Detweiler's redshift invariant for spinning particles along circular orbits on a Schwarzschild background
Donato Bini, Thibault Damour, Andrea Geralico, Chris Kavanagh
aa r X i v : . [ g r- q c ] J a n Detweiler’s redshift invariant for spinning particles along circular orbits on aSchwarzschild background
Donato Bini , Thibault Damour , Andrea Geralico , Chris Kavanagh Istituto per le Applicazioni del Calcolo “M. Picone,” CNR, I-00185 Rome, Italy Institut des Hautes ´Etudes Scientifiques, 91440 Bures-sur-Yvette , France. (Dated: January 30, 2018)We study the metric perturbations induced by a classical spinning particle moving along a circularorbit on a Schwarzschild background, limiting the analysis to effects which are first order in spin.The particle is assumed to move on the equatorial plane and has its spin aligned with the z -axis. Themetric perturbations are obtained by using two different approaches, i.e., by working in two differentgauges: the Regge-Wheeler gauge (using the Regge-Wheeler-Zerilli formalism) and a radiation gauge(using the Teukolsky formalism). We then compute the linear-in-spin contribution to the first-orderself-force contribution to Detweiler’s redshift invariant up to the 8.5 post-Newtonian order. Wecheck that our result is the same in both gauges, as appropriate for a gauge-invariant quantity, andagrees with the currently known 3.5 post-Newtonian results. I. INTRODUCTION
In the new field of gravitational wave astrophysics, aninteresting potential source are extreme mass ratio in-spirals, where a small compact body of mass µ orbitsand eventually coalesces with a much more massive blackhole of mass M , where µ/M ∼ − . These systemsare most commonly modelled using the gravitational self-force (GSF) approach. In this approach, in order to accu-rately model the inspiral waveform, one needs to accountcorrectly for both dissipation of the orbital parametersand conservative shifts, which grow secularly when takenin conjunction with the dissipation. A significant fo-cus of conservative GSF calculations has been on gauge-invariant, physical effects localized on the small mass µ .These were initiated by Detweiler [1] who defined, andcomputed, a redshift variable for a particle on a circu-lar orbit in Schwarzschild spacetime (i.e., the linear-in-mass-ratio contribution to u t , the time component of theparticle’s 4-velocity). This provided the first identifiedconservative, gauge-invariant GSF effect (though it wasnot, initially, related to the dynamics of small-mass-ratiosystems). Soon after, the GSF computation of shifts inthe innermost stable circular orbit, and of precession ofthe periapsis [2, 3] provided other conservative, gauge-invariant GSF effects (of more direct dynamical signifi-cance).Detweiler’s redshift computations were pushed to highnumerical accuracy, and compared to the third post-Newtonian (3PN) analytical knowledge of comparable-mass binary systems [4, 5]. Moreover, the later discov-ery of the “First Law of Binary Black Hole Mechanics”[6], allowed one to extract the dynamical significance ofGSF redshift computations [7, 8]. The first complete analytic self-force computation of Detweiler’s redshift in-variant at the fourth post-Newtonian (4PN) was per- For earlier computations of the logarithmic 4PN-level contribu-tion, see Refs. [9] and [5]. formed by Bini and Damour [10], who showed how tocombine the Regge-Wheeler-Zerilli [11, 12] (RWZ) for-malism for the Schwarzschild gravitational perturbationswith the hypergeometric-expansion analytical solutionsof the RWZ radial equation obtained by Mano, Suzukiand Takasugi [13, 14] (MST). The methodology given inRef. [10] allowed the extension to higher PN levels: in-deed, results were soon derived at the 6PN level [15],the 8.5PN one [16], the 9.5PN one [17], ending, with aconsiderable jump, at the 22.5PN one [18].In the meantime the GSF community became inter-ested in computing other gauge invariant quantities, as-sociated with spin precession along circular orbits inSchwarzschild [19–21] and tidal invariants (quadrupolar,octupolar) again along circular orbits in Schwarzschild[22–24]; while most of these works contained strong fieldnumerics or analytic PN calculations, other conceptuallyuseful methods were also introduced (e.g., the PSLQ re-construction of fractions, see Ref. [25]).In addition to defining new invariants, considerablework has been ongoing in extending GSF computationstowards more astrophysically relevant scenarios. For ex-ample, the redshift and spin precession invariants alongeccentric (equatorial) orbits in Schwarzschild have beenstudied [26–33]. Including for the first time spin on theprimary black hole, Abhay Shah gave in 2015 the first(4PN) GSF computation of the redshift invariant alongcircular orbits in Kerr spacetime [34, 35]. This PN calcu-lation was then extended in Refs. [36, 37] and calculatedfor eccentric orbits in Ref. [38]. While formulations havebeen provided for spin precession in Kerr spacetime [39],the practical calculation of further gauge invariants orthe generalisation to inclined orbits have been halted bytechnical difficulties in the regularization procedures andmetric completion of the non-radiative multipoles. How-ever, significant recent work, including numerical calcu-lations of the full self-force for generic inclined eccentricorbits in Kerr, show that these issues are in principlesolved [40–43].One of the strong motivations for the analytic GSF-PNcomputational effort has been the possibility to convertsuch high PN-order GSF information into other approx-imation formalisms useful for computing (comparable-mass) binary inspirals, such as the Effective-One-Body(EOB) model [44–46]. For example, Damour [9] showedhow to compute some combinations of EOB radial po-tentials from GSF data. Further use of the first lawof mechanics [6] allowed the computation of individualEOB radial potentials [8, 47]. Following this, high-orderPN computations of these potentials were actually ac-complished [29, 31, 36, 38]. It has been shown that theknowledge of the eccentric redshift invariant maps com-pletely the non-spinning effective-one-body Hamiltonian[28]. Transcription of information from GSF to the spin-ning EOB Hamiltonian remains ongoing.The aim of this paper is to provide a generalisa-tion of Detweiler’s redshift in Schwarzschild spacetimeto the case where the small body µ has a small but non-negligible spin s , and to provide an 8.5PN-accurate post-Newtonian expansion valid to linear order in both themass-ratio and the spin. Test spinning particles no longermove on geodesics of the spacetime but experience a forcedue to the coupling of the spin of the body and the Rie-mann curvature tensor of the background which must beincluded, according to the Mathisson-Papapetrou-Dixon(MPD) model [48–50]. Hence, we shall consider a par-ticle moving along an accelerated circular orbit. Metricperturbations generated by spinning particles (both inSchwarzschild and in Kerr spacetimes) have been consid-ered, e.g., in Refs. [51–53] (see also the review by Sasakiand Tagoshi [54]), with the aim of computing the emittedfluxes of gravitational wave. Similarly, PN calculationsinvolving spinning bodies also exist [55, 56].To our knowledge our study is the first analytic cal-culation of a conservative effect of the self-force for aspinning particle. To internally validate our results weperform all calculations both in the Regge-Wheeler (RW)gauge (solving the RWZ equations) and in the (outgoing)radiation gauge (using the Teukolsky approach). The ex-ception to this is the low-multipole problem, for which weuse the RWZ approach in both cases. As an importantside result of our work, we explicitly give the complete(interior and exterior) metric perturbations for ℓ = 0 , µ (instead of m ) for the small mass, and M (instead of m ) for the large mass.] The metric signature is chosento be − + ++ and units are such that c = G = 1 un-less differently specified. Greek indices run from 0 to 3,whereas Latin ones from 1 to 3. II. SPINNING PARTICLE MOTION IN THEBACKGROUND SCHWARZSCHILD SPACETIME
Our background Schwarzschild spacetime has a line el-ement, written in standard coordinates ( t, r, θ, φ ), givenby ds = ¯ g αβ dx α dx β (2.1)= − f dt + f − dr + r ( dθ + sin θdφ ) , where f = 1 − M/r . Let us first introduce an orthonor-mal frame adapted to the static observers, namely thoseat rest with respect to the space coordinates e ˆ t = f − / ∂ t , e ˆ r = f / ∂ r ,e ˆ θ = 1 r ∂ θ , e ˆ φ = 1 r sin θ ∂ φ , (2.2)where { ∂ α } is the coordinate frame. As a convention,the physical (orthonormal) component along − ∂ θ whichis perpendicular to the equatorial plane will be referredto as “along the positive z -axis” and will be indicated bythe index ˆ z , when convenient: e ˆ z = − e ˆ θ . Furthermore,we indicate with a bar background quantities to be distin-guished from corresponding perturbed spacetime quanti-ties.The Mathisson-Papapetrou-Dixon (MPD) equations[48–50] governing the motion of a spinning test particlein a given gravitational background readD ¯ P µ d ¯ τ = − R µναβ ¯ U ν S αβ , (2.3)D S µν d ¯ τ = 2 ¯ P [ µ ¯ U ν ] , (2.4)where ¯ P µ ≡ µ ¯ u µ (with ¯ u · ¯ u = −
1) is the total 4-momentum of the particle with mass µ , S µν is a (antisym-metric) spin tensor, and ¯ U µ = dz µ /d ¯ τ is the timelike unittangent vector of the “center of mass line” (with para-metric equations x µ = z µ (¯ τ )) used to make the multipolereduction, parametrized by the proper time ¯ τ . In orderfor the model to be mathematically self-consistent certainadditional conditions should be imposed. As is standard,we adopt here the Tulczyjew-Dixon conditions [50, 59],i.e., S µν ¯ P ν = µ S µν ¯ u ν = 0 . (2.5)Consequently, the spin tensor can be fully represented bya spatial vector (with respect to ¯ u ), S (¯ u ) α = 12 η (¯ u ) αβγ S βγ , (2.6)where η (¯ u ) αβγ = η µαβγ ¯ u µ is the spatial unit volume 3-form (with respect to ¯ u ) built from the unit volume 4-form η αβγδ = √− ¯ g ǫ αβγδ , with ǫ αβγδ ( ǫ = 1) beingthe Levi-Civita alternating symbol and ¯ g the determinantof the metric.Both the mass µ ≡ ( − ¯ P α ¯ P α ) , and the the magnitude s of the spin vector s = S (¯ u ) β S (¯ u ) β = 12 S µν S µν , (2.7)are constant along the trajectory of a spinning parti-cle, as follows from Eqs. (2.3), (2.4), when using Eq.(2.5). We shall endow here the spin magnitude s witha positive (negative) sign if its orbital angular momen-tum is parallel (respectively, antiparallel) to e ˆ z = − e ˆ θ .A requirement which is essential for the validity of theMathisson-Papapetrou-Dixon model (and of the test par-ticle approach) is that the characteristic length scale | s | /µ associated with the particle’s internal structure be smallcompared to the natural length scale M associated withthe background field. Hence the following condition mustbe assumed: | ˆ s | ≡ | s | / ( µM ) ≪
