Computing inspirals in Kerr in the adiabatic regime. I. The

a r X i v :g r -q c /0505075v 2 28 J u n 2005Computing inspirals in Kerr in the adiabatic regime.I.The scalar case Steve Drasco ?,′Eanna ′E.Flanagan ??,Scott A.Hughes § ?Center for Radiophysics and Space Research,Cornell University,Ithaca,NY 14853?Laboratory for Elementary Particle Physics,Cornell University,Ithaca,NY 14853§Department of Physics,MIT,77Massachusetts Ave.,Cambridge,MA 02139 MIT Kavli Institute for Astrophysics and Space Research,MIT,77Massachusetts Ave.,Cambridge,MA 02139Abstract.A key source for LISA will be the inspiral of compact objects into massive black holes.Recently Mino has shown that in the adiabatic limit,gravitational waveforms for these sources can be computed by using for the radiation reaction force the gradient of one half the di?erence between the retarded and advanced metric aeef4fd076eeaeaad1f33010ing post-Newtonian expansions,we argue that the resulting waveforms should be su?ciently accurate for signal detection with LISA.Data-analysis templates will require higher accuracy,going beyond adiabaticity;this remains a signi?cant challenge.We describe an explicit computational procedure for obtaining waveforms based on Mino’s result,for the case of a point particle coupled to a scalar ?eld.We derive an explicit expression for the time-averaged time derivative of the Carter constant,and verify that the expression correctly predicts that circular orbits remain circular while evolving under the in?uence of radiation reaction.The derivation uses detailed properties of mode expansions,Green’s functions and bound geodesic orbits in the Kerr spacetime,which we review in detail.This paper is about three quarters review and one quarter new material.The intent is to give a complete and self-contained treatment of scalar radiation reaction in the Kerr spacetime,in a single uni?ed notation,starting with the Kerr metric,and ending with formulae for the time evolution of all three constants of the motion that are su?ciently explicit to be used immediately in a numerical code.

Computing inspirals in Kerr in the adiabatic regime2 1.Summary and overview

1.1.Gravitational radiation reaction of point particles

A key unsolved problem in general relativity is to compute the gravitational radiation produced by a small object spiralling into a much larger black hole.This problem is of direct observational relevance.Inspirals of compact objects into intermediate

mass black holes(M～102?103M⊙)may be observed by LIGO and other ground based interferometers[1];recent observations suggest the existence of black holes in

this mass range[2,3].In addition,a key source for the space-based gravitational wave detector LISA is the?nal epoch of inspiral of a stellar-mass compact object into a massive(M～106M⊙)black hole at the center of a galaxy.Gair et.al.[4]have estimated that LISA should see over a thousand such inspiral events during its multi-year mission lifetime,based on Monte Carlo simulations of the dynamics of stellar cusps by Freitag[5].

Observations of these signals will have several major scienti?c payo?s[6]:?From the observed waveform,one can measure the mass and spin of the central black hole with fractional accuracies of order10?4[7,8].The spin can provide useful information about the growth history(mergers versus accretion)of the black hole[9].

?Likewise,one obtains a census of the inspiralling objects’masses with precision ～10?4,teaching us about the stellar mass function and mass segregation in the central parsec of galactic nuclei.

?The measured event rate will teach us about dynamics in the central parsec of galaxies.

?The gravitational waves will be wonderful tools for probing the nature of black holes in galactic nuclei,allowing us to observationally map,for the?rst time,the spacetime geometry of a black hole,and providing a high precision test of general relativity in the strong?eld regime[10,11].

To be in the LISA waveband,the mass M of the black hole must be in the range 105M⊙～

Realizing the above science goals will require accurate theoretical models (templates)of the gravitational waves.This is because the method of matched?ltering will be used both to detect the signals buried in the detector noise,and to measure the parameters characterizing detected signals.The accuracy requirement is roughly that the template should gain or lose no more than～1cycle of phase compared to the true waveform over the～105cycles of inspiral.These sources must therefore be modeled with a fractional accuracy～10?5.The past several years have seen a signi?cant research e?ort in the relativity community aimed at providing these accurate templates.

To date,there have been several approaches to generating waveforms.The foundation for all the approaches is the fact that,sinceμ/M?1,the?eld of the compact object can be treated as a linear perturbation to the large black hole’s gravitational?eld.On short timescales,the compact object moves on a geodesic of

Computing inspirals in Kerr in the adiabatic regime3 the Kerr geometry,characterized by its conserved energy E,z-component of angular momentum L z,and Carter constant Q.Over longer timescales,radiation reaction causes the parameters E,L z and Q to evolve and the orbit to shrink.

The various approaches are:

(i)Use of post-Newtonian methods:Fairly crude waveforms can be obtained using

post-Newtonian methods[13,8].These have been used to approximately scope out LISA’s ability to detect inspiral events[4]and to measure the waveform’s parameters[8].However,since the orbital speeds are a substantial fraction of the speed of light,these waveforms are insu?ciently accurate for the eventual detection and data analysis of real signals.

