG . (74)The fact that the Hertz potentials yield solutions of the homogeneous linearisedEinstein equations was succinctly summarised by Wald [199] using the method ofadjoints: since the operators satisfy the identity S E = OT , by taking the adjointand using the fact that E is self-adjoint we find that E S † = T † O † so we have ahomogeneous solution of the linearised Einstein equations provided the Hertz po-tential satisfies the (adjoint) homogeneous Teukolsky equation.Finally, we note that in addition to imposing conditions on the stress-energy, thestandard radiation gauge reconstruction procedure fails to reproduce certain “com-pletion” portions of the metric perturbation associated with small shifts in the centralmass and angular momentum, and gauge. A more generally valid metric perturba-tion may be obtained by supplementing the reconstructed piece described here withcompletion pieces and with a “corrector” tensor x αβ that is designed to eliminatedany restrictions on the stress-energy, h αβ = ℜ ( S † Ψ ) αβ + x αβ + ˙ g αβ + ( £ X g ) αβ . (75)The interested reader may refer to [199, 104, 36] for the original derivations of thereconstruction procedure, to [8] for an analysis of the sourced equation satisfied bythe Hertz potential, to [124, 42] for details of metric completion, and to [81, 90] fora thorough explanation of the corrector tensor approach. In order to determine gravitational wave strain, we require the metric perturbationfar from the source. If we consider the metric perturbation reconstructed in radiationgauge, then to leading order in a large-distance expansion from the source the com-ponents h mm and h ¯ m ¯ m dominate, with both falling of as ( distance ) − . It is commonto write these in terms of the two gravitational wave polarizations, h mm = h + + ih × , h ¯ m ¯ m = h + − ih × . (76)Furthermore, at large radius the operator T of Eq. (55b) reduces to a second deriva-tive along the l µ null direction, leading to a simple relationship between ψ and thesecond time derivative of the strain, ψ ∼ −
12 ¨ h ¯ m ¯ m . (77)This gives us a straightforward way to determine the strain by computing two timeintegrals of ψ . Further mathematical details on the relationship between ψ andoutgoing gravitational radiation are given in Refs. [132, 133, 185], on the equivalentrelationship between ψ and incoming radiation are given in Ref. [201], and onnumerical implementation considerations in Refs. [168, 112]. We now give explicit expressions for the various quantities defined in the previoussections specialized to Kerr spacetime. The spin coefficients are tetrad dependent.When working with the Carter tetrad, the non-zero spin coefficients have a particu-larly symmetric form given by ρ = − ρ (cid:48) = − ζ (cid:114) ∆ Σ , τ = τ (cid:48) = − ia sin θζ √ Σ , β = β (cid:48) = − i ζ a + ir cos θ θ √ Σ , ε = − ε (cid:48) = Mr − a − ia ( r − M ) cos θ ζ √ Σ ∆ , (78)while for the Kinnersley tetrad they are given by ρ K = − ζ , ρ (cid:48) K = ∆ ζ ¯ ζ , τ K = − ia sin θ √ ζ ¯ ζ , τ (cid:48) K = − ia sin θ √ ζ , β K = cot θ √ ζ , β (cid:48) K = cot θ √ ζ − ia sin θ √ ζ , ε K = , ε (cid:48) K = ∆ ζ ¯ ζ − r − M ζ ¯ ζ . (79)The commuting GHP operators have the same form in both tetrads, £ ξ = ∂ t , £ η = a ∂ t + a ∂ ϕ . (80) In additional to decoupling the equations, Teukolsky further showed that the Teukol-sky equations are fully separable using a mode ansatz. The specific form of theansatz depends on the choice of null tetrad. Teukolsky worked with the Kinnersleytetrad [107], in which case the Teukolsky equations are separable using the ansatz A similar separability result also holds when working with the Carter tetrad by replacing the lefthand sides as follows: ψ → ζ ∆ − ψ , ζ ψ → ζ ∆ψ , ( S T ) → ζ ∆ − ( S T ) , ζ ( S T ) → ζ ∆ ( S T ) . ψ = (cid:90) ∞ − ∞ ∞ ∑ (cid:96) = (cid:96) ∑ m = − (cid:96) ψ (cid:96) (cid:109) ω ( r ) S (cid:96) (cid:109) ( θ , φ ; a ω ) e − i ω t d ω , (81) ζ ψ = (cid:90) ∞ − ∞ ∞ ∑ (cid:96) = (cid:96) ∑ m = − (cid:96) − ψ (cid:96) (cid:109) ω ( r ) − S (cid:96) (cid:109) ( θ , φ ; a ω ) e − i ω t d ω , (82)with the functions s ψ (cid:96) (cid:109) ω ( r ) and s S (cid:96) (cid:109) ( θ , φ ; a ω ) satisfying the spin-weighted spheroidalharmonic and Teukolsky radial equations, respectively, (cid:20) dd χ (cid:18) ( − χ ) dd χ (cid:19) + a ω χ − ( m + s χ ) − χ − as ω χ + s + A (cid:21) s S (cid:96) (cid:109) = , (83)and (cid:20) ∆ − s ddr (cid:18) ∆ s + ddr (cid:19) + K − is ( r − M ) K ∆ + is ω r − s λ (cid:96) (cid:109) (cid:21) s ψ (cid:96) (cid:109) ω = s T (cid:96) (cid:109) ω , (84)where χ : = cos θ , A : = s λ (cid:96) (cid:109) + am ω − a ω and K : = ( r + a ) ω − am , and wherethe eigenvalue s λ (cid:96) (cid:109) depends on the value of a ω . As with the standard spherical har-monics, the dependence of the spin-weighted spheroidal harmonics on the azimuthalcoordinate is exclusively through an overall complex exponential factor, s S (cid:96) (cid:109) ( θ , φ ; a ω ) = s S (cid:96) (cid:109) ( θ , a ω ) e i (cid:109) φ . (85)The spin-weighted spheroidal harmonics satisfy the symmetry identities s S (cid:96) (cid:109) ( θ , φ ; a ω ) = ( − ) (cid:96) + (cid:109) − s S (cid:96) (cid:109) ( π − θ , φ ; a ω ) , (86a) s S (cid:96) (cid:109) ( θ , φ ; a ω ) = ( − ) (cid:96) + s s ¯ S (cid:96) − (cid:109) ( π − θ , φ ; − a ω ) (86b)which can be combined to obtain the useful identity s S (cid:96) (cid:109) ( θ , φ ; a ω ) = ( − ) (cid:109) + s − s ¯ S (cid:96) − (cid:109) ( θ , φ ; − a ω ) , (87)which relates an ( s , (cid:96), (cid:109) , a ω ) harmonic to the conjugate of an ( − s , (cid:96), − (cid:109) , − a ω ) harmonic. Similarly, the eigenvalue satisfies the identities s λ (cid:96) (cid:109) ( a ω ) = − s λ (cid:96) (cid:109) ( a ω ) − s , (88a) s λ (cid:96) (cid:109) ( a ω ) = s λ (cid:96) − (cid:109) ( − a ω ) (88b)which can be combined to obtain s λ (cid:96) (cid:109) ( a ω ) = − s λ (cid:96) − (cid:109) ( − a ω ) − s . (89)The sources for the radial Teukolsky equation are defined by The factors of ∆ here are not required for separability, but are included so that the radial functionsare consistent with Teukolsky’s original radial functions.2 π ( S T ) = − Σ (cid:90) ∞ − ∞ ∞ ∑ (cid:96) = (cid:96) ∑ (cid:109) = − (cid:96) T (cid:96) (cid:109) ω ( r ) S (cid:96) (cid:109) ( θ , ϕ ; a ω ) e − i ω t d ω , (90a)8 πζ ( S T ) = − Σ (cid:90) ∞ − ∞ ∞ ∑ (cid:96) = (cid:96) ∑ (cid:109) = − (cid:96) − T (cid:96) (cid:109) ω ( r ) − S (cid:96) (cid:109) ( θ , ϕ ; a ω ) e − i ω t d ω . (90b)Finally, when acting on a single mode of the mode-decomposed Weyl scalars thesymmetry operators yield (cid:83) ψ = − | | λ (cid:96) (cid:109) ψ , (cid:82) ψ = − | | λ (cid:96) (cid:109) ψ + ζ ¯ ζ S T , (cid:83) (cid:48) ψ = − |− | λ (cid:96) (cid:109) ψ , (cid:82) (cid:48) ψ = − | | λ (cid:96) (cid:109) ψ + ζ ¯ ζ S T , (91)where | s | λ (cid:96) (cid:109) ω : = s λ (cid:96) (cid:109) ω + | s | + s is independent of the sign of s . Solutions to the radial Teukolsky equation may be written in terms of a pair ofhomogeneous mode basis functions chosen according to their asymptotic behaviorat the four null boundaries to the spacetime. For radiative ( ω (cid:54) =
0) modes, the fourcommon choices are denoted• “in”: representing waves coming in from I − then partially falling into the hori-zon and partially scattering back out to I + ; these modes are purely ingoing intothe horizon;• “up”: representing waves coming up from H − then partially travelling out to I + and partially scattering back into H + ; these modes are purely outgoing atinfinity;• “out”: representing waves coming from I − and H − then travelling out to I + ;these modes are purely outgoing from the horizon;• “down”: representing waves coming from I − and H − then travelling down to H + ; these modes are purely incoming at infinity;These have asymptotic behaviour given by s R in (cid:96) (cid:109) ω ( r ) ∼ (cid:110) s R in,ref (cid:96) (cid:109) ω r − − s e + i ω r ∗ ++ s R in,trans (cid:96) (cid:109) ω ∆ − s e − ikr ∗ s R in,inc (cid:96) (cid:109) ω r − e − i ω r ∗ r → r + r → ∞ (92a) s R up (cid:96) (cid:109) ω ( r ) ∼ (cid:110) s R up,inc (cid:96) (cid:109) ω e + ikr ∗ s R up,trans (cid:96) (cid:109) ω r − − s e + i ω r ∗ ++ s R up,ref (cid:96) (cid:109) ω ∆ − s e − ikr ∗ r → r + r → ∞ (92b) s R out (cid:96) (cid:109) ω ( r ) ∼ (cid:110) s R out,trans (cid:96) (cid:109) ω e + ikr ∗ s R out,inc (cid:96) (cid:109) ω r − − s e + i ω r ∗ ++ s R out,ref (cid:96) (cid:109) ω r − e − i ω r ∗ r → r + r → ∞ (92c) s R down (cid:96) (cid:109) ω ( r ) ∼ (cid:110) s R down,ref (cid:96) (cid:109) ω e + ikr ∗ ++ s R down,inc (cid:96) (cid:109) ω ∆ − s e − ikr ∗ s R down,trans (cid:96) (cid:109) ω r − e − i ω r ∗ r → r + r → ∞ (92d) This is distinct from Chandrasekhar’s eigenvalue which is given in Eq. (104).3
IH IH - -+ +
IH IH - -+ +
IH IH - -+ +
IH IH - -+ +
Fig. 1
Left to right: boundary conditions satisfied by the “in”, “up”, “out” and “down” solutions. where k : = ω − m Ω + with Ω + : = a Mr + the angular velocity of the horizon, andwhere r ∗ : = r + κ + ln r − r + M + κ − ln r − r − M with κ ± : = r ± − r ∓ ( r ± + a ) the surface gravity onthe outer/inner horizon. This behaviour is depicted graphically in Fig. 1.Inhomogeneous solutions of the radial Teukolsky equation can then be written interms of a linear combination of the basis functions, ψ (cid:96) (cid:109) ω ( r ) = C in (cid:96) (cid:109) ω ( r ) R in (cid:96) (cid:109) ω ( r ) + C up (cid:96) (cid:109) ω ( r ) R up (cid:96) (cid:109) ω ( r ) , (93) − ψ (cid:96) (cid:109) ω ( r ) = − C in (cid:96) (cid:109) ω ( r ) − R in (cid:96) (cid:109) ω ( r ) + − C up (cid:96) (cid:109) ω ( r ) − R up (cid:96) (cid:109) ω ( r ) , (94)where the weighting coefficients are determined by variation of parameters, s C in (cid:96) (cid:109) ω ( r ) = (cid:90) ∞ r s R up (cid:96) (cid:109) ω ( r (cid:48) ) W ( r (cid:48) ) ∆ s T (cid:96) (cid:109) ω ( r (cid:48) ) dr (cid:48) , (95a) s C up (cid:96) (cid:109) ω ( r ) = (cid:90) rr + s R in (cid:96) (cid:109) ω ( r (cid:48) ) W ( r (cid:48) ) ∆ s T (cid:96) (cid:109) ω ( r (cid:48) ) dr (cid:48) , (95b)with W ( r ) = s R in (cid:96) (cid:109) ω ( r ) ∂ r [ s R up (cid:96) (cid:109) ω ( r )] − s R up (cid:96) (cid:109) ω ( r ) ∂ r [ s R in (cid:96) (cid:109) ω ( r )] the Wronskian [inpractice, it is convenient to use the fact that ∆ s + W ( r ) = const].If one computes the “in” and “up” mode functions with normalisation such thattransmission coefficients are unity, s R in,trans (cid:96) (cid:109) ω = = s R up,trans (cid:96) (cid:109) ω , then the gravitationalwave strain can be determined directly from ψ using Eq. (77) to givelim r → ∞ r ( h + − ih × ) = (cid:90) ∞ − ∞ ∞ ∑ (cid:96) = (cid:96) ∑ m = − (cid:96) − C up (cid:96) (cid:109) ω ω − S (cid:96) (cid:109) ( θ , φ ; a ω ) e − i ω ( t − r ∗ ) d ω , (96)where the weighting coefficient − C up (cid:96) (cid:109) ω is to be evaluated in the limit r → ∞ .Similarly, the time averaged flux of energy carried by gravitational waves passingthrough infinity and the horizon can be computed from the “in” and “up” normal-ization coefficients [95], Strictly speaking, the horizon fluxes given here have been derived from the rates of change ofthe black hole parameters due to shear of the horizon generators [188]. It is generally assumed thatthese are equivalent to the gravitational wave fluxes, although this has not, to our knowledge, beenshown explicitly.4 F H E = lim r → r + ∑ (cid:96) (cid:109) ω πα (cid:96) (cid:109) ω ω | − C in (cid:96) (cid:109) ω | , (97) F I E = lim r → ∞ ∑ (cid:96) (cid:109) ω πω | − C up (cid:96) (cid:109) ω | , (98)where α (cid:96) (cid:109) ω : = ( Mr + ) k ( k + ε )( k + ε ) ω | C (cid:96) (cid:109) ω | with ε : = √ M − a / ( Mr + ) . Simi-larly, the flux of angular momentum is given by F H L z = lim r → r + ∑ (cid:96) (cid:109) ω π (cid:109) α (cid:96) (cid:109) ω ω | − C in (cid:96) (cid:109) ω | , (99) F I L z = lim r → ∞ ∑ (cid:96) (cid:109) ω π (cid:109) ω | − C up (cid:96) (cid:109) ω | . (100)Similar expressions can be obtained in terms of the modes ψ (cid:96) (cid:109) ω of ψ by using theTeukolsky-Starobinsky identities to relate − C up (cid:96) (cid:109) ω to C up (cid:96) (cid:109) ω . The necessary detailsof how these asymptotic amplitudes are related can be found in Refs. [134, 123].When decomposed into modes, each of the Teukolsky-Starobinsky identities sep-arate to yield identities relating the positive spin-weight spheroidal and radial func-tions to the negative spin-weight ones, D ( − ψ (cid:96) (cid:109) ω ) = C (cid:96) (cid:109) ω ψ (cid:96) (cid:109) ω , (101a) ∆ ( D †0 ) ( ∆ ψ (cid:96) (cid:109) ω ) = C (cid:96) (cid:109) ω − ψ (cid:96) (cid:109) ω , (101b) L − L L L ( S (cid:96) (cid:109) ω ) = D − S (cid:96) (cid:109) ω , (101c) L † − L †0 L †1 L †2 ( − S (cid:96) (cid:109) ω ) = D S (cid:96) (cid:109) ω , (101d)where D n : = ∂ r − iK ∆ + n r − M ∆ , D † n : = ∂ r + iK ∆ + n r − M ∆ , (102a) L n : = ∂ θ + Q + n cot θ , L † n : = ∂ θ − Q + n cot θ , (102b)(with K defined above and Q : = − a ω sin θ + m csc θ ) are essentially mode versionsof the GHP differential operators. The constants of proportionality are given by C (cid:96) (cid:109) ω = D + ( − ) (cid:96) + m iM ω , (103a) D = ( s λ Ch (cid:96) (cid:109) ) ( s λ Ch (cid:96) (cid:109) − ) + a ω ( m − a ω )( s λ Ch (cid:96) (cid:109) − )( s λ Ch (cid:96) (cid:109) − )+ ( a ω ) (cid:2) ( s λ Ch (cid:96) (cid:109) − ) + ( m − a ω ) (cid:3) , (103b)where s λ Ch (cid:96) (cid:109) ω : = s λ (cid:96) (cid:109) ω + s + s (104) is the eigenvalue used by Chandrasekhar [34]. This particular choice of C (cid:96) (cid:109) ω en-sures that the s = + s = − Finally, when written in terms of modes the homogeneous radiation gauge angu-lar inversion equations can be algebraically inverted to give the modes of the Hertzpotentials in terms of the modes of the Weyl scalar, ψ ORG (cid:96) (cid:109) ω = ( − ) m D ¯ ψ − ω (cid:96) − m + iM ω ψ (cid:96) (cid:109) ω | C (cid:96) (cid:109) ω | , (105) ψ IRG (cid:96) (cid:109) ω = ( − ) m D − ¯ ψ − ω (cid:96) − m − iM ω − ψ (cid:96) (cid:109) ω | C (cid:96) (cid:109) ω | . (106)where the separability ansatz for the Hertz potentials differs by a factor of ζ − fromthat of the Weyl scalars, ζ − ψ ORG = (cid:90) ∞ − ∞ ∞ ∑ (cid:96) = (cid:96) ∑ m = − (cid:96) ψ ORG (cid:96) (cid:109) ω ( r ) S (cid:96) (cid:109) ( θ , φ ; a ω ) e − i ω t d ω , (107) ψ IRG = (cid:90) ∞ − ∞ ∞ ∑ (cid:96) = (cid:96) ∑ m = − (cid:96) ψ IRG (cid:96) (cid:109) ω ( r ) − S (cid:96) (cid:109) ( θ , φ ; a ω ) e − i ω t d ω . (108)Alternatively, one can use the radial inversion equations to relate the asymptotic am-plitudes of ψ IRG (cid:96) (cid:109) ω to the asymptotic amplitudes of ψ (cid:96) (cid:109) ω and to relate the asymp-totic amplitudes of ψ ORG (cid:96) (cid:109) ω to the asymptotic amplitudes of − ψ (cid:96) (cid:109) ω . Further detailsare given in [134] for the IRG case and in [123] for the ORG case.Note that in order to transform back to the time-domain solution, as a final stepwe must perform an inverse Fourier transform. This poses a challenge in gravita-tional self-force calculations, where non-smoothness of the solutions in the vicinityof the worldline lead to the Gibbs phenomenon of non-convergence of the inverseFourier transform. Resolutions to this problem typically rely on avoiding directlytransforming the inhomogeneous solution by using the methods of extended homo-geneous or extended particular solutions. For further details, see [91, 92]. In numerical implementations, the Teukolsky equation can be problematic to workwith due to the presence of a long-ranged potential. One approach to this problemis to transform to an alternative master function that satisfies an equation with amore short-ranged potential. The Sasaki-Nakamura transformation is designed todo exactly this. It introduces a new function of the form An alternative proportionality constant can be derived such that the s = + s = − X ∼ (cid:40) ¯ ζ ζ ( r + a ) / r Þ (cid:48) Þ (cid:48) r ζ ψ ( r + a ) / r ÞÞ r ζ ψ , (109)where the factors of ζ ensure that these are purely radial operators. There is con-siderable freedom to rescale these expressions by inserting appropriate functionsof r , for more details see Ref. [94] in which case the X given here corresponds to √ r + a r J − J − r R in the s = − √ r + a r J + J + ∆ r R in the s = + On a Schwarzschild background spacetime, separability is readily achieved withouthaving to rely on the Teukolsky formalism. Writing the metric perturbation in termsof its null tetrad components, they have GHP type s = h ln : { , } , h m ¯ m : { , } , h ll : { , } , h nn : {− , − } s = ± h lm : { , } , h l ¯ m : { , } , h nm : { , − } , h n ¯ m : {− , } s = ± h mm : { , − } , h ¯ m ¯ m : {− , } . Here we have gathered the components into scalar ( s = s = ±
1) andtensor ( s = ±
2) sectors.In some instances, it is convenient to work with the trace-reversed metric pertur-bation, ¯ h αβ = h αβ − h g αβ . In terms of null tetrad components, the trace is givenby h = − ( h ln − h m ¯ m ) so a trace reversal simply corresponds to the interchange h ln ↔ h m ¯ m : ¯ h ln = h m ¯ m and ¯ h m ¯ m = h ln , with all other components unchanged.The tetrad components may be decomposed into a basis of spin-weighted spher-ical harmonics h ab = ∞ ∑ (cid:96) = | s | (cid:96) ∑ m = − (cid:96) h (cid:96) (cid:109) ab ( t , r ) s Y (cid:96) (cid:109) ( θ , φ ) (110)where s = h ln , h m ¯ m , h ll and h nn ; s = + h lm and h nm ; s = − h l ¯ m and h n ¯ m ; s = + h mm ; and s = − h ¯ m ¯ m . Here, we have introduced the spin-weightedspherical harmonics s Y (cid:96) (cid:109) ( θ , φ ) = s S (cid:96) (cid:109) ω ( θ , φ ; 0 ) with the associated eigenvalue s λ (cid:96) : = s λ (cid:96) (cid:109) ( a ω = ) = (cid:96) ( (cid:96) + ) − s ( s + ) .In the Schwarzschild case the GHP derivative operators split into operators that(up to an overall factor of r ) act only on the two-sphere, ð = √ r ( ∂ θ + i csc θ ∂ φ − s cot θ ) , ð (cid:48) = √ r ( ∂ θ − i csc θ ∂ φ + s cot θ ) . (111) This expression is appropriate when working with the Kinnersley tetrad; for the Carter tetradboth definitions for X need to be scaled by a common factor of ¯ ζζ to obtain a radial operator.7 and operators that act only in the t − r subspace Þ = √ f (cid:20) ∂ t + f ∂ r − bMr (cid:21) , Þ (cid:48) = √ f (cid:20) ∂ t − f ∂ r − bMr (cid:21) . (113)The two-sphere operators act as spin-raising and lowering operators to relate spin-weighted spherical harmonics of different spin-weight √ r ð (cid:2) s Y (cid:96) (cid:109) ( θ , φ ) (cid:3) = − (cid:2) (cid:96) ( (cid:96) + ) − s ( s + ) (cid:3) / s + Y (cid:96) (cid:109) ( θ , φ ) , (114a) √ r ð (cid:48) (cid:2) s Y (cid:96) (cid:109) ( θ , φ ) (cid:3) = (cid:2) (cid:96) ( (cid:96) + ) − s ( s − ) (cid:3) / s − Y (cid:96) (cid:109) ( θ , φ ) . (114b)In particular, this provides a relationship between the spin-weighted spherical har-monics and the scalar spherical harmonics.It is convenient to split the six vector and tensor sector components of the metricperturbation into real (even parity) and imaginary (odd parity) parts, representingwhether they are even or odd under the transformation ( θ , φ ) → ( π − θ , φ + π ) : h (cid:96) (cid:109) lm = h (cid:96) (cid:109) l , even + i h (cid:96) (cid:109) l , odd , h (cid:96) (cid:109) l ¯ m = − h (cid:96) (cid:109) l , even + i h (cid:96) (cid:109) l , odd , h (cid:96) (cid:109) nm = h (cid:96) (cid:109) n , even + i h (cid:96) (cid:109) n , odd , h (cid:96) (cid:109) n ¯ m = − h (cid:96) (cid:109) n , even + i h (cid:96) (cid:109) n , odd , h (cid:96) (cid:109) mm = h (cid:96) (cid:109) , even + i h (cid:96) (cid:109) , odd , h (cid:96) (cid:109) ¯ m ¯ m = h (cid:96) (cid:109) , even − i h (cid:96) (cid:109) , odd . The four scalar sector components are necessarily even parity, so we therefore haveseven fields in the even-parity sector and three in the odd-parity sector. The even andodd parity sectors decouple, meaning that they can be solved for independently. Ininstances where there is symmetry under reflection about the equatorial plane thisdecoupling is explicit in that the even parity sector only contributes for (cid:96) + (cid:109) evenand the odd parity sector only contributes for (cid:96) + (cid:109) odd.Finally, we can also optionally further decompose into the frequency domain, h (cid:96) (cid:109) ab ( t , r ) = (cid:90) ∞ − ∞ h (cid:96) (cid:109) ω ab ( r ) e − i ω t d ω (115)in order to obtain functions of r only. This has the advantage of reducing the problemof computing the metric perturbation to that of solving systems of 7 + ( (cid:96), (cid:109) , ω ) . These expressions are obtained when working with the Carter tetrad. The equivalent operatorsfor the Kinnersley tetrad are Þ = f − ∂ t + ∂ r , Þ (cid:48) = ( ∂ t − f ∂ r − bM / r ) . (112)8 There is some freedom in the specific choice of basis into which tensors are de-composed. In particular, the relative scaling of the l µ and n µ tetrad vectors leadsto a slightly different basis if one works with the Kinnersley tetrad rather than theCarter tetrad. It is also possible to work with alternative basis vectors spanning the t − r space. In some instances it is convenient to work with coordinate basis vectors δ µ t and δ µ r rather than null vectors. One can also choose to omit the factor of r inthe definition of m µ and ¯ m µ . The choice of basis does not have a fundamental im-pact, but some choices lead to more straightforward or natural interpretations of theresulting equations.Additionally, as an alternative to a spin-weighted harmonic basis, one couldequivalently work with a basis of vector and tensor spherical harmonics, which arerelated to the spin-weighted spherical harmonics by Z (cid:96) (cid:109) A : = (cid:2) (cid:96) ( (cid:96) + )] − / D A Y (cid:96) (cid:109) = √ (cid:16) − Y (cid:96) (cid:109) m A − Y (cid:96) (cid:109) ¯ m A (cid:17) (116a) Z (cid:96) (cid:109) AB : = (cid:20) ( (cid:96) − ) ! ( (cid:96) + ) ! (cid:21) / (cid:2) D A D B + (cid:96) ( (cid:96) + ) Ω AB (cid:3) Y (cid:96) (cid:109) = √ (cid:16) − Y (cid:96) (cid:109) m A m B + Y (cid:96) (cid:109) ¯ m A ¯ m B (cid:17) . (116b)for the even-parity sector and X (cid:96) (cid:109) A : = − (cid:2) (cid:96) ( (cid:96) + )] − / ε AB D B Y (cid:96) (cid:109) = − i √ (cid:16) − Y (cid:96) (cid:109) m A + Y (cid:96) (cid:109) ¯ m A (cid:17) (117a) X (cid:96) (cid:109) AB : = − (cid:20) ( (cid:96) − ) ! ( (cid:96) + ) ! (cid:21) / ε ( AC D B ) D C Y (cid:96) (cid:109) = − i √ (cid:16) − Y (cid:96) (cid:109) m A m B − Y (cid:96) (cid:109) ¯ m A ¯ m B (cid:17) . (117b)for the odd-parity sector. Here, m A = √ [ , i sin θ ] and ¯ m A form a complex orthonor-mal basis on the two-sphere and are related to the two-sphere components of thetetrad vectors m α and ¯ m α by a factor of r . The differential operator D is the covari-ant derivate on the two-sphere with metric Ω AB = diag ( , sin θ ) .For simplicity we opt to work with the Carter tetrad spin-weighted harmonicbasis exclusively for the remainder of this discussion, but point out that equivalentresults hold for a scalar-vector-tensor basis. In particular, the expressions that followcan be transformed to the commonly-used Barack-Lousto-Sago basis [206] usingthe relations h (cid:96) (cid:109) ll = r f (cid:104) h ( ) (cid:96) (cid:109) + h ( ) (cid:96) (cid:109) (cid:105) , (118a) h (cid:96) (cid:109) nn = r f (cid:104) h ( ) (cid:96) (cid:109) − h ( ) (cid:96) (cid:109) (cid:105) , (118b) h (cid:96) (cid:109) ln = r h ( ) (cid:96) (cid:109) , (118c) h (cid:96) (cid:109) m ¯ m = r h ( ) (cid:96) (cid:109) , (118d) h (cid:96) (cid:109) lm = − r √ f (cid:112) (cid:96) ( (cid:96) + ) (cid:104) h ( ) (cid:96) (cid:109) + h ( ) (cid:96) (cid:109) − i ( h ( ) (cid:96) (cid:109) + h ( ) (cid:96) (cid:109) ) (cid:105) , (118e) h (cid:96) (cid:109) l ¯ m = r √ f (cid:112) (cid:96) ( (cid:96) + ) (cid:104) h ( ) (cid:96) (cid:109) + h ( ) (cid:96) (cid:109) + i ( h ( ) (cid:96) (cid:109) + h ( ) (cid:96) (cid:109) ) (cid:105) , (118f) h (cid:96) (cid:109) nm = − r √ f (cid:112) (cid:96) ( (cid:96) + ) (cid:104) h ( ) (cid:96) (cid:109) − h ( ) (cid:96) (cid:109) − i ( h ( ) (cid:96) (cid:109) − h ( ) (cid:96) (cid:109) ) (cid:105) , (118g) h (cid:96) (cid:109) n ¯ m = r √ f (cid:112) (cid:96) ( (cid:96) + ) (cid:104) h ( ) (cid:96) (cid:109) − h ( ) (cid:96) (cid:109) + i ( h ( ) (cid:96) (cid:109) − h ( ) (cid:96) (cid:109) ) (cid:105) , (118h) h (cid:96) (cid:109) mm = r (cid:112) ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) (cid:104) h ( ) (cid:96) (cid:109) − ih ( ) (cid:96) (cid:109) (cid:105) , (118i) h (cid:96) (cid:109) ¯ m ¯ m = r (cid:112) ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) (cid:104) h ( ) (cid:96) (cid:109) + ih ( ) (cid:96) (cid:109) (cid:105) . (118j)Note that a trace reversal in the Barack-Lousto-Sago basis corresponds to the in-terchange h ( ) (cid:96) (cid:109) ↔ h ( ) (cid:96) (cid:109) , consistent with the trace reversal in the null tetrad basiscorresponding to the interchange h (cid:96) (cid:109) ln ↔ h (cid:96) (cid:109) m ¯ m . Other common choices of scalar-vector-tensor basis are described in Refs. [116, 190] and can be transformed to thespin-weighted harmonic basis using similar relations. The Regge-Wheeler formalism is based on the idea of constructing solutions tothe linearised Einstein equations from solutions to the scalar wave equation with apotential. In the case of the Regge-Wheeler master function, it is a solution of (cid:20) (cid:3) + Ms r (cid:21) ψ RW s = S s , (119)where s is the spin of the field ( s = s = s = ψ RW s = ∞ ∑ (cid:96) = (cid:96) ∑ m = − (cid:96) r ψ RW s (cid:96) (cid:109) ( t , r ) Y (cid:96) (cid:109) ( θ , φ ) , (120)with ψ RW s (cid:96) (cid:109) ( t , r ) satisfying the Regge-Wheeler equation, (cid:20) ∂∂ r (cid:18) f ∂∂ r (cid:19) − f ∂∂ t − (cid:18) (cid:96) ( (cid:96) + ) r + M ( − s ) r (cid:19)(cid:21) ψ RW s (cid:96) (cid:109) = S RW s (cid:96) (cid:109) . (121)In order to study metric perturbations of Schwarzschild spacetime, we considerthe s = ψ RW2 (cid:96) (cid:109) : = fr (cid:20) r h (cid:96) (cid:109) r , odd (cid:112) (cid:96) ( (cid:96) + ) − r ∂ r h (cid:96) (cid:109) , odd (cid:112) ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) (cid:21) . (122)It satisfies the s = S RW2 (cid:96) (cid:109) = − π (cid:20) f T (cid:96) (cid:109) r , odd (cid:112) (cid:96) ( (cid:96) + ) − r ∂ r ( f T (cid:96) (cid:109) , odd ) (cid:112) ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) (cid:21) . (123)Rather than working with the Regge-Wheeler master function itself, it is of-ten preferable to introduce two closely related functions: the Cunningham-Price-Moncrief (CPM) master function defined by ψ CPM (cid:96) (cid:109) : = r ( (cid:96) − ) (cid:112) (cid:96) ( (cid:96) + )( (cid:96) + ) (cid:20) ∂ t h (cid:96) (cid:109) r , odd − ∂ r h (cid:96) (cid:109) t , odd + r h (cid:96) (cid:109) t , odd (cid:21) , (124)and the Zerilli-Moncrief (ZM) master function defined by ψ ZM (cid:96) (cid:109) : = r (cid:96) ( (cid:96) + ) (cid:20) ˜ K (cid:96) (cid:109) + Λ ( f ˜ h (cid:96) (cid:109) rr − r f ∂ r ˜ K (cid:96) (cid:109) ) (cid:21) , (125)where Λ : = ( (cid:96) − )( (cid:96) + ) + Mr and To maintain contact with the existing literature, here and through the rest of this section wemake use of a coordinate basis based on δ α t and δ α r rather than the null tetrad basis. It is, however,straightforward to translate between the t – r based components and l – n based components using thedefinition of the tetrad, e.g. h (cid:96) (cid:109) t = (cid:112) f / ( h (cid:96) (cid:109) l + h (cid:96) (cid:109) n ) and h (cid:96) (cid:109) r = / √ f ( h (cid:96) (cid:109) l − h (cid:96) (cid:109) n ) gives thetranslation of the vector sector to Carter tetrad components. We do, however, continue to maintainthe factor of 1 / r in the vectors m µ and ¯ m µ spanning the two-sphere so the expressions here maydiffer from those given elsewhere by appropriate factors of r ˜ K (cid:96) (cid:109) : = h (cid:96) (cid:109) m ¯ m − f h (cid:96) (cid:109) r , even (cid:112) (cid:96) ( (cid:96) + ) + (cid:104) (cid:96) ( (cid:96) + ) − r f ∂ r (cid:105) h (cid:96) (cid:109) , even (cid:112) ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) , (126a)˜ h (cid:96) (cid:109) rr : = h (cid:96) (cid:109) rr − ∂ r ( r h (cid:96) (cid:109) r , even ) (cid:112) (cid:96) ( (cid:96) + ) + ∂ r (cid:0) r ∂ r h even2 (cid:1)(cid:112) ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) (126b)are gauge invariant fields.The CPM master function satisfies the same s = S CPM (cid:96) (cid:109) = − r ( (cid:96) − ) (cid:112) (cid:96) ( (cid:96) + )( (cid:96) + ) (cid:20) ∂ t ( r T (cid:96) (cid:109) r , odd ) − ∂ r ( r T (cid:96) (cid:109) t , odd ) (cid:21) . (127)The two master functions are related by a time derivative (plus source terms), ψ RW2 (cid:96) (cid:109) = ∂ t ψ CPM (cid:96) (cid:109) + f r T (cid:96) (cid:109) r , odd ( (cid:96) − ) (cid:112) (cid:96) ( (cid:96) + )( (cid:96) + ) . (128)The ZM master function satisfies the Zerilli equation (the Regge-Wheeler equa-tion with a different potential), (cid:20) ∂∂ r (cid:18) f ∂∂ r (cid:19) − f ∂∂ t − V ZM (cid:21) ψ ZM (cid:96) (cid:109) = S ZM (cid:96) (cid:109) , (129)where V ZM = (cid:96) ( (cid:96) + ) r − Mr + M f Λ r − M ( r − M ) Λ r , (130)and where the ZM source is S ZM (cid:96) (cid:109) = f Λ π (cid:112) (cid:96) ( (cid:96) + ) T (cid:96) (cid:109) r , even − π √ (cid:112) ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) r T (cid:96) (cid:109) , even + (cid:96) ( (cid:96) + ) Λ (cid:110)(cid:104) r Λ (cid:16) ( (cid:96) − )( (cid:96) + )( l + l − ) + ( (cid:96) + (cid:96) − ) Mr + M r (cid:17) − r f ∂ r (cid:105) ( f T (cid:96) (cid:109) rr − f − T (cid:96) (cid:109) tt ) + M Λ f T (cid:96) (cid:109) rr + r f T (cid:96) (cid:109) m ¯ m (cid:111) . (131) Transforming to the frequency domain, the Regge-Wheeler and Zerilli equationsbecome a set of ordinary differential equations, one for each ( (cid:96), (cid:109) , ω ) mode. So-lutions to these equations may be written in terms of a pair of homogeneous modebasis functions chosen according to their asymptotic behavior at the four null bound-aries to the spacetime. For radiative ( ω (cid:54) =