1. This leads one to con-sider only the terms of first order in the spin in Eqs. (2.3)and (2.4) and to neglect higher order terms. As a result,the 4-momentum ¯ P is parallel to ¯ U to first order in ˆ s ,i.e., ¯ P = µ ¯ U + O (ˆ s ), and the spin tensor is parallel-transported along the path (from Eq. (2.4)). In particu-lar under these assumptions we can identify ¯ U µ ≡ ¯ u µ .Finally, when the background spacetime has Killingvectors, there are conserved quantities along the motion[60]. For example, in the case of stationary axisymmetricspacetimes with coordinates adapted to the spacetimesymmetries, ξ = ∂ t is the timelike Killing vector and η = ∂ φ is the azimuthal Killing vector. The correspondingconserved quantities are the energy ¯ E and the angularmomentum ¯ J of the particle, namely¯ E = − ξ α ¯ P α + 12 S αβ ∇ β ξ α , ¯ J = η α ¯ P α − S αβ ∇ β η α , (2.8)where ∇ β ξ α = − Mr δ trαβ and ∇ β η α = r sin θ δ φrαβ . A. Solution for a spinning test particle in circularmotion in the Schwarzschild spacetime
The MPD equations admit (to linear order in ˆ s ) the fol-lowing solution for a spinning test particle moving alonga circular orbit on the equatorial plane with spin vector S ( ¯ U ) = S ˆ θ e ˆ θ = s e ˆ z orthogonal to it (see, e.g., Ref. [61]):¯ U = ¯ u t ( ∂ t + Ω ∂ φ ) , (2.9)with normalization factor¯ u t = 1 √ − u (cid:18) −
32 ˆ s u / − u (cid:19) , (2.10)and angular velocity M Ω = u / (cid:18) −
32 ˆ su / (cid:19) , (2.11) where u = M/r is the dimensionless inverse radial dis-tance and ˆ s = s/ ( µM ) is the dimensionless spin param-eter introduced above. A spatial triad adapted to ¯ U canbe built with E = e ˆ r , E = e ˆ θ , E = r Ω f / ¯ u t (cid:18) ∂ t + fr Ω ∂ φ (cid:19) . (2.12)These will be useful below in the definition of the stresstensor.A key component of defining gauge invariant func-tions is to consider gauge-invariant quantities as func-tions of gauge invariant arguments. We shall use asgauge-invariant argument (to parametrize circular orbits)the dimensionless frequency variable y = ( M Ω) / , sothat from Eq. (2.11) we have (to first order in ˆ s ) y = u (cid:18) −
32 ˆ su / (cid:19) / = u (cid:0) ˆ su / (cid:1) / , (2.13)with inverse u = y (cid:18) sy / (cid:19) / = y (cid:0) − ˆ sy / (cid:1) / . (2.14)Finally, the conserved quantities (2.8), in terms of theoriginal (inverse) radial variable u and in terms of the(invariant) frequency variable y read (to first order in ˆ s )¯ Eµ = 1 √ − u (cid:20) − u − ˆ s u / − u ) (cid:21) = 1 − y √ − y − ˆ s y / √ − y , ¯ JµM = 1 p u (1 − u ) (cid:20) s √ u − u − u (cid:18) − u (cid:19)(cid:21) = 1 p y (1 − y ) + ˆ s − y √ − y . (2.15) III. DETWEILER’S REDSHIFT INVARIANT z FOR A SPINNING PARTICLE
The aim of the present paper is to compute Detweiler’sredshift invariant associated with a spinning particle tofirst order in spin, i.e., the linear-in-mass-ratio pertur-bation in the time component of the particle’s 4-velocityto first order in both parameters q ≡ µ/M ≪ s ≪
1. We now consider a particle moving (accordingto the MPD equations) along an accelerated circular or-bit but in a perturbed Schwarzschild spacetime (see Ap-pendix B).Let g R αβ = ¯ g αβ + qh R αβ be the regularized perturbedmetric (in the Detweiler-Whiting sense), where h R αβ is theregularized metric perturbation sourced by the spinningparticle, which can be written as a sum of non-spinningand spinning parts, namely h Rαβ = h (0) αβ + ˆ sh (ˆ s ) αβ . (3.1)The (perturbed) particle 4-velocity is given by U = u t ( ∂ t + Ω ∂ φ ) = u t k , k = ∂ t + Ω ∂ φ . (3.2)We wish to find an expression for the gauge invariantredshift z ≡ /u t . The unit normalization of the 4-velocity in the perturbed spacetime gives the condition − ( u t ) − = ¯ g tt + ¯ g φφ Ω + qh Rkk = − (cid:18) − Mr (cid:19) + r Ω + qh kk , (3.3)where (hereafter, we remove the label R for simplicity) h kk = h kk ( y ) = h αβ k α k β | u = y +ˆ sy / = h kk (0) ( y ) + ˆ sh kk ˆ s ( y ) (3.4)The redshift invariant thus reads z ( y ) = 1 u t ( y ) = (cid:18) − u − y u − qh kk ( y ) (cid:19) / . (3.5)However, the right-handside (rhs) of this equation stillcontains the gauge dependent radius u = M/r , whichmust be expressed in terms of the gauge invariant variable y . The perturbed relation between the variables u and y is now given by u = y (cid:0) − ˆ sy / (cid:1) / + qf ( y ) , (3.6)as a consequence of the MPD equations in the perturbedspacetime (see Appendix B), where f ( y ) = f ( y ) + ˆ sf ˆ s ( y ) , (3.7) f ( y ) = 16 y M [ ∂ r h kk (0) ] R ( y ) , (3.8)and f ˆ s ( y ) will be specified in Appendix B (see, e.g., Eq.(28) of Ref. [1] for the derivation of f ( y )).Substituting the relation (3.6) and expanding to firstorder in both q and ˆ s we get z ( y ) = p − y − q √ − y (cid:2) h kk (0) ( y )+ˆ sh kk ˆ s ( y ) + 6ˆ sy / f ( y ) i ≡ z (0)1 ( y ) + q (cid:16) z (1)ˆ s ( y ) + ˆ sz (1)ˆ s ( y ) (cid:17) , (3.9)where the explicit forms of the spin-independent, andspin-linear, 1SF contributions to z ( y ) (defined in thelast line) are respectively given by z (1)ˆ s ( y ) = − √ − y h kk (0) ( y ) , (3.10)and z (1)ˆ s ( y ) = − √ − y h h kk ˆ s ( y ) + M y / ∂ r h kk (0) ( y ) i . (3.11) Two things should be noted. First, the spin-linear con-tribution f ˆ s ( y ) to the O ( q ) term qf ( y ) in the u ↔ y functional link (3.6) has dropped out of the final results.