(ii)Use of conservation laws:In this approach[14,15,16,17]one uses the Teukolsky-Sasaki-Nakamura(TSN)formalism[18,19,20]to compute the?uxes of energy E and angular momentum L z to in?nity and down the black hole horizon generated by a a compact object on a geodesic orbit.Imposing global conservation of energy and angular momentum,one infers the rates of change of the orbital energy and angular momentum.For certain special classes of orbits(circular and equatorial orbits)this provides enough information that one can also infer the rate of change of the Carter constant Q,and thus the inspiralling trajectory.

(iii)Direct computation of the self-force:In this more fundamental approach one computes the self-force or radiation-reaction force arising from the interaction of the compact object with its own gravitational?eld.A formal expression for this force in a general vacuum spacetime in terms of the retarded Green’s function was computed several years ago[21,22].Translating this expression into a practical computational scheme for Kerr black holes is very di?cult and is still in progress.

Roughly100papers devoted to this problem have appeared in the last few years;

see,for example Poisson[23]for an overview and references.

(iv)Time-domain numerical simulations:Another technique is to numerically integrate the Teukolsky equation as a2+1PDE in the time domain[24,25, 26,27,28,29,30],and to model the compact object as a?nite-sized source.This approach faces considerable challenges:(i)There is a separation of timescales —the orbital period is much shorter than the radiation reaction timescale.(ii) There is a separation of lengthscales—the compact object is much smaller than the black hole.(iii)The self-?eld of the small object must be computed with extremely high accuracy,as the piece of the self-?eld responsible for the self-force is a tiny fraction(～μ/M)of the pergent self-?eld.This approach may eventually be competitive with(ii)and(iii)but is currently somewhat far from being competitive[?ux accuracies are～10%as compared to～10?6for(ii)].

1.2.Radiation reaction in the adiabatic regime

A key feature of these systems is that they evolve adiabatically:the radiation reaction timescale～M2/μis much longer than the orbital timescale～M,by a factor of the inverse M/μof the mass ratio.This has implications for the nature of the signal.The self-acceleration a of the compact object can be expanded in powers of the mass ratio as

a=μ

M

a2,diss+

μ

M2 .(1.1)

Computing inspirals in Kerr in the adiabatic regime4 Here a1,diss and a1,cons are the dissipative and conservative pieces of the leading-order self-acceleration computed in Refs.[21,22].Similarly, a2,diss and a2,cons are the corresponding pieces of the?rst correction to the self-acceleration,which has not yet been computed(although see Ref.[31]for work in this direction).

The e?ect of the dissipative pieces of the self force will accumulate secularly,while the e?ect of the conservative pieces will not.Hence the e?ect of the dissipative pieces on the phase of the orbit will be larger than that of the conservative pieces by a factor of the number of cycles of inspiral,～M/μ.Consider now,for example,the azimuthal phaseφ(t)of the orbit.This can be expanded in powers of the mass ratio using a two-timescale expansion as[32,33]

φ(t)=M

M

φ2(t,t1)+O μ2

Computing inspirals in Kerr in the adiabatic regime5 endowed with a scalar charge,coupled to a scalar?eld.This computation is useful as a warm up exercise for the more complicated case of a particle emitting gravitational waves[40].

We start in Sec.2by describing the model of a point particle coupled to a scalar ?eld,and review how the self-force causes both an acceleration of the particle’s motion and also an evolution of the renormalized rest mass of the particle[41].In Sec.3we review the properties of generic bound geodesic orbits in the Kerr spacetime.A crucial result we use later is that the r andθmotions are periodic when expressed as functions of a particular time parameter we call Mino time[34],and that the t andφmotions consist of linearly growing terms plus terms that are periodic with the period of the r-motion,plus terms that are periodic with the period of theθ-motion[42].

Section4reviews Mino’s derivation of the half-retarded-minus-half-advanced prescription for radiation reaction in the adiabatic limit[34].In Sec.5we discuss a convenient basis of modes for solutions of the scalar wave equation in Kerr,namely the“in”,“out”,“up”and“down”modes used by Chrzanowski[43]and Gal’tsov[36]. We also review several key properties of these modes including symmetry relations and relations between the various re?ection and transmission coe?cients that appear in the de?nitions of the modes.Section6reviews the standard derivation of the mode expansion of the retarded Green’s function,and Sec.7reviews the derivation of the mode expansion of the radiative Green’s function given by Gal’tsov[36],correcting several typos in Gal’tsov which are detailed in Appendix B.

Next,in Sec.8we turn to the source term in the wave equation for the scalar ?eld.We review the derivation of Drasco and Hughes of the harmonic decomposition of this source term in terms of a discrete sum over frequencies in which harmonics of three di?erent fundamental frequencies occur[42].We derive expressions for the mode coe?cients in the expansion of the retarded?eld near future null in?nity and near the future event horizon.These coe?cients are expressed as integrals over a torus in phase space,which is?lled ergodically by the geodesic motion.

