Conservative dynamics of two-body systems at the fourth post-Newtonian approximation of general relativity

0.44 MB·

by  Thibault Damour

Conservative dynamics of two-body systems at the
fourth post-Newtonian approximation of general relativity
Thibault Damour∗
Institut des Hautes Etudes Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France
Piotr Jaranowski†
Faculty of Physics, University of Bial ystok, Cio lkowskiego 1L, 15–245 Bial ystok, Poland
Gerhard Scha¨fer‡
Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit¨at Jena, Max-Wien-Pl. 1, 07743 Jena, Germany
(Dated: 8 March, 2016)
Thefourthpost-Newtonian(4PN)two-bodydynamicshasbeenrecentlytackledbyseveraldiffer-
ent approaches: effective field theory, Arnowitt-Deser-Misner Hamiltonian, action-angle-Delaunay
averaging, effective-one-body,gravitational self-force, first law of dynamics, and Fokkeraction. We
review theachievementsof theseapproaches and discuss thecomplementarity of their results. Our
mainconclusionsare: (i)theresultsofthefirstcompletederivationofthe4PNdynamics[T.Damour,
6
P.Jaranowski,andG.Sch¨afer,Phys.Rev.D89,064058(2014)]havebeen,piecewise,fullyconfirmed
1
by several subsequent works; (ii) the results of the Delaunay-averaging technique [T. Damour, P.
0
Jaranowski, and G.Sch¨afer, Phys.Rev.D91, 084024 (2015)] havebeen confirmedbyseveral inde-
2
pendentworks;and(iii)severalclaimsinarecentharmonic-coordinatesFokker-actioncomputation

r [L.Bernardetal.,arXiv:1512.02876v2[gr-qc]]areincorrect,butcanbecorrectedbytheadditionof
p
acoupleofambiguityparameterslinkedtosubtletiesintheregularizationofinfraredandultraviolet
A
divergences.