[This follows from the usual fact that the unperturbedvalue of the rhs of Eq. (3.5) is extremal with respect to u (a consequence of the geodesic character of non-spinningcircular orbits).] We therefore, do not need to explic-itly compute f ˆ s ( y ) (for completeness we provide, how-ever, its formal expression in terms of regularized metriccomponents and their derivatives in Appendix B). Sec-ond, when considering the spin-linear 1SF contribution z (1)ˆ s ( y ) to z ( y ), there appears, besides the naively ex-pected h kk ˆ s ( y ) contribution, an extra term proportionalto ∂ r h kk (0) . This extra term is needed to ensure thegauge-invariance of z (1)ˆ s ( y ), and its origin is the back-side of what we just explained concerning the disappear-ance of f ˆ s ( y ) in z ( y ). Indeed, as a spinning particleno longer follows a geodesic, the previous cancellationno longer (fully) operates, and this gives rise to the lastcontribution in Eq. (3.11).In the following, we shall focus on the new, spin-linearredshift contribution Eq. (3.11), and on the computationof its regularized value z (1)ˆ s ( y ) = − √ − y (cid:2) [ h kk ˆ s ] R ( y )+ M y / [ ∂ r h kk (0) ] R ( y ) i . (3.12)Its determination requires the two separate GSF com-putations: h kk ˆ s ( y ) and ∂ r h kk (0) . The term involving ∂ r h kk (0) comes from the non-spinning sector, which hasbeen discussed by the authors in previous works [17].Thus for the next sections we will focus on the computa-tion of h kk ˆ s ( y ) (and of its regularization). IV. SPIN-DEPENDENCE OF THE METRICPERTURBATION AND h kk All of our results will be computed both in the Regge-Wheeler-Zerilli and radiation-gauge frameworks. The de-tails of the RWZ procedure are given in Appendix C, theultimate outcome of which are the spherical harmonic ℓ modes, h ℓkk , of h kk . The details of the radiation-gaugemetric reconstruction will be given in a future work bysome of the authors [62]. The outcome there are thetensor harmonic modes of the full metric perturbation,from which h ℓkk is easily computed. In both calculations,the main difference with the non-spinning case lies in thestress-energy tensor, which we review next. A. The energy-momentum tensor associated withthe spinning particle
The energy momentum tensor of the spinning particleis given by T αβ = T αβµ + T αβs , (4.1)where T αβµ = µ Z dτ √− g U α U β δ ,T αβs = − Z dτ ∇ γ (cid:20) √− g S γ ( α U β ) δ (cid:21) , (4.2)with S αβ = ˆ sµM [ E ∧ E ] αβ . (4.3)Here δ denotes the 4-dimensional delta function centeredon the particle’s world line, i.e., δ ≡ δ ( x α − x α ( τ ))= δ ( t − u t τ ) δ ( r − r ) δ ( θ − π/ δ ( φ − Ω t ) ≡ δ ( t − u t τ ) δ . (4.4) We find then T αβµ = µr u t U α U β δ ,T αβs = −∇ γ (cid:20) u t S γ ( α U β ) r δ (cid:21) , (4.5)so that the total energy-momentum tensor finally reads T αβ = µ h X (0) αβ + ˆ sM X ( s ) αβ i δ +ˆ sµM h Y ( s ) αβ δ r + Z ( s ) αβ δ φ i , (4.6)where δ r = δ ′ ( r − r ) δ ( θ − π/ δ ( φ − Ω t ) ,δ φ = δ ( r − r ) δ ( θ − π/ δ ′ ( φ − Ω t ) . (4.7)The various contributions are given by X (0) αβ = µu t r f − r f Ω0 0 0 00 0 0 0 − r f Ω 0 0 r Ω , (4.8) X ( s ) αβ = Γ K f Ω K r ( − r + 7 M ) 0 0 − M r − Ω K f r Γ K − M r f Ω K , (4.9) Y ( s ) αβ = Γ K f − f Ω K r r − M r r − M r − r Ω K , Z ( s ) αβ = 12 r Γ K − − r Ω K f r Ω K f , (4.10)where terms of the form f ( r ) δ ′ ( r − r ) have been replacedby f ( r ) δ ′ ( r − r ) − f ′ ( r ) δ ( r − r ). In the spin contri-butions (and only in them), the orbital frequency Ω hasbeen replaced (consistently with the linear in spin ap-proximation) by Ω K . Here, the subscript K denotes thecorresponding Keplerian (geodesic) values of u t and Ωcorresponding to a spinless particle, i.e.,Γ K = 1 q − Mr , Ω K = s Mr , (4.11)and f = f ( r ). Decomposing the energy momentum tensor (4.6) onthe tensor harmonic basis and Fourier transforming (intime), then leads to the source terms S (even / odd) lmω ( r ) en-tering the Regge-Wheeler equation governing both even-type and odd-type perturbations. V. GSF-PN EXPANSION OF THE SPINNINGREDSHIFT
The bulk of this section will be devoted to the mainnew result of this paper, a post-Newtonian expansion ofthe spin dependence of h kk ( y ), the metric perturbationtwice contracted with the helical Killing vector, consid-ered as a function of the orbital-frequency parameter y . A. Retarded and Regularized h kk The outcome of the post-Newtonian RWZ andradiation-gauge approaches are the ℓ -modes of the re-tarded value of h kk , labeled h ℓkk for ℓ ≥
2. Specifically,as detailed in previous works, we obtain explicit PN seriesfor certain low values of ℓ = 2 , . . . ,
6, and generic-formsolutions as a function of ℓ that are valid for all values ℓ ≥
6. These, when supplemented by the low multipoles ℓ = 0 , h kk = ∞ X ℓ =0 h ℓkk . (5.1)This sum is found to diverge due to the singular natureof the (spinning point particle) source. Though we arediscussing here a quantity which does not involve deriva-tives of the metric, we would a priori expect the large- ℓ behavior of the modes to take the form h ℓkk ∼ ± A ∞ (2 ℓ + 1) + B ∞ + O ( ℓ − ) , (5.2)because the source of h µν contains (for a spinning parti-cle) the derivative of a δ function. Here, the sign of the A -term depends, as usual, whether the involved radiallimit is taken from above or from below. Our explicitcomputations found that the value of the A ∞ -coefficienthappened to be zero both in Regge-Wheeler gauge, andin radiation gauge.The expected large- ℓ behavior (5.2) suggests to evalu-ate the regularized value h R kk of h kk by working with theaverage between the two radial limits, namely h R kk = X ℓ (cid:20)