0) modes, the four common choices aredenoted “in”, “up”, “out” and “down”, with the same interpretation as described in Sec. 4.1.7 for the Teukolsky equation. These have asymptotic behaviour given by s X in (cid:96) (cid:109) ω ( r ) ∼ (cid:110) s X in,ref (cid:96) (cid:109) ω e + i ω r ∗ ++ s X in,trans (cid:96) (cid:109) ω e − i ω r ∗ s X in,inc (cid:96) (cid:109) ω e − i ω r ∗ r → Mr → ∞ (132a) s X up (cid:96) (cid:109) ω ( r ) ∼ (cid:110) s X up,inc (cid:96) (cid:109) ω e + i ω r ∗ s X up,trans (cid:96) (cid:109) ω e + i ω r ∗ ++ s X up,ref (cid:96) (cid:109) ω e − i ω r ∗ r → Mr → ∞ (132b) s X out (cid:96) (cid:109) ω ( r ) ∼ (cid:110) s X out,trans (cid:96) (cid:109) ω e + i ω r ∗ s X out,inc (cid:96) (cid:109) ω e + i ω r ∗ ++ s X out,ref (cid:96) (cid:109) ω e − i ω r ∗ r → Mr → ∞ (132c) s X down (cid:96) (cid:109) ω ( r ) ∼ (cid:110) s X down,ref (cid:96) (cid:109) ω e + i ω r ∗ ++ s X down,inc (cid:96) (cid:109) ω e − i ω r ∗ s X down,trans (cid:96) (cid:109) ω e − i ω r ∗ r → Mr → ∞ (132d)where r ∗ = r + M ln ( r M − ) is the Regge-Wheeler tortoise coordinate.Inhomogeneous solutions of the Regge-Wheeler equation can then be written interms of a linear combination of the basis functions, s ψ (cid:96) (cid:109) ω ( r ) = s C in (cid:96) (cid:109) ω ( r ) s X in (cid:96) (cid:109) ω ( r ) + s C up (cid:96) (cid:109) ω ( r ) s X up (cid:96) (cid:109) ω ( r ) , (133)where the weighting coefficients are determined by variation of parameters, s C in (cid:96) (cid:109) ω ( r ) = (cid:90) ∞ r s X up (cid:96) (cid:109) ω ( r (cid:48) ) W ( r (cid:48) ) f s S (cid:96) (cid:109) ω ( r (cid:48) ) dr (cid:48) , (134a) s C up (cid:96) (cid:109) ω ( r ) = (cid:90) r M s X in (cid:96) (cid:109) ω ( r (cid:48) ) W ( r (cid:48) ) f s S (cid:96) (cid:109) ω ( r (cid:48) ) dr (cid:48) , (134b)with W ( r ) = s X in (cid:96) (cid:109) ω ( r ) ∂ r [ s X up (cid:96) (cid:109) ω ( r )] − s X up (cid:96) (cid:109) ω ( r ) ∂ r [ s X in (cid:96) (cid:109) ω ( r )] the Wronskian [inpractice, it is convenient to use the fact that f ( r ) W ( r ) = const]. Homogeneous solutions to the Zerilli equation can be obtained from homogeneoussolutions to the Regge-Wheeler equation by applying differential operators, X ZM,up (cid:96) (cid:109) ω = (cid:104) ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) + M fr Λ (cid:105) X RW,up (cid:96) (cid:109) ω + M f d X RW,up (cid:96) (cid:109) ω dr ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) + i ω M , (135a) X ZM,in (cid:96) (cid:109) ω = (cid:104) ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) + M fr Λ (cid:105) X RW,in (cid:96) (cid:109) ω + M f d X RW,in (cid:96) (cid:109) ω dr ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) − i ω M . (135b)The constant of proportionality here is such that the transmission coefficients of thetwo Zerilli solutions is the same as that of the Regge-Wheeler solution. The modes of the CPM master function are related to the modes of the Teukolskyradial function by the Chandrasekhar transformation, ψ (cid:96) (cid:109) ω = − iD r D † D † (cid:0) r ψ CPM (cid:96) (cid:109) ω (cid:1) , (136a) − ψ (cid:96) (cid:109) ω = − iD r f DD (cid:0) r ψ CPM (cid:96) (cid:109) ω (cid:1) , (136b)where D = (cid:112) ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) is the Schwarzschild limit of the constant thatappears in the Teukolsky-Starobinsky identities, Eq. (103b). In the absence ofsources this can be inverted to give ψ CPM (cid:96) (cid:109) ω = D C (cid:96) (cid:109) ω r D † D † (cid:0) r − ψ (cid:96) (cid:109) ω (cid:1) , (137) ψ CPM (cid:96) (cid:109) ω = D C † (cid:96) (cid:109) ω r f − D r f D (cid:0) r f ψ (cid:96) (cid:109) ω (cid:1) , (138)where C (cid:96) (cid:109) ω = D − iM ω is the Schwarzschild limit of the second constant thatappears in the Teukolsky-Starobinsky identities, Eq. (103a). As in the radiation gauge case, the gravitational wave strain can be determined di-rectly from ψ ZM and ψ CPM . There is a slight subtlety in that the Regge-Wheeler-Zerilli gauge in which the metric is typically reconstructed is not compatible withthe transverse-traceless gauge in which gravitational waves are normally defined (itis easy to see this since h mm = = h ¯ m ¯ m in the Regge-Wheeler-Zerilli gauge). In-stead, we can use the Chandrasekhar transformation in Eq. (136) to first transformto ψ and then compute the strain from that as we did in radiation gauge. Doing sowe have r ( h + − ih × ) = ∞ ∑ (cid:96) = (cid:96) ∑ m = − (cid:96) D ( ψ ZM (cid:96) (cid:109) − i ψ CPM (cid:96) (cid:109) ) − Y (cid:96) (cid:109) ( θ , φ ) , (139)where it is understood that equality holds in the limit r → ∞ (at fixed u = t − r ∗ ).If we work in the frequency domain and compute the “in” and “up” mode func-tions with normalisation such that transmission coefficients are unity, s X in,trans (cid:96) (cid:109) ω = = s X up,trans (cid:96) (cid:109) ω , then ψ ZM (cid:96) (cid:109) and ψ CPM (cid:96) (cid:109) are given by the “up” weighting coefficients C ZM,up (cid:96) (cid:109) ω and C CPM,up (cid:96) (cid:109) ω evaluated in the limit r → ∞ . Similarly, the time averaged fluxof energy carried by gravitational waves passing through infinity and the horizoncan be computed from the “in” and “up” weighting coefficients, F H E = lim r → M ∑ (cid:96) (cid:109) ω D π ω (cid:20) | C ZM,in (cid:96) (cid:109) ω | + | C CPM,in (cid:96) (cid:109) ω | (cid:21) , (140a) F I E = lim r → ∞ ∑ (cid:96) (cid:109) ω D π ω (cid:20) | C ZM,up (cid:96) (cid:109) ω | + | C CPM,up (cid:96) (cid:109) ω | (cid:21) . (140b)Similarly, the flux of angular momentum through infinity and the horizon can becomputed from the “in” and “up” normalization coefficients, F H L z = lim r → M ∑ (cid:96) (cid:109) ω D π (cid:109) ω (cid:20) | C ZM,in (cid:96) (cid:109) ω | + | C CPM,in (cid:96) (cid:109) ω | (cid:21) , (141a) F I L z = lim r → ∞ ∑ (cid:96) (cid:109) ω D π (cid:109) ω (cid:20) | C ZM,up (cid:96) (cid:109) ω | + | C CPM,up (cid:96) (cid:109) ω | (cid:21) . (141b) Much like in the Teukolsky formalism, the CPM master function is gauge invariantand may be used to reconstruct the metric perturbation in a chosen gauge. In theRegge-Wheeler gauge, defined by the choice h (cid:96) (cid:109) a , even = h (cid:96) (cid:109) , even = h (cid:96) (cid:109) , odd =
0, the oddparity metric perturbation is given by h (cid:96) (cid:109) l , odd = √ f (cid:112) (cid:96) ( (cid:96) + ) √ r (cid:0) ∂ r + f − ∂ t (cid:1)(cid:0) r ψ CPM (cid:96) (cid:109) (cid:1) + π r ( (cid:96) − )( (cid:96) + ) T (cid:96) (cid:109) l , odd , (142) h (cid:96) (cid:109) n , odd = √ f (cid:112) (cid:96) ( (cid:96) + ) √ r (cid:0) ∂ r − f − ∂ t (cid:1)(cid:0) r ψ CPM (cid:96) (cid:109) (cid:1) + π r ( (cid:96) − )( (cid:96) + ) T (cid:96) (cid:109) n , odd , (143)and the even parity metric perturbation is given by [91] h (cid:96) (cid:109) m ¯ m = f ∂ r ψ ZM (cid:96) (cid:109) + A ψ ZM (cid:96) (cid:109) − π r (cid:96) ( (cid:96) + ) Λ T (cid:96) (cid:109) tt , h (cid:96) (cid:109) rr = Λ f (cid:20) (cid:96) ( (cid:96) + ) r ψ ZM (cid:96) (cid:109) − h (cid:96) (cid:109) m ¯ m (cid:21) + rf ∂ r h (cid:96) (cid:109) m ¯ m , h (cid:96) (cid:109) tr = r ∂ t ∂ r ψ ZM (cid:96) (cid:109) + rB ∂ t ψ ZM (cid:96) (cid:109) + π r (cid:96) ( (cid:96) + ) (cid:20) T (cid:96) (cid:109) tr − rf Λ ∂ t T (cid:96) (cid:109) tt (cid:21) , h (cid:96) (cid:109) tt = f h rr + π f (cid:112) ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) T (cid:96) (cid:109) , even , (144)where A ( r ) : = r Λ (cid:20) ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) + M r (cid:16) ( (cid:96) − )( (cid:96) + ) + Mr (cid:17)(cid:21) , (145) B ( r ) : = r f Λ (cid:20) ( (cid:96) − )( (cid:96) + ) (cid:18) − Mr (cid:19) − M r (cid:21) . (146) As in the Teukolsky case, in order to transform back to the time-domain solution,as a final step we must perform an inverse Fourier transform. This poses a challengein gravitational self-force calculations, where non-smoothness of the solutions inthe vicinity of the worldline lead to the Gibbs phenomenon of non-convergence ofthe inverse Fourier transform. Resolutions to this problem typically rely on avoidingdirectly transforming the inhomogeneous solution by using the methods of extendedhomogeneous or extended particular solutions. For further details, see [91, 92].
In the case of perturbations of a Schwarzschild black hole, the equations for the met-ric perturbation itself are separable. This makes it practical to work in the Lorenzgauge and to directly solve the Lorenz gauge field equations for the metric pertur-bation.Rewriting the Lorenz gauge condition, Eq. (11), in terms of null tetrad compo-nents we have four gauge equations, ( Þ (cid:48) − ρ (cid:48) ) h ll + ( Þ − ρ ) h m ¯ m − ρ h ln − ( ð h l ¯ m + ð (cid:48) h lm ) = , (147a) ( Þ − ρ ) h nn + ( Þ (cid:48) − ρ (cid:48) ) h m ¯ m − ρ (cid:48) h ln − ( ð (cid:48) h nm + ð h n ¯ m ) = , (147b) ( Þ (cid:48) − ρ (cid:48) ) h lm + ( Þ − ρ ) h nm − ð h ln − ð (cid:48) h mm = , (147c) ( Þ (cid:48) − ρ (cid:48) ) h l ¯ m + ( Þ − ρ ) h n ¯ m − ð (cid:48) h ln − ð h ¯ m ¯ m = . (147d)These decouple into 3 even parity equations (the first two and the real part of ei-ther the third or fourth) and 1 odd-parity equation (the imaginary part of either thethird or fourth equation). Similarly, the Lorenz gauge linearised Einstein equation,Eq. (12), yields ten field equations (7 even and 3 odd) given by ˆ (cid:3) ( h m ¯ m − h ln ) = π T , (148a) ( ˆ (cid:3) − ψ + ρρ (cid:48) )( h ln + h m ¯ m ) + ρ h nn + ρ (cid:48) h ll + ρ ( ð h n ¯ m + ð (cid:48) h nm ) + ρ (cid:48) ( ð h l ¯ m + ð (cid:48) h lm ) = − π ( T ln + T m ¯ m ) , (148b) ( ˆ (cid:3) + ρρ (cid:48) ) h ll + ρ ( h ln + h m ¯ m ) + ρ ( ð h l ¯ m + ð (cid:48) h lm ) = − π T ll , (148c) ( ˆ (cid:3) (cid:48) + ρρ (cid:48) ) h nn + ρ (cid:48) ( h ln + h m ¯ m ) + ρ (cid:48) ( ð (cid:48) h nm + ð h n ¯ m ) = − π T nn , (148d) ( ˆ (cid:3) − ψ + ρρ (cid:48) ) h lm + ρ h nm + ρ ð ( h ln + h m ¯ m )+ ρ (cid:48) ð h ll + ρ ð (cid:48) h mm = − π T lm , (148e) ( ¯ˆ (cid:3) − ψ + ρρ (cid:48) ) h l ¯ m + ρ h n ¯ m + ρ ð (cid:48) ( h ln + h m ¯ m )+ ρ (cid:48) ð (cid:48) h ll + ρ ð h ¯ m ¯ m = − π T l ¯ m , (148f) ( ¯ˆ (cid:3) (cid:48) − ψ + ρρ (cid:48) ) h nm + ρ (cid:48) h lm + ρ (cid:48) ð ( h ln + h m ¯ m )+ ρ ð h nn + ρ (cid:48) ð (cid:48) h mm = − π T nm , (148g) ( ˆ (cid:3) (cid:48) − ψ + ρρ (cid:48) ) h n ¯ m + ρ (cid:48) h l ¯ m + ρ (cid:48) ð (cid:48) ( h ln + h m ¯ m )+ ρ ð (cid:48) h nn + ρ (cid:48) ð h ¯ m ¯ m = − π T n ¯ m , (148h)ˆ (cid:3) h mm + ρ ð h nm + ρ (cid:48) ð h lm = − π T mm , (148i)¯ˆ (cid:3) h ¯ m ¯ m + ρ ð (cid:48) h n ¯ m + ρ (cid:48) ð (cid:48) h l ¯ m = − π T ¯ m ¯ m (148j)where the operatorsˆ (cid:3) : = − ÞÞ (cid:48) + ρ (cid:48) Þ + ρ Þ (cid:48) + ðð (cid:48) , ˆ (cid:3) (cid:48) : = − Þ (cid:48) Þ + ρ Þ (cid:48) + ρ (cid:48) Þ + ð (cid:48) ð , ¯ˆ (cid:3) : = − ÞÞ (cid:48) + ρ (cid:48) Þ + ρ Þ (cid:48) + ð (cid:48) ð , ¯ˆ (cid:3) (cid:48) : = − Þ (cid:48) Þ + ρ Þ (cid:48) + ρ (cid:48) Þ + ðð (cid:48) , all coincide with the scalar wave operator when acting on type { , } objects (butdiffer when acting on objects of generic GHP type). Note that we have chosen hereto work with the non-trace-reversed metric perturbation; equivalent equations forthe trace-reversed perturbation can be obtained by noting that a trace-reversal cor-responds to the interchange h ln ↔ h m ¯ m .The Lorenz gauge equations can be decomposed into the same basis of spin-weighted spherical harmonics as for the metric perturbation itself. The mode de-composed equations follow immediately from the above GHP expressions alongwith Eqs. (114) and either (112) or (113) for the GHP derivative operators (the spe-cific form for the mode decomposed equations depends on the choice of tetrad). Following a procedure much like in the Regge-Wheeler and Teukolsky cases, onecan construct solutions to the Lorenz gauge equations by working in the frequencydomain and solving ordinary differential equations. The only additional complexityis that for each ( (cid:96), (cid:109) , ω ) mode we must now work with a system of k coupled sec- ond order radial equations with 2 k linearly independent homogeneous solutions. As we did in the Regge-Wheeler and Teukolsky cases, it is natural to divide theseinto k “in” solutions and k “up” solutions satisfying appropriate boundary conditionsat the horizon or radial infinity. Then, using variation of parameters the inhomoge-neous solutions are given by h ( i ) (cid:96) (cid:109) ω ( r ) = C in (cid:96) (cid:109) ω ( r ) · h ( i ) , in (cid:96) (cid:109) ω ( r ) + C up (cid:96) (cid:109) ω ( r ) · h ( i ) , up (cid:96) (cid:109) ω ( r ) (149)where i = , . . . , k represent the k components of the metric perturbation and where h ( i ) , in (cid:96) (cid:109) ω ( r ) are vectors of k linearly independent homogeneous solutions for a given i .To compute the weighting coefficient vectors C in / up (cid:96) (cid:109) ω ( r ) we define a 2 k × k matrixof homogeneous solutions by Φ ( r ) = − h ( i ) , in (cid:96) (cid:109) ω h ( i ) , up (cid:96) (cid:109) ω − ∂ r h ( i ) , in (cid:96) (cid:109) ω ∂ r h ( i ) , up (cid:96) (cid:109) ω . (150)The vectors of weighting coefficients are then obtained with the standard variationof parameters prescription: (cid:18) C in ( r ) C up ( r ) (cid:19) = (cid:90) Φ − ( r (cid:48) ) (cid:18) ( r (cid:48) ) (cid:19) dr (cid:48) , (151)where T ( r (cid:48) ) represents the vector of k sources constructed from the components ofthe stress energy tensor projected onto the basis and decomposed into modes. Thelimits on the integral depend upon whether the “in” or “up” weighting coefficient arebeing solved for, in the same way as for the Regge-Wheeler and Teukolsky cases.As in the Regge-Wheeler and Teukolsky cases, in order to transform back tothe time-domain solution, as a final step we must perform an inverse Fourier trans-form. This poses a challenge in gravitational self-force calculations, where non-smoothness of the solutions in the vicinity of the worldline lead to the Gibbs phe-nomenon of non-convergence of the inverse Fourier transform. Resolutions to thisproblem typically rely on avoiding directly transforming the inhomogeneous solu-tion by using the methods of extended homogeneous or extended particular solu-tions. For further details, see [91, 92]. As an alternative to directly solving the 7 + There are k = k = + + ticular solutions to the s =
0, 1 and 2 Regge-Wheeler-Zerilli equations, along witha fourth field obtained by solving the s = s = ω = h (cid:96) (cid:109) l , odd = − √ f (cid:112) (cid:96) ( (cid:96) + ) ( i ω ) r (cid:20) r D (cid:18) ψ RW1 (cid:96) (cid:109) r (cid:19) + λ r D (cid:0) r ψ RW2 (cid:96) (cid:109) (cid:1)(cid:21) + π f ( i ω ) ( T (cid:96) (cid:109) l , odd − T (cid:96) (cid:109) n , odd ) , (152) h (cid:96) (cid:109) n , odd = √ f (cid:112) (cid:96) ( (cid:96) + ) ( i ω ) r (cid:20) r D †0 (cid:18) ψ RW1 (cid:96) (cid:109) r (cid:19) + λ r D †0 (cid:0) r ψ RW2 (cid:96) (cid:109) (cid:1)(cid:21) − π f ( i ω ) ( T (cid:96) (cid:109) l , odd − T (cid:96) (cid:109) n , odd ) , (153) h (cid:96) (cid:109) , odd = (cid:112) ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + )( i ω ) r (cid:104) ψ RW1 (cid:96) (cid:109) + f ∂ r (cid:0) r ψ RW2 (cid:96) (cid:109) (cid:1) + λ ψ RW2 (cid:96) (cid:109) (cid:105) + π f ( i ω ) T (cid:96) (cid:109) , odd , (154)where D and D †0 are the operators defined in Eq. (102) specialized to the Schwarzschild( a =
0) case. Equivalent expressions for the even sector are significantly more com-plicated and are given in Appendix A of Ref. [17], while expressions for low multi-poles and ω = As in the Regge-Wheeler-Zerilli and Teukolsky cases, the flux of gravitational waveenergy and angular momentum may be computed from the asymptotic values ofthe fields. In the Lorenz gauge case where one solves for the metric perturbationdirectly, the gravitational wave strain is simply given by h mm as in Eq. (76), r ( h + + ih × ) = r h mm = ∞ ∑ (cid:96) = (cid:96) ∑ (cid:109) = − (cid:96) (cid:90) ∞ − ∞ r h (cid:96) (cid:109) ω mm Y (cid:96) (cid:109) ( θ , φ ) e − i ω ( t − r ∗ ) d ω , (155)where it is understand that the equality holds in the limit r → ∞ . Similarly, theenergy fluxes are given explicitly by [12] F I E = lim r → ∞ ∑ (cid:96) (cid:109) ω ω r π | h (cid:96) (cid:109) ω mm | , (156a) F H E = lim r → M ∑ (cid:96) (cid:109) ω π M ( + M ω ) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:112) ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) r f h (cid:96) (cid:109) ω ll − (cid:112) ( (cid:96) − )( (cid:96) + )( + iM ω ) r (cid:112) f h (cid:96) (cid:109) ω lm + iM ω ( + iM ω ) rh (cid:96) (cid:109) ω mm (cid:12)(cid:12)(cid:12)(cid:12) . (156b)and the angular momentum fluxes are given by F I L z = lim r → ∞ ∑ (cid:96) (cid:109) ω (cid:109) ω r π | h (cid:96) (cid:109) ω mm | , (157a) F H L z = lim r → M ∑ (cid:96) (cid:109) ω (cid:109) π M ω ( + M ω ) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:112) ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) r f h (cid:96) (cid:109) ω ll − (cid:112) ( (cid:96) − )( (cid:96) + )( + iM ω ) r (cid:112) f h (cid:96) (cid:109) ω lm + iM ω ( + iM ω ) rh (cid:96) (cid:109) ω mm (cid:12)(cid:12)(cid:12)(cid:12) . (157b) In the previous section we reviewed black hole perturbation theory with a genericsource term. In this section, we consider how to formulate the source describing asmall object. This is the local problem in self-force theory: In a spacetime perturbedby a small body, what are the sources in the field equations (7) and (8)? Moreover,if the body’s bulk motion is described by an equation of motion (13), what are theforces on the right-hand side?The result of the analysis is (i) a skeletonization of the small body, in whichthe body is reduced to a singularity equipped with the body’s multipole moments,together with (ii) an equation of motion governing the singularity’s trajectory. Thesetting here is very general: the background can be any vacuum spacetime. Ourcoverage of the subject is terse, and we refer to Refs. [148, 156] for detailed reviewsor to Ref. [11] for a non-expert introduction. For simplicity, we assume that outside of the small object, the spacetime is vacuum,and that the perturbations are due solely to the object. Over most of the spacetime,the metric is well described by the external background metric g αβ . However, verynear the object, in a region comparable to the object’s own size, the object’s gravitydominates. In this region, which we call the body zone , the approximation (1) breaksdown.This problem is usually overcome in one of two ways: using effective field the-ory [73] (common in post-Newtonian and post-Minkowskian theory [149]) or usingthe method of matched asymptotic expansions (see, e.g., Refs. [60, 106] for broadintroductions, Refs. [39, 69, 144] for applications in post-Newtonian theory, andRefs. [43, 103, 192, 130, 131, 46, 142, 47, 80, 151, 44, 154, 79, 158] and the re-views [148, 156] for the work most relevant here). Here we adopt the latter approach.We let ε = m / R , where m is the small object’s mass and R is a characteristic lengthscale of the external universe; in a small-mass-ratio binary, R will be the mass M ofthe primary, while in a weak-field binary it can be the orbital separation. We then as-sume Eq. (1), which we dub the outer expansion , is accurate outside the body zone.Near the object, we assume the metric is well approximated by a second expansion,called an inner expansion , that effectively zooms in on the body zone. To make this“zooming in” precise, we first choose some measure, (cid:114) , of radial distance from theobject, with (cid:114) an order-1 function of the external coordinates x α . We then define thescaled distance ˜ (cid:114) : = (cid:114) / ε . The body zone corresponds to (cid:114) ∼ ε R , but to ˜ (cid:114) ∼ R .The outer expansion (1) is an approximation in the limit ε → (cid:114) . The inner expansion is instead an approximation inthe limit ε → (cid:114) , g exact µν ( ˜ (cid:114) , ε ) = g obj µν ( ˜ (cid:114) ) + ε H ( ) µν ( ˜ (cid:114) ) + ε H ( ) µν ( ˜ (cid:114) ) + O ( ε ) . (158)(We suppress other coordinate dependence.) In the body zone, the coefficients g obj µν and H ( n ) µν are order unity. The background metric g obj µν represents the metric of thesmall object’s spacetime as if it were isolated, and the perturbations H ( n ) µν arise fromthe tidal fields of the external universe and nonlinear interactions between those tidalfields and the body’s own gravity.In our construction of the inner expansion, we have assumed that there is onlyone scale that sets the size of the body zone: the object’s mass m . This implicitlyassumes that the object is compact, such that its typical diameter d is comparable to m . That in turn implies that the object’s (cid:96) th multipole moment scales as md (cid:96) ∼ m (cid:96) + = ε (cid:96) + R (cid:96) + . (159)For a noncompact object, we would need to introduce additional perturbation pa-rameters in the outer expansion, and additional scales in the inner expansion. Our inner expansion also assumes that while there is a small length scale asso-ciated with the object, there is no analogous time scale; in other words, the objectis not undergoing changes on its own internal time scale ∼ m . This is equivalentto assuming the object is in quasi-equilibrium with its surroundings. In practice itcorresponds to a spatial derivative near the object dominating over a time deriva-tive by one power of (cid:114) (in the outer expansion) or by one power of ε (in the innerexpansion).To date, inner expansions have been calculated for tidally perturbed Schwarz-schild and Kerr black holes as well as nonrotating or slowly rotating neutron stars;see Refs. [41, 23, 108, 143, 139, 138, 109, 146, 111, 144] for recent examples ofsuch work. These calculations represent one of the major applications of the meth-ods of black hole perturbation theory reviewed in the previous section, and they formpart of an ongoing endeavour to include tidal effects in gravitational-wave templatesand to infer properties of neutron stars from observed signals [62, 209].However, in self-force applications we require only a minimal amount of infor-mation from the inner expansion, often much less than is provided in the abovereferences. The necessary information is extracted from a matching condition : be-cause the two expansions are expansions of the same metric, they must match oneanother when appropriately compared. The most pragmatic formulation of this con-dition is that the inner and outer expansions must commute. If we perform an outerexpansion of the inner expansion (or equivalently, re-expand it for (cid:114) (cid:29) ε R ), and ifwe perform an inner expansion of the outer expansion (or equivalently, expand for (cid:114) (cid:28) R ), and express the end results as functions of r , then both procedures yield adouble series for small ε and small (cid:114) . We assume that these two double expansionsagree with one another, order by order in ε and (cid:114) . A primary consequence of thismatching condition is that near the small object, the metric perturbations in the outerexpansions must behave as h ( n ) µν = h ( n , − n ) µν (cid:114) n + h ( n , − n + ) µν (cid:114) n − + h ( n , − n + ) µν (cid:114) n − + . . . , (160)growing large at small (cid:114) . If h ( n ) µν grew more rapidly (for example, if h ( n ) µν ∼ (cid:114) n + ), thenthe outer expansion could not match an inner expansion. Moreover, the coefficientof (cid:114) n matches a term in the (cid:114) (cid:29) ε R expansion of g obj µν : g obj µν = η µν + ε h ( , − ) µν (cid:114) + ε h ( , − ) µν (cid:114) + ε h ( , − ) µν (cid:114) + . . . , (161)where η µν is the metric of flat spacetime. The terms in this series are in one-to-onecorrespondence with the multipole moments of g obj µν , which in turn can be interpretedas the multipole moments of the object itself. This allows us to write h ( n , − n ) µν in termsof the object’s first n moments; one new moment arises at each new order in ε , justas one would expect from the scaling (159). The moments, together with the general Ref. [145] alerts readers to a significant error in some of the work on slowly rotating bodies.2 form (160), are all we require from the inner expansion. After it is obtained, we caneffectively “integrate out” the body zone from the problem, as described below.To intuitively understand the meanings of the double expansions, and of expres-sions such as (160) and (161), we can interpret them as being valid in the bufferregion ε R (cid:28) (cid:114) (cid:28) R . This region is the large- ˜ (cid:114) limit of the body zone but thesmall- (cid:114) limit of the external universe. To determine more than just the general form of the perturbations, we substituteEq. (160) into the Einstein equations (7)–(8) and then solve order by order in ε and (cid:114) . These types of local calculations are carried out using two tools: covariantnear-coincidence expansions and expansions in local coordinate systems. Ref. [148]contains a thorough, pedagogical introduction to both methods. Here we summarizeonly the basic ingredients.Covariant expansions are based on Synge’s world function, σ ( x α , x α (cid:48) ) = (cid:18) (cid:90) β ds (cid:19) , (162)which is equal to 1/2 the square of the proper distance s (as measured in g µν ) be-tween the points x α (cid:48) and x α along the unique geodesic β connecting the two points;for a given x α (cid:48) , this is a well-defined function of x α so long as x α is within the convexnormal neighbourhood of x α (cid:48) . The other necessary tool is the bitensor g µ (cid:48) µ ( x α , x α (cid:48) ) ,which parallel propagates vectors from x α (cid:48) to x α . A smooth tensor field A µ ν at x α can be expanded around x α (cid:48) as A µ ν ( x α ) = g µ (cid:48) µ g νν (cid:48) (cid:104) A µ (cid:48) ν (cid:48) − A µ (cid:48) ν (cid:48) ; α (cid:48) σ α (cid:48) + A µ (cid:48) ν (cid:48) ; α (cid:48) β (cid:48) σ α (cid:48) σ β (cid:48) + O ( λ ) (cid:105) , (163)where we use λ : = x α (cid:48) and x α . The vector σ α (cid:48) : = ∇ α (cid:48) σ is tangent to β and has a magnitude √ σ equal to the proper dis-tance between x α (cid:48) and x α . The (perhaps unexpected) minus sign in Eq. (163) arisesbecause σ α (cid:48) points away from x α rather than toward it.When a derivative, either at x α or at x α (cid:48) , acts on an expansion like (163), it in-volves derivatives of g µµ (cid:48) and σ α (cid:48) . These can then be re-expanded using, for example, g µµ (cid:48) ; ν = g µρ (cid:48) g ν (cid:48) ν R ρ (cid:48) µ (cid:48) ν (cid:48) δ (cid:48) σ δ (cid:48) + O ( λ ) (164)and σ ; µµ (cid:48) = − g ν (cid:48) µ (cid:16) g µ (cid:48) ν (cid:48) + R µ (cid:48) α (cid:48) ν (cid:48) β (cid:48) σ α (cid:48) σ β (cid:48) (cid:17) + O ( λ ) ; (165)see Eqs. (6.7)–(6.11) of Ref. [148]. To make use of these tools, we install a curve γ , with coordinates z α ( τ ) , in thespacetime of g µν , which will be a representative worldline for the small object. Re-call that τ is proper time as measured in g µν . If the object is a material body, theworldline will be in its physical interior. If the object is a black hole, the worldlinewill only serve as a reference point for the field outside the black hole; mathemati-cally, γ resides in the manifold on which g µν lives, not the manifold on which g obj µν lives. In either case, we only analyze the metric in the object’s exterior, never in itsinterior.A suitable measure of distance from γ is s ( x α , x α (cid:48) ) : = (cid:113) P µ (cid:48) ν (cid:48) σ µ (cid:48) σ ν (cid:48) , (166)where x α (cid:48) = z α ( τ (cid:48) ) is a point on γ near x α , and P µν : = g µν + u µ u ν projects or-thogonally to γ . Here we have introduced γ ’s four-velocity u µ = dz µ d τ , normalizedto g µν u µ u ν = −