Section9combines the results of the preceding sections to derive expressions for the time-averaged rates of change of two of the conserved quantities of geodesic motion,namely the energy E and the angular momentum L z.The expressions are derived from the radiative self force,following Gal’tsov[36].Gal’tsov also shows that identical expressions are obtained by using the?uxes of energy and angular momentum to in?nity and down the black hole horizon.[This result has recently been independently derived for circular,equatorial orbits in Ref.[44].]We also show that the time-averaged rate of change of the renormalized rest mass of the particle vanishes.

In section10we derive an expression for the time-averaged rate of change of the Carter constant,using the radiative self-force and the mode expansion of the radiative Green’s function.This expression[Eq.(10.1)below]is the main new result in this paper.It involves two new amplitudes that are computed in terms of integrals over the torus in phase space,just as for the amplitudes appearing in the energy and angular momentum?uxes.Finally,in Sec.11we show that our result correctly predicts the known result that circular orbits remain circular while evolving under the in?uence of radiation reaction.This prediction serves as a check of our result.

As apparent from the above summary,about25percent of this paper is new material,and the remaining75percent is review.The intent is to give a complete and self-contained treatment of scalar radiation reaction in the Kerr spacetime,in a single uni?ed notation,starting with the Kerr metric,and ending with formulae for the evolution of all three constants of the motion that are su?ciently explicit to be

Computing inspirals in Kerr in the adiabatic regime6 used immediately in a numerical code.

2.The scalar?eld model

We consider a point particle of scalar charge q coupled to a scalar?eldΦ.We denote byμ0the bare rest mass of the particle,to be distinguished from a renormalized rest massμwhich will occur below.The particle moves in a spacetime with metric gαβwhich is?xed;we neglect the gravitational waves generated by the particle.The worldline of the particle is xα=zα(τ),whereτis proper time.The action is taken to be

S=?1

?g(?Φ)2? dτ{μ0?qΦ[z(τ)]}.(2.1)

Varying this action with respect to the worldline yields the equation of motion

(μ0?qΦ)aα=q(gαβ+uαuβ)?βΦ,(2.2) where uαis the4-velocity and aαis the4-acceleration.Following Poisson[23],we de?ne the renormalized massμby?

μ(τ)=μ0?qΦ[z(τ)].(2.3) The equation of motion(2.2)can then be written as

aα=

q

dτ

(τ)=?quα?αΦ.(2.5) The phenomenon of evolution of renormalized rest-mass is discussed further in Refs. [45,46,47].Equations(2.4)and(2.5)can also be combined to give the expression fα=uβ?β(μuα)=q?αΦ.(2.6) for the total self force.The components of fαparallel to and perpendicular to the four velocity yield the quantity dμ/dτandμtimes the self-acceleration,respectively.

Varying the action(2.1)with respect to the?eldΦgives the equation of motion Φ(x)=T(x),(2.7) where the scalar source is

T(x)=?q ∞?∞dτδ(4)[x,z(τ)].(2.8) Here

δ(4)(x,x′)=δ(4)(x?x′)/

√

?g(x′)G ret(x,x′)T(x′)=?q dτG ret[x,z(τ)].(2.10)

?If we endow the particle with an electric charge e and add the electromagnetic coupling term e dτuαAαto the action(2.1),the equation of motion(2.4)becomesμaα=q(gαβ+uαuβ)?βΦ+ eFαβuβ,where Fαβis the Faraday tensor.This shows that the renormalized mass(2.3)is the mass that would be measured by coupling to other?elds.

Computing inspirals in Kerr in the adiabatic regime7 Here G ret(x,x′)is the retarded Green’s function for the scalar wave equation(2.7).

The retarded?eld(2.10)must of course be regularized before being inserted into the expression(2.4)for the self-acceleration and into the expression(2.5)for dμ/dτ, or else pergent results will be obtained.The appropriate regularization prescription has been derived by Quinn[41].The regularized self-acceleration at the point zα(τ) is[41]

aα(τ)=q2

6

Rβγuγ?limε→0 τ?ε?∞dτ′?βG ret[z(τ),z(τ′)] ,(2.11)

and the regularized expression for dμ/dτis

dμ

12R+q2lim

ε→0 τ?ε?∞dτ′uα?αG ret[z(τ),z(τ′)].(2.12)

As discussed in the introduction,we specialize in this paper to bound motion about a Kerr black hole of mass M with M?μ.In this case the terms in Eqs.(2.11) and(2.12)involving the Ricci tensor Rαβvanish.Also,as long as the orbit is not very close to the innermost stable orbit,the evolution of the orbit is adiabatic:the orbital evolution timescale～M2/μis much longer than the orbital timescale～M. This adiabaticity allows a signi?cant simpli?cation of the formulae(2.11)and(2.12) for the self-acceleration and for the rate of change of mass dμ/dτ,as shown by Mino [34].Namely,these formulae reduce to the simple expressions(2.4)and(2.5),but withΦreplaced by the radiative?eld