12 ( h ℓkk (+) + h ℓkk ( − ) ) − B ∞ (cid:21) , (5.3)where h lkk ( ± ) are the left and right contributions.Here, we have reasoned as if we were working in agauge which is regularly related to the Lorenz gauge,and as if we were using a decomposition in scalar spheri-cal harmonics (in which cases the results (5.2) and (5.3)would follow from well-known GSF results). Actually,there are two subtleties: (i) the gauges we use are notregularly related to the Lorenz gauge, and (ii) we usea decomposition in tensorial spherical harmonics. Con-cerning the first point, we are relying on the fact thatwe are computing a gauge-invariant quantity, which wecould have, in principle, computed in a Lorenz gauge,and concerning the second point, we are relying on thefact that working with the averaged value of h kk effec-tively reduces the problem to the regularization of a fieldhaving a simpler singularity structure, which is regular-ized by an ℓ -independent B ∞ -type subtraction. [For a recent discussion of these subtleties in the case of thespin-precession invariant, see, e.g., Sec. III E of Ref. [33],and references therein.] Pending a rigorous formal justi-fication of our procedure, we wish to note here that weshall provide two different checks of our regularizationprocedure: (1) our two independent calculations in twodifferent gauges have yielded the same final results; and(2) the first three terms of our final results agree withindependently calculated results in the post-Newtonianliterature.As a sample we give the form of the generic- ℓ resultsfrom the RWZ approach for some low-PN orders. Split-ting the two contributions due to mass and spin, i.e., h ℓkk ( ± ) ( y ) = h ℓkk (0) ( ± ) ( y ) + ˆ s h ℓkk ˆ s ( ± ) ( y ) , (5.4)for ℓ ≥ h ℓkk (0) (+) = h ℓkk (0) ( − ) = 2 y − (26 ℓ + 26 ℓ + 3)(2 ℓ − ℓ + 3) y +3 (6 ℓ + 18 ℓ + 98 ℓ + 166 ℓ + 761 ℓ + 681 ℓ − ℓ − ℓ − ℓ ( ℓ + 1)(2 ℓ + 3)(2 ℓ + 5) y + O ( y ) , (5.5)and h ℓkk ˆ s (+) = h ℓkk ˆ s ( − ) = 3 ( ℓ + ℓ + 3)(2 ℓ − ℓ + 3) y / − ℓ + 30 ℓ + 21 ℓ − ℓ + 414 ℓ + 423 ℓ + 720)2(2 ℓ − ℓ − ℓ ( ℓ + 1)(2 ℓ + 3)(2 ℓ + 5) y / + O ( y / ) . (5.6)Our B ∞ is given by expanding these about ℓ = ∞ , orderby order in the PN expansion.
1. Low multipoles ℓ = 0 , When the source is a non-spinning point particle, Zer-illi [12] has shown long ago how to compute both theexterior and the interior metric perturbations by explic-itly solving the inhomogeneous RWZ field equations. [Seealso Ref. [63] for the corresponding exterior metric com-putation in the case of a Kerr perturbation.] Here, wehave generalized the work of Zerilli to the case of a spin-ning particle, and we have determined both the exteriorand the interior metric perturbations in a RW-like gauge.Our derivation, and our explicit results, are given in Ap-pendix A. Let us highlight here the most important as-pects of our results. In addition, having analytically derived regularization parame-ters would be numerically useful by providing explicit strong-fieldsubtraction terms. We count here the term of order y / that cancels out in the finalresult, after appearing in intermediate calculations. First, the relevant components of the exterior metricperturbation are found (as expected) to come from theadditional (conserved) energy and angular momentumcontribution of the spinning particle, namely h ℓ =0 , tt (+) = 2 δMr , h ℓ =0 , tφ (+) = − δJr , (5.7)where δM ≡ ¯ E and δJ ≡ ¯ J are given by the Killing en-ergy and angular momentum (2.15) of the spinning par-ticle, respectively (see Appendix A for details).The unsubtracted contribution to h kk (+) at the parti-cle’s location due to low multipoles is then given by h ℓ =0 , kk (+) = h ℓ =0 , tt (+) + 2Ω h ℓ =0 , tφ (+) = 2 δMr − r δJ = 2 u (1 − u ) √ − u − ˆ s u / (4 − u + 54 u )(1 − u ) / = 2 y (1 − y ) √ − y − sy / p − y , (5.8)to first order in ˆ s . To determine the needed left-rightaverage (cid:16) h ℓ =0 , kk (+) + h ℓ =0 , kk ( − ) (cid:17) , we further need to deter-mine the interior metric perturbation. This is done inAppendix A. Let us cite here the corresponding jump ofthe metric components across r = r . The RWZ equa-tions for ℓ = 0 and ℓ = 1-odd are found to imply[ h ℓ =0 , kk ] = h ℓ =0 , kk (+) − h ℓ =0 , kk ( − ) = − s y / √ − y , (5.9)whereas ℓ = 1-even is a gauge mode having no contribu-tion to h kk (see Appendix A for details).The final result is then12 (cid:16) h ℓ =0 , kk (+) + h ℓ =0 , kk ( − ) (cid:17) = 2 y (1 − y ) √ − y − ˆ s y / (1 − y ) √ − y , (5.10) which should still be subtracted as for the other ℓ ≥ B. Final results for h kk in the two gauges The subtraction term in the RW gauge is found to be B ∞ = B (0) + ˆ sB ˆ s , (5.11)with B (0) = 2 y − y + 932 y + 83128 y + 123618192 y + 11616332768 y + 4409649524288 y + 422674112097152 y + 26189878473536870912 y + O ( y ) , (5.12)and B ˆ s = 34 y / − y / − y / − y / − y / − y / − y / + O ( y / ) . (5.13)After regularization, using the PN solution for ℓ > ℓ = 2 , , , , h R kk = h R kk (0) + ˆ sh R kk ˆ s , (5.14)with h R kk ˆ s = − y / + 92 y / − y / + (cid:18) π (cid:19) y / + (cid:18) γ + 406415 ln(2) + 3365 ln( y ) − π (cid:19) y / + (cid:18) − π − γ + 21877 ln(3) − − y ) (cid:19) y / + 2174241575 πy + (cid:18) − − − γ − − y )+ 182650175221184 π + 105221565536 π (cid:19) y / − πy + (cid:18) − γ + 4548127007727650 ln( y )+ 7042553383779625 ln(2) + 157132768967167772160 π + 1182637137191165150720 π − − y ) − γ + 2176 ζ (3) − γ ln( y ) − y ) − γ ln(2) (cid:19) y / + O ( y ) . (5.15)When doing the computation in the radiation gauge,we find that the subtraction terms are identical. Theregularized value of h kk ˆ s is, however, different. Let usgive here the difference ∆ h Rkk ˆ s = h R, RG kk ˆ s − h Rkk ˆ s , whereRG labels the radiation gauge result:∆ h Rkk ˆ s = (cid:0) − π (cid:1) y / + (cid:18) − π (cid:19) y / + (cid:18) − π (cid:19) y / − πy (cid:18) − π