1. Note that s remains positive regardless of whether x α (cid:48) and x α are connected by a spacelike, timelike, or null geodesic. In terms of these covariantquantities, the expansion (160) can be written more concretely as h ( n ) µν ( x α ) = g µ (cid:48) µ g ν (cid:48) ν h ( n , − n ) µ (cid:48) ν (cid:48) ( x α (cid:48) , σ α (cid:48) / s ) s n + h ( n , − n + ) µ (cid:48) ν (cid:48) ( x α (cid:48) , σ α (cid:48) / s ) s n − + O ( λ − n + ) . (167) s represents the distance from x α (cid:48) to x α [playing the role of (cid:114) in (160)], and the vec-tor σ α (cid:48) / s represents the direction of the geodesic connecting x α (cid:48) to x α . Generically,log ( s ) terms also appear [153], but we suppress them for simplicity.Rather than directly substituting an ansatz of the form (167) into the vacuum fieldequations and solving for the coefficients h ( n , p ) µ (cid:48) ν (cid:48) , it is typically more convenient toadopt a local coordinate system centred on γ and afterward recover (167). Here weadopt Fermi-Walker coordinates ( τ , x a ) , which are quasi-Cartesian coordinates con-structed from a tetrad ( u α , e α a ) on γ . The spatial triad e α a is Fermi-Walker transportedalong the worldline according to De α a d τ = a a u α , (168)where Dd τ : = u µ ∇ µ . a a : = a µ e µ a is a spatial component of the covariant accelera-tion a µ : = Du µ d τ ; this will eventually become the left-hand side of Eq. (13). At eachvalue of proper time τ , we send a space-filling family of geodesics orthogonallyoutward from ¯ x α = z α ( τ ) , generating a spatial hypersurface Σ τ . Each such surfaceis labelled with a coordinate time τ , and each point on the surface is labeled withspatial coordinates x a = − e a ¯ α σ ¯ α , (169)where σ ¯ α : = ∇ ¯ α σ is tangent to Σ ¯ τ , satisfying σ ¯ α u ¯ α =
0. The magnitude of thesecoordinates, given by s : = (cid:112) δ ab x a x b = (cid:113) g ¯ α ¯ β σ ¯ α σ ¯ β , is the proper distance from ¯ x α to x α . In the special case that x α (cid:48) = ¯ x α , s and s are identical. The analog of Eq. (163)is the coordinate Taylor series A µ ν ( τ , x a ) = A µ ν ( τ , ) + A µ ν , a ( τ , ) x a + A µ ν , ab ( τ , ) x a x b + O ( s ) . (170)In these coordinates, the four-velocity reduces to u µ = ( , , , ) , and the accel-eration to a µ = ( , a i ) . The external background metric, which is smooth at x a = g ττ = − − a i x i − ( R τ i τ j + a i a j ) x i x j + O ( s ) , (171a) g τ a = − R τ ia j x i x j + O ( s ) , (171b) g ab = δ ab − R aib j x i x j + O ( s ) , (171c)reducing to the Minkowski metric on γ , and the only nonzero Christoffel symbolson γ are Γ a ττ = a a and Γ ττ a = a a . If the worldline is not accelerating, the coordinatesbecome inertial along γ .The Riemann tensor components in Eq. (171) are evaluated on the worldline.Higher powers of x a in the expansion come with higher powers of the acceleration,derivatives of the Riemann tensor, and nonlinear combinations of the Riemann ten-sor. In a vacuum background, the Riemann tensor on the worldline is commonlydecomposed into tidal moments. The quadrupolar moments are defined as E ab : = R τ a τ b , (172) B ab : = ε pq ( a R b ) τ pq . (173)Higher moments involve derivatives of the Riemann tensor. Equations (44)–(48)of Ref. [159] display the background metric (171) through order s and the oc-tupolar tidal moments. Ref. [147] presents the background metric in an alterna-tive, lightcone-based coordinate system through order λ and the hexadecapolarmoments.Given the local Fermi-Walker coordinates, one can adopt a coordinate analog ofEq. (167), h ( n ) µν = h ( n , − n ) µν ( τ , n a ) s n + h ( n , − n + ) µν ( τ , n a ) s n − + h ( n , − n + ) µν ( τ , n a ) s n − + O ( s − n + ) . (174)Here n a = x a s = δ ab ∂ b s is a radial unit vector. To facilitate solving the field equations,we can expand the coefficients in angular harmonics: h ( n , p ) µν ( τ , n a ) = ∑ (cid:96) ≥ h ( n , p ,(cid:96) ) µν L ( τ ) ˆ n L , (175)where L : = i · · · i (cid:96) is a multi-index, and ˆ n L : = n (cid:104) L (cid:105) , where n L : = n i · · · n i (cid:96) . The angu-lar brackets denote the symmetric, trace-free (STF) combination of indices, wherethe trace is defined with δ ab . This is equivalent to expanding the coefficients h ( n , p ) µν in scalar spherical harmonics: h ( n , p ) µν ( τ , n a ) = ∞ ∑ (cid:96) = (cid:96) ∑ m = − (cid:96) h ( n , p ,(cid:96) (cid:109) ) µν ( τ ) Y (cid:96) (cid:109) ( ϑ , ϕ ) , (176)where the angles ( ϑ , ϕ ) are defined in the natural way from n a = ( sin ϑ cos ϕ , sin ϑ sin ϕ , cos ϑ ) . (177)Like spherical harmonics, ˆ n L is an eigenfunction of the flat-space Laplacian, satis-fying δ ab ∂ a ∂ b ˆ n L = − (cid:96) ( (cid:96) + ) s ˆ n L . One can further decompose h ( n , p ,(cid:96) ) µν L into irreducibleSTF pieces that are in one-to-one correspondence with the coefficients in a tensorspherical harmonic decomposition. We refer the reader to Appendix A of Ref. [24]for a detailed introduction to such expansions and a collection of useful identities.The general local solution in the buffer region can be found by substituting theexpansions (171) and (174) into the vacuum field equations and working order byorder in ε and s . Because spatial derivatives increase the power of 1 / s , dominatingover τ derivatives, this process reduces to solving a sequence of stationary fieldequations.An alternative approach is to instead solve for the perturbations H ( n ) µν in the innerexpansion, starting with a large- ˜ (cid:114) ansatz complementary to Eq. (160), and thentranslate the results into the small- (cid:114) expansions for h ( n ) µν . This approach can draw onexisting, high-order inner expansions (e.g., Refs. [147, 146, 144]), though doing sooften requires transformations of the coordinates and of the perturbative gauge toarrive at a practical form for the outer expansion (see, e.g., Ref. [158]). The general solutions for h ( ) µν and h ( ) µν in the buffer region are known to varyingorders in ε and (cid:114) in a variety of gauges, including classes of “rest gauges” (termi-nology from Ref. [158]), “P smooth” gauges [79], “highly regular” gauges [158](in which no 1 / s term appears in h ( ) µν ), radiation gauges [162], and the Regge-Wheeler-Zerilli gauge [191]. (In the last two cases, the gauge choices are restrictedto particular classes of external backgrounds.) However, nearly all covariant expres-sions, and expansions to the highest order in (cid:114) , are in the Lorenz gauge. Ref. [153]provides an algorithm for generating the local solution in the Lorenz gauge, and alarge class of similar gauges, to arbitrary order in ε .In all gauges, the general solution is typically divided into two pieces: h ( n ) µν = h S ( n ) µν + h R ( n ) µν . (178) This is akin to the usual split of a general solution into a particular and a homo-geneous solution. h S µν = ∑ n ε n h S ( n ) µν is the object’s self-field , encoding all the localinformation about the object’s multipole structure (including the entirety of g obj µν ).Although this field is defined only outside the object, it would be singular at s = R (cid:29) s >
0; it contains all the negativepowers of s in (174), as well as all non-negative powers of s with finite differentia-bility (e.g., all terms proportional to s p n L with p ≥ p (cid:54) = (cid:96) ). For that reason, itis also known as the singular field .The second piece of the general solution, h R µν = ∑ n ε n h R ( n ) µν , encodes effectively external information linked to global boundary conditions. It takes the form of apower series, h R ( n ) µν = ∑ (cid:96) c ( n ) µν L ( τ ) x L ; (179)unlike h S µν , which involves the locally determined multipole moments, every co-efficient c ( n ) µν L is an unknown that can only be determined when external bound-ary conditions are imposed. Although, once again, the field is defined outside theobject, we can identify ∑ (cid:96) c ( n ) µν L ( τ ) x L with a Taylor series, where the coefficients c ( n ) µν L ( τ ) = (cid:96) ! ∂ L h R ( n ) µν ( τ , ) define h R ( n ) µν and its derivatives on the worldline. More-over, h R µν can be combined with the external background to form an effective metric ˘ g µν : = g µν + h R µν (180)that is a vacuum solution, satisfying G µν [ ˘ g ] = γ . ˘ g µν characterizes the object’s rest frame and local tidal environment.Because h R µν is smooth at x a =
0, it is also referred to as the regular field .This type of division of the local solution into h S µν and h R µν was first emphasizedby Detweiler and Whiting at first order in ε [46, 49]. There is considerable free-dom in the specific division, as smooth vacuum perturbations can be interchangedbetween the two pieces, and multiple distinct choices have been made in practice,particularly beyond linear order [170, 84, 154, 79]. However, one can always choosethe division such that (i) ˘ g µν is a smooth vacuum metric, and (ii) ˘ g µν is effectivelythe “external” metric, in the sense that the object moves as a test body in it, as de-scribed in the next section. Here for concreteness we adopt the choice introduced inRef. [154] (see also Refs. [153, 159, 156]), and we provide the explicit forms of thefirst- and second-order self-fields in the Lorenz gauge, as presented in Ref. [159].For the purpose of explicitly displaying factors of the object’s multipole mo-ments, from this point forward we take ε to be a formal counting parameter that canbe set equal to unity.At first order, the self-field is determined by the object’s mass. It is given inFermi-Walker coordinates by h S ( ) ττ = ms + ma i n i + ms E ab ˆ n ab + O ( s ) , (182a) h S ( ) τ a = ms (cid:16) B bc ε acd ˆ n bd − ˙ a a (cid:17) + O ( s ) , (182b) h S ( ) ab = m δ ab s − m δ ab a i n i + ms (cid:16) E ( ac ˆ n b ) c − E ab − E cd δ ab ˆ n cd (cid:17) + O ( s ) (182c)and in covariant form by h S ( ) µν = m λ s g α (cid:48) µ g β (cid:48) ν (cid:0) g α (cid:48) β (cid:48) + u α (cid:48) u β (cid:48) (cid:1) + m λ s g α (cid:48) µ g β (cid:48) ν (cid:2)(cid:0) s − r (cid:1) a σ ( g α (cid:48) β (cid:48) + u α (cid:48) u β (cid:48) )+ rs a ( α (cid:48) u β (cid:48) ) (cid:3) + λ mg α (cid:48) µ g β (cid:48) ν s (cid:104)(cid:0) r − s (cid:1) (cid:0) g α (cid:48) β (cid:48) + u α (cid:48) u β (cid:48) (cid:1) R u σ u σ − s R α (cid:48) u β (cid:48) u − rs u ( α (cid:48) R β (cid:48) ) u σ u + s ( r + s ) ˙ a ( α (cid:48) u β (cid:48) ) + r ( s − r ) ˙ a σ ( g α (cid:48) β (cid:48) + u α (cid:48) u β (cid:48) ) (cid:105) + O ( λ ) , (183)where x α (cid:48) is an arbitrary point on γ near the field point x α . In the covariant expres-sions we have adopted the notation a σ : = a α (cid:48) σ α (cid:48) , R u σ u σ : = R µ (cid:48) α (cid:48) ν (cid:48) β (cid:48) u µ (cid:48) σ α (cid:48) u ν (cid:48) σ β (cid:48) ,etc. The quantity r : = u µ (cid:48) σ µ (cid:48) is a measure of the proper time between x α (cid:48) and x α .Equations (182) and (183) are given in Ref. [159] through order λ . Equation (4.7)of Ref. [86] presents the covariant expansion of h S ( ) µν through order λ (omittingacceleration terms).At second order, the self-field involves both the mass and spin of the object. Itcan be written as the sum of three pieces, h S ( ) µν = h SS µν + h SR µν + h spin µν . (184)The spin contribution is h spin τ a = S ai n i s + O ( s ) , (185)where other components are O ( s ) , and where S ab = ε abi S i is the spin tensor and S i the spin vector. The other two pieces are either quadratic in the mass, h SS ττ = − m s − m E ab ˆ n ab + O ( s ln s ) , (186a) h SS τ a = − m B bc ε acd ˆ n bd + O ( s ln s ) , (186b) h SS ab = m δ ab − m ˆ n ab s + m (cid:16) E c ( a ˆ n b ) c − E cd δ ab ˆ n cd + E cd ˆ n abcd (cid:17) − m E ab ln s + O ( s ln s ) , (186c)or involve products of the mass with the regular field, Ref. [159] further divides h SR µν into two pieces, labeled h SR µν and h δ m µν .8 h SR ττ = − ms (cid:16) h R1 ab ˆ n ab + h R1 ab δ ab + h R1 ττ (cid:17) + O ( s ) , (187a) h SR τ a = − ms (cid:16) h R1 τ b ˆ n ab + h R1 τ a (cid:17) + O ( s ) , (187b) h SR ab = ms (cid:104) h R1 c ( a ˆ n b ) c − δ ab h R1 cd ˆ n cd − (cid:0) h R1 i j δ i j + h R1 ττ (cid:1) ˆ n ab + h R1 ab + δ ab h R1 cd δ cd + δ ab h R1 ττ (cid:105) + O ( s ) . (187c)On the right, the components of h R1 µν are evaluated at s =
0. At order s , h SR µν alsodepends on first derivatives of h R1 µν evaluated at s =
0; at order s , it depends on secondderivatives of h R1 µν evaluated at s =
0; and so on. h S ( ) αβ is given in Fermi coordinatesthrough order s in Appendix D of Ref. [153].In covariant form, these fields are h spin µν = g α (cid:48) µ g β (cid:48) ν u ( α (cid:48) S β (cid:48) ) γ (cid:48) σ γ (cid:48) λ s + O ( λ ) , (188)with S α (cid:48) β (cid:48) : = e a α (cid:48) e b β (cid:48) S ab , h SS µν = m λ s g α (cid:48) µ g β (cid:48) ν (cid:110) s g α (cid:48) β (cid:48) − σ α (cid:48) σ β (cid:48) − r σ ( α (cid:48) u β (cid:48) ) − ( r − s ) u α (cid:48) u β (cid:48) (cid:3)(cid:111) − m g α (cid:48) µ g β (cid:48) ν ln ( λ s ) R α (cid:48) u β (cid:48) u + m λ s g α (cid:48) µ g β (cid:48) ν (cid:26) s g α (cid:48) β (cid:48) (cid:0) r + s (cid:1) R σ u σ u + rs (cid:2) r σ ( α (cid:48) R β (cid:48) ) u σ u + (cid:0) r − s (cid:1) u ( α (cid:48) R β (cid:48) ) u σ u − s R σ ( α (cid:48) β (cid:48) ) u (cid:3) + s R α (cid:48) σβ (cid:48) σ − rs σ ( α (cid:48) R β (cid:48) ) σ u σ − s (cid:0) r − s (cid:1) u ( α (cid:48) R β (cid:48) ) σ u σ + s (cid:0) r + s (cid:1) R α (cid:48) u β (cid:48) u − (cid:2)(cid:0) r − s (cid:1) σ α (cid:48) σ β (cid:48) + r (cid:0) r − s (cid:1) u ( α (cid:48) σ β (cid:48) ) (cid:3) R σ u σ u − (cid:0) r − r s − s (cid:1) u α (cid:48) u β (cid:48) R σ u σ u (cid:27) + O ( λ ln λ ) , (189)and h SR µν = m λ s g α (cid:48) µ g β (cid:48) ν (cid:40) g α (cid:48) β (cid:48) (cid:20) s h R1 µ (cid:48) ν (cid:48) g µ (cid:48) ν (cid:48) − (cid:0) r − s (cid:1) h R1 uu − h R1 σσ − r h R1 u σ (cid:21) + s δ m α (cid:48) β (cid:48) − h R1 α (cid:48) β (cid:48) s + h R1 σ ( α (cid:48) σ β (cid:48) ) + r h R1 σ ( α (cid:48) u β (cid:48) ) − h R1 σσ u α (cid:48) u β (cid:48) − h R1 µ (cid:48) ν (cid:48) g µ (cid:48) ν (cid:48) (cid:104) σ α (cid:48) σ β (cid:48) + r σ ( α (cid:48) u β (cid:48) ) + ( r − s ) u α (cid:48) u β (cid:48) (cid:105) + r h R1 u ( α (cid:48) σ β (cid:48) ) + ( r − s ) h R1 u ( α (cid:48) u β (cid:48) ) + h R1 u σ σ ( α (cid:48) u β (cid:48) ) − h R1 uu σ α (cid:48) σ β (cid:48) (cid:41) + O (cid:0) λ (cid:1) , (190)where δ m αβ = m (cid:16) h R1 αβ + g αβ g µν h R1 µν (cid:17) + mu ( α h R1 β ) µ u µ + m ( g αβ + u α u β ) u µ u ν h R1 µν . (191)The covariant expressions for h SS µν and h SR µν are known through order λ [159] andare available upon request to the authors. The covariant expansion of h spin µν appearsexplicitly here for the first time, but it is known to higher order in λ [117].In this section, we have stated results from the so-called self-consistent expansionof the metric [151]. In this framework, the metric is not expanded in an ordinaryTaylor series in ε . Instead, it takes the form g exact µν ( x α , ε ) = g µν ( x α ) + ε h ( ) µν ( x α , P ) + ε h ( ) µν ( x α , P ) + O ( ε ) , (192)where P represents a list of system parameters: the worldline γ and multipolemoments of the small object, along with any evolving external parameters. If thesmall object is orbiting a black hole that is approximately Kerr, the external param-eters will consist of small, slowly evolving corrections to the black hole’s mass andspin [126]. These parameters all evolve with time in an ε -dependent way, meaningthat Eq. (192) is not a Taylor series; this allowance for ε -dependent coefficients isa hallmark of singular perturbation theory [106]. In the self-force problem, it mustbe allowed in order to construct a uniformly accurate approximation on large timescales [152]. It will lead naturally into the multiscale expansion described in thelater sections of this review.If we use an ordinary Taylor series in place of Eq. (192), then z µ is replacedwith the series expansion z µ ( τ , ε ) = z µ ( τ ) + ε z µ ( τ ) + . . . (referred to as a Gralla-Wald expansion after the authors of Ref. [80]). Here z µ is a geodesic of the externalbackground spacetime, and the local analysis described above is carried out withseries expansions in powers of distance from this geodesic. The acceleration a µ inthis approach is thus set to zero in all the above formulas. The ε dependence of z µ then manifests itself in h S ( ) µν through an additional term, h dipole µν = m i n i ( g µν + u µ u ν ) s + O ( / s ) (193) = g α (cid:48) µ g β (cid:48) ν (cid:34) − m µ (cid:48) σ µ (cid:48) λ s ( g α (cid:48) β (cid:48) + u α (cid:48) u β (cid:48) ) + O ( / λ ) (cid:35) , (194)proportional to a mass dipole moment m α = e α a m a = mz α . m α describes the positionof the object’s center of mass relative to z µ . It appears in the second-order metricperturbation in the outer expansion but in the zeroth-order inner metric, g obj µν . Bysetting m i to zero in the self-consistent expansion, one defines γ to be the centerof mass at this order. A correction to m i generically appears in h ( ) µν and in g obj µν + ε H ( ) µν , and it is likewise set to zero in a self-consistent expansion [154, 158]. Ina Gralla-Wald expansion, m i and corrections to it are allowed to be nonzero; forthat case, h dipole µν is given through order λ in Fermi coordinates in Sec. IVC of Ref. [151] (where m i is denoted M i ). Explicit expressions through order λ , in bothFermi-coordinate and covariant form, are known through order λ [159] and areavailable upon request.In the context of a binary, the small object inspirals, eventually moving very farfrom any initially nearby background geodesic. This causes z µ and higher correc-tions to grow large with time, spelling the breakdown of the Gralla-Wald expansion.For this reason, we have focused on the self-consistent formulation in this review.Refs. [151, 155, 156] provide detailed explications of the relationship between thetwo types of expansions. Along with the local form of the metric perturbations, the Einstein equations deter-mine the motion of the small object and the evolution of its mass and spin. Specifi-cally, if we let γ be the object’s center of mass (by setting the mass dipole momentin h ( ) µν to zero), then the vacuum field equations uniquely determine the first-orderequations of motion [130, 80, 151] D z α d τ = − P αδ (cid:16) h R ( ) δβ ; γ − h R ( ) βγ ; δ (cid:17) u β u γ − m R α βγδ u β S γδ + O ( ε ) (195)and dmd τ = O ( ε ) and DS αβ d τ = O ( ε ) . (196)The first term on the right of Eq. (195) is referred to as the first-order gravita-tional self-force (per unit mass) or as the MiSaTaQuWa force (after the authorsof Refs. [130, 166]); the second term on the right is the Mathisson-Papapetrou spinforce [118, 140].Equation (195) represents the leading correction to geodesic motion for a gravi-tating, extended, compact object. However, these equations are equivalent to thoseof a test body, not in the background or in the physical spacetime but in the effectivemetric ˘ g µν . In particular, Eq. (195) can be rewritten as˘ D z µ d ˘ τ = − m ˘ R α βγδ ˘ u β S γδ + O ( ε ) , (197)where ˘ τ is proper time in ˘ g µν , ˘ Dd ˘ τ : = ˘ u α ˘ ∇ α , ˘ ∇ is a covariant derivative compatiblewith ˘ g µν , and ˘ u µ = dz µ d ˘ τ . This is the equation of motion of a spinning test particle.Similarly, the evolution equations (196) are the equations of a test mass and spin,which are constant and parallel propagated, respectively. For a non-compact object, finite-size effects from higher multipole moments will dominate overself-force effects.1
If we specialize to a spherical, nonspinning object (and set the subleading massdipole moment to zero), the field equations determine the second-order equation ofmotion [154, 158] D z α d τ = − P αµ (cid:16) g µ δ − h R δµ (cid:17)(cid:16) h R δβ ; γ − h R βγ ; δ (cid:17) u β u γ + O ( ε ) (198)and dmd τ = O ( ε ) . This can be rewritten as the geodesic equation in ˘ g µν ,˘ D z µ d ˘ τ = O ( ε ) . (199)See Sec. IIIA of Ref. [155] for the (simple) steps involved in rewriting Eq. (198) asEq. (199).For a generic compact object, the spin and quadrupole moments will both appearin Eq. (198). Although the second-order equations of motion have not been deriveddirectly from the field equations in that case, it is known that at least through thisorder, the motion remains that of a test body in some effective metric [192]. Atleast for a material body, this remains true even in the fully nonlinear setting [84].The spin’s evolution and its contribution to the acceleration through second order,extracted from the nonlinear results for a material body, are given in Eq. (2.11) ofRef. [3].In this section we have again presented results for the self-consistent expansion.In the Gralla-Wald approach, one instead obtains evolution equations for the massdipole moment. Such equations are derived at first order in Refs. [80, 151, 78] andat second order in Ref. [79] (see also Ref. [155], which derives such second-orderequations in a more compact, parametrization-invariant form).We stress that the equations in this section follow directly from the vacuum Ein-stein equations, together with a center-of-mass condition, outside the small object.There is no assumption about the object’s internal composition, nor is there any reg-ularization of singular quantities. We refer to Refs. [46, 171, 44] for variants of theapproach described here and to Refs. [166, 73, 84] for alternatives to the matched-expansions approach. After having derived the local form of the metric, and the equations of motion, wecan effectively remove the body zone from the problem. We do so by allowing thelocal forms (179)–(190) to hold all the way down to γ . This causes the self-field todiverge at γ , artificially introducing a singular field. However, this does not alter thephysics in the buffer region or external universe, and the singularity is more easilyhandled than the small-scale physics of the small object.Once the fields have been extended to γ , one can solve the field equationsthroughout the spacetime using either a puncture scheme or point-particle meth- ods. The puncture scheme is the more general of the two approaches. We define the puncture field h P ( n ) µν : = h S ( n ) µν W (200)as the local expansion of h S ( n ) µν truncated at some order λ k , multiplied by a windowfunction W that is equal to 1 in a neighbourhood of z α and transitions to zero atsome finite distance from z α . This implies that h P ( n ) µν = h S ( n ) µν + O ( λ k + ) . We thendefine the residual field h R ( n ) µν : = h ( n ) µν − h P ( n ) µν , (201)which satisfies h R ( n ) µν = h R ( n ) µν + O ( λ k + ) , making h R ( n ) µν a C k field at γ . Outside thesupport of h P ( n ) µν , h R ( n ) µν becomes identical to the full field h ( n ) µν .Moving h P ( ) µν to the right-hand side of the vacuum field equations, we obtainfield equations for h R ( n ) µν : G ( ) µν [ h R ( ) ] = − G ( ) µν [ h P ( ) ] : = S eff ( ) µν , (202) G ( ) µν [ h R ( ) ] = − G ( ) µν [ h ( ) , h ( ) ] − G ( ) µν [ h P ( ) ] : = S eff ( ) µν . (203)These equations hold at all points off γ . The C k behaviour of the solution is then en-forced by defining the effective sources S eff ( n ) µν as ordinary integrable functions at γ ,rather than treating G ( ) µν [ h P ( n ) ] in the distributional sense of a linear operator actingon an integrable function; this distinction is important to rule out delta functions inthe source, which would create spurious singularities in the residual field.If k ≥
1, then we can replace h R ( n ) µν with h R ( n ) µν in the equations of motion (195) and(198). The total field h ( n ) µν = h R ( n ) µν + h P ( n ) µν is also guaranteed to satisfy the physicalboundary condition in the buffer region (i.e., the matching condition) and at theouter boundaries of the problem.An alternative to the puncture scheme is to solve directly for the total fields h ( n ) µν .Once extended to γ , they satisfy G ( ) µν [ ε h ( ) + ε h ( ) ] + ε G ( ) µν [ h ( ) , h ( ) ] = π T µν + O ( ε ) , (204) Our description may seem (incorrectly) to imply that the puncture field is only defined in aconvex normal neighbourhood of the body. For numerical purposes, the puncture is extended overa region of any convenient size. Typically this is done by converting the local, covariant expressionsin terms of Synge’s world function into expansions in coordinate distance, using, e.g., the Boyer-Lindquist coordinates of the background spacetime. The punctures can then be extended as thesecoordinate functions. The end result for the combined field h ( n ) µν = h R ( n ) µν + h P ( n ) µν is insensitive tothe choice of extension. In the self-consistent approach, some care is required in formulating these equations. Specifi-cally, they can only be split into a sequence of equations, one at each order in ε , after imposinga gauge condition [151]; this is required in order to allow the puncture to move on an acceler-ated trajectory. We do not belabour this point because we ultimately formulate the equations in asomewhat different, multiscale form tailored to binary inspirals.3 where here we do interpret each term on the left-hand side in a distributional sense.The stress-energy tensor is then defined by the left-hand side. Through second or-der, it can be shown to be the stress-energy of a spinning particle in the effectivemetric [43, 80, 151, 153, 193]: T µν = m (cid:90) γ ˘ u µ ˘ u ν ˘ δ ( x , z ( ˘ τ )) d ˘ τ + (cid:90) γ ˘ u ( µ S ν ) α ˘ ∇ α ˘ δ ( x , z ( ˘ τ )) d ˘ τ , (205)where ˘ δ ( x , x (cid:48) ) = δ ( x α − x (cid:48) α ) √− ˘ g and ˘ u µ : = ˘ g µν ˘ u ν . We refer to this point-particle stress-energy as the Detweiler stress-energy after the author of Ref. [44]. Like the equa-tions of motion, the point-particle approximation is a derived consequence of thevacuum Einstein equations and the matching condition, rather than an input.In cases where the point-particle method is well defined, it and the puncturescheme yield identical full fields h ( n ) µν . However, unlike a puncture scheme, a point-particle method does not yield the regular fields h R ( n ) µν as output. The regular fields,and self-forces, must instead be extracted from h ( n ) µν . This is most often done usingthe method of mode-sum regularization [14, 10] reviewed in detail in Refs. [6, 205]and sketched in Sec. 7.2 below.We will refer to both the effective sources S eff ( n ) µν and the point-particle source T µν as skeleton sources . This terminology follows Mathisson’s notion [118, 55] ofa “gravitational skeleton” (see also Refs. [52, 53, 54]): an extended body can berepresented by a singularity equipped with an infinite set of multipole moments.Punctures provide a generalization of this concept to settings where the singulari-ties are too strong to be represented by distributions. For that reason, although thenomenclature of punctures and effective sources originated from methods of solvingthe first-order field equations in Refs. [15, 195], punctures have a more fundamentalrole at second and higher orders [170, 171, 44, 79, 153, 156]. For the same reason,we have presented punctures as a more primitive concept than the point-particlestress-energy.In either approach, the skeleton sources presented here apply equally for all com-pact objects, whether black holes or material bodies. The only distinguishing featureof a material body would be a spin that surpasses the Kerr bound (i.e., | S i | > m ).However, at third order in perturbation theory, the quadrupole moment will appearin the perturbation h ( ) µν . Unlike the mass and dipole moments, the quadrupole mo-ment is not governed by the Einstein equation [54, 84, 85], and its evolution mustbe determined from the object’s equation of state. Hence, at third order the interiorcomposition of the object begins to influence the external metric, and we can beginto distinguish between black holes and material bodies. But note that the quadrupolemoments of compact objects differ primarily due to their differing tidal deformabil-ity, and this difference is suppressed by an additional five powers of ε [23], suggest-ing it is almost certainly irrelevant for small-mass-ratio binaries. At second order, this is true in a class of highly regular gauges. In other gauges, it requiresa direct use of the puncture via a particular distributional definition of the nonlinear quantity G ( ) µν [ h ( ) , h ( ) ] [193].4 The previous section summarized the local problem in self-force theory: the reduc-tion of an extended body to a skeleton source in the Einstein equations, along withan equation of motion for that source. In the remaining sections, we turn to the global problem : solving the perturbative Einstein equations, coupled to the equationof motion (195) or (198), globally in a specific background metric.In the context of a small-mass-ratio binary, the background geometry is the Kerrspacetime of the central black hole. According to the equations of motion, the smallbody in the binary is only slightly accelerated away from geodesic motion in thatbackground. This section summarizes (i) properties of bound geodesic motion inKerr spacetime and (ii) how to exploit those properties to analyze accelerated orbits.We emphasize action-angle methods that mesh specifically with our treatment ofthe Einstein equations in the final section of this review. However, much of ourtreatment is valid for a more generic acceleration.We warn the reader that the notation in this section differs in several ways fromthat of the preceding section. The differences are noted in the first subsubsectionbelow.