Φrad=

1

μ

(gαβ+uαuβ)?βΦrad.(2.14) together with the expression for the evolution of rest mass

dμ

Computing inspirals in Kerr in the adiabatic regime8 3.1.The Kerr spacetime

In Boyer-Lindquist coordinates(t,r,θ,φ),the Kerr metric is

ds2=? 1?2MrΣdtdφ+(?4??a2sin2θ)sin2θ

?

dr2.(3.1) Here

Σ≡r2+a2cos2θ,(3.2)

?≡r2+a2?2Mr,(3.3)

?≡

?g=Σsinθ,(3.5) and the wave operator is given by

Σ Φ=? ?4?Φ,tφ+ 1? Φ,φφ

+(?Φ,r),r+

1

??t+?r+

a

2Σ

?t??2Σ?φ,(3.8) and

m=

1

2(r+ia cosθ) ia sinθ?t+?θ+i

?

dr,(3.10)

n=??

2Σ

dφ?

1

√

Computing inspirals in Kerr in the adiabatic regime9 3.2.Constants of the motion

We de?ne the conserved energy per unit rest massμ

E=? u·

?

?φ

,(3.15) and Carter constant pided byμ2M2

Q=u2θ?a2cos2θE2+cot2θL2z+a2cos2θ.(3.16) [From now on we will for simplicity call these dimensionless quantities“energy”,

“angular momentum”and

“Carter constant”.]

The

geodesic equations

can then be written in the form[42]

dr

dλ 2=Q?cot2θL2z?a2cos2θ(1?E2)≡Vθ(θ),(3.18) dφ

??1

?a2L z

dλ

=E (r2+a2)2? ≡V t(r,θ).(3.20) Hereλis the Mino time parameter[34],related to proper timeτby

dλ=

1

?

(?2E?aL z)2??u2r?r2.(3.26)§These quantities were denoted R(r),Θ(θ),Φ(r,θ)and T(r,θ)in Ref.[42].We no not use this notation here since it would clash with the functions R,ΘandΦde?ned in Eq.(5.1)below.

Computing inspirals in Kerr in the adiabatic regime10 Solving this for u r gives

u r=± ?2(?2E?aL z)2?r2?;(3.27) this formula will be useful later.

3.3.Parameterization of solutions

Following Mino[34],we parameterize any geodesic by seven parameters:

E,L z,Q,λr0,λθ0,t0,φ0.(3.28) Here t0andφ0are the values of t(λ)andφ(λ)atλ=0.The quantityλr0is the value ofλnearest toλ=0for which r(λ)=r min,where r min is the minimum value of r attained on the geodesic.Similarlyλθ0is the value ofλnearest toλ=0for which θ(λ)=θmin,whereθmin is the minimum value ofθattained on the geodesic.This parameterization is degenerate because of the freedom to reparametrize the geodesic viaλ→λ+?λ.We discuss this degeneracy further in Sec.3.7.

Frequently in this paper we will focus on the?ducial geodesic associated with the constants E,L z and Q,namely the geodesic with

λr0=λθ0=t0=φ0=0.(3.29) 3.4.Motions in r andθ

It follows from the geodesic equations(3.17)and(3.18)that the functions r(λ)and θ(λ)are periodic.We denote the periods byΛr andΛθ,respectively,so

r(λ+Λr)=r(λ),θ(λ+Λθ)=θ(λ).(3.30) Using the initial condition r(λr0)=r min we can write the solution r(λ)to Eq.(3.17) explicitly as

r(λ)=?r(λ?λr0),(3.31) where the function?r(λ)is de?ned by

?r(λ)r min dr V r(r)=λ.(3.32)

Similarly we can write the solutionθ(λ)to Eq.(3.18)explicitly as

θ(λ)=?θ(λ?λθ0),(3.33) where the function?θ(λ)is de?ned by

?θ(λ)θmin dθVθ(θ)=λ.(3.34) The functions?r(λ)and?θ(λ)are just the r andθmotions for the?ducial geodesic.