36 + 16 π (cid:19) y / + O (cid:0) y (cid:1) . (5.16) This difference is, however, a gauge effect that will disap-pear when computing the gauge-invariant quantity z (1)ˆ s . C. Final results for ∂ r h kk (0) in the two gauges The computation of ∂ r h kk (0) proceeds exactly as in thecase of h kk ˆ s . We then skip all unnecessary details anddisplay only the final result, which in the Regge-Wheelergauge is: M [ ∂ r h kk (0) ] R ( y ) = y − y + 758 y + (cid:18) − π − (cid:19) y + (cid:18) − γ − − y ) + 1661512 π (cid:19) y + (cid:18) − γ + 9207 ln( y ) − π (cid:19) y − πy / + (cid:18) − π − π + 36116722835 γ + 18058362835 ln( y )+ 10644082835 ln(2) + 1701 ln(3) (cid:19) y + 3538981225 πy / + (cid:18) γ ln( y ) − − π + 24503278316777216 π − γ − − y ) − γ + 109568525 ln( y ) + 1753088525 ln(2) − ζ (3) + 876544525 ln(2) ln( y ) + 1753088525 γ ln(2) + 20855431768697683391184640000 (cid:19) y + 39234389693274425 πy / + O ( y ) . (5.17)The subtraction term in this case turns out to be (in both gauges) B ∞ = − y + 114 y + 2764 y + 199256 y + 2278316384 y + 15547565536 y + 38995471048576 y + 203184634194304 y + O ( y ) . (5.18)Again, defining the difference with the radiation gauge as ∆ ∂ r h Rkk (0) = ∂ r h R, RG kk (0) − ∂ r h Rkk (0) , we find M ∆ ∂ r h Rkk (0) = − (cid:0) − π (cid:1) y − (cid:18) − π (cid:19) y − (cid:18) − π (cid:19) y + 128 πy / − (cid:18) − π
36 + 16 π (cid:19) y + O (cid:16) y / (cid:17) . (5.19)Importantly, we note that this is exactly − y − / ∆ h Rkk ˆ s . In view of Eq. (3.11) this will ensure the gauge-independenceof our final result for z (1)ˆ s .0 D. Final result for z (1)ˆ s The linear in spin correction to Detweiler’s gauge-invariant redshift function finally reads z (1)ˆ s ( y ) = y / − y / − y / + (cid:18) − π − γ − − y ) (cid:19) y / + (cid:18) π − − − γ −
265 ln( y ) (cid:19) y / − πy + (cid:18) − γ + 1287935 ln(3) + 381421 ln( y ) + 297761947393216 π − π (cid:19) y / − πy + (cid:18) − γ + 3406817181819125 ln(2) − − ζ (3) − π − π + 342425 γ + 58208105 ln(2) + 8696961575 γ ln(2) − y ) + 342425 γ ln( y )+ 4348481575 ln(2) ln( y ) + 85625 ln( y ) + 4031091580999922500 (cid:19) y / + O ( y ) . (5.20) FIG. 1: Some of the various PN approximants to the linear-in-spin 1SF contribution to the redshift function z ( y ) for a spin-ning particle moving along a circular orbit in a Schwarzschildspacetime. This is the main result of the present paper. Impor-tantly, as we already said, this final result is (as expectedfor a gauge-invariant quantity) identical between the twogauges we have worked in.We show in Fig. 1 the behavior of the various PN ap-proximants to z (1)ˆ s ( y ), which becomes more and morenegative as the light-ring is approached, thereby suggest-ing a negative power-law divergence there. E. Comparison with PN results
In PN theory the linear-in-spin part of the Hamilto-nian (and therefore, using Ref. [57], the correspondinglinear-in-spin part, z (1)ˆ s , of the redshift z = ∂H/∂m ),is known up to the next-to-next-to-leading order [58].Using the results of [58], we have computed z (1)ˆ s as afunction of x ≡ (( M + µ )Ω) / , with the following result(corresponding to the 3.5PN order): z (1)ˆ s ( x ) = ∞ X k =2 C (2 k +1) / χ ( ν ; ln x ) χ x (2 k +1) / , (5.21)where the coefficients are given by C / χ = 13 ν ∆ − ν + 23 ν ,C / χ = − ν ∆ + 1918 ν − ν ∆ − ν + 12 ν ,C / χ = − ν + 1124 ν ∆ + 278 ν − ν − ν ∆ − ν − ν ∆ . (5.22)Here χ ≡ S /µ , ν ≡ µM/M and ∆ ≡ ( M − µ ) /M tot = √ − ν , with M tot = M + µ . To con-vert this result into the 1SF contribution to z ( y ), weuse: S = µM ˆ s , q = µ/M , µ = M tot (1 − ∆) / M = M tot (1 + ∆) /
2, and x = (1 + q ) / y . The firstterm C / χ does not contribute at the first order in q ,i.e., at the first order in SF expansion, while the last two1terms yield z (1)ˆ s = ˆ sy / (cid:0) − y + O ( y ) (cid:1) q + O ( q ) . (5.23)This agrees with the first two terms of (5.20), therebyproviding an independent (partial) check of our result. VI. CONCLUDING REMARKS
The original contribution of this paper is the formu-lation of the generalization of Detweiler’s redshift func-tion z ( y ) for a spinning particle on a circular orbit inSchwarzschild, and its first computation at a high PN-order (8.5PN, instead of the currently known 3.5PN or-der). The spinning particle moves here along an accel-erated orbit, deviating from a timelike circular geodesicsbecause of the spin itself which couples to the Riemanntensor of the background. We have shown how this non-geodesic character of the orbit induces in the spin-linearcontribution to z ( y ) a (gauge-dependent) term propor-tional to the radial gradient of h kk which plays a crucialrole in ensuring the gauge-invariance of the final result.We have checked the gauge-invariance of our result byproviding a dual calculation, in two different gauges, andin verifying that the final results agree. Our formulationopens the way to strong field numerical studies and pro-vides a benchmark for their results. It would also be ofinterest to have independent investigations of the regu-larization procedure we use.Another original result of this work (essential to ac-complish the first result) has been the “completion” ofthe perturbed metric by the explicit computation (in theRegge-Wheeler gauge) of the contribution of the non-radiative multipoles to both the interior and exterior metric generated by a spinning particle. We expect thisresult to play a useful role in future applications.Finally, using available PN results, we have checkedthe first terms of our final result. Appendix A: Low multipoles l = 0 , We give below the solutions for the non-radiativemodes ( ℓ = 0 and ℓ = 1 odd) needed for the comple-tion of the full metric perturbation. Our approach is ageneralization of well-known results of Zerilli [12] to thecase of a spinning particle. The ℓ = 1 even mode is essen-tially a gauge mode that describes a shift of the center ofmomentum of the system. We have checked that it doesnot contribute to the present calculation.