Geodesics in Kerr spacetime are integrable, with three constants of motion associ-ated with the spacetime’s three Killing symmetries: (specific) energy E = − u α ξ α ,(specific) azimuthal angular momentum L z = u α δ αφ , and the Carter constant Q = u α u β ( (cid:63)(cid:63) K αβ − a η α η β ) . Inverting these three equations, together with g αβ u α u β = −
1, for the four-velocity components, we obtain [65] Σ (cid:18) drd τ (cid:19) = R ( r ) , (206) Σ (cid:18) dzd τ (cid:19) = Z ( z ) , (207) Σ dtd τ = T r ( r ) + T z ( z ) + aL z : = (cid:102) t , (208) Σ d φ d τ = Φ r ( r ) + Φ z ( z ) − aE : = (cid:102) φ . (209) The constant K = u α u β (cid:63)(cid:63) K αβ is also sometimes referred to as Carter’s constant.5 Here ( t , r , z : = cos θ , φ ) refer to Boyer-Lindquist coordinates, and R ( r ) : = [ P ( r )] − ∆ (cid:2) r + ( aE − L z ) + Q (cid:3) , (210) Z ( z ) : = Q − (cid:0) Q + a γ + L z (cid:1) z + a γ z , (211) T r ( r ) : = r + a ∆ P ( r ) , (212) T z ( z ) : = − a E ( − z ) , (213) Φ r ( r ) : = a ∆ P ( r ) , (214) Φ z ( z ) : = L z − z , (215)with P ( r ) : = E ( r + a ) − aL z and γ : = − E . We opt to use z rather than θ through-out this section.The equations for r ( τ ) and z ( τ ) are coupled, but they are immediately decoupledby adopting a new parameter λ , called Mino time [128], that satisfies d λ d τ = Σ − . (216)(This is not to be confused with the bookkeeping parameter used in the local ex-pansions of the previous section.) The equations also take a hierarchical form: once r ( λ ) and z ( λ ) are known, Eqs. (208) and (209) can be straightforwardly integratedto obtain t ( λ ) and φ ( λ ) .Given this hierarchical form, we will focus on the r – z dynamics. In Eq. (206), R ( r ) is a fourth-order polynomial in r , meaning it can also be written as R ( r ) = − γ ( r − r )( r − r )( r − r )( r − r ) , with r ≥ r ≥ r ≥ r . Similarly, in Eq. (207), Z ( z ) = a γ ( z − z )( z − z ) , with | z | > | z | . For bound orbits, the radial motionoscillates between the turning points r a = r (apoapsis) and r p = r (periapsis),and the polar motion between z max = | z | and z min = −| z | . Hence, the geodesicis confined to a torus-like region r p ≤ r ≤ r a , | z | < z max . If Q =
0, the motion isconfined to the equatorial plane z =
0. If a = z =
0. However, a generic orbit ergodically fillsthe torus-like region.For convenience in the remaining sections, we use lowercase Latin indices fromthe beginning of the alphabet ( a , b , c ) to denote r or z and define xxx = ( r , z ) . However,repeated indices, as in an expression such as f a x a , are not summed over; instead,such sums will be written as fff · xxx : = f r x r + f z x z . An expression such as f a ( x a ) willdenote either one of f r ( r ) or f z ( z ) , while an expression such as f a ( xxx ) will denoteeither one of f r ( r , z ) or f z ( r , z ) . f α ( x β ) will denote f α ( t , r , z , φ ) . Refs. [57, 65] and many other references instead define z as cos θ , with analogous differencesin their definitions of the roots z n defined below. The other roots ( r , r , and z ) do not correspond to physical turning points. In particular, | z | > We use lowercase Latin indices from the middle of the alphabet ( i , j , k ) to la-bel elements of a set of orbital parameters. For example, P i = ( E , L z , Q ) . For theseindices, unlike a , b , c , we use Einstein summation.We use f throughout this section to denote a generic function, not the specificfunction f ( r ) that appears in the Schwarzschild metric (14). An overdot will denotea derivative with respect to λ .Finally, we preemptively refer the reader to Refs. [178, 128, 57, 58, 65, 203, 182]for additional details about geodesic orbits in Kerr. Unlike Keplerian orbits, generic geodesics in Kerr do not close; the periods of ra-dial, polar, and azimuthal motion are all, generically, incommensurate. Neverthe-less, because of their doubly oscillatory nature, it is often useful in applications toexpress the geodesic trajectories in a quasi-Keplerian form, replacing the constants { E , L z , Q } with an alternative set { p , e , z max } . In terms of these, r and z can be writ-ten in the manifestly periodic form [57] r ( ψ r ) = pM + e cos ψ r , (217) z ( ψ z ) = z max cos ψ z , (218)where, for a bound orbit, 0 ≤ e <
1. The phases ( ψ r , ψ z ) are multiples of 2 π atperiapsis and at z = z max , respectively. Unlike r and z , which change direction everyhalf cycle, ψ r and ψ z grow monotonically, leading to better numerical behavior atthe turning points.Because none of the periods are commensurate, ψ r and ψ z evolve independently(of each other and of φ ). Using d ψ a d λ = dx a d λ / dx a d ψ a , one finds [57] d ψ r d λ = M (cid:112) γ [( p − p ) − e ( p + p cos ψ r )][( p − p ) + e ( p − p cos ψ r )] − e : = (cid:102) r , (219) d ψ z d λ = (cid:113) a γ ( z − z cos ψ z ) : = (cid:102) z , (220)where p : = r ( − e ) / M and p : = r ( + e ) / M . These can be integrated subject toarbitrary choices of initial phase ψ a ( ) = ψ a .The parameters { p , e , z max } , unlike { E , L z , Q } , are related directly to the coor-dinate shape of the orbit, specifically to its turning points. Equation (217) is theformula for an ellipse, and it implicitly defines p and e to be the semi-latus rectumand eccentricity of that ellipse, related to the periapsis and apoapsis by r p = pM + e and r a = pM − e . (221) As stated above, z max = z , but to further the analogy with Keplerian orbits, we canalso define an inclination angle ι such that z = z max = sin ι . (222)The remaining roots of R ( r ) and Z ( z ) are also compactly expressed in terms ofthese parameters [65]: r = (cid:16) α + (cid:112) α − β (cid:17) and r = β / r , (223)where α : = M / γ − ( r a + r p ) and β : = a Q / ( γ r a r p ) , and z = (cid:115) Qa γ z . (224)These expressions are in a “mixed” form that involves both sets of constants. How-ever, { E , L z , Q } can be written in terms of { p , e , ι } as [58] E = | d , g , h | − | d , h , f | − χ (cid:112) | d , g , h | + | h , d , g , h , f | + | h , d , h , g , f || f , h | + | f , g , h | , (225) L z = − g p MEh p + M χ (cid:115) g p E h p + f p E − d p h p , (226) Q = z (cid:18) a γ + L z cos ι (cid:19) , (227)where χ : = sgn ( L z ) is + − d ( r ) : = ∆ ( r + z a ) / M , (228) f ( r ) : = ( r / M ) + a [ r ( r + M ) + z ∆ ] / M , (229) g ( r ) : = ar / M , (230) h ( r ) : = [ r ( r − M ) + ∆ tan ι ] / M , (231)and a subscript a or p indicates evaluation at r a or r p . The quantities | · | appearingin E are determinants or products of determinants that we define recursively as | f , g | : = f p g a − f a g p and | f , g , . . . | : = | f , g || g , . . . | .Given the parametrizations (217) and (218) and the equations of motion (208)and (209), t ( λ ) and φ ( λ ) can be written as Ref. [57] and some other authors use the alternative, inequivalent definition cos ι = L z √ L z + Q . Thisdoes not describe the maximum coordinate inclination angle but has other useful properties [97]. Note that r and r have the opposite meaning in Ref. [58] than their meaning here. Our notationfor the roots r n follows Ref. [65].8 t ( λ ) = t + t r ( ψ r ( λ )) + t z ( ψ z ( λ )) + aL z λ , (232) φ ( λ ) = φ + φ r ( ψ r ( λ )) + φ z ( ψ z ( λ )) − aE λ , (233)with t a ( ψ a ) = (cid:90) ψ a ψ a T a ( ψ (cid:48) a ) (cid:102) a ( ψ (cid:48) a ) d ψ (cid:48) a and φ a ( ψ a ) = (cid:90) ψ a ψ a Φ a ( ψ (cid:48) a ) (cid:102) a ( ψ (cid:48) a ) d ψ (cid:48) a . (234)Here t and φ are integration constants.This completes the quasi-Keplerian description of geodesic orbital motion. Tra-jectories are described by the three constants of motion p i : = ( p , e , ι ) and the foursecularly growing phase variables ψ α : = ( t , ψ r , ψ z , φ ) . A given trajectory is uniquelyspecified by the set of seven constants { p , e , ι , t , ψ r , ψ z , φ } , called orbital ele-ments. The solution to the geodesic equation can also be put in closed, analyticalform [65] by expressing ψ α ( λ ) in terms of elliptic integrals and their inverses (theJacobi elliptic functions). It is often essential to decompose quantities on the worldline into Fourier series, par-ticularly when solving the perturbative Einstein equations in the frequency domain.This procedure is expedited by knowing the orbit’s fundamental frequencies. In thissection, we summarize the calculation of frequencies and of phase variables (actionangles) associated with those frequencies. Unlike the phases ψ α , the angle variablesare strictly linear in λ , facilitating Fourier expansions in that time variable.In the right-hand sides of Eqs. (208), (209), (219), and (220), we have definedthe “frequencies” (cid:102) α ( ψψψ ) as the rates of change of ψ α , d ψ α d λ = (cid:102) α ( ψψψ ) . (235)The true frequencies ϒ α associated with λ are the average rates of change of ψ α , ϒ α = (cid:104) (cid:102) α (cid:105) λ , (236)and the corresponding action angles are q α = ϒ α λ + q α , (237)with arbitrary constants q α . For a function f [ r ( λ ) , z ( λ )] on the worldline, the aver-age is defined as (cid:104) f (cid:105) λ : = lim Λ → ∞ Λ (cid:90) Λ − Λ f d λ . (238) For a generic, nonresonant orbit, this average agrees with the torus average (cid:104) f (cid:105) qqq = ( π ) (cid:73) f d q , (239)We use (cid:72) d q to denote (cid:82) π dq r (cid:82) π dq z and (cid:72) d ψ for the analogous integral over ψψψ . To simplify the analysis, we choose our phase space coordinates qqq such that q r vanishes at some periapsis and q z vanishes at some z = z max . We furthermore choose q t , q φ , our spacetime coordinates t and φ , and our parameter λ such that they allvanish at some particular passage through periapsis. These choices, which do notrepresent any loss of generality, imply q α = ψ α = α = t , r , φ , (240a) q z = − ϒ z λ z , (240b)where λ z is the first value of λ at which z = z max . ψ z can be inferred from q z . Onecan easily do without these specifications if desired.With our choices, q a represents the mean growth of ψ a from the first radial orpolar turning point, and we can express it in terms of ψ a as q a ( ψ a ) = ϒ a (cid:90) ψ a d ψ (cid:48) a (cid:102) a ( ψ (cid:48) a ) . (241)This allows us to straightforwardly write the torus average as as an integral over ψψψ , (cid:104) f (cid:105) qqq = Λ r Λ z (cid:73) f d ψ (cid:102) r ( ψ r ) (cid:102) z ( ψ z ) , (242)where Λ a = (cid:82) π d ψ a (cid:102) a ( ψ a ) is the radial or polar period with respect to λ . Althoughthey agree generically, (cid:104) f (cid:105) λ and (cid:104) f (cid:105) qqq differ in the special case of resonant orbits,discussed in later sections.For the r and z motion, the frequencies reduce to ϒ a = π / Λ a , which can beanalytically evaluated to [65] ϒ r = π (cid:112) γ ( r a − r )( r p − r ) K ( k r ) , (243) ϒ z = π (cid:112) a γ z K ( k z ) , (244)where We focus only on functions of r and z , which are automatically periodic functions of the intrinsicphases ψψψ and qqq . The averaging operation immediately generalizes in the natural way to functions f [ z α ( λ )] that are periodic in t and φ .0 K ( x ) : = (cid:90) π / d θ (cid:112) − x sin θ (245)is the complete elliptic integral of the first kind, and its arguments are k r : = r a − r p r a − r r − r r p − r and k z : = ( z max / z ) .The frequencies of t and φ motion can also be found analytically. Because of theadditive forms of dtd λ and d φ d λ in (208) and (209), the averages reduce to a sum ofone-dimensional integrals. Evaluating those integrals leads to [203] ϒ t = E [ r ( r a + r p + r ) − r a r p + ( r a + r p + r + r ) F r + ( r a − r )( r p − r ) G r ]+ Mr + − r − (cid:20) ( M E − aL z ) r + − Ma Er − r + (cid:18) − F + r p − r + (cid:19) − (+ ↔ − ) (cid:21) + M E + EQ ( − G z ) γ z + ME ( r + F r ) , (246) ϒ φ = ar + − r − (cid:20) MEr + − aL z r − r + (cid:18) − F + r p − r + (cid:19) − (+ ↔ − ) (cid:21) + L z Π ( z , k z ) K ( k z ) , (247)where E ( x ) : = (cid:90) π / d θ (cid:112) − x sin θ , (248) Π ( x , y ) : = (cid:90) π / d θ ( − x sin θ ) (cid:112) − y sin θ (249)are the complete elliptic integrals of the second and third kind, respectively. Wehave also introduced G a : = E ( k a ) K ( k a ) and F A : = ( r p − r ) Π ( h A , k r ) K ( k r ) for A = { r , + , −} , with h r = r a − r p r a − r and h ± : = ( r a − r p )( r − r ± )( r a − r )( r p − r ± ) . Here r ± = M ± √ M − a denote the inner andouter horizon radii.In terms of the angle variables, a quantity f ( r , z ) on the worldline can be ex-panded in the Fourier series f [ r ( λ ) , z ( λ )] = ∑ kkk f ( q ) kkk e − iq kkk ( λ ) , (250)where q kkk : = kkk · qqq = k r q r + k z q z , and unless stated otherwise, sums range over kkk ∈ Z .The coefficients are given by f ( q ) kkk = ( π ) (cid:73) f e iq kkk d q , (251)which can also be calculated as f ( q ) kkk = ϒ r ϒ z ( π ) (cid:73) f e iq kkk ( ψψψ ) (cid:102) r ( ψ r ) (cid:102) z ( ψ z ) d ψ (252)with q kkk ( ψψψ ) = k r q r ( ψ r ) + ik z q z ( ψ z ) given by Eq. (241). The torus average of thefunction (and infinite λ average for nonresonant orbits) is the zero mode in theFourier series: (cid:104) f (cid:105) qqq = f ( q ) .Using such Fourier expansions, we can invert Eq. (241) to write the phases ψ α in terms of the angle variables. The transformation q α → ψ α ( q β ) must satisfy ∂ψ α ∂ q β ϒ β = d ψ α d λ = (cid:102) α together with our choice ψ α ( q β = ) =
0. The solution is thesum of a secular and a purely oscillatory piece: ψ α ( q β ) = q α − ∆ ψ α ( ) + ∆ ψ α ( qqq ) , (253)where ∆ ψ a = ∑ k (cid:54) = (cid:102) ka e − ikq a − ik ϒ a , (254) ∆ t = ∆ t r + ∆ t z : = ∑ k (cid:54) = (cid:18) T kr e − ikq r − ik ϒ r + T kz e − ikq z − ik ϒ z (cid:19) , (255) ∆ φ = ∆ φ r + ∆ φ z : = ∑ k (cid:54) = (cid:18) Φ kr e − ikq r − ik ϒ r + Φ kz e − ikq z − ik ϒ z (cid:19) . (256) T a and Φ a are given in Eqs. (212)–(215), and we have written them, along with (cid:102) a ( q a ) , as one-dimensional Fourier series in q a ; e.g., T a = ∑ k T ka e − ikq a . We will con-sistently use ∆ to indicate that a quantity is periodic in qqq with zero average.We can conveniently write the coordinate trajectory in terms of the action anglesas the sum of a secular term and an oscillatory term: z α ( λ ) = z α sec [ q β ( λ )] + ∆ z α [ qqq ( λ )] , (257)where the secular piece is z α sec ( q β ) = ( q t , , , q φ ) + (cid:104) − ∆ t ( ) , r ( q ) , z ( q ) , − ∆ φ ( ) (cid:105) , (258)and the purely oscillatory pieces are given by Eqs. (255), (256), and ∆ x a ( q a ) = ∑ k (cid:54) = x a ( q ) k e − ikq a , (259)with coefficients readily calculated from Eq. (252). Ref. [65] gives x a ( q a ) in closedform in their Eqs. (26) and (38), ∆ t as the sum of their t ( r ) and t ( θ ) in their Eqs. (28)and (39), and ∆ φ as the sum of their φ ( r ) and φ ( θ ) in those same equations. (Wecaution the reader that the notation in Ref. [65] differs from ours in several ways.) Our description here has followed the constructive approach of Refs. [57, 58, 65],finding the frequencies and angle variables by directly solving the geodesic equa-tion. There is an alternative, historically prior approach [178] based on the Hamil-tonian description of geodesics, which builds on Carter’s original proof [32] of in-tegrability using the Hamilton-Jacobi equation. That approach derives action an-gles and their associated fundamental frequencies from a canonical transformation ( z α , u α ) → ( q α , J α ) , where the actions J α are the canonical momenta conjugate tothe action angles q α . For the purpose of decomposing fields, such as the metric perturbation, into Fouriermodes, it is more useful to know the frequencies with respect to coordinate time t .These are the frequencies observed at infinity and that appear in the gravitationalwaveform. They are given by Ω α = ϒ α ϒ t . (260)The angle variables associated with them are ϕ α = Ω α t + ϕ α (261)with Ω t =
1. We choose the origin of this phase space in analogy with Eq. (240): ϕ α = α = t , r , φ , and ϕ z = − Ω z t z , (262)where t z is the first value of t at which z = z max .These new angle variables are related to q α by a transformation that must satisfy ∂ϕ α ∂ q β ϒ β = d ϕ α d λ = Ω α (cid:102) t ( qqq ) . Such a transformation, with our choice of origin ϕ α ( q β = ) =
0, is ϕ α ( q β ) = q α − Ω α ∆ t ( ) + Ω α ∆ t ( qqq ) (263)with ∆ t given by Eq. (255).In analogy with Eq. (250), a function of r and z on the worldline can be expandedin a Fourier series f [ r ( t ) , z ( t )] = ∑ kkk f ( ϕ ) kkk e − i ϕ kkk ( t ) , (264)with ϕ kkk : = kkk · ϕϕϕ = k r ϕ r + k z ϕ z and with coefficients given by the analog of Eq. (251).Using the Jacobian det (cid:16) ∂ϕ a ∂ q b (cid:17) = (cid:102) t / ϒ t , we can also write the coefficients as integralsover qqq , f ( ϕ ) kkk = e − i Ω kkk ∆ t ( ) ( π ) ϒ t (cid:73) (cid:102) t e iq kkk + i Ω kkk ∆ t ( qqq ) f d q , (265)where Ω kkk : = k r Ω r + k z Ω z . Or we can write them as integrals over ψψψ , f ( ϕ ) kkk = ϒ r ϒ z ( π ) ϒ t (cid:73) (cid:102) t ( ψψψ ) e iq kkk ( ψψψ )+ i Ω kkk [ δ t r ( ψ r )+ δ t z ( ψ z )] f (cid:102) r ( ψ r ) (cid:102) z ( ψ z ) d ψ , (266)where q a ( ψ a ) is given by Eq. (241), and δ t a ( ψ a ) : = ∆ t a [ q a ( ψ a )] − ∆ t a ( ) = (cid:90) ψ a T a ( ψ (cid:48) a ) − (cid:104) T a (cid:105) λ (cid:102) a ( ψ (cid:48) a ) d ψ (cid:48) a , (267)with T a given by Eq. (212) and (213). If f is separable [i.e., if it can be written asa sum of products of the form f r ( r ) f z ( z ) ], then expressing the integrals in termsof qqq or ψψψ allows one to evaluate the two-dimensional integral as a product of one-dimensional integrals.We can further define an average over t , (cid:104) f [ r ( t ) , z ( t )] (cid:105) t : = lim T → ∞ T (cid:90) ∞ − ∞ f dt , (268)which for nonresonant orbits is equal to the torus average (cid:104) f (cid:105) ϕϕϕ : = ( π ) (cid:73) f d ϕ = f ( ϕ ) . (269)Note that the meaning of a time average (and associated torus average) inherentlydepends on one’s choice of time parameter [160], and that (cid:104) f (cid:105) t differs from (cid:104) f (cid:105) λ : (cid:104) f (cid:105) t = ϒ t (cid:104) (cid:102) t f (cid:105) λ = (cid:104) f (cid:105) λ + ϒ t ∑ kkk (cid:54) = (cid:102) ( q ) tkkk f ( q ) − kkk . (270)The relevance of each average depends on context.Using these Fourier expansions, we can express the phases ψ α in terms of ϕ α .The two are related by a transformation satisfying ∂ψ α ∂ϕ β Ω β = d ψ α dt . With our choiceof origin ψ α ( ϕ β = ) =
0, the solution is ψ α ( ϕ β ) = ϕ α − ∆ ϕ ψ α ( ) + ∆ ϕ ψ α ( ϕϕϕ ) . (271)The oscillatory terms are ∆ ϕ ψ t = ∆ ϕ ψ α = ∑ kkk (cid:54) = (cid:16) d ψ α dt (cid:17) ( ϕ ) kkk − i Ω kkk e − i ϕ kkk for α = r , z , φ . (272)Here we use ∆ ϕ rather than ∆ to indicate that a quantity is purely oscillatory (i.e.,periodic with zero mean) with respect to ϕϕϕ rather than qqq . ( d ψ α / dt ) ( ϕ ) kkk can be cal-culated using Eq. (266) with d ψ α / dt = (cid:102) α / (cid:102) t .Just as we did with q α , we can express the coordinate trajectory in terms of ϕ α as the sum of a secular and an oscillatory piece, z α ( t ) = z α ( ϕ ) sec [ ϕ β ( t )] + ∆ ϕ z α [ ϕϕϕ ( t )] , (273)where the secular piece is z α ( ϕ ) sec ( ϕ β ) = (cid:0) ϕ t , , , ϕ φ (cid:1) − (cid:104) , r ( ϕ ) , z ( ϕ ) , − ∆ ϕ φ ( ) (cid:105) , (274)and the oscillatory pieces are ∆ ϕ t = ∆ ϕ φ given by Eq. (272) (recalling ψ φ : = φ ),and ∆ ϕ x a ( ϕϕϕ ) = ∑ kkk (cid:54) = x a ( ϕ ) kkk e − i ϕ kkk , (275)with coefficients calculated from Eq. (266). Recall that the the radial and polar motions are restricted to a torus-like region r p ≤ r ≤ r a and | z | ≤ z max in physical space. If the periods of radial and polar motionare incommensurate, then the orbit ergodically fills this region. The transformation x a → q a maps the r – z motion onto the surface of a torus in phase space, which theangles q a ergodically cover. However, for some values of the orbital parameters, theperiods are commensurate, meaning k res r ϒ r + k res z ϒ z = ( k res r , k res z ) . [Since integer multiples of ( k res r , k res z ) will also satisfy this condition, wetake ( k res r , k res z ) to be the smallest two such integers.] In these cases, rather thanhaving two independent frequencies, the r – z motion has a single frequency, ϒ = ϒ z / | k res r | = ϒ r / | k res z | , and rather than ergodically covering the torus, the orbit closeson the torus and in the r – z plane. The actual shape of this closed orbit is not uniquelyspecified by its frequencies but depends strongly on the relative initial phase ψ r − ψ z .Such orbits are referred to as resonant [63]. For resonant orbits, the average overthe torus no longer represents a meaningful average over the orbit. Rather than hav-ing the single stationary mode f ( q ) , a function f ( r , z ) on the worldline has an infiniteset of stationary modes corresponding to all integer multiples of kkk res . The infiniteMino-time average in Eq. (238) is then (cid:104) f [ r ( λ ) , z ( λ )] (cid:105) λ = lim Λ → ∞ Λ (cid:90) Λ − Λ f d λ = ∞ ∑ N = − ∞ f ( q ) Nkkk res ; (276)for a resonant orbit, the infinite λ average does not, generically, agree with the torusaverage f ( q ) .More broadly, the Fourier series (250) becomes degenerate: q kkk ( λ ) = q kkk + Nkkk res ( λ ) for all integers N . However, since there is a common period, we can replace the twoaction angles q a with a single angle q ( λ ) = ϒ λ and rewrite the two-dimensionalFourier series (250) as a non-degenerate one-dimensional one, f [ r ( λ ) , z ( λ )] = ∑ k ∈ Z f ( q ) k e − kq ( λ ) . (277)The coefficients are related to those in Eq. (250) by f ( q ) k = ∑ kkk f ( q ) kkk , where the sumranges over all ( k r , k z ) satisfying k r | k res z | + k z | k res r | = k . We then have (cid:104) f (cid:105) λ = f ( q ) .The set of resonant orbits is dense in the space of frequencies, though it is ofmeasure zero. A given resonant ratio ϒ r / ϒ z = | k res z | / | k res r | describes a surface in theparameter space spanned by p i . We refer to Ref. [27] for the characterization of thelocations of these surfaces and to Refs. [82, 64, 119, 28] for further discussion ofresonant geodesic orbits. We now consider an accelerated orbit satisfying the equation of motion (13), whichwe write compactly as D z α d τ = f α . (278)The normalization u α u α = − f α is orthogonal to the worldline: f α u α = E = − u t , L z = u φ , and Q = u α u β (cid:63)(cid:63) K αβ − a ( u ˜ φ ) on theaccelerated orbit, then dEd τ = − f t , dL z d τ = f φ , dQd τ = (cid:63)(cid:63) K αβ u α f β − a u ˜ φ f ˜ φ , (279)where f ˜ φ = a ( f φ + a f t ) and u ˜ φ = a ( L z − aE ) . In other words, the “constants” ofmotion are no longer constant. However, if f α is small, each parameter will changeonly slowly or oscillate slightly around a slowly varying mean.Our treatment of accelerated orbits mirrors that of geodesics, beginning withquasi-Keplerian methods and then describing the calculation of fundamental fre-quencies and perturbed angle variables. In the quasi-Keplerian treatment we placeno restriction on f α , and in particular we do not assume it is small. In the treatmentof perturbed angle variables we restrict the analysis to a small perturbing force,setting f µ ( ) = In celestial mechanics, perturbed Keplerian orbits have historically been describedusing the method of osculating orbits. The idea in this method is, given an ex-act solution to the unperturbed problem in terms of a set of orbital elements p i = { p , e , ι , . . . } , to write the perturbed orbit in precisely the same form but topromote the orbital elements to functions of time. At each instant t , the perturbedorbit with elements { p ( t ) , e ( t ) , ι ( t ) , . . . } is tangent to a Keplerian ellipse (called theosculating orbit) with those same values of orbital elements.In general relativity, this idea is referred to as the method of osculating geodesics[128, 161, 72, 204]. Our geodesics in Kerr are described by Eqs. (217), (218), (232),and (233), which involve the seven orbital elements I A = { p , e , ι , t , ψ r , ψ z , φ } . Ifwe let z α G ( I A , λ ) denote a geodesic with these orbital elements, and ˙ z α G ( I A , λ ) = ∂ z α G / ∂ λ its tangent vector, then the osculation conditions are z α ( λ ) = z α G [ I A ( λ ) , λ ] and dz α d λ ( λ ) = ˙ z α G [ I A ( λ ) , λ ] . (280)These conditions define a one-to-one transformation ( z α , ˙ z α ) → I A . Such a transfor-mation is possible because the number of orbital elements is equal to the numberof degrees of freedom on the orbit: the eight degrees of freedom { z α , ˙ z α } minus theconstraint ˙ z α f α = I A ( λ ) . Appealing to the chain rule dz α d λ = ∂ z α G ∂ I A dI A d λ + ∂ z α G ∂λ ,to the geodesic equation for z α G (in terms of the non-affine parameter λ ) , and to theequation of motion (278) for z α (converted to the non-affine parametrization), wefind [161] ∂ z α G ∂ I A dI A d λ = , (281) ∂ ˙ z α G ∂ I A dI A d λ = f α (cid:18) d τ d λ (cid:19) + [ κ ( λ ) − κ G ( λ )] ˙ z α G , (282)where κ = (cid:0) d τ d λ (cid:1) − d τ d λ . If we define λ to satisfy Eq. (216) on both the geodesic andaccelerated orbit, then κ = κ G = Σ − d Σ d λ , simplifying Eq. (282) to ∂ ˙ z α G ∂ I A dI A d λ = Σ f α . (283)These equations are exact, and f α need not be small.Equations (281) and (283) can be straightforwardly inverted to solve for dI A d λ , pro-viding a system of first-order ordinary differential equations for the orbital elements.However, working with the initial phases { t , ψ r , ψ z , φ } is cumbersome in practice.In the above evolution equations, the phases ψ a are given by their geodesic values,meaning the solutions to Eqs. (219) and (220) with fixed I A . That is, at each value of λ , in Eqs. (219) and (220) we replace d ψ a / d λ with d ψ a / d λ (cid:48) , then integrate from λ (cid:48) =