Computing inspirals in Kerr in the adiabatic regime 11

3.5.Motion in t

Next,the function V t (r,θ)that appears on the right hand side of Eq.(3.20)is a sum of a function of r and a function of θ:

V t (r,θ)=V tr (r )+V tθ(θ),

(3.35)

where V tr (r )=E?4/?+aL z (1??2/?)and V tθ(θ)=?a 2E sin 2θ.Therefore using t (0)=t 0we obtain

t (λ)=t 0+ λ

dλ′{V tr [r (λ′)]+V tθ[θ(λ′)]}.(3.36)

Next we de?ne the averaged value V tr of V tr to be

V tr =1

Λr Λr

dλV tr [?r (λ)].(3.37)

Here the second equality follows from the representation (3.31)of the r motion together with the periodicity condition (3.30).Similarly we de?ne

V tθ =

1Λθ Λθ

dλV tθ[?θ(λ)].(3.38)Inserting these de?nitions into Eq.(3.36)allows us to write t (λ)as a sum of a linear term and terms that are periodic [42]:

t (λ)=t 0+Γλ+?t r (λ)+?t θ(λ)

(3.39)≡t 0+Γλ+?t (λ).

(3.40)Here we have de?ned the constant Γ= V tr + V tθ ,

(3.41)and the functions

?t r (λ)=

λ

0dλ′

V tr [r (λ′

)]? V tr

,

(3.42)?t θ(λ)=

λ0

dλ′ V tθ[θ(λ′)]? V tθ .

(3.43)

The key property of these functions is that they are periodic:

?t r (λ+Λr )=?t r (λ),(3.44)?t θ(λ+Λθ)=?t θ(λ);

(3.45)

this follows from the de?nitions (3.42)and (3.43)together with the periodicity condition (3.30).We can exhibit the dependence of these functions on the parameters λr 0and λθ0by substituting the formulae (3.31)and (3.33)for r (λ)and θ(λ)into Eqs.(3.42)and (3.43).The result is

?t r (λ)=?t r (λ?λr 0)??t r (?λr 0),(3.46)

?t θ(λ)=?t

θ(λ?λθ0)??t θ(?λθ0),(3.47)

where the functions ?t

r (λ)and ?t θ(λ)are de?ned by ?t

r (λ)= λ

0dλ′ V tr [?r (λ′)]? V tr ,(3.48)?t

θ(λ)=

λ0

dλ′ V tθ[?θ

(λ′)]? V tθ .(3.49)

Computing inspirals in Kerr in the adiabatic regime

12

3.6.Motion in φThe motion in φcan be analyzed in exactly the same way as the motion in t .First,the function V φ(r,θ)that appears on the right hand side of Eq.(3.19)is a sum of a function of r and a function of θ:V φ(r,θ)=V φr (r )+V φθ(θ),(3.50)where V φr (r )=aE (?2/??1)?a 2L z /?and V φθ(θ)=csc 2θL z .Therefore using φ(0)=φ0we obtain φ(λ)=φ0+ λ0

dλ′{V φr [r (λ′)]+V φθ[θ(λ′)]}.(3.51)Next we de?ne the averaged value V φr of V φr to be V φr =1Λr Λr 0

dλV φr [?r (λ)].(3.52)Here the second equality follows from the representation (3.31)of the r motion together with the periodicity condition (3.30).Similarly we de?ne V φθ =1Λθ Λθ0

dλV φθ[?θ(λ)].(3.53)Inserting these into Eq.(3.51)and using φ(0)=φ0allows us to write φ(λ)as a sum of a linear term and terms that are periodic [42]:

φ(λ)=φ0+Υφλ+?φr (λ)+?φθ(λ)(3.54)

≡φ0+Υφλ+?φ(λ).(3.55)

Here we have de?ned the constant

Υφ= V φr + V φθ ,(3.56)and the functions ?φr (λ)= λ

0dλ′ V φr [r (λ′)]? V φr ,

(3.57)?φθ(λ)= λ0

dλ′

V φθ[θ(λ′)]? V φθ .(3.58)The key property of these functions is that they are periodic:

?φr (λ+Λr )=?φr (λ),

(3.59)?φθ(λ+Λθ)=?φθ(λ);(3.60)

this follows from the de?nitions (3.57)and (3.58)together with the periodicity condition (3.30).We can exhibit the dependence of these functions on the parameters λr 0and λθ0by substituting the formulae (3.31)and (3.33)for r (λ)and θ(λ)into Eqs.(3.57)and (3.58).The result is

?φr (λ)=?φr (λ?λr 0)??φr (?λr 0),(3.61)

?φθ(λ)=?φ

θ(λ?λθ0)??φθ(?λθ0),(3.62)where the functions ?φr and ?φθare de?ned by ?φr (λ)= λ0dλ′ V φr [?

r (λ′)]? V φr ,(3.63)?φθ(λ)= λ

0dλ′ V φθ[?θ(λ′)]? V φθ .(3.64)

Computing inspirals in Kerr in the adiabatic regime13 3.7.Re-parameterization freedom

Not all of the parameters E,L z,Q,λr0,λθ0,t0andφ0that characterize the geodesic are independent.This is because of the freedom to change the dependent variable λviaλ→?λ=λ+?λ.Under this change of variable the parametersλr0andλθ0 transform as

λr0→?λr0=λr0+?λ,(3.65)

λθ0→?λθ0=λθ0+?λ.(3.66) We can compute how the parameters t0andφ0transform as aeef4fd076eeaeaad1f33010bining Eqs.