1. The ℓ = 0 mode The ℓ = 0 mode is of even parity, is independent oftime and represents the perturbation in the total mass-energy of the system. This was shown by Zerilli for thecase of a non-spinning test particle, and our explicit cal-culations below show that this extends to the case of aspinning particle if one uses as additional contributionto the mass of the system the conserved Killing energy δM ≡ ¯ E , Eq. (2.8), (2.15), of the spinning particle. Notethat our derivation directly solves the inhomogeneousRegge-Wheeler-Zerilli equations, without using Komar-type surface integrals.For this mode there are two gauge degrees of freedomand one can set H = 0 = K . The remaining perturba-tion functions H and H satisfy the following equations dH dr + H rf = 2 √ πµ u t r (cid:20) δ ( r − r ) − M Ω K ˆ s (cid:18) r − Mr f δ ( r − r ) + r δ ′ ( r − r ) (cid:19)(cid:21) ,dH dr + H rf = 2 √ πµ M Ω K r Γ K f ˆ sδ ( r − r ) , (A1)to first order in ˆ s , with solution H = 2 √ πµu t (cid:20) r (1 − sM Ω K ) θ ( r − r ) + f rf (cid:18) s M Ω K r f (cid:19) θ ( r − r ) (cid:21) ,H = 2 √ πµu t (cid:20) f rf (cid:18) s M Ω K r f (cid:19) θ ( r − r ) − ˆ sM Ω K δ ( r − r ) (cid:21) . (A2)The nonvanishing metric components to first order in ˆ s can then be written (in terms of δM ≡ ¯ E , Eq. (2.15)) as h tt = f H √ π = 2 µ δMr (cid:20) rfr f (cid:18) − r − Mr f M Ω K ˆ s (cid:19) θ ( r − r ) + θ ( r − r ) (cid:21) ,h rr = H √ πf = 2 µ δMrf θ ( r − r ) − µ δMf M Ω K ˆ sδ ( r − r ) . (A3)
2. The ℓ = 1 odd mode Similarly, we have explicitly shown, by solving theRegge-Wheeler-Zerilli field equations, that the ℓ = 1 odd mode represents the angular momentum perturba-2tion δJ = ¯ J , Eq. (2.8), (2.15), added by the spinningparticle to the system.The perturbation equations for this case assume h (odd)1 = 0, whereas h (odd)0 is such that d h (odd)0 dr − h (odd)0 r = − µ √ πu t Ω h δ ( r − r ) − M Ω K ˆ s (cid:16) δ ( r − r ) + r M ( r − M ) δ ′ ( r − r ) (cid:17)i , (A4)to first order in ˆ s , with solution h (odd)0 = 2 r π µu t Ω r (cid:20) r r (cid:18) −
12 ˆ s ( r + M )Ω K (cid:19) θ ( r − r ) + r r (1 + ˆ sr Ω K f ) θ ( r − r ) (cid:21) δ m, , (A5)and only the m = 0 mode is nonzero. The only nonvanishing metric component is then given (in terms of δJ ≡ ¯ J ,Eq. (2.15)) by h tφ = − r π h (odd)0 sin θ = − µ δJr (cid:20) r r (cid:18) −
32 ( r − M )Ω K ˆ s (cid:19) θ ( r − r ) + θ ( r − r ) (cid:21) sin θ , (A6)to first order in ˆ s . Appendix B: MPD equations in the perturbedspacetime
In this section we briefly discuss the MPD equationsin the perturbed spacetime to first-order in spin. Thiscomplementary material is left here for convenience andit will be of use in future works. Working to the firstorder in spin, Eqs. (2.3) and (2.4) reduce to µ D U µ dτ = − R µναβ U ν S αβ , (B1)D S µν dτ = 0 , (B2)where we recall that U µ ≡ dz µ /dτ , and where we haveused the property P = µU + O ( s ) for the momentum ofthe particle.Assuming that the background metric admits theKilling vector k = ∂ t + Ω ∂ φ and that the body’s orbitis aligned with k , U = u t k , implying − ( u t ) − = k · k = − f + Ω r + h kk , (B3)the MPD equations become µu t ∇ k k µ = − R µναβ k ν S αβ , ∇ U S µν = 0 . (B4)Defining then the spin vector (orthogonal to both U and u ≡ P/µ at the first order in spin) by spatial duality (seeEq. (2.6)) S γ = 12 u t k σ η σγαβ S αβ , S γ k γ = 0 , (B5) one finds immediately that the spin vector is parallel-propagated along U , ∇ U S γ = 0. The equations of motioninstead can be cast in the form µu t ∇ k k µ = −
12 ( ∇ µβ k α ) S αβ = −
12 ( ∇ µ K βα ) S αβ , (B6)where the (antisymmetric) tensor K αβ is given by K αβ = ∇ α k β = ∂ [ α k β ] . (B7)Finally, we require that the spin vector be orthogonal tothe equatorial plane, i.e., S = − se ˆ θ , e ˆ θ = 1 r (cid:18) − r h θθ (cid:19) ∂ θ . (B8)The equations of motion then imply the following solu-tion for Ω M Ω = M Ω K (cid:20) −