0, with initial values ψ a ( λ ) , up to λ (cid:48) = λ . Similarly, in Eqs. (234), the integralsare evaluated with fixed orbital elements in the integrands. The evolution equationsalso involve derivatives of these integrals with respect to the orbital elements. Evaluating all these integrals at every time step would be computationally expen-sive. In applications, it is therefore preferable to work with the variables { p , e , ι , ψ α } instead of I A . We write a geodesic trajectory and its tangent vector as z α G [ p i , ψ β ( λ )] and ˙ z α G [ p i , ψ β ( λ )] = ˙ ψ G β ∂ z α G ∂ψ β , where ˙ ψ G α = (cid:102) α ( p i , ψψψ ) are the geodesic “frequencies”given by Eqs. (208), (209), (219), and (220). The osculation conditions then read z α ( λ ) = z α G [ p i ( λ ) , ψ β ( λ )] and dz α d λ ( λ ) = ˙ z α G [ p i ( λ ) , ψ β ( λ )] . (284)Appealing to the chain rule dz α d λ = d ψ β d λ ∂ z α G ∂ψ β + dp i d λ ∂ z α G ∂ p i , to the geodesic equation for z α G (in terms of λ ), and to the equation of motion (278) for z α (in terms of λ ), wefind that the osculation conditions imply ∂ z α G ∂ p i d p i d λ + ∂ z α G ∂ ψψψ · δδδ (cid:102) = , (285) ∂ ˙ z α G ∂ p i d p i d λ + ∂ ˙ z α G ∂ ψψψ · δδδ (cid:102) = Σ f α . (286)Here ∂ z α G ∂ψψψ · δδδ (cid:102) = ∂ z α G ∂ψ r δ (cid:102) r + ∂ z α G ∂ψ z δ (cid:102) z , and we have defined δ (cid:102) a : = d ψ a d λ − ˙ ψ Ga . (287)Eq. (285) and (286) provide eight equations for the seven derivatives d ψ α d λ and dp i d λ ; any one of the four equations represented by (286) may be eliminated using f α u α =
0. The t and φ components of Eq. (285) are simply the osculation conditions d ψ α d λ = (cid:102) α ( ψψψ , p i ) for α = t , φ . (288)The r and z components of Eq. (285) can be rearranged to obtain δ (cid:102) a = − ∂ z aG / ∂ p i ∂ z aG / ∂ ψ a d p i d λ , (289)where we have used the fact that r is independent of ψ z and that z is independent of ψ r . Substituting this into Eq. (286) yields d p i d λ L i ( z α G ) = Σ f α , (290)where L i ( x ) : = ∂ ˙ x ∂ p i − ∂ r / ∂ p i ∂ r / ∂ ψ r ∂ ˙ x ∂ ψ r − ∂ z / ∂ p i ∂ z / ∂ ψ z ∂ ˙ x ∂ ψ z . (291)One can easily invert Eq. (290) to obtain equations for dp i d λ : d pd λ = Σ D (cid:8) [ L e ( z ) , L ι ( φ )] f r + [ L e ( φ ) , L ι ( r )] f z + [ L e ( r ) , L ι ( z )] f φ (cid:9) , (292) ded λ = Σ D (cid:8) [ L ι ( z ) , L p ( φ )] f r + [ L ι ( φ ) , L p ( r )] f z + [ L ι ( r ) , L p ( z )] f φ (cid:9) , (293) d ι d λ = Σ D (cid:8) [ L p ( z ) , L e ( φ )] f r + [ L p ( φ ) , L e ( r )] f z + [ L p ( r ) , L e ( z )] f φ (cid:9) , (294)with [ L i ( x ) , L j ( y )] : = L i ( x ) L j ( y ) − L j ( x ) L i ( y ) and D : = L p ( r )[ L e ( z ) , L ι ( φ )] + L e ( r )[ L ι ( z ) , L p ( φ )] + L ι ( r )[ L p ( z ) , L e ( φ )] . (295)Finally, the evolution equations for ψ a are obtained by substituting Eqs. (292)–(294) into Eq. (289), yielding d ψ a d λ = (cid:102) a ( p i , ψψψ ) + δ (cid:102) a ( p i , ψψψ ) , (296)where δ (cid:102) r = − pe sin ψ r (cid:20) ( + e cos ψ r ) d pd λ − p ded λ cos ψ r (cid:21) , (297) δ (cid:102) z = d ι d λ cot ι cot ψ z . (298)There are superficial singularities in these formulas when ψ a is an integer multipleof π . However, the divergences are analytically eliminated when the formulas areexplicitly evaluated.The full set of evolution equations is given by Eqs. (292)–(294), (296), and (288).In these equations, x a ( λ ) = x aG [ p i ( λ ) , ψ a ( λ )] is given by Eqs. (217)–(218), ˙ x a ( λ ) = ˙ x aG [ p i ( λ ) , ψ a ( λ )] by˙ r = pMe (cid:102) r sin ψ r ( + e cos ψ r ) and ˙ z = − z max (cid:102) z sin ψ z , (299)with Eqs. (219)–(220) for (cid:102) a , and ( ˙ t , ˙ φ ) = ( (cid:102) t , (cid:102) φ ) by Eqs. (208) and (209). Wher-ever E , L z , and Q appear, they are given in terms of p i by their geodesic expres-sions (225)–(227). The quantities [ L i ( x ) , L j ( y )] , when explicitly evaluated, consti-tute lengthy analytical formulas in terms of p i and ψψψ . However, for several L i ( x ) ,the second and third term vanish in Eq. (291). Specifically, L ι ( r ) = ∂ ˙ r ∂ι = ∂ r ∂ψ r ∂ (cid:102) r ∂ι ,and L j ( z ) = ∂ ˙ z ∂ p j = ∂ z ∂ψ z ∂ (cid:102) z ∂ j for j = p , e .The evolution can be slightly simplified by adopting ψ r or ψ z as the parameteralong the trajectory. That is easily done by using, e.g., dp i d ψ a = d ψ a / d λ dp i d λ . How-ever, for a sufficiently large perturbing force, d ψ a d λ can vanish at some points inthe evolution, making ψ a an invalid parameter. In that case, we can split ψ a into ψ a = ψ Ga − ψ a , where d ψ Ga / d λ = (cid:102) a and d ψ a / d λ = − δ (cid:102) a . ψ Ga is then a conve- nient, monotonic parameter along the worldline. Alternatively, t can be used, apply-ing, e.g., dp i dt = (cid:102) − t dp i d λ .The evolution equations simplify more dramatically in the special case of equato-rial orbits, for which z = f z =
0. In this case, ι and ψ z do not appear, and Eqs. (292)–(293) reduce to d pd λ = r (cid:2) L e ( φ ) f r − L e ( r ) f φ (cid:3) L p ( r ) L e ( φ ) − L e ( r ) L p ( φ ) , (300) ded λ = r (cid:2) L p ( r ) f φ − L p ( φ ) f r (cid:3) L p ( r ) L e ( φ ) − L e ( r ) L p ( φ ) , (301)If ψ Gr is used as the independent parameter along the orbit, then the other threeevolution equations are d ψ r / d ψ Gr = − δ (cid:102) r / (cid:102) r and Eq. (288) for t ( ψ Gr ) and φ ( ψ Gr ) .In our treatment we have left the evolution equations in a highly inexplicit formeven in the relatively simple equatorial case. Refs. [161] and [204] provide explicitformulas in the cases of planar and nonplanar orbits in Schwarzschild spacetime.Ref. [72] details the generic case in Kerr spacetime and describes several alternativeformulations of the osculating evolution.Before proceeding, we note again that the equations in this section are valid forarbitrary forces, though the orbital elements are most meaningful when the forceis small and the orbit is close to a geodesic. In the next section, we restrict to thecase of a small perturbing force. In the unperturbed case, the equations of geodesic motion could be written in termsof the orbital parameters and angle variables as dq α d λ = ϒ α ( p i ) , (302) d p i d λ = . (303)If the perturbing force is small, with an expansion f α = ε f α ( ) ( z µ , ˙ z µ ) + ε f α ( ) ( z µ , ˙ z µ ) + O ( ε ) , (304)and is periodic in t and φ , then the equations of forced motion can still be written interms of orbital parameters and angle variables: However, the method has most often been applied [202, 137, 204, 194] in the context of anapproximation in which the self-force at each instant is approximated by the value it would take ifthe particle had spent its entire prior history moving on the osculating geodesic. Since the force isthen constructed from the field generated by the osculating geodesic particle, this approximationmight more properly be dubbed the method of osculating sources.0 dq α d λ = ϒ ( ) α ( p jq ) + εϒ ( ) α ( p jq ) + O ( ε ) , (305) d p iq d λ = ε G i ( ) ( p jq ) + ε G i ( ) ( p jq ) + O ( ε ) . (306)Note that the subscript q on the orbital parameters p iq = ( p q , e q , ι q ) is a label, not anindex. The form (307) is mildly restrictive, and it does not include the Mathisson-Papapetrou spin force, for example; for a spinning body, we must introduce ad-ditional parameters and action angles describing the spin’s magnitude and direc-tion [173, 208]. For our purposes we adopt a more restrictive form, f α = ε f α ( ) ( xxx , ˙ z µ ) + ε f α ( ) ( xxx , ˙ z µ ) + O ( ε ) , (307)which assumes that the force inherits the background spacetime’s symmetries. Weexplain in Sec. 7.1.2 that the form (307) still needs further, minor alteration to de-scribe the self-force, but it is sufficiently general as a starting point.Unlike in the unperturbed case, the orbital parameters p iq and frequencies areno longer constant; they evolve slowly with time. However, the variables ( q α , p iq ) cleanly separate the two scales in the orbit’s evolution: the variables p iq only changeslowly, over the long time scale ∼ / ε , while the angle variables q α change on theorbital time scale ∼ π / ϒ ( ) α . In the context of a small-mass-ratio binary, wherethe inspiral is driven by gravitational-wave emission, the long time scale ∼ / ε isreferred to as the radiation-reaction time .The division of the orbital dynamics into slowly and rapidly varying functionshas the same utility as in the geodesic case: it enables convenient Fourier expansionsof functions on the worldline, which mesh with a Fourier expansion of the fieldequations (described in the final section of this chapter). Functions f ( r , z ) on theaccelerated worldline can be expanded in the Fourier series f [ r ( λ ) , z ( λ )] = ∑ kkk f ( q ) kkk ( p jq ) e − iq kkk ( λ ) , (308)with a clean separation between slowly varying amplitudes and rapidly varyingphases. The coefficients remain given by Eq. (251). By eliminating oscillatory driv-ing terms in the orbital evolution equations, the transformation to ( q α , p iq ) also facil-itates more rapid numerical evolutions [194] and, ultimately, more rapid generationof waveforms [38]. In this and the next section, for visual simplicity we shall omitthe label “ ( q ) ” from mode coefficients associated with qqq .Now, to put the equations of motion in the form (305)–(306), we begin with theevolution equations (288), (292)–(294), and (296). Given the expansion (307), theseequations take the form d ψ α d λ = (cid:102) ( ) α ( ψψψ , p j ) + ε (cid:102) ( ) α ( ψψψ , p j ) + O ( ε ) , (309) d p i d λ = ε g i ( ) ( ψψψ , p j ) + ε g i ( ) ( ψψψ , p j ) + O ( ε ) . (310) Here g i ( n ) is given by Eqs. (292)–(294) with f α → f α ( n ) . We have renamed (cid:102) α to (cid:102) ( ) α , (cid:102) ( ) a is given by δ (cid:102) a with f α → f α ( ) , and (cid:102) ( ) α = α = t , φ . In this form of theequations, every term on the right is a periodic, oscillatory function of the phases.However, one can transform to the new variables ( q α , p iq ) , which have no oscillatorydriving terms, using an averaging transformation [106, 194], ψ α ( q β , p jq , ε ) = ψ ( ) α ( q β , p jq ) + εψ ( ) α ( q β , p jq ) + O ( ε ) , (311) p i ( q β , p jq , ε ) = p iq + ε p i ( ) ( qqq , p jq ) + ε p i ( ) ( qqq , p jq ) + O ( ε ) , (312)where ψ ( ) α ( q β , p jq ) = q α − ∆ ψ ( ) α ( , p iq ) + ∆ ψ ( ) α ( qqq , p jq ) (313)is the geodesic relationship, and the corrections ψ ( n ) α and p i ( n ) for n > π -periodic in each q a (with a potentially nonzero mean). In analogy with the geodesiccase, we have chosen the origin of phase space such that ψ ( ) α ( q β = ) =
0. Thischoice will ensure that at fixed p iq , ψ ( ) α and q α satisfy all the relationships inSec. 6.1.3. Note that we could replace ∆ ψ ( ) α ( , p iq ) with any other q α -independentfunction of p iq ; this would still represent a geodesic relationship between ψ ( ) α and q α , but with different choices of initial phases for different values of p iq . For conve-nience in later expressions, we define A α ( p iq ) : = − ∆ ψ ( ) α ( , p iq ) . (314)By substituting the expansions (311) and (312) into Eqs. (309) and (310), appeal-ing to (305) and (306), and equating coefficients of powers of ε , one can solve forthe frequencies ϒ ( n ) α and driving forces G i ( n ) , as well as for the functions in the av-eraging transformation. Explicitly, the leading-order terms in Eqs. (309) and (310)are ∂ ψ ( ) α ∂ q β ϒ ( ) β = ϒ ( ) α + ϒϒϒ ( ) · ∂ ∆ ψ ( ) α ∂ qqq = (cid:102) ( ) α ( ψψψ ( ) , p jq ) , (315) G i ( ) + ϒϒϒ ( ) · ∂ p i ( ) ∂ qqq = g i ( ) ( ψψψ ( ) , p jq ) . (316)Equation (315) is simply the geodesic relationship between ψ α and q α . It followsthat we can use the geodesic solution (241) for q a ( ψ ( ) a , p iq ) . Concretely, we maywrite q a ( ψ ( ) a , p iq ) = ϒ ( ) a ( p iq ) (cid:90) ψ ( ) a d ψ (cid:48) a (cid:102) ( ) a ( ψ (cid:48) a , p iq ) , (317) Here we combine a near-identity averaging transformation with a zeroth-order one.2 implying that the Fourier coefficients in Eq. (308) can be computed as the integralsover ψψψ ( ) in Eq. (251), with the replacements ϒ a → ϒ ( ) a and ψ a → ψ ( ) a . This relieson our particular choice of A α in Eq. (314); different choices would lead to different p iq -dependent lower limits of integration in Eq. (317), which in turn would lead to p iq -dependent phase factors appearing in Eq. (251).Using either of the forms (251) or (252), we can easily decompose Eqs. (315)and (316) into Fourier series, with ∆ ψ ( ) α = ∑ kkk (cid:54) = ∆ ψ ( , kkk ) α ( p jq ) e − iq kkk and p i ( ) = ∑ kkk p i ( , kkk ) ( p jq ) e − iq kkk . From the kkk = ϒ ( ) α ( p jq ) = (cid:68) (cid:102) ( ) α ( ψψψ ( ) , p jq ) (cid:69) qqq , (318) G i ( ) ( p jq ) = (cid:68) g i ( ) ( ψψψ ( ) , p jq ) (cid:69) qqq , (319)and from the kkk (cid:54) = ∆ ψ ( , kkk ) α ( p jq ) = − (cid:102) ( , kkk ) α ( p jq ) i ϒ ( ) kkk ( p jq ) , (320) p i ( , kkk ) ( p jq ) = − g i ( , kkk ) ( p jq ) i ϒ ( ) kkk ( p jq ) , (321)where ϒ ( ) kkk : = kkk · ϒϒϒ ( ) = k r ϒ ( ) r + k z ϒ ( ) z . Note that these equations leave p i ( , ) ar-bitrary.As foreshadowed above, Eqs. (318) and (320) are precisely the same as thegeodesic formulas (236) and (253) (with the replacement p i → p iq ). The only changeis that the orbital parameters p iq , which determine the frequencies and amplitudes,now adiabatically evolve with time.Importantly, Eq. (321) requires ϒ ( ) kkk (cid:54) =
0. This condition fails at resonances,where ϒ ( ) kkk res =
0. Therefore, the averaging transformation is impossible when thereis a resonance. We discuss this resonant case in the next section. Eq. (320) also su-perficially appears to encounter a singularity at resonance, but this is skirted by theparticular form of (cid:102) ( , kkk ) α , as we see from the more explicit formula (253).Moving onto the first subleading order in Eqs. (309) and (310), we have ϒ ( ) α + G j ( ) ∂ ψ ( ) α ∂ p jq + ϒϒϒ ( ) · ∂ ∆ ψ ( ) α ∂ qqq + ϒϒϒ ( ) · ∂ ψ ( ) α ∂ qqq = (cid:102) ( ) α + p j ( ) ∂ (cid:102) ( ) α ∂ p jq + ψψψ ( ) · ∂ (cid:102) ( ) α ∂ ψψψ ( ) , (322) G i ( ) + G j ( ) ∂ p i ( ) ∂ p jq + ϒϒϒ ( ) · ∂ p i ( ) ∂ qqq + ϒϒϒ ( ) · ∂ p i ( ) ∂ qqq = g i ( ) + p j ( ) ∂ g i ( ) ∂ p jq + ψψψ ( ) · ∂ g i ( ) ∂ ψψψ ( ) , (323)where all quantities on the left are functions of ( qqq , p jq ) , and all those on the right arefunctions of ( ψψψ ( ) , p jq ) . Taking the average of these equations yields ϒ ( ) α ( p iq ) = (cid:68) (cid:102) ( ) α (cid:69) qqq + (cid:42) p i ( ) ∂ (cid:102) ( ) α ∂ p iq + ψψψ ( ) · ∂ (cid:102) ( ) α ∂ ψψψ ( ) (cid:43) qqq − G j ( ) ∂ A α ∂ p jq , (324) G i ( ) ( p jq ) = (cid:68) g i ( ) (cid:69) qqq + (cid:42) p j ( ) ∂ g i ( ) ∂ p jq + ψψψ ( ) · ∂ g i ( ) ∂ ψψψ ( ) (cid:43) qqq − G j ( ) ∂ p i ( , ) ∂ p jq . (325)We see from Eq. (324) that a judicious choice of p i ( , ) allows us to set ϒ ( ) α = α = r , z , φ . (326)Such a p i ( , ) is determined from p i ( , ) ∂ϒ ( ) α ∂ p iq = − (cid:68) (cid:102) ( ) α (cid:69) qqq − ∑ kkk (cid:54) = p i ( , kkk ) ∂ (cid:102) ( , − kkk ) α ∂ p iq − (cid:42) ψψψ ( ) · ∂ (cid:102) ( ) α ∂ ψψψ ( ) (cid:43) qqq + G j ( ) ∂ A α ∂ p jq (327)for α = r , z , φ . We could alternatively set ϒ ( ) α = A α ;i.e., the functions A α and p i ( , ) in the averaging transformation are degenerate with ϒ ( ) α . Ref. [194] provides a more thorough discussion of the freedom within near-identity averaging transformations.The averages in Eqs. (324)–(325) involve ψ ( ) a , which can be obtained fromEq. (322). A 2 π -biperiodic solution to that equation is This seems to be the unique 2 π -biperiodic solution. Any other solution can only differ by a ho-mogeneous solution to Eq. (322), which must take the form exp ( (cid:82) (cid:102) (cid:48) a dq a / ϒ ( ) a ) f ( q b − q a ϒ ( ) b / ϒ ( ) a ) for some function f , with b (cid:54) = a . It appears that such a function cannot simultaneously be 2 π peri-odic in both q a and q b .4 ψ ( ) a ( qqq , p jq ) = Y a ( q a , p jq ) ∑ kkk ∑ k S kkka ( p jq ) Y ka ( p jq ) − i ϒ ( ) kkk − ik ϒ ( ) a − (cid:104) (cid:102) (cid:48) a (cid:105) qqq e − iq kkk − ikq a , (328)where Y a ( q a , p jq ) : = exp [ − F a ( q a , p jq ) / ϒ ( ) a ( p jq )] = ∑ k Y ka e − ikq a , F a : = ∑ k (cid:54) = (cid:102) (cid:48) ka − ik e − ikq a is the antiderivative of the purely oscillatory part of (cid:102) (cid:48) a : = ∂ (cid:102) ( ) a / ∂ ψ ( ) a , and S a ( qqq , p iq ) : = − G j ( ) ∂ ψ ( ) a ∂ p jq + (cid:102) ( ) a + p j ( ) ∂ (cid:102) ( ) a ∂ p jq = ∑ kkk S kkka ( p iq ) e − iq kkk . (329)The remaining pieces of Eqs. (322) and (323) determine the purely oscillatoryparts of ψ ( ) t , ψ ( ) φ , and p i ( ) . Specifically, ψ ( , kkk ) α = − i ϒ ( ) kkk (cid:32) (cid:102) ( , kkk ) α + P kkk α − G j ( ) ∂ ∆ ψ ( , kkk ) α ∂ p jq (cid:33) , (330) p i ( , kkk ) = − i ϒ ( ) kkk (cid:32) g i ( , kkk ) + Q ikkk − G j ( ) ∂ p i ( , kkk ) ∂ p jq (cid:33) (331)for α = t , φ , where P α : = p j ( ) ∂ (cid:102) ( ) α ∂ p jq + ψψψ ( ) · ∂ (cid:102) ( ) α ∂ψψψ ( ) and Q i : = p j ( ) ∂ g i ( ) ∂ p jq + ψψψ ( ) · ∂ g i ( ) ∂ψψψ ( ) .This averaging transformation can be carried to any order. Analogous calcula-tions also apply if we use P i = ( E , L z , Q ) rather than p i = ( p , e , ι ) . Ultimately, thecoordinate trajectory z α can be expressed in terms of ( q α , p iq ) as z α ( q β , p iq ) = z α ( ) ( q β , p iq ) + ε z α ( ) ( qqq , p iq ) + O ( ε ) . (332)The leading-order trajectory has the same dependence on q α and p iq as a geodesic.That is, if we write a geodesic as z α G ( q β , p i ) , given by Eq. (257), then z α ( ) ( q β , p iq ) = z α G ( q β , p iq ) . Wherever the geodesic expressions involve P i , they are here evaluatedat P iq = ( E q , L q , Q q ) , which are related to p iq by the geodesic relationships. (We sup-press the subscript z on L z .) The difference between the geodesic and the acceleratedtrajectory lies entirely in the evolution of their arguments: rather than evolving ac-cording to Eqs. (302) and (303), q α and p iq now evolve according to Eqs. (305) and(306).In the context of a binary, the small corrections ε z α ( ) ( qqq , p iq ) to the trajectory re-main uniformly small over the entire inspiral until the transition to plunge [135];because they are periodic functions of qqq , they have no large secular terms. t ( ) and φ ( ) are given by Eq. (330), with t ( , ) and φ ( , ) left arbitrary. r ( ) and z ( ) aregiven by x a ( ) = ψψψ ( ) · ∂ x aG ∂ ψψψ ( ) + p i ( ) ∂ x aG ∂ p iq , (333) with ψ ( ) a given by Eq. (328), the oscillatory part of p i ( ) by Eq. (321), and p i ( , ) byEq. (327).Refs. [88, 119, 68, 101, 194] contain more detailed action-angle treatments ofperturbed orbits. With the exception of Ref. [194], these treatments have not begunwith equations of the form (309) and (310). Instead, they began with approximateangle variables, which we will denote ˆ q α and which satisfy ˆ q α = q α + O ( ε ) . Theequations of motion then take the form d ˆ q α d λ = ϒ ( ) α ( p j ) + ε U ( ) α ( ˆ qqq , p j ) + O ( ε ) , (334) d p i d λ = ε F i ( ) ( ˆ qqq , p j ) + ε F i ( ) ( ˆ qqq , p j ) + O ( ε ) . (335)Ref. [88] derives concrete equations of this form in the case that proper time τ is used instead of Mino time and that { E , L z , Q } are used instead of { p , e , ι } . [The driving forces F i ( n ) are then given by Eq. (279).] The averaging transformation ( ˆ q α , p i ) → ( q α , p iq ) can be found as we did above, with substantial simplificationsarising from the fact that d ˆ q α d λ is constant at leading order; the transformation is givenby Eqs. (383) and (384) below (without the restriction kkk (cid:54) = Nkkk res in the nonresonantcase).Our particular construction in this section and the next is instead designed to linkthe action-angle description with the quasi-Keplerian one. It appears here for thefirst time. However, Ref. [194] considers more general sets of coupled differentialequations that involve variables analogous to our ψ α as well as variables analogousto ˆ q α , though without providing a solution analogous to our (328). Because we solve field equations and extract waveforms using Boyer-Lindquist time t , it is once again useful to construct variables ( ϕ α , p i ϕ ) associated with t , where ϕ t = ψ t = t and p i ϕ = ( p ϕ , e ϕ , ι ϕ ) . The construction of the variables ( ϕ α , p i ϕ ) (and of theirevolution equations) is analogous to the construction based on λ : the osculating-geodesic equations for d ψ α / dt and d p i / dt have the same form as Eqs. (309) and(310), simply with (cid:102) ( n ) α → (cid:102) ( n ) α / (cid:102) ( ) t and g i ( n ) → g i ( n ) / (cid:102) ( ) t , and after a near-identityaveraging transformation we arrive at the equations of motion d ϕ α dt = Ω ( ) α ( p j ϕ ) , (336) d p i ϕ dt = εΓ i ( ) ( p j ϕ ) + ε Γ i ( ) ( p j ϕ ) + O ( ε ) . (337) The notation in Ref. [88] differs in several significant ways from ours. In particular, Ref. [88]uses λ to denote a rescaled τ , q α to denote an analogue of our ˆ q α (and associated with τ ratherthan Mino time), and ψ α to denote an analogue of our q α (again associated with τ ).6 Ω ( ) α are the geodesic frequencies, and Ω ( ) t =
1. The corrections Ω ( n ) α for α = r , z , φ and n > Ω ( n ) t = n > d ϕ t / dt = ( ϕ α , p i ϕ ) and ( q α , p iq ) are related by a transformation ϕ α ( q β , p jq , ε ) = ϕ ( ) α ( q α , p iq ) + εΦ ( ) α ( qqq , p jq ) + O ( ε ) , (338) p i ϕ ( q β , p jq , ε ) = p iq + επ i ( ) ( qqq , p jq ) + O ( ε ) , (339)where the leading term in ϕ α is given by the geodesic relationship (263), which werestate as ϕ ( ) ( q α , p iq ) : = q α + B α ( p iq ) + Ω ( ) α ( p iq ) ∆ t ( ) ( qqq , p jq ) , (340)defining B α ( p iq ) : = − Ω ( ) α ( p iq ) ∆ t ( ) ( , p jq ) . (341)Like in the geodesic case, this value for B α imposes that ϕ α and q α (and ψ ( ) α )have the same origin in phase space. As discussed around Eqs. (314) and (317),this means that we can immediately utilize all the relationships from Sec. 6.1.4. Thecorrections Φ ( ) α and π i ( ) are 2 π -biperiodic in qqq .The terms in this transformation, as well as the driving forces Γ i ( n ) , can be derivedby substituting Eqs. (338) and (339) into Eqs. (336) and (337). Taking the averageof the resulting equations and appealing to Eqs. (305) and (306), we obtain Ω ( ) α = ϒ ( ) α ϒ ( ) t and Γ i ( ) = G i ( ) ϒ ( ) t (342)at leading order and Ω ( ) α = = − ϒ ( ) t (cid:16) G j ( ) ∂ p j B α + (cid:104) R i (cid:105) qqq ∂ p i Ω ( ) α + (cid:104) P t (cid:105) qqq Ω ( ) α (cid:17) , (343) Γ i ( ) = ϒ ( ) t (cid:16) G i ( ) + G j ( ) ∂ p j (cid:104) π i ( ) (cid:105) qqq − (cid:104) R j (cid:105) qqq ∂ p j Γ i ( ) − (cid:104) P t (cid:105) qqq Γ i ( ) (cid:17) (344)at the first subleading order, where R i : = (cid:102) ( ) t π i ( ) and P t = ψψψ ( ) · ∂ ψψψ (cid:102) ( ) t + p i ( ) ∂ p i (cid:102) ( ) t .The average (cid:104) π i ( ) (cid:105) qqq is chosen to enforce Ω ( ) α = ∆ t ( , kkk ) = (cid:102) ( , kkk ) t − i ϒ ( ) kkk , (345) π i ( , kkk ) = (cid:102) ( , kkk ) t − i ϒ ( ) kkk Γ i ( ) (346) at leading order and Φ ( , kkk ) α = − i ϒ ( ) kkk (cid:16) R ikkk ∂ p i Ω ( ) α + P kkkt Ω ( ) α − G i ( ) ∂ i ∆ ϕ ( , kkk ) α (cid:17) (347)at the first subleading order. In all of the above expressions, kkk refers to a Fourierdecomposition into e − iq kkk modes. All functions of p i are evaluated at p iq , and insidethe integrals (251), all functions of ψψψ are evaluated at ψψψ ( ) ( qqq , p iq ) .When solving the field equations, we shall require Fourier decompositions withrespect to ϕϕϕ , f [ r ( t ) , z ( t )] = ∑ kkk f ( ϕ ) kkk ( p j ϕ ) e − i ϕ kkk . (348)We can calculate the coefficients as integrals over qqq using the transformation (338).However, it is simpler to use the geodesic change of variables defined by the leadingterm in the transformation. The coefficients are then given by Eq. (265) with thereplacements p i → p i ϕ , ϒ α → ϒ ( ) α , (cid:102) α → (cid:102) ( ) α , or by Eq. (266) with the additionalreplacement ψψψ → ψψψ ( ) .We will also require the transformation from ( ψ α , p i ) to ( ϕ α , p i ϕ ) : ψ α ( ϕ β , p i ϕ , ε ) = ψ ( ) α ( ϕ β , p i ϕ ) + εψ ( ϕ , ) α ( ϕϕϕ , p i ϕ ) + O ( ε ) , (349) p i ( ϕ α , p j ϕ , ε ) = p i ϕ + ε p i ( ϕ , ) ( ϕϕϕ , p j ϕ ) + O ( ε ) . (350)Following the same steps as in the previous section, at leading order we recoverthe geodesic frequencies and find ψ ( ) α ( ϕ β , p i ϕ ) is given by the geodesic relation-ship (271). Solving the subleading-order equations is made difficult because theanalogue of Eq. (322) has the form ΩΩΩ ( ) · ∂ ψ ( ϕ , ) α ∂ ϕϕϕ − ψψψ ( ϕ , ) · ∂∂ ψψψ ( ) (cid:32) (cid:102) ( ) α (cid:102) ( ) t (cid:33) = . . . (351)The α = r , z components of this equation, unlike those of Eq. (322), are coupled par-tial differential equations for ψψψ ( ϕ , ) , which do not have a solution of the form (328).However, we can find the subleading terms in Eqs. (349) and (350) by combiningour knowledge of ( ϕ α , p i ϕ ) and ( ψ α , p i ) as functions of ( q α , p iq ) . Substituting theexpansions (338) and (339) into the right-hand sides of Eqs. (349) and (350) andequating the results with Eqs. (311) and (312), we find ψ ( ϕ , ) α ( ϕ ( ) β , p i ϕ ) = ψ ( ) α ( q β , p i ϕ ) − ϕ ( ) β ∂ ψ ( ) α ∂ ϕ β ( ϕ ( ) γ , p i ϕ ) − π i ( ) ∂ ψ ( ) α ∂ p i ϕ ( ϕ ( ) γ , p i ϕ ) , (352) p i ( ϕ , ) ( ϕϕϕ ( ) , p j ϕ ) = p i ( ) ( qqq , p i ϕ ) − π i ( ) ( ϕϕϕ ( ) , p j ϕ ) , (353) where ϕ ( ) α is given by Eq. (340) with p iq → p i ϕ . The inverse transformation, whichwe will also need, is ϕ α ( ψ β , p i , ε ) = ϕ ( ) α ( ψ β , p i ) + εΦ ( ) α ( qqq ( ) , p i ) − εψ ( ) β ∂ ϕ ( ) α ∂ ψ β − ε p i ( ) ∂ ϕ ( ) α ∂ p i , (354) p i ϕ ( ψ β , p i , ε ) = p i + επ i ( ) ( ϕϕϕ ( ) , p j ) − ε p i ( ) ( qqq ( ) , p j ) , (355)where ϕ ( ) α and q ( ) α are the geodesic functions of ψ α and p i .Finally, we can reconstruct the coordinate trajectory z α in the form z α ( ϕ β , p i ϕ ) = z α ( ) ( ϕ β , p i ϕ ) + ε z α ( ϕ , ) ( ϕϕϕ , p i ϕ ) + O ( ε ) . (356)The zeroth-order trajectory has the same functional dependence as a geodesic; thatis, z α ( ) ( ϕ β , p i ϕ ) = z α G ( ϕ β , p i ϕ ) , where z α G ( ϕ β , p i ) is given in Eq. (273). In analogywith the Mino-time solution, wherever the geodesic expressions involve P i , theyare here evaluated at P i ϕ = ( E ϕ , L ϕ , Q ϕ ) , which are related to p i ϕ by the geodesicrelationships. The first-order corrections z α ( ϕ , ) are t ( ϕ , ) = φ ( ϕ , ) = ψ ( ϕ , ) φ givenby Eq. (352), and x a ( ϕ , ) = ψψψ ( ϕ , ) · ∂ x aG ∂ ψψψ ( ) + p i ( ϕ , ) ∂ x aG ∂ p i ϕ , (357)with ψψψ ( ϕ , ) and p i ( ϕ , ) given by Eqs. (352) and (353). Self-accelerated orbits are often described with a multiscale (or two-timescale) ex-pansion [160, 88, 129, 150, 157, 96, 25, 126]. This is essentially equivalent to theaveraging transformation described above.To illustrate the method, we return to Eqs. (309) and (310). We introduce a slowtime variable ˜ λ : = ελ ; this changes by an amount ∼ ε on the time scale ∼ / ε . Inplace of the transformations (311) and (312), we adopt expansions ψ α ( q β , ˜ λ , ε ) = q α + ˜ A α ( ˜ λ ) + ∆ ˜ ψ ( ) α ( qqq , ˜ λ ) + ε ˜ ψ ( ) α ( qqq , ˜ λ ) + O ( ε ) , (358) p i ( q α , ˜ λ , ε ) = ˜ p i ( ) ( ˜ λ ) + ε ˜ p i ( ) ( qqq , ˜ λ ) + O ( ε ) , (359)where q α satisfies dq α d λ = ˜ ϒ ( ) α ( ελ ) + ε ˜ ϒ ( ) α ( ελ ) + O ( ε ) : = ˜ ϒ α ( ελ , ε ) . (360) We then substitute these expansions into Eqs. (309) and (310), applying the chainrule dd λ = ˜ ϒ α ∂∂ q α + ε ∂∂ ˜ λ . (361) q α and ˜ λ are then treated as independent variables, making Eqs. (309) and (310)into a sequence of equations, one set at each order in ε . These equations are essen-tially equivalent to (315) and (316) at leading order and to Eqs. (322) and (323) atfirst subleading order, with tildes placed over all quantities and the following re-placements: p iq → ˜ p i ( ) , G i ( n ) → d ˜ p i ( n − ) / d ˜ λ , G i ( ) ∂ψ ( ) α ∂ p jq → d ( ˜ A α + ∆ ˜ ψ ( ) α ) / d ˜ λ , and G i ( ) ∂ p i ( ) ∂ p jq →