7
2
I. INTRODUCTION presence of a unique ambiguity, of infrared (IR) origin,

] parametrized by the parameter α, and (ii) the need to
c
q The general-relativistic two-body problem has ac- choose the value α = αB3FM = 811/672 to reproduce
- quired a renewed importance in view of the impending the analytically known [13] 4PN interaction energy for
r prospectofdetectingthegravitationalwavesignalsemit- circular orbits. Moreover, they stated that their 4PN
g
[ tedbyinspiralingandcoalescingcompactbinaries. After dynamics disagrees with the one recently derived by us
the successfulcompletion,15yearsago,ofthe derivation [14, 16, 17] and that part of the discrepancy comes from
2 of the third post-Newtonian (3PN) two-body dynam- ourtreatmentofthenonlocalcontributiontothedynam-
v
ics [1] (see Refs. [2–5] for later rederivations), a natural ics.
3
8 challenge was to tackle the (conservative) fourth post- The firstaims of the presentpaper will be: 1)to show
2 Newtonian (4PN) dynamics. Several partial steps in the that the claims, denoted as (i) and (ii) above, of Ref.
1 derivationofthe4PNdynamicshavebeenperformed[6– [21] are incorrect; 2) to suggest that the discrepancies
0 14]1, and culminated in the recent first full derivation, between their result and ours is due both to their incor-
1. within the Arnowitt-Deser-Misner (ADM) Hamiltonian rect evaluation of the conserved energy of the nonlocal
0 formulation of general relativity, of the 4PN dynamics 4PN dynamics and to the need to complete their result
6 in Ref. [16], completed by an action-angle-Delaunay av- by at least one further contribution parametrized by an
1 eraging study in Ref. [17]. A remarkable feature of the additional ambiguity parameter, denoted a below. We,
:
v conservative 4PN dynamics of Ref. [16] is the presence however,emphasizethatoursuggestedcorrectionsrepre-
i of a time-symmetric nonlocal-in-time interaction, which sentonlytworatherminoradjustments amonghundreds
X
is directly related to the 4PN-level tail-transported (re- of terms that agree between two very difficult (four-loop
r tarded) interaction first discussed in Ref. [18] (see also level!) independent calculations,using differentmethods
a
Refs. [19, 20] for recent rediscussions). anddifferentgauges. Theotheraimsofourpaperwillbe:
In a recent preprint [21] Bernard et al. reported the 3)toexplaininmoredetailthaninouroriginalpaper[17]
computation of the Fokker action describing the 4PN the logical basis and consistency of our reduction of the
dynamics in harmonic coordinates. They made several nonlocal 4PN dynamics to a formally local action-angle
claims in (version 2 of) their preprint, notably (i) the Hamiltonian;and4)tosummarizethemanyindependent
results that have confirmed, piecemeal, all the elements
of our 4PN dynamics [16, 17].
Regardingthelastitem,wewishtoemphasizethat,be-
∗ [email protected]
† [email protected] sides the straightforward PN (or, equivalently, effective-
‡ [email protected] field-theory)calculationsofthetwo-bodydynamics,cru-
1 See also Ref. [15] for the O(G1) interaction, valid to all PN- cial information about the two-body dynamics has been
orders. acquired (as will be detailed below) through combining
2
(invariousways)severalotherapproaches: the effective- Thelengthscalesenteringtheactionabove4isarbitrary
one-body (EOB) formalism [22–25], the first law of bi- because (as shown in Ref. [16]) the s dependence of the
nary mechanics [26–28], andgravitationalself-force(SF) nonlocalaction(2.2)cancelsagainstthe sdependence of
theory,especiallywhencombinedwiththeMano-Suzuki- the local action (2.3).
Takasugi [29, 30] hypergeometric-expansionapproach to In our second paper, Ref. [17], we combined different
Regge-Wheeler-Zerilli theory. We shall point out when techniques (which will be explained in detail below) for
needed the multi-way consistency checks between these transformingthe (center-of-mass-framereductionofthe)
approaches that have been obtained at the 4PN level. nonlocal ADM action (2.1) into an equivalent ordinary
Our notation for two-body systems will follow the one action, say
used in our previous works: the two masses are de-
nµotedmmm1,/m(m2, +anmd )w,eνtheµn/Mden=otme Mm /≡(mm1++mm)22., SDJS′ = pidqi−HDJS′(q,p)dt , (2.5)
≡ 1 2 1 2 ≡ 1 2 1 2 Z h i
We indicate powers of G and c when it is pedagogically
useful, but we sometimes set G= c = 1 when it is more where HDJS′(q,p) is formally given by an expansion in
(even) powers of qip . [Reference [17] gave an exact for-
convenient. i
mula for the action-angle version of Snonloc, and the ex-
plicit form of the expansion of HDJS′(q,p) in powers of
qip through(qip )6.] The ordinaryaction(2.5)is equiv-
II. EXTANT PIECEWISE CONFIRMATIONS i i