(3.40),(3.46)and(3.47)gives the following formula for the t motion:

t=t0+Γλ+?t r(λ?λr0)??t r(?λr0)+?tθ(λ?λθ0)??tθ(?λθ0).(3.67) Rewriting the right hand side in terms of?λ,?λr0and?λθ0yields

t=t0+Γ?λ?Γ?λ+?t r(?λ??λr0)??t r(?λr0)+?tθ(?λ??λθ0)??tθ(?λθ0)

≡?t0+Γ?λ+?t r(?λ??λr0)??t r(??λr0)+?tθ(?λ??λθ0)??tθ(??λθ0).(3.68) Comparing the?rst and second lines here allows us to read o?the value of?t0:

?t

=t0?Γ?λ+?t r(?λr0??λ)??t r(?λr0)+?tθ(?λθ0??λ)??tθ(?λθ0).(3.69) 0

Similarly we obtain

φ0→?φ0=φ0?Υφ?λ+?φr(?λr0??λ)??φr(?λr0)+?φθ(?λθ0??λ)??φθ(?λθ0).(3.70) In Sec.8.4below we will explicitly show that all of the amplitudes and waveforms we compute are invariant under these transformations.

3.8.Averages over Mino time in terms of angular variables

We will often encounter functions Fθ(λ)of Mino timeλthat are periodic with period Λθ.For such functions the average over Mino time is given by

Fθ λ=1

.(3.73)

1?z

Hereβ=a2(1?E2)and z?and z+are the two zeros of Vθ(z).They are ordered such that0≤z?≤1≤z+.The variableχis de?ned by

cos2?θ(λ)=z?cos2χ,(3.74) together with the requirement thatχincreases monotonically asλincreases.From the de?nition(3.34)of?θ(λ)together with the formula(3.73)for Vθ(z)we get

dχ

β(z+?z?cos2χ).(3.75)

Computing inspirals in Kerr in the adiabatic regime14 Therefore the average(3.71)can be written as

Fθ λ=1

Λr Λr0F r(λ)dλ.(3.77) Now the potential V r(r)in Eq.(3.17)can be written as[42]

V r(r)=(1?E2)(r1?r)(r?r2)(r?r3)(r?r4),(3.78) where r1,r2,r3and r4are the four roots of the quartic V r(r)=0,ordered such that r4≤r3≤r2≤r1.For stable orbits the motion takes place in r2≤r≤r1.The orbital eccentricityεand semi-latus rectum p are de?ned by

r1=

p

1+ε

.(3.80) We also de?ne the parameters p3and p4by

r3=p3

1+ε

.(3.81)

Then following Ref.[14]we de?ne the parameterψby

?r(λ)=

p

dλ

=P(ψ),(3.83)

where P(ψ)=√

1?ε2

[p(1?ε)?p3(1+εcosψ)]1/2[p(1+ε)?p4(1+εcosψ)]1/2.(3.84)

The average(3.77)overλis therefore

F r λ=1P(ψ)= 2π0dψF r(λ)/P(ψ)

Computing inspirals in Kerr in the adiabatic regime 15

3.8.1.Bi-periodic functions Finally suppose we have a function F (λr ,λθ)of two parameters λr and λθwhich is biperiodic:

F (λr +Λr ,λθ)=F (λr ,λθ)

F (λr ,λθ+Λθ)=F (λr ,λθ).

(3.86)

The average value of this function is

F λ=1Λr Λθ

2π0dχ 2π0

dψF [λr (ψ),λθ(χ)]β(z +?z ?cos 2χ)P (ψ).(3.88)This formula will be used in later sections.4.Mino’s derivation of half retarded minus half advanced prescription In this section we review the proof by Mino

[34]that for computing radiation reaction in the adiabatic limit in Kerr,one can use the “half retarded minus half advanced”prescription.We specialize to the scalar case.

We start by de?ning some notation for self-forces.Suppose that we have a particle with scalar charge q at a point P in a spacetime M with metric g αβ.Suppose that the 4-velocity of the particle at P is u α,and that we are given a solution Φ(not necessarily the retarded solution)of the wave equation (2.7)for the scalar ?eld,for which the source is a delta function on the geodesic determined by P and u α.The self-force on the particle is then some functional of P ,u α,g αβand Φ,which we write as

f α[P ,u α,

g αβ,Φ].(4.1)

Here we suppress the trivial dependence on q .Note that this functional does not depend on a choice of time orientation for the manifold,and also it is invariant under u α→?u α.