32 ˆ sM Ω K + q ( ˜Ω + ˆ s ˜Ω s ) (cid:21) , (B9)where Ω K ≡ q Mr , as defined in Eq. (4.11) above,˜Ω = − M u [ ∂ r h (0) kk ] r = M/u , (B10)and3˜Ω s = − u / − u ) h (0) kk + (5 − u ) u / h (0) rr − u (3 − u )(1 − u )2 M (1 − u ) h (0) tφ − (1 − u )(2 − u + 4 u ) u / M (1 − u ) h (0) φφ − M u − / ∂ rr h (0) kk ] r = M/u − M u (1 − u )[ ∂ rr h (0) φk ] r = M/u + M u (1 − u )[ ∂ r ¯ φ h (0) rk ] r = M/u −
14 (1 − u )[ ∂ ¯ φ h (0) rk ] r = M/u + M u / − u ) [ ∂ r h (0) kk ] r = M/u − M u [ ∂ r h (1) kk ] r = M/u + M − u )(1 − u ) u − / [ ∂ r h (0) rr ] r = M/u + (1 − u )4(1 − u ) [ ∂ r h (0) tφ ] r = M/u + (1 − u )(2 − u ) u / M (1 − u ) [ ∂ r h (0) φφ ] r = M/u . (B11)Introducing the dimensionless frequency parameter y = ( M Ω) / gives the relation y = u − ˆ su / + q F ( u ) , (B12)where F ( u ) = F ( u ) + ˆ s F ˆ s ( u ), with F ( u ) = 23 u ˜Ω ( u ) , F ˆ s ( u ) = 13 u / ˜Ω ( u ) + 23 u ˜Ω s ( u ) . (B13)This relation can be inverted to give (see Eq. (3.6)) u = y (cid:0) − ˆ sy / (cid:1) / + qf ( y ) , (B14)where f ( y ) = f ( y ) + ˆ sf ˆ s ( y ), with f ( y ) = −F ( y ) ,f ˆ s ( y ) = −F ˆ s ( y ) − y / F ( y ) − y / F ′ ( y ) . (B15)Substituting then into Eq. (B3) finally yields Eq. (3.9). Appendix C: Metric reconstruction in theRegge-Wheeler gauge1. Solving the RWZ equations
The perturbation functions of both parity can be ex-pressed in terms of a single unknown for each sector,satisfying the same Regge-Wheeler equation L ( r )(RW) [ R (even / odd) ℓmω ] = S (even / odd) ℓmω ( r ) , (C1)where L ( r )(RW) denotes the RW operator L ( r )(RW) = f ( r ) d dr + 2 Mr f ( r ) ddr + [ ω − V (RW) ( r )]= d dr ∗ + [ ω − V (RW) ( r )] , (C2) with d/dr ∗ = f ( r ) d/dr , and the RW potential V (RW) ( r ) = f ( r ) (cid:18) ℓ ( ℓ + 1) r − Mr (cid:19) . (C3)The source terms have the form S (even / odd) ℓmω ( r ) = c ℓmω δ ( r − r ) + c ℓmω δ ′ ( r − r )+ c ℓmω δ ′′ ( r − r ) + c ℓmω δ ′′′ ( r − r ) . (C4)The coefficients c ℓmωk , k = 0 . . . r and have the general form c ℓmωk = ˜ c ℓmωk δ ( ω − m Ω) , (C5)with c ℓmω ≡ X in ℓω and X up ℓω of theRW operator as G ( r, r ′ ) = G (in) ( r, r ′ ) H ( r ′ − r ) + G (up) ( r, r ′ ) H ( r − r ′ ) , where G (in) ( r, r ′ ) = X in ℓω ( r ) X up ℓω ( r ′ ) W ℓω ,G (up) ( r, r ′ ) = X in ℓω ( r ′ ) X up ℓω ( r ) W ℓω . (C6)Here W ℓω denotes the (constant) Wronskian W ℓω = f ( r ) (cid:20) X in ℓω ( r ) ddr X up ℓω ( r ) − ddr X in ℓω ( r ) X up ℓω ( r ) (cid:21) = const . (C7)and H ( x ) is the Heaviside step function. Both even-parity and odd-parity solutions are then given by inte-grals over the corresponding (distributional) sources as R (even / odd) ℓmω ( r ) = Z dr ′ G ( r, r ′ ) f ( r ′ ) S (even / odd) ℓmω ( r ′ ) . (C8)Once the radial function is known for both parities,the perturbed metric components are then computedby Fourier anti-transforming, multiplying by the angu-lar part and summing over m (between − ℓ and + ℓ ), andthen over ℓ (between 0 and + ∞ ).4
2. Computing h kk Let us consider the quantity h kk ≡ h αβ k α k β , where k = ∂ t + Ω ∂ φ . In the RW gauge we have h kk = X ℓm h ℓmkk = X ℓm ( h ℓm (even) kk + h ℓm (odd) kk ) , (C9)where the even and odd contributions (for ℓ ≥
2) are ofthe form h ℓm (even) kk ( r ) = (cid:12)(cid:12)(cid:12) Y ℓm (cid:16) π , (cid:17)(cid:12)(cid:12)(cid:12) A even ℓm ( r ) J in ( r ) J up ( r ) ,h ℓm (odd) kk ( r ) = (cid:12)(cid:12)(cid:12) ∂ θ Y ℓm (cid:16) π , (cid:17)(cid:12)(cid:12)(cid:12) A odd ℓm ( r ) ˜ J in ( r ) ˜ J up ( r ) , once evaluated along the world line of the particle r = r , θ = π/ φ = Ω t . The coefficients A even / odd ℓm ( r ) and J in / up ( r ) = α in / up ( r ) X in / up ℓω ( r )+ β in / up ( r ) dX in / up ℓω dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = r , ˜ J in / up ( r ) = ˜ α in / up ( r ) X in / up ℓω ( r )+ ˜ β in / up ( r ) dX in / up ℓω dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = r , (C10) all depend on ˆ s and are known functions of r , with ω = m Ω.Expanding all terms to first order in ˆ s and combiningthe odd and even contributions leads to h ℓmkk = h ℓmkk (0) + ˆ sh ℓmkk ˆ s . (C11)Performing the m -summation yields by definition h ℓkk ≡ X m h ℓmkk . (C12)One must then finally express r in terms of y and ˆ s toget h ℓkk ( y, ˆ s ). Acknowledgments
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