0. These equations can be solved just as we solved Eqs. (315), (316),(322), and (323).The only difference between this expansion and Eqs. (311) and (312) is how eachparameterizes the orbit’s slow evolution, whether with slowly evolving parameters p iq or with slow time ˜ λ . Indeed, the solutions are easily related. The solutions toEqs. (305) and (306) can be expanded as q α ( ˜ λ , ε ) = ε (cid:104) ˜ q ( ) α ( ˜ λ ) + ε ˜ q ( ) α ( ˜ λ ) + O ( ε ) (cid:105) , (362) p iq ( ˜ λ , ε ) = ˜ p iq ( ) ( ˜ λ ) + ε ˜ p iq ( ) ( ˜ λ ) + O ( ε ) , (363)where ˜ q ( n ) α ( ˜ λ ) = (cid:82) ˜ λ ˜ ϒ ( n ) α ( ˜ λ (cid:48) ) d ˜ λ (cid:48) + ˜ q ( n ) α ( ) with˜ ϒ ( ) α ( ˜ λ ) = ϒ ( ) α , (364)˜ ϒ ( ) α ( ˜ λ ) = ϒ ( ) α + ˜ p iq ( ) ( ˜ λ ) ∂ p i ϒ ( ) α . (365)On the right, ϒ ( n ) α and its derivatives are evaluated at ˜ p iq ( ) ( ˜ λ ) . Substituting theseexpansions into Eqs. (311) and (312) and comparing to Eqs. (358) and (359), weread off ∆ ˜ ψ ( ) α ( qqq , ˜ λ ) = ∆ ψ ( ) α , (366)˜ p i ( ) ( ˜ λ ) = ˜ p iq ( ) ( ˜ λ ) , (367)˜ ψ ( ) α ( qqq , ˜ λ ) = ψ ( ) α + ˜ p jq ( ) ( ˜ λ ) ∂ p j ψ ( ) α , (368)˜ p i ( ) ( qqq , ˜ λ ) = p i ( ) + ˜ p iq ( ) ( ˜ λ ) . (369)Here ψ ( ) α , p i ( ) , ψ ( ) α , and their derivatives are evaluated at [ qqq , ˜ p jq ( ) ( ˜ λ )] . These par-ticular relationships rely on choosing ˜ A α ( ˜ λ ) = A α [ ˜ p iq ( ) ( ˜ λ )] . Just as the averagingtransformation did, the multiscale expansion has considerable degeneracy between˜ A α , ˜ ϒ ( ) α , and (cid:104) ˜ p i ( ) (cid:105) . If different choices are made, then we cannot identify q α be-tween the two methods. However, regardless of choices, both methods will ulti- mately output identical solutions of the form ψ α ( ˜ λ , ε ) and p i ( ˜ λ , ε ) (assuming iden-tical initial conditions), and when written in that form they can be unambiguouslyrelated.All the same relationships apply if we instead use t -based variables with a slowtime ˜ t : = ε t . When considering the multiscale expansion of the Einstein equation, itwill be useful to have at hand the expansions ϕ α ( ˜ t , ε ) = ε (cid:104) ˜ ϕ ( ) α ( ˜ t ) + ε ˜ ϕ ( ) α ( ˜ t ) + O ( ε ) (cid:105) , (370) p i ϕ ( ˜ t , ε ) = ˜ p i ϕ ( ) ( ˜ t ) + ε ˜ p i ϕ ( ) ( ˜ t ) + O ( ε ) . (371)It follows from Eqs. (336) and (337) that the coefficients in these expansions satisfy d ˜ ϕ ( ) α d ˜ t = Ω ( ) α ( ˜ p j ϕ ( ) ) , (372) d ˜ p i ϕ ( ) d ˜ t = Γ i ( ) ( ˜ p j ϕ ( ) ) , (373) d ˜ ϕ ( ) α d ˜ t = ˜ p j ϕ ( ) ∂ j Ω ( ) α ( ˜ p j ϕ ( ) ) , (374) d ˜ p i ϕ ( ) d ˜ t = Γ i ( ) ( ˜ p j ϕ ( ) ) + ˜ p j ϕ ( ) ∂ j Γ i ( ) ( ˜ p j ϕ ( ) ) . (375)We can also write Eq. (370) as ϕ α ( ˜ t , ε ) = ε (cid:90) ˜ t Ω α ( ˜ t (cid:48) , ε ) d ˜ t (cid:48) + ϕ α ( , ε ) , (376)with Ω α ( ˜ t , ε ) = ˜ Ω ( ) α ( ˜ t ) + ε ˜ Ω ( ) α ( ˜ t ) + O ( ε ) , (377)where ˜ Ω ( ) α ( ˜ t ) = Ω ( ) α [ ˜ p i ϕ ( ) ( ˜ t )] and ˜ Ω ( ) α ( ˜ t ) = ˜ p j ϕ ( ) ( ˜ t ) ∂ j Ω ( ) α [ ˜ p j ϕ ( ) ( ˜ t )] .There is a tradeoff in solving Eqs. (372)–(375) rather than Eqs. (336) and (337):Eqs. (372)–(375) double the number of numerical variables, but they are indepen-dent of ε , meaning they can be solved for all values of ε simultaneously. Eqs. (336)and (337) have half as many variables, but they cannot be solved without first spec-ifying a value of ε .Since the waveform phase in a binary is directly related to the orbital phase, theexpansion (370) provides a simple means of assessing the level of accuracy of agiven approximation. The approximation that includes only the first term, ˜ ϕ ( ) α , iscalled the adiabatic approximation (denoted 0PA); it consists of the coupled equa-tions (372) and (373), which describe a slow evolution of the geodesic frequencies.An approximation that includes all terms through ˜ ϕ ( n ) α is called an nth post-adiabaticapproximation (denoted n PA); it consists of the coupled equations (372)–(375). Wereturn to the efficacy of 0PA and 1PA approximations in the final section of thisreview. Ref. [88] determined what inputs are required for a 0PA or 1PA approxima-tion. To describe these inputs, we define the time-reversal ψ α → − ψ α , f α ( ψψψ ) → ε α f α ( − ψψψ ) , where f α is the accelerating force, ε α : = ( − , , , − ) , and there isno summation over α . We then define the dissipative and conservative pieces of theforce: f α diss = f α ( ψψψ ) − ε α f α ( − ψψψ ) , (378) f α con = f α ( ψψψ ) + ε α f α ( − ψψψ ) . (379)These definitions imply that under time reversal, f α diss → − f α diss and f α con → + f α con . Itis straightforward to see from Eqs. (292)–(298) and the definition of L i that d p i / dt only receives a direct contribution from f α diss , while d ψ a / dt only receives a directcontribution from f α con . At 0PA order, f α enters the evolution through Eq. (373), inthe quantity Γ i ( ) = (cid:42) d p i dt (cid:12)(cid:12)(cid:12)(cid:12) f α → f α ( ) (cid:43) ϕϕϕ = ϒ ( ) t (cid:42) d p i d λ (cid:12)(cid:12)(cid:12)(cid:12) f α → f α ( ) (cid:43) qqq . (380)Hence, the 0PA approximation only requires f α ( ) diss . At 1PA, f α enters the evolutionthrough both Γ i ( ) and Γ i ( ) in Eq. (375), where Γ i ( ) is given by Eq. (344) with (325),(321), (328), and with p i ( , ) chosen such that ϒ ( ) α =
0. These quantities involve f α ( ) diss [via (cid:104) g i ( ) (cid:105) qqq = (cid:104) dp i d λ | f α → f α ( ) (cid:105) qqq in Eq. (325)], f α ( ) con (via ψ ( ) α and p i ( , ) , whichare both partially determined by (cid:102) ( ) a = δ (cid:102) a | f α → f α ( ) ), and f α ( ) diss . Hence, the 1PAapproximation requires the entirety of f α ( ) as well as f α ( ) diss .The fact that dissipative effects dominate over conservative ones on the longtime scale of an inspiral is important in practical simulations of binaries. At leastat first order, the dissipative self-force is substantially easier to compute than theconservative self-force. We discuss this in the final section of this review.We refer to Ref. [106] for a pedagogical introduction to multiscale expansionsin more general contexts. Ref. [88] contains a detailed discussion of the multiscaleapproximation for self-accelerated orbits in Kerr spacetime. Refs. [160, 150] presentvariants of the method in simpler binary scenarios. Given that the orbital frequencies slowly evolve, they will occasionally encountera resonance. Typically [119], the frequencies will continue to evolve, transitioningout of the resonance. These transient resonances have significant impact on orbitalevolution. The near-identity averaging transformation ( ψ α , p i ) → ( q α , p iq ) becomes singu-lar at a resonance, as described below Eq. (321). Specifically, it becomes singularfor the mode numbers Nkkk res for which ϒ Nkkk res =
0. To assess the effect of a resonance,we start from the equations of motion in the form (334)–(335).The driving forces F i ( n ) and “frequencies” U ( ) α can be expanded in Fourier seriessuch as F i ( n ) = ∑ kkk F i ( n , kkk ) ( p j ) e − i ˆ q kkk . However, near a resonance, a set of apparentlyoscillatory terms becomes approximately stationary. Specifically, near a resonancewhere ϒ ( ) res : = k res r ϒ ( ) r + k res z ϒ ( ) z =
0, the phase q res = kkk res · ˆ qqq , and all integer mul-tiples of it, ceases to evolve on the orbital time scale. To see this, suppose the reso-nance occurs at a time λ res . Near that time, q res can be expanded in a Taylor series q res ( λ ) = q res ( λ res ) + ˙ q res ( λ res )( λ − λ res ) +
12 ¨ q res ( λ res )( λ − λ res ) + . . . (381)Since ˙ q res ( λ res ) ≈ ϒ ( ) res ( λ res ) = q res ≈ d ϒ ( ) res / d λ , we see that q res changes onthe time scale δ λ = (cid:115) d ϒ ( ) res / d λ ∼ √ ε , (382)which is much longer than the orbital time scale (but much shorter than theradiation-reaction time).During the passage through a resonance, these additional quasistationary drivingforces cause secular changes to the orbital parameters. We isolate these effects byperforming a partial near-identity averaging transformation that eliminates all oscil-lations from the evolution equations except those depending on the resonant anglevariable q res . An appropriate transformation, through 1PA order, is given byˆ q α ( qqq , p jq , ε ) = q α + ε B α β ( p iq ) q β + ε ∑ kkk (cid:54) = Nkkk res U ( , kkk ) α − F j ( , kkk ) i ϒ ( ) kkk ∂ϒ ( ) α ∂ p jq − i ϒ ( ) res ( p jq ) e − iq kkk , (383) p i ( qqq , p jq , ε ) = p iq + ε C i ( p jq ) + ε ∑ kkk (cid:54) = Nkkk res F i ( , kkk ) − i ϒ ( ) res e − iq kkk , (384)where functions of p j inside the sums are evaluated at p jq , and B α β and C i are anyfunctions satisfying B α β ϒ ( ) β = C j ∂ p jq ϒ ( ) α + (cid:104) U ( ) α (cid:105) qqq . (385)These transformations satisfy the analogs of Eqs. (316) and (322) but with reso-nant modes excluded, with B α β and C i chosen to eliminate the frequency correc-tions ϒ ( ) α , and with the simplifications that (cid:102) ( ) a is replaced by ϒ ( ) a and ∆ ψ ( ) a by0 (consequences of q ( ) α already being a leading-order action angle). Together, thetransformations (383) and (384) bring the equations of motion to the form dq α d λ = ϒ ( ) α ( p jq ) + O ( ε ) , (386) d p iq d λ = ε G i ( ) ( p jq ) + ε ∑ N (cid:54) = F i ( , Nkkk res ) ( p jq ) e − iNq res + O ( ε ) , (387)with G i ( ) = (cid:104) F i ( ) (cid:105) qqq . For simplicity, we suppress 1PA terms.How much does the second term in Eq. (387) contribute to the evolution of p iq ?Far from resonance, the additional term averages to zero. If we denote the term as εδ G i , then across resonance, it contributes an amount δ p iq = ε (cid:82) δ Gd λ . Applyingthe stationary phase approximation to the integral, we find δ p i = ∑ N (cid:54) = F i ( , Nkkk res ) (cid:115) πε | N ˙ ϒ ( ) res | exp (cid:20) sgn (cid:16) N ˙ ϒ ( ) res (cid:17) i π + iNq res ( λ res ) (cid:21) + o ( √ ε ) , (388)where we use a dot to denote a derivative with respect to ˜ λ , ˙ ϒ ( ) res : = d ϒ ( ) res d ˜ λ = G i ( ) ∂ϒ ( ) res ∂ p iq , and both ˙ ϒ ( ) res and F i ( , Nkkk res ) are evaluated at p jq ( λ res ) . The magnitude of δ p i is ∼ √ ε ; intuitively, this corresponds to a quasistationary driving force of size ∼ ε multiplied by the resonance-crossing time δ λ ∼ / √ ε . But δ p i is not a simpleproduct of the two; each quasistationary driving force is weighted by a phase factor,such that δ p i depends sensitively on the value of the resonant phase at resonance, q res ( λ res ) . This implies that in order to determine δ p i at leading order, one mustknow the 1PA phase evolution prior to resonance.A proper accounting of the passage through resonance requires matching a near-resonance expansion to an off-resonance, multiscale expansion; see Sec. III ofRef. [119] or Appendix B of Ref. [18] for demonstrations of this matching pro-cedure. Because a resonance shifts the orbital parameters by an amount ∼ √ ε , andthe shifted parameters subsequently evolve over the long time scale ∼ / ε , the res-onance introduces half-integer powers into the multiscale expansion. For example,after a resonance, Eqs. (370)–(371) become ϕ α ( ˜ t , ε ) = ε (cid:104) ˜ ϕ ( ) α ( ˜ t ) + ε / ˜ ϕ ( / ) α ( ˜ t ) + ε ˜ ϕ ( ) α ( ˜ t ) + O ( ε / ) (cid:105) , (389) p i ϕ ( ˜ t , ε ) = ˜ p i ϕ ( ) ( ˜ t ) + ε / ˜ p i ϕ ( / ) ( ˜ t ) + ε ˜ p i ϕ ( ) ( ˜ t ) + O ( ε / ) . (390)The effect of a single resonance therefore dominates over all other post-adiabaticeffects. However, determining the resonant corrections ˜ ϕ ( / ) α and p i ϕ ( / ) requiresthe shifts (388), which in turn require the resonant phase q res through 1PA order.This means that the 1/2-post-adiabatic-order corrections can be thought of as outsize1PA corrections.Further discussions of transient resonances can be found in Refs. [63, 70, 64,172, 119, 99, 18, 125, 101]. Because resonances are dense in the parameter space,an inspiraling body will pass through an infinite number of them. However, becausethe forcing coefficients F i ( , kkk ) decay with increasing kkk , the only resonances with significant impact are “low-order” resonances, such as ϒ r / ϒ z = /
2. A large fractionof inspiraling orbits will encounter such a resonance in the late inspiral [172, 18],but neglecting the effect of resonance in EMRIs may lead to only a small loss ofdetectable signals [18].In addition to the intrinsic r – z orbital resonances discussed here, resonances canalso occur due to a variety of other effects. There can be extrinsic resonances inwhich k res r Ω r + k res z Ω z + k res φ Ω φ = ( k res r , k res z , k res φ ) ; these lead tonon-isotropic emission of gravitational waves, causing possibly observable kicks tothe system’s center of mass [89, 120], but their effects are subdominant relative to r – z resonances. If the secondary is spinning, its spin can also create resonances [211],as can the presence of external matter source such as a third body [25, 210]. In this section we describe how to combine the methods of the previous sectionsto model small-mass-ratio binaries. This consists of solving the global problem ina Kerr background: the perturbative Einstein equations with a skeleton source (i.e.,a point particle or effective source) moving on a trajectory governed by Eq. (195)or (198). The first part of the section summarizes a multiscale expansion of thefield equations, building directly on our treatment of orbital dynamics. At adiabaticorder, waveforms can be generated by solving the linearized Einstein or Teukolskyequation with a point particle source and calculating the dissipative first-order self-force. At 1PA order, one must solve the second-order Einstein equation and computethe first-order conservative self-force and second-order dissipative self-force. These1PA calculations require, as a central ingredient, a mode decomposition of the sin-gular field; this is the subject of the second part of the section.For simplicity, we assume the small object is spherical and nonspinning and thatit does not encounter any significant orbital resonances.
Like the orbital dynamics, the metric in a binary has two distinct time scales: theorbital periods T α = π / Ω α and the long radiation-reaction time ∼ / ( ε T α ) . Theevolution on the orbital time scale is characterized by periodic dependence on theorbital action angles ϕ α , which satisfy Eq. (336). The evolution on the radiation-reaction time is characterized by a slow change of the orbital parameters p i ϕ , gov-erned by Eqs. (337), and of the central black hole parameters ( M BH , J BH ) , whichevolve due to absorption of energy and angular momentum according to Eqs. (97) and (99). If we did not neglect the small object’s spin and higher moments, theywould come with additional parameters and phases [173, 208].The black hole parameters change at a rate F H ∝ | h | ∼ ε . Over the radiation-reaction time, this accumulates to a change ∼ ε , allowing us to write the evolvingparameters as M BH = M + εδ M and J BH = J + εδ J , where M and J are constant and M A : = ( δ M , δ J ) evolve on the radiation-reaction time. We then work on the fixedKerr background with parameters M and a = J / M , with a set of slowly evolvingsystem parameters P α = { p i ϕ , M A } .We will use the split into action angles ϕ α and system parameters P α to expandthe metric perturbation and stress-energy as h µν = ∑ n = ∑ (cid:109) , kkk ε n h ( n (cid:109) kkk ) µν ( P α , xxx ) e i (cid:109) φ − i ϕ (cid:109) kkk + O ( ε ) , (391) T µν = ∑ n = ∑ (cid:109) , kkk ε n T ( n (cid:109) kkk ) µν ( P α , xxx ) e i (cid:109) φ − i ϕ (cid:109) kkk + O ( ε ) , (392)where (cid:109) , k r , k z all run from − ∞ to ∞ , xxx = ( r , z ) , and ϕ (cid:109) kkk : = (cid:109) ϕ φ + k r ϕ r + k z ϕ z .Here ϕ α and P α are functions of t and ε governed by Eqs. (336), (337), and dM A dt = ε F ( ) A ( p i ϕ ) + O ( ε ) , (393)where F ( ) A = ( F H E / ε , F H L z / ε ) is given by any of Eqs. (97) and (99), Eqs. (140a)and (141a), or Eqs. (156b) and (157b); the reason this depends only on p i ϕ at lead-ing order is explained in Sec. 7.1.4 below. The decomposition into azimuthal modes e i (cid:109) φ is not strictly necessary here, but it simplifies the analysis of the stress-energyin the next subsection, and it dovetails with the decompositions into angular har-monics in Sec. 4, as all the bases of harmonics involve φ only through the factor e i (cid:109) φ .The expansions (391) and (392) differ slightly from the “self-consistent expan-sion” (192) in that the parameters P in the self-consistent expansion include thecomplete trajectory z µ and its derivatives. We can therefore move from Eqs. (192)and (205) to Eqs. (391) and (392) by substituting the expansion of z α ( t ) fromEq. (356). To fully motivate our multiscale expansion, we work through this ex-pansion of T µν in the next subsection.But first, we focus on the overall structure and efficacy of the multiscale expan-sion. Given Eqs. (391) and (392), the perturbative field equations become equationsfor the Fourier coefficients h ( n (cid:109) kkk ) µν . These are identical, at leading order, to the usualfrequency-domain field equations of black hole perturbation theory, with discretefrequencies d ϕ (cid:109) kkk dt = ω (cid:109) kkk ( p i ϕ ) : = k r Ω ( ) r ( p i ϕ ) + k z Ω ( ) z ( p i ϕ ) + (cid:109) Ω ( ) φ ( p i ϕ ) . (394) More concretely, if we substitute the expansions (391) and (392) into the Einsteinequations, then t derivatives act as ∂∂ t = Ω ( ) α ( p j ϕ ) ∂∂ ϕ α + d P α dt ∂∂ P α (395) → − i ω (cid:109) kkk ( p j ϕ ) + ε (cid:34) Γ i ( ) ( p j ϕ ) ∂∂ p i ϕ + F ( ) A ( p j ϕ ) ∂∂ M A (cid:35) + O ( ε ) . (396)Using this, we can write covariant derivatives as ∇ α → ˜ ∇ (cid:109) kkk α + εδ t α ˜ ∂ (cid:109) kkkt + O ( ε ) , (397)where ˜ ∇ (cid:109) kkk α is an ordinary covariant derivative with ∂ φ → i (cid:109) and ∂ t → − i ω (cid:109) kkk ,and ˜ ∂ (cid:109) kkkt is the operator in square brackets in Eq. (396). If we then treat ϕ α and P α as independent variables, we can split the field equations into coefficients of e i (cid:109) φ − i ϕ (cid:109) kkk and of explicit powers of ε . This results in a sequence of differentialequations in ( r , z ) for the coefficients h ( n (cid:109) kkk ) µν : G ( (cid:109) kkk ) µν [ h ( (cid:109) kkk ) ] = π T ( (cid:109) kkk ) µν , (398) G ( (cid:109) kkk ) µν [ h ( (cid:109) kkk ) ] = π T ( (cid:109) kkk ) µν − ∑ (cid:109) (cid:48) (cid:109) (cid:48)(cid:48) ∑ kkk (cid:48) kkk (cid:48)(cid:48) G ( (cid:109) kkk ) µν [ h ( (cid:109) (cid:48) kkk (cid:48) ) , h ( (cid:109) (cid:48)(cid:48) kkk (cid:48)(cid:48) ) ] − Γ i ( ) ˙ G ( (cid:109) kkk ) µν [ ∂ p i ϕ h ( (cid:109) kkk ) ] − F ( ) A ˙ G ( (cid:109) kkk ) µν [ ∂ M A h ( (cid:109) kkk ) ] . (399)Here G ( (cid:109) kkk ) µν and G ( (cid:109) kkk ) µν are the linearized and quadratic Einstein tensors (3) and(4) with the replacement ∇ α → ˜ ∇ (cid:109) kkk α . ˙ G ( (cid:109) kkk ) µν is the piece of G ( ) µν that, after apply-ing the rule (397), is linear in ˜ ∂ (cid:109) kkkt . Explicit expressions for these quantities canbe found in Sec. VC of Ref. [126] in a Schwarzschild background in the Lorenzgauge. The left-hand side of the field equations (398) and (399) is identical to what itwould be if we expanded h µν in Fourier modes e i (cid:109) φ − i ω (cid:109) kkk t . Such a Fourier ex-pansion is what has been implemented historically in first-order frequency-domaincalculations with geodesic sources (e.g., [45, 95, 98, 58, 48, 67, 1, 91, 105, 180, 64,92, 2, 136, 93, 123, 121]), and we can now immediately re-interpret those computa- The field equations in Ref. [126] are further specialized to quasicircular orbits, with frequencies ω (cid:109) = (cid:109) Ω ( ) φ , but they remain valid under the replacement ω (cid:109) → ω (cid:109) kkk . In Sec. VC of Ref. [126]they also include frequency corrections Ω ( ) φ , which we have eliminated here with our choice ofaveraged variables ( ϕ α , p i ϕ ) ; the analogue of our choice is described in their Appendix A. Beyondthese minor differences, they more substantially differ by allowing the phases and system variablesto depend on r in addition to t . We discuss the reason for this in Sec. 7.1.3.7 tions as leading-order implementations of the expansion (391). This is a principaladvantage of using the variables ( ϕ α , p i ϕ ) instead of ( q α , p iq ) .Importantly, Eqs. (398) and (399) can be solved for any values of the parameters P α , without having to simulate complete inspirals. At each point in the parameterspace, the solution, comprising the set of amplitudes h ( n (cid:109) kkk ) µν , can loosely be thoughtof as a “snapshot” of the spacetime in the frequency domain. These solutions canbe used to calculate the driving forces in the evolution equations (337) and (393)for d P α / dt . After populating the space of snapshots, one can then use these evo-lution equations, together with the phase evolution equation (336), to evolve anyparticular binary spacetime through the space. Note that even though each snapshotis determined by an “instantaneous” value P α , each snapshot fully accounts fordissipation and for the nongeodesic past history of the binary: because the evolutionis slow compared to the orbital time scale, these effects are suppressed by a powerof ε and are incorporated through the ˙ G ( (cid:109) kkk ) µν source terms in Eq. (399).What would go wrong if, rather than using this multiscale expansion, we wereto actually use h ( ) µν = ∑ (cid:109) kkk h ( (cid:109) kkk ) µν ( r , z ) e i (cid:109) φ − i ω (cid:109) kkk t as our first-order metric perturba-tion? This would be approximating the trajectory of the companion as a geodesicof the background black hole spacetime. As explained in the discussion aroundEq. (193), such an approximation would accumulate large errors with time: the“small” corrections to the trajectory would grow large as the object spirals inward.The growing correction, represented by z α = m α / m in Eq. (193), would manifestitself as a dipole term in h ( ) µν that would grow until h ( ) µν became larger than h ( ) µν ,spelling the breakdown of regular perturbation theory.We can now understand this behavior directly from the orbital phases. If we wereto use the geodesic phases ω (cid:109) kkk t , we would be implicitly expanding the phase ϕ α ( t , ε ) = (cid:90) t Ω ( ) α [ p j ϕ ( ε t (cid:48) , ε )] dt (cid:48) + ϕ α (400)in powers of ε , as ϕ α ( t , ε ) = ˜ Ω ( ) α ( ) t + ε (cid:34) d ˜ Ω ( ) α d ˜ t ( ) t + ˜ Ω ( ) α ( ) t (cid:35) + O ( ε ) + ϕ α , (401)where we have used Eq. (377). Such an expansion would be accurate on the orbitaltimescale but would accumulate large errors on the dephasing time ∼ / √ ε , whichis much shorter than the radiation reaction time. Moreover, the order- ε terms inthis expansion would appear as non-oscillatory, linear- and quadratic-in- t terms in h ( ) µν , implying that h ( ) µν would not admit a discrete Fourier expansion or correctlydescribe the system’s approximate triperiodicity. The multiscale expansion avoids First-order implementations in the time domain [9, 12, 184, 13, 56, 83, 8] do not mesh quite soreadily with a multiscale expansion. We discuss their utility within a multiscale expansion in latersubsections.8 these errors and maintains uniform accuracy prior to the transition to plunge (andexcluding resonances).The basic idea of this multiscale expansion of the field equations was first putforward in Ref. [88]. It is described in detail in Ref. [126] for the special case ofquasicircular orbits in Schwarzschild spacetime. Our presentation here, building onour particular treatment of orbital motion in the preceding section, is the most com-plete description to date of the generic case. We provide additional details below. Athorough description is in preparation [61].