Next,we de?ne the retarded self-force as

f αret [P ,u α,

g αβ]=f α[P ,u α,g αβ,Φret ],(4.2)

where Φret is the retarded solution to the wave equation (2.7)using the time orientation that is determined by demanding that u αbe future directed.Similarly we de?ne the advanced self force by

f αadv [P ,u α,

g αβ]=f α[P ,u α,g αβ,Φadv ],(4.3)

where Φadv is the advanced solution.It follows from these de?nitions that f αret [P ,?u α,g αβ]=f αadv [P ,u α,g αβ].(4.4)

Computing inspirals in Kerr in the adiabatic regime16 4.1.Properties of the self-force

The derivation of the half-retarded-minus-half-advanced prescription in the adiabatic limit rests on two properties of the self-force.The?rst property is the fact that the self-force can be computed by subtracting from the pergent?eldΦa locally constructed singular?eldΦsing that depends only on the spacetime geometry in the vicinity of the point P and on the four velocity at P.This property is implicit in the work of Quinn[41]and Quinn and Wald[22],and was proved explicitly by Detweiler and Whiting[49].We can write this property as

fα[P,uα,gαβ,Φ]=q?α(Φ?Φsing[P,uα,gαβ]).(4.5) The second property is a covariance property.Suppose thatψis a di?eomorphism from the manifold M to another manifold N that takes P toψ(P).We denote byψ?the natural mapping of tensors over the tangent space at P to tensors over the tangent space atψ(P).We also denote byψ?the associated natural mapping of tensor?elds on M to tensor?elds on N.Then the covariance property is

fα[ψ(P),ψ?uα,ψ?gαβ,ψ?Φ]=ψ?fα[P,uα,gαβ,Φ].(4.6) This expresses the fact that the self-force does not depend on quantities such as a choice of coordinates.It follows from the de?nition(4.2)that the retarded self-force fαret satis?es a similar covariance relation:

fαret[ψ(P),ψ?uα,ψ?gαβ]=ψ?fαret[P,uα,gαβ].(4.7) 4.2.Property of generic bound geodesics in Kerr

Next we review a property of bound geodesics in Kerr upon which the proof depends. This is the fact that for generic bound geodesics there exists isometriesψof the Kerr spacetime of the form

t→2t1?t,φ→2φ1?φ,(4.8) where t1andφ1are constants,which come arbitrarily close to mapping the geodesic onto itself.

To see this,note that if a geodesic is mapped onto itself by the mapping(4.8), then the point[t,r(t),θ(t),φ(t)]is mapped onto[2t1?t,r(t),θ(t),2φ1?φ(t)],which must equal[t′,r(t′),θ(t′),φ(t′)]where t′=2t1?t.If we specialize to t=t1then we see thatφ(t1)=φ1.We denote this point by Q=(t1,r1,θ1,φ1).Next,since the geodesic is determined by Q and by the four-velocity uαat Q,the geodesic is mapped onto itself byψif and only ifψ?uα=?uαat Q.This will be true if and only if the r andθcomponents of uαvanish.Thus,a geodesic will be invariant under a map of the form(4.8)if and only if it contains a point Q which is simultaneously a turning point of the r andθmotions.

Consider now a generic bound geodesic.Such a geodesic can be characterized by the parameters E,L z,Q,λr0,λθ0,t0,φ0in the notation of Sec.3.3.The turning points of the r motion occur atλ=λr,n≡λr0+nΛr/2,where n is an integer. Similarly the turning points of theθmotion occur atλ=λθ,m≡λθ0+mΛθ/2,where m is an integer.Since genericallyΛr andΛθwill be incommensurate,for anyε>0 we can?nd values of the integers m and n so that

|λr,n?λθ,m|≤ε.(4.9)

Computing inspirals in Kerr in the adiabatic regime 17Thus we can ?nd a point on the geodesic that is arbitrarily to close to being a turning

point

for

both

the

r

and

θmotions,and hence there is an isometry of the form (4.8)that comes arbitrarily close to mapping the geodesic onto itself.

Below we will simply assume that the isometry does map the geodesic onto itself;the error associated with this assumption can be made arbitrarily small in the adiabatic limit .

4.3.Property of the adiabatic limit

The self-acceleration a αat a point P on a geodesic has three independent components,since a αu α=0.These three components are determined by the time derivatives of the three conserved quantities

dE

dt ,dK

dt

t , dL z dt t

,(4.10)where dE 2T T ?T

dt dE

dt (t )=Ξα(t )f αret (t )(4.15)

Note however that real inspirals will consist of ?nite curves which are approximately locally geodesic;the corresponding error for such curves will contribute to the post-adiabatic correction to the inspiral,i.e.,to the second term in Eq.(1.2).