We illustrate, and further motivate, the multiscale expansion by examining the mul-tiscale form of the source terms in the coupled equations (204) and (198): the De-tweiler stress-energy and the self-force.We start with the stress-energy (205). Writing the trajectory as z α ( t ) = [ t , r o ( t ) , z o ( t ) , φ o ( t )] (402)(where the subscript stands for “object’s orbit”), setting the spin to zero, using the δ function to evaluate the integral, and expanding the factors of √− ˘ g and d τ d ˘ τ , weexpress T µν as T µν = m ˘ g µα ˘ g νβ u α u β u t Σ (cid:104) + ε (cid:16) u γ u δ − g γδ (cid:17) h R ( ) γδ (cid:105) × δ [ xxx − xxx o ( t )] δ [ φ − φ o ( t )] + O ( ε ) . (403)We now take as a given our multiscale expansion (356) of z α ( t ) ; this assumed theform (307) for the force, which we return to below. Substituting (356) and using u µ = ˙ z µ / Σ , we obtain the coefficients in the expansion T µν = ε T ( ) µν ( ϕ α , p i ϕ ) + ε T ( ) µν ( ϕ α , p i ϕ ) + O ( ε ) . (404)The leading term is T ( ) µν ( ϕ α , p i ϕ ) = m ˙ z ( ) µ ˙ z ( ) ν (cid:102) ( ) t Σ ( ) δ [ xxx − xxx ( ) ( ϕϕϕ , p i ϕ )] δ [ φ − φ ( ) ( ϕ α , p i ϕ )] , (405)where Σ ( ) : = r ( ) + a z ( ) , ˙ z ( ) µ : = g µν ( xxx ( ) ) ˙ z ν ( ) , ˙ z µ ( ) = (cid:102) ( ) µ ( ψψψ ( ) , p i ϕ ) for µ = t , φ ,and ˙ xxx ( ) is given by Eq. (299) with p i → p i ϕ and ψ a → ψ ( ) a . The second-order termis T ( ) µν ( ϕ α , p i ϕ ) = m (cid:102) ( ) t Σ ( ) (cid:20) z ( )( µ h R ( ) ν ) β ˙ z β ( ) +
12 ˙ z ( ) µ ˙ z ( ) ν (cid:16) u γ ( ) u δ ( ) − g γδ ( ) (cid:17) h R ( ) γδ (cid:21) × δ [ xxx − xxx ( ) ( ϕϕϕ , p i ϕ )] δ [ φ − φ ( ) ( ϕ α , p i ϕ )] + z α ( ϕ , ) ∂ T ( ) µν ∂ x α ( ) , (406)where u µ ( ) = ˙ z µ ( ) / Σ ( ) , g γδ ( ) : = g γδ ( xxx ( ) ) , and h R ( ) γδ is evaluated at z α ( ) . The last termin Eq. (406) involves the action of z α ( ϕ , ) ∂∂ x α ( ) on ˙ z µ ( ) ( ψψψ ( ) , p i ϕ ) ; this can be evaluatedusing xxx ( ϕ , ) · ∂∂ xxx ( ) = ψψψ ( ϕ , ) · ∂∂ ψψψ ( ) + p i ( ϕ , ) ∂∂ p i ϕ , (407)with ψψψ ( ϕ , ) and p i ( ϕ , ) given by Eqs. (352) and (353).Next, we consider the mode decomposition of the expanded stress-energy. Wefirst define the mode coefficients T ( n (cid:109)(cid:109) (cid:48) kkk ) µν : = ( π ) (cid:73) T ( n ) µν e i ϕ (cid:109) kkk − i (cid:109) (cid:48) φ d ϕ d ϕ φ d φ , (408)which assume no relationship between the dependence on φ and ϕ φ . Substituting T ( ) µν from Eq. (405), using the azimuthal δ function to evaluate the integral over φ ,inserting Eq. (271) for φ ( ) , and using (cid:72) e i ( (cid:109) − (cid:109) (cid:48) ) ϕ φ d ϕ φ = πδ (cid:109)(cid:109) (cid:48) , we obtain T ( (cid:109)(cid:109) (cid:48) kkk ) µν = δ (cid:109)(cid:109) (cid:48) ( π ) (cid:73) m ˙ z ( ) µ ˙ z ( ) ν (cid:102) ( ) t Σ ( ) δ ( xxx − xxx ( ) ) e i ϕ kkk − i (cid:109) ∆ ϕ φ ( ) d ϕ . (409)This enforces (cid:109) (cid:48) = (cid:109) , establishing that the stress-energy only depends on φ and ϕ φ in the combination e i (cid:109) ( φ − ϕ φ ) . We can now do away with the (cid:109) (cid:48) label and eval-uate the integral in Eq. (409) in the form (265) or (266). The result is T ( (cid:109) kkk ) µν = m ϒ ( ) r ϒ ( ) z ( π ) ϒ ( ) t ∑ σ r = ± σ z = ± ˙ z ( ) µ ( ψψψ σ ) ˙ z ( ) ν ( ψψψ σ ) Σ ( xxx ) ˙ r ( ) ( ψ σ r r ) ˙ z ( ) ( ψ σ z z ) e i ϕ kkk ( ψψψ σ ) − i (cid:109) ∆ ϕ φ ( ) ( ψψψ σ ) × [ θ ( r − r p ) − θ ( r − r a )] θ ( z max − | z | ) , (410)where the various quantities have been defined as functions of the field point xxx =( r , z ) , and σ a = ± refers to a portion of the orbit in which x a is increasing ( σ a = + )or decreasing ( σ a = − ). ψ ± r ( r ) is the value of ψ r satisfying Eq. (217) (with p i → p i ϕ ) on an outgoing ( + ) or ingoing ( − ) leg of the radial motion; ψ ± z ( z ) is definedanalogously from Eq. (218). ϕ kkk ( ψψψ σ ) is given by ϕ kkk ( ψψψ σ ) = q kkk ( ψψψ σ ) + Ω ( ) kkk · [ δ t r ( ψ σ r r ) + δ t z ( ψ σ z z )] , (411) with q a ( ψ a ) by Eq. (241), and δ t a ( ψ a ) by Eq. (267). ∆ ϕ φ ( ) ( ψψψ σ ) is given byEq. (272). The Mino-time velocities are ˙ z ( ) µ ( ψψψ σ ) = g µν ( xxx ) ˙ z ν ( ) ( ψψψ σ ) with ˙ x a ( ) ( ψψψ ) given by Eq. (299) [or (206) and (207)] and ˙ t and ˙ φ by Eqs. (208) and (209). Wecan also use ˙ z ( ) t = − E ϕ Σ and ˙ z ( ) φ = L ϕ Σ ; recall that we suppress the subscript z on L z in P i ϕ = ( E ϕ , L ϕ , Q ϕ ) .This calculation demonstrates how T ( ) µν inherits the form (392) from the trajec-tory z µ . Given this form of T ( ) µν , the linearized Einstein equation preserves it (in anappropriate class of gauges), justifying our ansatz for h ( ) µν . Given that form of h ( ) µν ,the second-order stress-energy (406) inherits the same form, as do the other sourcesin Eq. (399), and so, finally, does h ( ) µν .All of this relies on the presumed form (307) for the force, from which we derivedthe form (356) for z µ . Our force on the right-hand side of Eq. (198) is not quiteof that form. To derive its form, first note that, assuming Eq. (356), the puncturefield h P µν has a form analogous to Eq. (391), and therefore h R µν does as well. If wewrite this as h R µν ( P α , xxx , ϕ φ − φ , ϕϕϕ , ε ) , apply a covariant derivative using (397), andevaluate the result on the trajectory z µ ( t ) , then the right-hand side of Eq. (198) takesthe form f µ ( P α , xxx o , ˙ z α , ϕϕϕ , ε ) , (412)where we have used ϕ φ − φ o = − ∆ ϕ φ ( ) ( ϕϕϕ ) − εφ ( ϕ , ) ( ϕϕϕ ) to eliminate dependenceon ϕ φ . This differs from Eq. (307) in two ways: it depends explicitly on ( ϕϕϕ , p i ϕ ) ,and it depends on the additional parameters M A .With respect to the first difference, we can use Eqs. (354) and (355) to write theforce in the form f µ = ε f µ ( ) ( ψψψ , p i , M A ) + ε f µ ( ) ( ψψψ , p i , M A ) + O ( ε ) . (413)The system of equations (309) and (310) thus becomes d ψ α d λ = (cid:102) ( ) α ( ψψψ , p j ) + ε (cid:102) ( ) α ( ψψψ , p j , M A ) + O ( ε ) , (414) d p i d λ = ε g i ( ) ( ψψψ , p j , M A ) + ε g i ( ) ( ψψψ , p j , M A ) + O ( ε ) , (415) dM A d λ = ε (cid:102) ( ) t ( ψψψ , p i ) F ( ) A ( p i ) + O ( ε ) . (416)The analysis of these equations then follows essentially without change as inSecs. 6.2.3–6.2.5. To see why the use of Eqs. (309) and (310) does not lead to vi-cious circularity, note that their subleading terms only affect f µ ( ) , which only entersinto the dynamics in Eq. (325). The nongeodesic functions appearing in Eqs. (309)and (310) are therefore determined from lower-order equations prior to requiring f µ ( ) .Finally, how does M A influence the orbital dynamics? It enters into the drivingforces g i ( n ) and (cid:102) ( ) α . However, it does not enter into (cid:104) g ( ) i (cid:105) . This follows from the fact that f µ ( ) con depends on M A but f µ ( ) diss does not, as explained in Sec. 7.1.4 below. M A therefore contributes to the action-angle dynamics at 1PA order via Eq. (325), aswell as to the coordinate trajectory correction z µ ( ϕ , ) at 1PA order through ψ ( ϕ , ) α and p i ( ϕ , ) . This is the only material change to our treatment of the orbital dynamics inSecs. 6.2.3–6.2.5.Together, the analyses of this section establish the consistency of our multiscaletreatments of the field equations and orbital motion. In the following sections, wedescribe more concretely how to utilize these treatments. Snapshot solutions, consisting of the mode amplitudes h n (cid:109) kkk µν ( P α , xxx ) , can be com-puted using any of the frequency-domain methods reviewed in Sec. 4. As an ex-ample, in this section we sketch how this is done at first order using the method ofmetric reconstruction in the radiation gauge, starting from the Teukolsky equation.This summarizes work from Ref. [121], which provided the first calculation of thefull first-order self-force for generic bound orbits in Kerr spacetime. Our summaryalso appeals to methods and results from Refs. [134, 58, 162, 124, 42].We first define leading-order Weyl scalars ψ and ψ related to the h ( ) µν ofEq. (391) by Eqs. (54)–(55b) with the replacements ∂ t → − i ω (cid:109) kkk and ∂ φ → i (cid:109) .For concreteness, we use ψ . In analogy with Eqs. (81) and (82), it can be writtenas ψ = ∞ ∑ (cid:96) = (cid:96) ∑ (cid:109) = − (cid:96) ∑ kkk ψ (cid:96) (cid:109) kkk ( p i ϕ , r ) S (cid:96) (cid:109) ( θ , φ ; a ω (cid:109) kkk ) e − i ϕ (cid:109) kkk . (417)Note that the radial coefficients depend on p i ϕ but not on M A ; this is because thelinearized ψ and ψ are insensitive to linear perturbations of the central black hole’smass or spin [198].The coefficients ψ (cid:96) (cid:109) kkk ( p i ϕ , r ) satisfy the radial Teukolsky equation (84) with ω → ω (cid:109) kkk and s ψ (cid:96) (cid:109) ω → s ψ (cid:96) (cid:109) kkk . The source in that equation is constructed fromthe stress-energy (405) or its modes (410) using the analog of Eq. (90a), T (cid:96) (cid:109) kkk = − π Σ (cid:90) z max − z max ( ˜ S (cid:109) kkk T )( r , z ) S (cid:96) (cid:109) ( θ , a ω (cid:109) kkk ) dz , (418)where the integral ranges over the support of T ( (cid:109) kkk ) µν , θ is related to z by z = cos θ ,and we have suppressed the dependence on p i ϕ . The source ˜ S (cid:109) kkk T in the inte-grand is given by Eq. (59) with T ll → T ( (cid:109) kkk ) ll (and the same for other tetrad compo-nents), ∂ t → − i ω (cid:109) kkk , and ∂ φ → i (cid:109) . What may appear to be an extra factor of 2 π inEq. (418) accounts for the factor of 1 / ( π ) introduced in the integration over φ inEq. (408).The retarded solution to the Teukolsky equation, as given in the variation-of-parameters form (93), is ψ (cid:96) (cid:109) kkk ( r ) = C in (cid:96) (cid:109) kkk ( r ) R in (cid:96) (cid:109) kkk ( r ) + C up (cid:96) (cid:109) kkk ( r ) R up (cid:96) (cid:109) kkk ( r ) , (419)where we have defined the homogeneous solutions R in/up (cid:96) (cid:109) kkk ( r ) : = R in/up (cid:96) (cid:109) ω (cid:109) kkk ( r ) . Theweighting coefficients are given by Eq. (95), which we restate here as C in (cid:96) (cid:109) kkk ( r ) : = (cid:90) r a r R up (cid:96) (cid:109) kkk ( r (cid:48) ) W ( r (cid:48) ) ∆ T (cid:96) (cid:109) kkk ( r (cid:48) ) dr (cid:48) , (420a) C up (cid:96) (cid:109) kkk ( r ) : = (cid:90) rr p R in (cid:96) (cid:109) kkk ( r (cid:48) ) W ( r (cid:48) ) ∆ T (cid:96) (cid:109) kkk ( r (cid:48) ) dr (cid:48) . (420b)In the vacuum regions r > r a and r < r p , outside the support of T (cid:96) (cid:109) kkk , the weightingcoefficients become constants, ˆ C in (cid:96) (cid:109) kkk = (cid:90) r a r p R up (cid:96) (cid:109) kkk ( r (cid:48) ) W ( r (cid:48) ) ∆ T (cid:96) (cid:109) kkk ( r (cid:48) ) dr (cid:48) , (421a) ˆ C up (cid:96) (cid:109) kkk = (cid:90) r a r p R in (cid:96) (cid:109) ω (cid:109) kkk ( r (cid:48) ) W ( r (cid:48) ) ∆ T (cid:96) (cid:109) kkk ( r (cid:48) ) dr (cid:48) , (421b)and in those regions the solution becomes ψ (cid:96) (cid:109) kkk ( r ) = (cid:40) ˆ C in (cid:96) (cid:109) kkk R in (cid:96) (cid:109) kkk ( r ) for r < r p ˆ C up (cid:96) (cid:109) kkk R up (cid:96) (cid:109) kkk ( r ) for r > r a . (422)We can evaluate the r and z integrals in C in/up (cid:96) (cid:109) kkk as integrals over ψ r and ψ z byusing appropriate changes of variables for each value of σ a in Eq. (410). For σ r = + and a generic function f ( r ) , the radial integrals are (cid:82) rr p f dr (cid:48) = (cid:82) ψ + r ( r ) f drd ψ r d ψ r and (cid:82) r a r f dr (cid:48) = (cid:82) πψ + r ( r ) f drd ψ r d ψ r ; for σ r = − , they are (cid:82) rr p f dr (cid:48) = − (cid:82) πψ − r ( r ) f drd ψ r d ψ r and (cid:82) r a r f dr (cid:48) = − (cid:82) ψ − r ( r ) π f drd ψ r d ψ r . The transformations for σ z = ± are analogous. Wecan also write the r and z derivatives in ˜ S (cid:109) kkk T as ∂∂ x a = ∂ψ a ∂ x a ∂∂ψ a (with no sum over a ). For more explicit formulas for the integrands, see Sec. 3B of Ref. [58]. See also,e.g., Refs. [67, 93] for discussion of practical methods of numerically evaluatingsuch integrals.The modes of ψ (or ψ ) are by themselves sufficient to calculate many quanti-ties, such as gravitational-wave fluxes. But for other purposes, such as the calcula-tion of the self-force and the needed input for the second-order field equations, onemust compute the entire metric perturbation. Starting from the modes of ψ or ψ ,this can be done using the method of metric reconstruction reviewed in Sec. 4.1.4.In the presence of a source, metric reconstruction typically yields a metric perturba-tion that has a gauge singularity extending in a “shadow” from the matter source tothe black hole horizon or from the matter to infinity [134, 81]. In the case of a pointparticle, this shadow becomes a string singularity. However, we can more usefullyreconstruct the metric perturbation in a “no-string” radiation gauge [162], in which it has no string but does have a jump discontinuity and radial δ function on a sphereof varying radius r = r ( ) ( t ) .To construct the no-string solution in practice, we first find a Hertz potential ψ IRG satisfying Eq. (68a) (at fixed p i ϕ ) in the disjoint vacuum regions r < r p and r > r p , subject to regularity at infinity and the horizon. The appropriate solution ineach region is given by Eq. (15) of Ref. [134]. In the libration region r p < r < r a ,the radial source T (cid:96) (cid:109) kkk is nonzero, as the Fourier decomposition smears the pointparticle source over the entire toroidal region { r p < r < r a , | z | < z max } . The solu-tion (15) of Ref. [134] therefore cannot be used in the libration region. However, itcan be analytically extended into that region, using Eq. (422) in place of Eq. (419).Because the time-domain solution is analytic everywhere except on the sphere at r ( ) ( t ) , the sum over (cid:109) kkk of the analytically continued functions from r < r p yieldsthe correct result for all r ≤ r ( ) ( t ) , and the sum over (cid:109) kkk of the analytically con-tinued functions from r > r a yields the correct result for all r ≥ r ( ) ( t ) [123, 121];this is the method of extended homogeneous solutions [16, 91]. As alluded toin Sec. 4.1.7, this method was originally devised to alleviate another problem thatarises in frequency-domain calculations for eccentric orbits: the sum over kkk modesof the inhomogeneous solution converges slowly within the libration region. In thecontext of metric reconstruction, the method allows one to avoid the complexitiesof nonvacuum reconstruction.From the extended modes of the Hertz potential, we can reconstruct modes of anincomplete metric perturbation, h ( (cid:109) kkk ) rec µν , using Eq. (67) (as ever, with ∂ t → − i ω (cid:109) kkk and ∂ φ → i (cid:109) ). To complete this perturbation, in the region r > r ( ) ( t ) we add massand spin perturbations, E ϕ ∂ g µν ∂ M and L ϕ ∂ g µν ∂ J , where the M derivative is taken at fixed J = Ma , and the J derivative at fixed M ; these account for the mass and spin that theparticle contributes to the spacetime [124, 42]. In general we must also add massand spin perturbations δ M ∂ g µν ∂ M and δ J ∂ g µν ∂ J throughout the spacetime (at fixed p i ϕ );these account for the slowly evolving corrections to the central black hole’s massand spin. Finally, in the region r < r ( ) ( t ) , we must add gauge perturbations thatensure the coordinates t and φ , and therefore the frequencies Ω ( ) α , have the samemeaning in the two regions r < r ( ) ( t ) and r > r ( ) ( t ) [179] (see also [20]).With the completed modes h ( (cid:109) kkk ) µν ( P α , xxx ) in hand, one can calculate any quan-tity of interest on the orbital timescale with fixed P α . In particular, one can calcu-late the first-order self-force and its dynamical effects using the mode-sum regular-ization formula derived in Ref. [162]; the formula in the no-string gauge is given byEq. (125) in that reference. See also Ref. [92] for a generalization of this method to problems with sources that are nowherevanishing. These corrections proportional to M A have not been added historically because for any specificsnapshot with parameters P α , call them P α , they can be absorbed with a redefinition M → M + εδ M and J → J + εδ J , setting M A =
0. However, in the context of an evolution, which movesthrough the space of P α values, they must always be included at 1PA order. Even at a single valueof P α where M A =
0, their time derivatives must be included in Eq. (399). See Ref. [126] for adiscussion.4
To date, Ref. [121] is the only work to carry out the entire calculation we havejust described for generic bound orbits in Kerr spacetime. However, for orbits inSchwarzschild spacetime and for equatorial orbits in Kerr, snapshot frequency-domain calculations of the complete h ( ) µν and f µ ( ) are now routine, whether in theLorenz gauge, Regge-Wheeler-Zerilli gauge, or no-string radiation gauge [12, 48,13, 1, 180, 56, 2, 136, 206, 123, 191]. Numerical implementations at second or-der, which are necessary for post-adiabatic accuracy, are still in an early stage buthave computed some physical quantities for quasicircular orbits in Schwarzschildspacetime [163].We can use the output of these snapshot calculations to obtain the true, evolvinggravitational waveforms. Once the snapshot mode amplitudes are calculated, fromthem we can calculate the inputs for the evolution equations (336), (337), and (393).In analogy with Eqs. (96), (139), and (155), the waveforms are then given by any of h + − ih × = ∑ (cid:96) (cid:109) kkk − ˆ C up1 (cid:96) (cid:109) kkk [ p i ϕ ( ˜ u , ε )] ω (cid:109) kkk − S (cid:96) (cid:109) ( θ , φ ; a ω (cid:109) kkk ) e − i ϕ (cid:109) kkk ( ˜ u , ε ) + O ( ε ) , (423a) = ∑ (cid:96) (cid:109) kkk D (cid:16) ˆ C ZM,up1 (cid:96) (cid:109) kkk [ p i ϕ ( ˜ u , ε )] − i ˆ C CPM,up1 (cid:96) (cid:109) kkk [ p i ϕ ( ˜ u , ε )] (cid:17) × − Y (cid:96) (cid:109) ( θ , φ ) e − i ϕ (cid:109) kkk ( ˜ u , ε ) + O ( ε ) , (423b) = ∑ (cid:96) (cid:109) kkk ˆ C (cid:96) (cid:109) kkkmm [ p i ϕ ( ˜ u , ε )] Y (cid:96) (cid:109) ( θ , φ ) e − i ϕ (cid:109) kkk ( ˜ u , ε ) + O ( ε ) , (423c)where D = (cid:112) ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + ) , and ω (cid:109) kkk = ω (cid:109) kkk [ p i ϕ ( ˜ u , ε )] . Here we have writ-ten the waveform in terms of ˜ u : = ε ( t − r ∗ ) ; we return to this point below. We alsonote that we have given the waveform in terms of modes of ψ rather than the lessnatural (for this purpose) ψ . In analogy with Eq. (421), we have defined the am-plitudes ˆ C ZM,up1 (cid:96) (cid:109) kkk and ˆ C CPM,up1 (cid:96) (cid:109) kkk as the relevant weighting coefficients for r > r p , andwe have defined the Lorenz-gauge amplitudes ˆ C (cid:96) (cid:109) kkkmm : = lim r → ∞ ( re i ω (cid:109) kkk r ∗ h (cid:96) (cid:109) kkkmm ) .We have also intentionally inserted a label “1” onto the mode amplitudes and omit-ted O ( ε ) amplitudes. This is because even if we determine the phase ϕ (cid:109) kkk ( ˜ u , ε ) through 1PA order, the second-order amplitudes do not increase the waveform’s or-der of accuracy; an order- ε amplitude in the waveform is indistinguishable from a2PA (order- ε ) correction to the phase.The waveform (423) is in the time domain, but it is almost trivially related tothe frequency-domain waveform. Defining h ( ω ) : = π (cid:82) ∞ − ∞ ( h + − ih × ) e i ω u du andapplying the stationary-phase approximation, we obtain, e.g., h ( ω ) = π ∑ (cid:96) (cid:109) kkk (cid:115) πε | d ω (cid:109) kkk / d ˜ t | ˆ C (cid:96) (cid:109) kkkmm [ ˜ t (cid:109) kkk ( ω )] Y (cid:96) (cid:109) × exp (cid:26) i [ ω ˜ t (cid:109) kkk ( ω ) − ϕ (cid:109) kkk ( ω )] + sgn ( d ω (cid:109) kkk / d ˜ t ) i π (cid:27) + o ( √ ε ) . (424)Here ˜ t (cid:109) kkk ( ω ) is the solution to ω = ω (cid:109) kkk ( ˜ t ) , and the phase as a function of ω is ϕ (cid:109) kkk ( ω ) = ϕ (cid:109) kkk [ ˜ t (cid:109) kkk ( ω )] .Before proceeding, we return to the dependence on ˜ u rather than ˜ t . In Eq. (423),all functions of ˜ u are the functions obtained by solving (336), (337), and (393),simply evaluated as a function of ˜ u . For example, from Eq. (400), ϕ α ( ˜ u , ε ) = ε (cid:90) ˜ u Ω ( ) α [ p j ϕ ( ˜ t (cid:48) , ε )] dt (cid:48) + ϕ α . (425)This dependence on u is not a trivial consequence of the multiscale expansion (391).To justify it, one must adopt a hyperboloidal choice of time that asymptotes to u at I + or perform a matched-expansions calculation, matching the solution (391) toan outgoing-wave solution near I + . Ref. [126] discusses these points along withseveral additional advantages of using a hyperboloidal slicing. To see why the re-placement t → u is intuitively sensible, note that with it, Eq. (423) correctly reducesto a snapshot waveform on the orbital timescale if we fix p i ϕ and replace ϕ (cid:109) kkk ( ˜ u , ε ) with its geodesic approximation Ω ( ) (cid:109) kkk u (with fixed Ω ( ) (cid:109) kkk ); without the replacement,the multiscale waveform would not correctly reduce in this way.In the next two subsections, we outline the steps required to generate multiscalewaveforms at adiabatic (0PA) and 1PA order, whether in the time or frequency do-main. At this stage we consider the evolution equations in the form (372)–(375). Thereis no difference between that form and Eqs. (336)–(337) at adiabatic order, but weadopt the notation of Eqs. (372)–(375) here for consistency with our discussion ofthe 1PA approximation in the next subsection.For convenience, we transcribe the adiabatic evolution equations (372) and (373): d ˜ ϕ ( ) α d ˜ t = Ω ( ) α ( ˜ p j ϕ ( ) ) , (426) d ˜ p i ϕ ( ) d ˜ t = Γ i ( ) ( ˜ p j ϕ ( ) ) . (427)An adiabatic waveform-generation scheme consists of the following steps:1. Solve the field equation (398) or the associated Teukolsky equation or Regge-Wheeler-Zerilli equations, on a grid of p i ϕ values. At each grid point in parameter space, compute and store two things: the driving forces Γ i ( ) ( p i ϕ ) in Eq. (373) andthe asymptotic mode amplitudes at I + [e.g., ± ˆ C up1 (cid:96) (cid:109) kkk ( p i ϕ ) in the Teukolskycase].2. Using the stored values of Γ i ( ) , evolve through the parameter space by solvingthe coupled equations (426) and (427) to obtain the adiabatic parameters ˜ p i ϕ ( ) and phases ˜ ϕ ( ) r , ˜ ϕ ( ) z , ˜ ϕ ( ) φ as functions of ˜ t = ε t .3. Construct the adiabatic waveform using, e.g., h + − ih × = ∑ (cid:96) (cid:109) kkk − ˆ C up1 (cid:96) (cid:109) kkk [ ˜ p i ϕ ( ) ( ˜ u )] ω (cid:109) kkk − S (cid:96) (cid:109) ( θ , φ ; a ω (cid:109) kkk ) e − i ϕ ( ) (cid:109) kkk ( ˜ u ) / ε , (428)where ω (cid:109) kkk = ω (cid:109) kkk [ ˜ p i ϕ ( ) ( ˜ u )] .Starting from seminal work in Refs. [74, 128], two groups of authors have devel-oped practical implementations of this scheme [95, 175, 176, 98, 59, 75, 96, 101].One of the convenient aspects of the adiabatic approximation is that it can be im-plemented entirely in terms of the Teukolsky equation with a point-particle source,with no requirement to calculate a reconstructed and completed metric or to extractthe regular fields h R ( n ) µν . The reason is that, as explained around Eqs. (378) and (379),only the first-order dissipative force f µ ( ) diss is needed to calculate the driving force Γ i ( ) . This force is entirely due to the half-retarded minus half-advanced piece of h ( ) µν [128], h ( ) rad µν = h ( ) ret µν − h ( ) adv µν . (429)Because h ( ) rad µν is a vacuum solution to the linearized Einstein equation, it can bereconstructed from the half-retarded minus half-advanced piece of ψ or ψ , usingthe radiation-gauge reconstruction method reviewed in Sec. 4.1.4 (as translated tothe multiscale expansion in the previous section). Again because it is a vacuum solu-tion, it is smooth at the particle, and it is equal there to the relevant part of h R ( ) µν thatcreates f µ ( ) diss . Furthermore, h ( ) rad µν can contain no stationary perturbations, imply-ing it cannot contain any contribution from the mass and spin perturbations M A , soit needs no completion. Hence, one can evolve the orbit and generate the waveformentirely from mode amplitudes of ψ or ψ .Concrete formulas for adiabatic driving forces in terms of Teukolsky amplitudeswere first derived in Ref. [74], which showed that the average rates of change of E and L z due to f α ( ) diss satisfy a balance law: d ˜ E ( ) ϕ dt = − F H E − F I E , (430) d ˜ L ( ) ϕ dt = − F H L z − F I L z , (431) where ˜ P i ϕ ( ) = ( ˜ E ( ) ϕ , ˜ L ( ) ϕ , ˜ Q ( ) ϕ ) are related to ˜ p i ϕ ( ) by the geodesic relationships (225)–(227) between P i and p i . The fluxes are those due to the retarded field, which wecan translate from Eqs. (97)–(100) as F H E = ∑ (cid:96) (cid:109) kkk πα (cid:96) (cid:109) ω (cid:109) kkk ω (cid:109) kkk | − ˆ C in1 (cid:96) (cid:109) kkk | : = ∑ (cid:96) (cid:109) kkk F H (cid:96) (cid:109) kkkE , (432) F I E = ∑ (cid:96) (cid:109) kkk πω (cid:109) kkk | − ˆ C up1 (cid:96) (cid:109) kkk | : = ∑ (cid:96) (cid:109) kkk F I (cid:96) (cid:109) kkkE , (433)and similarly for F H L z and F I L z .Equation (430) states that the change in the particle’s orbital energy is equalat leading order to the sum total of energy carried out of the system (into the blackhole and out to infinity). Equation (431) states the analog about the particle’s angularmomentum.Some time later, Ref. [176] derived a similar formula for the average rate ofchange of the Carter constant due to f α ( ) diss : d ˜ Q ( ) ϕ dt = − (cid:18) dQdt (cid:19) H − (cid:18) dQdt (cid:19) I , (434)(435)where (cid:18) dQdt (cid:19) (cid:63) = ∑ (cid:96) (cid:109) kkk L (cid:109) kkk + k z ˜ ϒ ( ) z ω (cid:109) kkk F (cid:63)(cid:96) (cid:109) kkkE (436)with L (cid:109) kkk = (cid:109) (cid:104) cot θ ( ) (cid:105) λ L ( ) ϕ − a ω (cid:109) kkk (cid:104) cos θ ( ) (cid:105) λ E ( ) ϕ . (437)While this evolution equation for Q superficially resembles those for E and L z , itis of fundamentally different character. The quantities F (cid:63) E and F (cid:63) L z are true physi-cal fluxes across the horizon and out to infinity; they are defined entirely in termsof the metric on the surfaces H + and I + . The quantities (cid:16) dQdt (cid:17) (cid:63) , on the otherhand, directly involve orbital parameters; they are not locally measurable fluxes.Thus, although Eq. (434) is sometimes referred to as a flux-balance law, there isno known sense in which it can be meaningfully described as such. However, theevolution equations for E , L z , and Q all share the same practical advantage: they canbe evaluated directly from the retarded solution to the Teukolsky equation with apoint-particle source, with no need to reconstruct the complete metric perturbationor to extract the regular field. Note that Ref. [176] uses C to denote our Q and Q to denote our K . We give here the expressionfor (cid:16) dQdt (cid:17) (cid:63) as presented in Ref. [64]. In all cases in the literature, expressions such as these arewritten in terms of averages (cid:104)·(cid:105) , which we can omit because we work with already averaged orbitalvariables.8 Combining Eqs. (430), (431), and (434), we can compute the adiabatic drivingforces Γ i ( ) ( ˜ p i ϕ ( ) ) = ∂ ˜ p i ϕ ( ) ∂ ˜ P j ϕ ( ) d ˜ P j ϕ ( ) dt (438)from the Teukolsky amplitudes − ˆ C in/up1 (cid:96) (cid:109) kkk given by Eq. (421). We can then followthe prescription outlined at the beginning of the section. Alternatively, we can ex-press the geodesic frequencies in terms of ˜ P i ϕ ( ) = ( ˜ E ( ) ϕ , ˜ L ( ) ϕ , ˜ Q ( ) ϕ ) and work directlywith those variables, treating ˜ p i ϕ ( ) as a function of ˜ P i ϕ ( ) by inverting the relation-ships (225)–(227)The adiabatic approximation has been used to evolve equatorial orbits in Kerrspacetime [66] and to generate waveforms in Schwarzschild spacetime [38]. Yet,despite the approximation’s efficient formulation, to date no adiabatic waveformshave been generated for orbits in Kerr spacetime, nor have orbital evolutions beenperformed for generic (eccentric and inclined) orbits. There are two main obstacles.One is generating sufficiently dense data on the p i ϕ space to perform accurate inter-polation or fitting. The second is the very large ( ∼ ) number of mode amplitudesthat are required to achieve an accurate waveform. Both obstacles are expected to besoon overcome [66, 38], but as of this writing, the gold standard for generic orbitsremains snapshot waveforms [59] that use geodesic phases. The 1PA evolution equations (372)–(375), as extended following the discussionaround Eqs. (414)–(416), are d ˜ ϕ ( ) α d ˜ t = Ω ( ) α ( ˜ p j ϕ ( ) ) , (439) d ˜ p i ϕ ( ) d ˜ t = Γ i ( ) ( ˜ p j ϕ ( ) ) , (440) d ˜ ϕ ( ) α d ˜ t = ˜ p j ϕ ( ) ∂ j Ω ( ) α ( ˜ p j ϕ ( ) ) , (441) d ˜ p i ϕ ( ) d ˜ t = Γ i ( ) ( ˜ p j ϕ ( ) , ˜ M ( ) A ) + ˜ p j ϕ ( ) ∂ j Γ i ( ) ( ˜ p j ϕ ( ) ) , (442) d ˜ M ( ) A d ˜ t = F ( ) A ( ˜ p j ϕ ( ) ) . (443)Here we have assumed M A = ˜ M ( ) A ( ˜ t ) + O ( ε ) . Because (i) M A only contributes sta-tionary modes to h ( (cid:109) kkk ) µν , (ii) any source term for h ( (cid:109) kkk ) µν that is quadratic in thesemodes will also be stationary, and (iii) a statiomary mode of h ( (cid:109) kkk ) µν will not con-tribute to Γ i ( ) , it follows that Γ i ( ) is linear in M A , implying we can write it in the form Γ i ( ) ( ˜ p j ϕ ( ) , ˜ M ( ) A ) = Γ i ( ) ( ˜ p j ϕ ( ) , ) + ˜ M ( ) A γ iA ( ˜ p j ϕ ( ) ) , (444)where A is summed over.A 1PA waveform-generation scheme then consists of the following steps:1. Solve the field equations (398) and (399) on a grid of p i ϕ values. At eachgrid point, compute and store the following: (i) the driving forces Γ i ( ) ( p i ϕ ) , Γ i ( ) ( p i ϕ , ) , and γ iA ( p i ϕ ) , (ii) the asymptotic first-order mode amplitudes at I + [e.g., − ˆ C up1 (cid:96) (cid:109) kkk ( p i ϕ ) ].2. Using the stored values of the driving forces, evolve through the parameter spaceby solving the coupled equations (439)–(443) to obtain ˜ p i ϕ ( ) and the phases ˜ ϕ ( ) α and ˜ ϕ ( ) α as functions of ˜ t = ε t .3. Construct the 1PA waveform h + − ih × = ∑ (cid:96) (cid:109) kkk − ˆ C up1 (cid:96) (cid:109) kkk [ ˜ p i ϕ ( ) ( ˜ u )] ω (cid:109) kkk − S (cid:96) (cid:109) ( θ , φ ; a ω (cid:109) kkk ) e − i (cid:104) ϕ ( ) (cid:109) kkk ( ˜ u )+ εϕ ( ) (cid:109) kkk ( ˜ u ) (cid:105) / ε , (445)where ω (cid:109) kkk = ω (cid:109) kkk [ ˜ p i ϕ ( ) ( ˜ u )] .We make two potentially clarifying remarks about these steps. First, even thoughthe 1PA dynamics depend on the black hole parameters M A , we need not includethese parameters in our storage grid. This is because the 1PA effect of M A is linearin M A , allowing us to only store its coefficient. However, note that the backgroundspin parameter a must be included in the grid (the background parameter M neednot be, as we can measure all lengths in units of M ). Our second remark is thatthough p i ϕ (cid:54) = ˜ p i ϕ ( ) at a given value of ˜ t and ε , we can still freely solve (398) and(399), working with p i ϕ , in order to determine the driving forces as functions; it isprecisely those functions, simply with p i ϕ → ˜ p i ϕ ( ) , that appear in Eqs. (439)–(443).A scheme of this sort was first sketched in Ref. [163] and detailed in Ref. [126]for the special case of quasicircular orbits into a Schwarzschild black hole. Fig. 3 ofRef. [126] gives a more thorough breakdown, though the structure of the multiscaleexpansion differs slightly from our formulation here. The general case for genericbound orbits in Kerr appears here for the first time.At its core, the scheme requires three key ingredients for each set of orbital pa-rameter values: the full first-order self-force, the asymptotic mode amplitudes of thefirst-order waveform, and the second-order dissipative self-force. As we summa-rized in Sec. 7.1.2, the first two ingredients have been calculated for generic boundorbits in Kerr spacetime [121] and are routinely calculated for less generic config-urations. The main obstacle to including these ingredients in an evolution schemeis the computational cost and runtime of sufficiently covering the parameter space.The third ingredient has not yet been calculated in even the simplest configurations,though there is ongoing development of a practical implementation [157, 127, 126],which led to the recent calculation of a second-order conservative effect [163]. In our description so far, we have largely glossed over what is the pivotal step in al-most all self-force calculations beyond the adiabatic approximation: the calculationof h R ( n ) µν and its derivatives, which are required for the conservative first-order self-force, the second-order self-force, as inputs for the second-order sources (whetherthe Detweiler stress-energy, the effective source, or the second-order Einsten ten-sor [127]), and as the essential ingredient in most dynamical quantities of interest.In order to compute h R ( n ) µν (either using a puncture, or the point-particle methodwith regularization) in a mode-decomposed calculation, a crucial component is amode-decomposed form for the puncture field. This can be obtained by expand-ing the puncture field into the same basis as is used in the calculation of the retardedfield and can typically be done analytically, or at least semi-analytically. The specificdetails depend on the context (e.g. choice of gauge, whether the mode decomposi-tion needs to be exact or if it can be an approximation, whether the harmonics arespheroidal or spherical and scalar, vector, tensor or spin-weighted). However, theessential ingredients in the method are common to all cases:1. Introduce a rotated angular coordinate system ( θ (cid:48) , φ (cid:48) ) such that the worldlineis instantaneously at the pole, θ (cid:48) =
0. This makes the mode decomposition inte-grals analytically tractable and in some instances reduces the number of sphericalharmonic (cid:109) modes that need to be considered.2. Expand the relevant quantity in a coordinate series about the worldline. In doingso, it is important to ensure that the series approximation is well behaved awayfrom the worldline, in particular at θ (cid:48) = π /
2. This can be achieved by multiplyingby an appropriate window function in the θ (cid:48) direction [206].The resulting expansion can always be algebraically manipulated into the formof a power series (including log terms in some cases) in ρ : = k χ / (cid:104) δ + − cos θ (cid:48) (cid:3) / . (446)Here, δ = k ∆ r χ , ∆ r : = r − r and χ : = − k sin φ (cid:48) , and k , k and k depend onthe orbital parameters and can be treated as constants in the mode decomposition.The coefficients in the power series contain powers of ∆ r and χ and also dependon the orbital parameters. Apart from that, the dependence on φ (cid:48) is only via oneof four possibilities: a. independent of φ (cid:48) ; b. cos φ (cid:48) sin φ (cid:48) ; c. cos φ (cid:48) ; d. sin φ (cid:48) . Theresulting dependence on φ (cid:48) will then combine in the next step with the e − i (cid:109) (cid:48) φ (cid:48) factor from the harmonic to produce a dependence on φ (cid:48) only via powers of χ .When decomposing tensors, certain tensor components may also include an over-all factor of sin θ (cid:48) , but only ever in such a way that it cancels a singularity in theharmonic at θ (cid:48) = ∆ r = In some instances (e.g. eccentric orbits) obtaining an expression for ρ in this form requiresthe definition of the rotated coordinates to include a dependence not just on the unrotated angularcoordinates, ( θ , φ ) , but also on the other coordinates (e.g. ∆ r for the eccentric case).01
3. Integrate against (the conjugate of) the relevant harmonic to obtain a mode de-composition in ( (cid:96), (cid:109) (cid:48) ) modes with respect to the rotated coordinate system. Inthe case of spin-weighted or vector and tensor harmonics, we must also be carefulto account for the rotation, R , of the frame, either by including the appropriatefactor of e is γ (cid:48) ( θ (cid:48) , φ (cid:48) , R ) in the spin-weighted case [26], or by including the appro-priate tensor transformation in the case of vector and tensor harmonics [206].In performing the integrals, we can exploit the fact that only certain integrals over φ (cid:48) are non-vanishing. In particular for the four possibilities listed in the previousstep:a. only contributes for (cid:109) (cid:48) even and only for the real part of e i (cid:109) (cid:48) φ (cid:48) ;b. only contributes for (cid:109) (cid:48) even and only for the imaginary part of e i (cid:109) (cid:48) φ (cid:48) ;c. only contributes for (cid:109) (cid:48) odd and only for the real part of e i (cid:109) (cid:48) φ (cid:48) ;d. only contributes for (cid:109) (cid:48) odd and only for the imaginary part of e i (cid:109) (cid:48) φ (cid:48) .The integrals over θ (cid:48) can all be done analytically and result in expressions of theform δ n + ( δ + ) ( n + ) / (cid:96) − n + ∑ i = a i δ i + log (cid:16) δ + δ (cid:17) (cid:96) + n + ∑ i = b i δ i n even (447) ( δ + ) ( n + ) / (cid:96) ∑ i = c i δ i + | δ | δ n + (cid:96) ∑ i = (cid:109) (cid:48) d i δ i n odd (448)where n is the power of ρ and where the coefficients a i , b i , c i and d i are (cid:96) -dependent rational numbers. The specific limits on the sums given here is forthe (cid:109) (cid:48) = ρ terms, for (cid:109) (cid:48) (cid:54) = i .The integrals over φ (cid:48) can also be done analytically and result in power series (forinteger powers of χ ), elliptic integrals (for half-integer powers), or the derivativeof a hypergeometric function with respect to its argument (for log terms). In allthree cases, they are functions of k and potentially also ∆ r .4. Transform back to the ( (cid:96), (cid:109) ) modes with respect to the unrotated ( θ , φ ) coordi-nate system using the Wigner-D matrix, D (cid:96) (cid:109)(cid:109) (cid:48) ( R ) . With a moving worldline therotation is time dependent, but this complication is not relevant in many cases,the notable exception being in the effective source method where it is necessaryto take time derivatives when computing the source from the puncture [127, 87].In many practical applications, an exact mode decomposition is not necessaryand an approximation is sufficient. For example, in the mode-sum regularizationscheme one is only interested in the modes of the puncture (or its radial derivativein the case of the self-force) evaluated in the limit ∆ r →
0. Similarly, in the effectivesource scheme a series expansion to some power in ∆ r suffices. Then, the exactexpression for the mode-decomposed puncture field has a series expansion in ∆ r ofthe form ∑ (cid:109) (cid:48) , i jk c , i ∆ r i + c , j ∆ r j | ∆ r | + c , k ∆ kr log ∆ r (449)where the coefficients c , i , c , j and c , k depend on the orbital parameters. In thosecases, the mode decomposition procedure simplifies significantly and one need onlycompute up to some maximum value for i , j and k . Similarly, another simplifica-tion arises from the fact that one may only be interested in the decomposition ofthe puncture accurate to some order in distance from the worldline in the angulardirections. This is is reflected in the number of (cid:109) (cid:48) modes that must be included: fora puncture accurate to n derivatives one must include up to | (cid:109) (cid:48) | = | s | ± n for thespin-weighted case (the vector and tensor cases similarly follow from their relationto the spin-weighed harmonics: | s | = | s | = (cid:109) and with ∆ r =
0. This leads to so-called mode-sum regularization formulas. For example, inthe case of the first-order gravitational self-force this is given by F α = ∑ (cid:96) ( F α ret − A ± ( (cid:96) + ) − B ) + D (450)where A ± , B and D are “regularization parameters” that depend on the orbital pa-rameters. Here, the value of A ± depends on whether the limit ∆ r → | ∆ r | piece of thepuncture. The parameters B does not have this property as it comes from the pieceof the puncture that does not | ∆ r | (in particular, for the self-force it comes fromthe derivative of the ∆ r piece of the puncture). The parameter D accounts for thepossibility that the subtraction does not exactly capture the behaviour of the contri-bution from the singular field (and only the singular field) and can often be set to 0by appropriately defining the subtraction [206, 162]. As a simple representative example, consider the problem of decomposing theleading-order piece of the Lorenz-gauge puncture [i.e. the first term in Eq. (183)]into the the spin-weighted spherical harmonic basis introduced in Sec. 4.2. Forconcreteness, we consider a circular geodesic orbit of radius r o with four-velocity u α = u t [ , , , Ω ] , where Ω = (cid:113) Mr o and u t = (cid:113) r o r o − M .As a first step, we expand the covariant expression in a coordinate series. Keep-ing only the leading term in the coordinate expansion, we have g α (cid:48) µ = δ αµ + O ( ∆ x ) and s = ρ + O ( ∆ x ) where ρ : = B ( δ + − cos θ (cid:48) ) , δ : = r o ∆ r B ( r o − M ) , χ : = − Mr o − M sin φ (cid:48) , B : = r o ( r o − M ) χ ( r o − M ) and ∆ r = r − r o . Here, we have made the stan- dard choice of identifying a point on the worldline with the point where the punctureis evaluated by setting ∆ t = t − t o = h ll = h nn = ρ r o − Mr o − M , h ln = Mr o − M h m ¯ m = ρ Mr o − M , h lm = h nm = − h l ¯ m = − h n ¯ m = − cos (cid:0) θ (cid:48) (cid:1) ρ i ( r o − M ) r o Ω √ f o ( r o − M ) , h mm = h ¯ m ¯ m = − cos (cid:0) θ (cid:48) (cid:1) ρ Mr o − M . (451)Note that we have included factors of cos (cid:0) θ (cid:48) (cid:1) and cos (cid:0) θ (cid:48) (cid:1) to ensure that thepuncture is sufficiently regular at θ (cid:48) = π while not altering its leading-order be-haviour near the worldline at θ (cid:48) = e is γ (cid:48) ( θ (cid:48) , φ (cid:48) , R ) ≈ i s e is φ (cid:48) + O ( θ (cid:48) ) .Since we are only interested in the leading-order behaviour near the worldline wewill only consider the modes (cid:109) (cid:48) + s = O ( ∆ r ) . Then,we encounter the following integrals over θ (cid:48) (cid:90) π ρ ¯ Y (cid:96) ( θ (cid:48) , ) sin θ (cid:48) d θ (cid:48) ≈ B (cid:112) π ( (cid:96) + ) (cid:20) − √ ( (cid:96) + ) | δ | (cid:21) , (452a) (cid:90) π cos (cid:0) θ (cid:48) (cid:1) ρ ¯ Y (cid:96), − ( θ (cid:48) , ) sin θ (cid:48) d θ (cid:48) = (cid:90) π cos (cid:0) θ (cid:48) (cid:1) ρ − ¯ Y (cid:96) ( θ (cid:48) , ) sin θ (cid:48) d θ (cid:48) ≈ − B (cid:112) π ( (cid:96) + ) (cid:20) Λ − √ ( (cid:96) + ) | δ | (cid:21) , (452b) (cid:90) π cos (cid:0) θ (cid:48) (cid:1) ρ ¯ Y (cid:96) ( θ (cid:48) , ) sin θ (cid:48) d θ (cid:48) = (cid:90) π cos (cid:0) θ (cid:48) (cid:1) ρ − ¯ Y (cid:96), − ( θ (cid:48) , ) sin θ (cid:48) d θ (cid:48) ≈ − B (cid:112) π ( (cid:96) + ) (cid:20) Λ − ( (cid:96) − )( (cid:96) + ) (cid:21) , (452c) (cid:90) π cos (cid:0) θ (cid:48) (cid:1) ρ ¯ Y (cid:96), − ( θ (cid:48) , ) sin θ (cid:48) d θ (cid:48) = (cid:90) π cos (cid:0) θ (cid:48) (cid:1) ρ − ¯ Y (cid:96) ( θ (cid:48) , ) sin θ (cid:48) d θ (cid:48) ≈ B (cid:112) π ( (cid:96) + ) (cid:20) Λ − √ ( (cid:96) + ) | δ | (cid:21) . (452d) (cid:90) π cos (cid:0) θ (cid:48) (cid:1) ρ ¯ Y (cid:96) ( θ (cid:48) , ) sin θ (cid:48) d θ (cid:48) = (cid:90) π cos (cid:0) θ (cid:48) (cid:1) ρ − ¯ Y (cid:96), − ( θ (cid:48) , ) sin θ (cid:48) d θ (cid:48) ≈ B (cid:112) π ( (cid:96) + ) (cid:20) Λ − ( (cid:96) − )( (cid:96) + ) (cid:21) , (452e)where Λ : = (cid:96) ( (cid:96) + )( (cid:96) − )( (cid:96) + ) and Λ : = ( (cid:96) − ) (cid:96) ( (cid:96) + )( (cid:96) + )( (cid:96) − )( (cid:96) − )( (cid:96) + )( (cid:96) + ) . Next, performing theintegrals over φ (cid:48) the integrands all involve integer (for the | δ | terms) and half-integer(for the δ terms) powers of χ , producing elliptic integrals or polynomial functionsof Mr o − M , respectively. Putting everything together, transforming to the frequencydomain (which in this case amounts to simply dividing by 2 π ) and transformingback to the unrotated frame using the Wigner-D matrix, we then obtain expressionsfor the mode-decomposed punctures, h (cid:96) (cid:109) ω ll = D (cid:96) (cid:109) , ( R ) (cid:112) ( (cid:96) + ) π (cid:20) K π r o (cid:114) r o − Mr o − M − ( (cid:96) + ) r / o √ r o − M | ∆ r | (cid:21) , (453a) h (cid:96) (cid:109) ω ln = D (cid:96) (cid:109) , ( R ) (cid:112) ( (cid:96) + ) π (cid:20) M K π r o √ r o − M √ r o − M − ( (cid:96) + ) Mr / o √ r o − M ( r o − M ) | ∆ r | (cid:21) , (453b) h (cid:96) (cid:109) ω lm = D (cid:96) (cid:109) , − ( R ) (cid:112) ( (cid:96) + ) π (cid:20) − Λ Ω K π (cid:114) r o r o − M + ( (cid:96) + ) Ω √ r o − M √ r o − M | ∆ r | (cid:21) + D (cid:96) (cid:109) , ( R ) (cid:112) ( (cid:96) + ) π (cid:20) Λ − ( (cid:96) − )( (cid:96) + ) (cid:21) [( r o − M ) K − ( r o − M ) E ] M / r o π √ r o − M , (453c) h (cid:96) (cid:109) ω mm = D (cid:96) (cid:109) , − ( R ) (cid:112) ( (cid:96) + ) π (cid:20) M Λ K π r o √ r o − M √ r o − M − ( (cid:96) + ) Mr / o √ r o − M ( r o − M ) | ∆ r | (cid:21) + D (cid:96) (cid:109) , ( R ) (cid:112) ( (cid:96) + ) π (cid:20) Λ − ( (cid:96) − )( (cid:96) + ) (cid:21) × [( r o − M )( r o − M ) K − ( r o − M )( r o − M ) E ] M π r o √ r o − M √ r o − M , (453d) with the other components given either by h (cid:96) (cid:109) ω ll = h (cid:96) (cid:109) ω nn , h (cid:96) (cid:109) ω m ¯ m = r o − MM h (cid:96) (cid:109) ω ln , h (cid:96) (cid:109) ω nm = h (cid:96) (cid:109) ω lm , or by complex conjugation. Here, K : = (cid:90) π (cid:18) − Mr o − M sin φ (cid:48) (cid:19) − / d φ (cid:48) , (454a) E : = (cid:90) π (cid:18) − Mr o − M sin φ (cid:48) (cid:19) / d φ (cid:48) , (454b)are complete elliptic integrals of the first and second kind, respectively.Higher order circular-orbit punctures including the contribution at O ( λ ) areavailable in Ref. [206]. Yet higher orders and punctures for more generic cases areavailable upon request to the authors. We stated in the introduction to this review that we aimed to summarize the keymethods of black hole perturbation theory and self-force theory rather than sum-marizing the status of the field, leaving that task to existing reviews. However, it isworth putting this review in the context of the field’s current state, and it is worthmentioning key topics that we did not cover.Regarding topics we neglected, we first state the obvious: we did not cover anyapplications of black hole perturbation theory other than small-mass-ratio binaries.Although the bulk of the review is intended to provide general treatments of blackhole perturbation theory, orbital dynamics in black hole spacetimes, and self-forcetheory in generic spacetimes, without specializing to binaries, it is undoubtedlyslanted toward our application of interest.For that reason, we will also focus exclusively on the state of small-mass-ratiobinary modelling. It is well established that 1PA waveforms are almost certainlyrequired to perform high-precision measurements of these binaries. Such measure-ments will require phase errors much smaller than 1 radian, while the errors in 0PAwaveforms will have errors of O ( ε ) [or O ( / √ ε ) in the case of a resonance], whichcould be 1 or many more radians. Ref. [122] has recently provided strong numericalevidence that a 0PA waveform will have significant errors for all mass ratios. Con-versely, the same reference shows that a 1PA waveform should be not only highlyaccurate for EMRIs and IMRIs, but reasonably accurate even for comparable-massbinaries. This bolsters a long line of evidence that perturbative self-force theory issurprisingly accurate well outside its expected domain of validity; see Ref. [169] forother recent evidence, as well as the reviews [110, 11].There are two main hurdles to overcome on the way to generating 1PA wave-forms. One is the difficulty of efficiently covering the parameter space. Once a re-gion is well covered by snapshots, recent advances make it possible to generate long,accurate waveforms extremely rapidly in that region, with generation times of a few tens of milliseconds for eccentric orbits in Schwarzschild spacetime [38]. However,covering the parameter space of generic orbits in Kerr is highly expensive even foradiabatic codes, let alone calculations of the first-order self-force.The second main hurdle is calculating the necessary second-order inputs for the1PA evolution. There has been steady progress in developing practical methods ofcomputing these inputs, but only recently have results begun to materialize [163]. Todate, these calculations have been restricted to quasicircular orbits in Schwarzschildspacetime; they must be extended to Kerr and to generic orbits.In lieu of accurate evolving waveforms, the development of data analysis meth-ods has so far been based on “kludge” waveforms constructed using a host of addi-tional approximations (primarily, post-Newtonian approximations for the fluxes) [77,7, 71, 5, 181, 37]. These kludges will be very far from accurate enough to enableprecise parameter estimation, but they are sufficiently similar to accurate waveformsto serve as testbeds for analysis methods. They may also be sufficiently accurate fordetection of loud signals. There is also ongoing work to improve the accuracy ofpost-Newtonian 0PA approximations to enable them to accurately fill out the weak-field region of the small-mass-ratio parameter space [174].Our summary of multiscale evolution has also omitted some important ingredi-ents in an accurate model. We must correctly account for passages through reso-nance, and we may need to include the transition to plunge for mass ratios ∼ f µ ( ) con , (ii) through the spin’s contribution to T ( ) µν in Eq. (205), which gen-erates a perturbation that contributes to f µ ( ) diss , and (iii) through a coupling between h R ( ) µν and the spin, which again contributes a second-order dissipative effect. We re-fer to Refs. [204, 3, 208, 211] for a sample of recent work on calculating these effectsand incorporating them into waveform-generation schemes. Specifically, Ref. [204]generated waveforms from inspirals into a Schwarzschild black hole including first-order conservative (but not second-order dissipative) spin effects; Ref. [3] derivedbalance laws incorporating spin; Ref. [208] derived the spin correction to the funda-mental frequencies; and Ref. [211] computed the spin’s contribution to fluxes fromspinning particles on generic orbits in Schwarzschild spacetime.We also note that while we have focused on a multiscale expansion built onfrequency-domain methods, there has been considerable development of time-domain snapshot calculations of h ( ) µν and f µ ( ) using fixed geodesic sources [9, 12,13, 56, 8]. The quantities h ( ) µν and f µ ( ) output from such computations cannot bedirectly fed into the second-order field equations (399) or into the multiscale evo-lution scheme. However, if we decompose the outputs into Fourier modes, as in h ( ) µν = ∑ (cid:109) kkk h ( (cid:109) kkk ) µν e im φ − ω (cid:109) kkk t , then the coefficients h ( (cid:109) kkk ) µν are identical to those ina multiscale expansion, and these can be used as inputs for the multiscale scheme.Moreover, any first-order quantity that depends only on P α will be identical in thetime domain with a geodesic source as in the multiscale expansion; this includesany quantity constructed as an average over the orbit, which includes most phys- ical quantities of interest [11]. Because time-domain methods are typically moreefficient than frequency-domain ones for highly eccentric orbits, certain dynamicalquantities entering into the evolution may be more usefully computed in the timedomain.Time-domain calculations also offer an alternative framework for waveform gen-eration: rather than using using Eq. (423), one can perform a multiscale evolution of P α to generate a self-accelerated trajectory and then solve the Teukolsky equationin the time domain with an accelerated point-particle source [184, 83]. This mayseem redundant, given that in the process of generating the multiscale evolution onemust already compute all the inputs for Eq. (423). However, it offers significantflexibility, in that it can take as input trajectories generated with any method, suchas inspirals which have been produced that include the full first-order self-force butomit second-order dissipative effects [202, 137, 194]. This gives it the additionaladvantage of being able to easily evolve through different dynamical regimes, suchas the the evolution from the adiabatic inspiral to the transition to plunge [186, 169].Beyond these alternative methods of wave generation, we have also passed overwhat has been the main application of self-force calculations. Although such cal-culations were originally motivated by modelling EMRI waveforms (and more re-cently, the prospect of using them to model IMRIs), they have also enabled thecalculation of numerous physical effects in binaries. These, in turn, have facilitateda rich interaction with other binary models: post-Newtonian and post-Minkowskiantheory, effective one body theory, and fully nonlinear numerical relativity [110].Sections 7 and 8 of Ref. [11] provide a summary of the physical effects that havebeen computed and the synergies with other models. We highlight Refs. [21, 40, 22]for more recent discussions of the power of such synergies and of the potential futureimpact of self-force calculations. Cross-References • Introduction to gravitational wave astronomy , N. Bishop•
Space-based laser interferometers , J. Gair, M. Hewitson, A. Petiteau•
The gravitational capture of compact objects by massive black holes , P. AmaroSeoane•
Post-Newtonian templates for gravitational waves from compact binary inspiral ,S. Isoyama, R. Sturani, H. Nakano•
Non-linear effects in EMRI dynamics and the imprints on gravitational waves ,G. Lukes-Gerakopoulos, V. Witzany, O. Semer´ak Acknowledgements
AP is grateful to Jordan Moxon and Eanna Flanagan for numerous helpful discus-sions. BW thanks Andrew Spiers for independently checking several equations. APalso acknowledges the support of a Royal Society University Research Fellowship.
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