Computing inspirals in Kerr in the adiabatic regime18 for someΞα.Below we will computeΞαexplicitly for the three di?erent conserved quantities and derive the key property that

ψ?Ξα(t)=?Ξα(2t1?t).(4.16) Now acting on Eq.(4.15)withψ?and using the relations(4.14)and(4.16)yields (Ξαfαret)(t)=?(Ξαfαadv)(2t1?t).(4.17) Taking a time average gives

Ξαfαret t=? Ξαfαadv t,(4.18)

and hence

d E2 Ξα(fαret?fαadv) t.(4.19)

Finally we use the explicit formula(4.5)for the self force.The contributions from the locally constructed singular?eldΦsing cancel,and we obtain

d E2 Ξα?α(Φret?Φadv) t.(4.20)

This can be written as

d E

2?α(Φret?Φadv)(4.22) is the self-force one obtains from the half retarded minus half advanced prescription, cf.Eqs.(2.14)and(2.15)above.What we have shown is that although the quantity fαrad di?ers from the true self-force fαret,the di?erence averages to zero when one takes a time average over the entire geodesic.

It remains to derive the formula(4.16).For energy and angular momentum,we can write the conserved quantity as the inner product of a Killing vectorξαwith the 4-velocity of the particle:

E=ξαuα.(4.23) Here ξ=??/?t for E=E,and ξ=?/?φfor E=L z.Taking a derivative with respect to proper timeτgives

d E

(δβα+uαuβ)ξβ.(4.25)

μV t(r,θ)

The formula(4.16)now follows fromψ?uα(t)=?uα(2t1?t)andψ?ξα=?ξα.

Computing inspirals in Kerr in the adiabatic regime19 Turn now to the Carter constant.Taking a time derivative of the expression (3.23)for K gives

dK

(δβα+uαuβ)Kβγuγ.(4.27)

μV t(r,θ)

The formula(4.16)now follows from the relationsψ?uα(t)=?uα(2t1?t)and ψ?Kαβ=Kαβ.

5.Separation of variables and basis of modes

5.1.Separation of variables for scalar wave equation

Separation of variables for the scalar wave equation in Kerr was?rst carried out by Dieter Brill and others in1972[50],and subsequently generalized to higher spins by Teukolsky[18,19].We substitute the ansatz

Φ(t,r,θ,φ)=R(r)Θ(θ)e imφe?iωt(5.1) into the homogeneous version of Eq.(2.7),and make use of the expression(3.6)for the wave operator.The result is the two equations

1

Θ+a2ω2cos2θΘ=?λΘ,(5.2)

sin2θ

and

1

(?R,r),r+

Computing inspirals in Kerr in the adiabatic regime20 Here P is the parity operator.

The radial equation(5.3)can now be simpli?ed by de?ning

u(r)=?R(r),(5.9) and by using the tortoise coordinate r?de?ned by

dr?=(?2/?)dr.(5.10) An explicit expression for r?is

r?=r+2r+

2?

2r?

2

,(5.11)

where r±=1±

√

dr?2

(r?)+Vωlm(r?)u(r?)=0,(5.12) where the potential is

Vωlm=??

?8

+ω2

+

1

1?a2.We?nd

Vωlm(r)→Vωlm(r+)=p2mω,where

p mω=ω?mω+,(5.15) and

ω+=

a

Computing inspirals in Kerr in the adiabatic regime21 5.2.1.“In”,“up”,“out”and“down”modes We now de?ne,following Gal’tsov,the following solution

u inωlm(r?)=αωlm τωlm|p mω|?1/2e?ip mωr?,r?→?∞

|ω|?1/2 e?iωr?+σωlm e iωr? ,r?→∞.(5.19) This equation serves to de?ne the mode as well as the complex transmission and re?ection coe?cientsτωlm andσωlm.The coe?cientαωlm is a normalization constant whose value is arbitrary;we will discuss a convenient choice ofαωlm later.We will often denote the set of subscriptsωlm collectively byΛ:thusσΛ,τΛ,and sometimes we will omit the subscript and use simplyσ,τ.

The“in”mode(5.19)is a mixture of outgoing and ingoing components at past and future null in?nity,since the mode function is multiplied by e?iωt.At the past and future event horizons,the mode is purely ingoing when the sign of p mωis the same as the sign ofω.However,from the de?nition(5.15)of p mωwe see that the sign ofωp mωcan be negative;this occurs for superradiant modes.Thus,at the future event horizon the“in”modes can be either ingoing or outgoing.

The crucial feature of the“in”modes is that they vanish on the past event horizon. This feature will be used later in constructing the various Green’s functions.A more precise statement of the result is that a solutionΦof the wave equation which is a linear combination of“in”modes with coe?cients cωlm,such that the coe?cients depend smoothly onω(a reasonable requirement),must vanish at the past event horizon.To see this,note from Eqs.(5.1)and(5.9)that the solution can be written as

Φ(t,r,θ,φ)= ∞?∞dω∞ l=0l m=?l e?iωt cωlm Sωlm(θ,φ)u inωlm(r?)

? ∞?∞dω lm e?iω(t+r?)cωlm Sωlm(θ,φ)αωlmτωlm|p mω|?1/2e imω+r?(5.21)

≡1

本文来源：https://www.bwwdw.com/article/h79q.html