alentto(thecenter-of-mass-framereductionof)Eq.(2.1)
OF THE 4PN ADM-DELAUNAY DYNAMICS
AND LIMITS ON POSSIBLE DEVIATIONS modulo some shifts of the phase-space coordinates that
areimplicitlydefinedbytheDelaunay-likereductionpro-
cedure of Ref. [17]. These shifts do not affect the gauge-
Let us start by recalling the structure of the 4PN re-
sults of Refs. [16, 17]. First, the Fokker-like reduced2 invariant observables deduced from either Eq. (2.1) or
Eq. (2.5) which we shall focus on in the following. Note
ADM action derived in Ref. [16] has the following form
alsoinpassingthattheuseoftheEOBformalisminRef.
(using the notation of Ref. [16]):
[17] is essentially a technical convenience, while the es-
sential conceptual step used there is the reduction of a
SDJS = paidxia−HDloJcS(xa,pa;s)dt nonlocal action to a local form by using Delaunay-like
Z hXa i averagingtechniques (see below).
+Snonloc(s), (2.1) The aim of this section is to summarize the current
existing confirmations of the correctness of the 4PN ac-
where the nonlocal piece of the ADM action3 is (with I tions (2.1) or (2.5). First, we wish to recall that the full
ij
denoting the quadrupole moment of the binary system, Poincar´einvariance of (2.1) (in a generalframe) was ex-
and I(3) d3I /dt3) plicitly checked in Ref. [16] (see also Ref. [14]). This is
ij ≡ ij a highly nontrivialcheck because the ADM derivationof
1G2M dtdt′ the action (contrary to the harmonic-coordinates one) is
Snonloc(s) = 5 c8 Pf2s/c t t′ Ii(j3)(t)Ii(j3)(t′), far from being manifestly Lorentz invariant.
ZZ | − | ToorganizetheotherconfirmationsofEq.(2.1)orEq.
(2.2)
(2.5), let us note that, in the center-of-mass frame, and
(Pf Partiefinie)andwherethe localpiece ofthe ADM
≡ whenusingsuitablyscaled5 dynamicalvariables[e.g.r =
action has the structure
rphys/(GM), p = p /µ = p /µ, S = Sphys/(GMµ),
1 2
−
Hloc (x ,p ;s)=Hloc (x ,p ;r ) H = (H Mc2)/µ] the rescaled Hamiltonian H has a
DJS a a DJS a a 12 −
polynomial structure in the symmetric mass ratio ν
r
+F[xa,pa]ln 12, (2.3) b b
s H =H +νH +ν2H +ν3H +ν4H , (2.6)
0 1 2 3 4
with
F[x ,p ]= 2G2M(I(3))2. (2.4) wthhee4rePHbN0d=ynba12mp2ic−s1bo/fra+tce−st2b(p·a·r·t)i+cle·b·i·n+tOhe(cfi−be1ld0)gdeensecrraibteeds
a a 5 c8 ij by thebmass M =m1+m2.
The contribution νH to the 4PN dynamics describes
1
the4PNapproximationtothefirst-orderself-force(1SF)
dynamical effects. Ovber the recent years, many works
2 SeeAppendixAforadiscussionofFokker-likereducedactions.
3 In Ref. [16] the necessity of adding the nonlocal contribution
Snonloc(s) was derived by combining the structure of the IR
divergence of the local ADM action [14] with the known exis-
tenceofa4PN-level,long-rangetail-transportedinteraction[18]. 4 Forclarity,wedonotuseheretheformulationwheresisreplaced
Prompted by an argument in Ref. [21], we discuss inAppendix byr12=|x1−x2|.
A the (limited) extent to which Snonloc(s) can be (formally) 5 Beware that we will often oscillate between using scaled or un-
directlyderivedfromtheADMaction. scaleddynamicalvariables.
3
havebeendevotedtoboththenumericalandtheanalyt- [6]] yields the precession function ρ(u) as a linear com-
ical computation of 1SF dynamical effects. To compare bination of a(u), a′(u), a′′(u) and d¯(u), where a(u) is
these results to the predictions following from our 4PN the 1SF correction to the main EOB radial potential
dynamics (2.1) or (2.5) we need bridges between PN re- A(u;ν) = 1 2u + νa(u) + O(ν2) [which generalizes
−
sultsandSFresults. TwosuchPN-SFbridgeshavebeen ASchwarzschild(u) = 1 2GM/c2r 1 2u], and where
particularly useful over the last years: the EOB formal- d¯(u) is the 1SF correc−tion to the s≡econ−d EOB radial po-
ism[22–25],andthefirstlawofbinarymechanics[26–28]. tential D¯(u) (A(u)B(u))−1 = 1+νd¯(u)+O(ν2). Let
≡
The first example of a PN-EOB-SF bridge was the usapplythisrelationtothe4PN-levelvaluesoftheEOB
derivation of the functional relation between the peri- potentials parametrizing the 4PN dynamics (2.5), as de-
astron precession of small-eccentricity orbits and the ra- rived in Ref. [17], namely (adding the third EOB poten-
dial potentials entering the EOB Hamiltonian [6]. At tial Q µ2Qˆ)
the 1SF level, this relation [see Eqs. (5.21)–(5.25) in ≡
94 41π2
A(u)=1 2u+2νu3+ νu4
− 3 − 32
(cid:18) (cid:19)
2275π2 4237 128 256 41π2 221 64
+ + γ + ln2 ν+ ν2+ ν lnu u5, (2.7a)
E
512 − 60 5 5 32 − 6 5
(cid:18)(cid:18)(cid:18)(cid:18) (cid:19) (cid:18) (cid:19) (cid:19)(cid:19)(cid:19)
D¯(u)=1+6νu2+ 52ν 6ν2 u3
−
533 (cid:0) 23761π2(cid:1) 1184 6496 2916 123π2 592
+ + γ ln2+ ln3 ν+ 260 ν2+ νlnu u4,
E
− 45 − 1536 15 − 15 5 16 − 15
(cid:18)(cid:18)(cid:18)(cid:18) (cid:19) (cid:18) (cid:19) (cid:19)(cid:19)(cid:19)
(2.7b)
5308 496256 33048
Qˆ(r′,p′)= 2(4 3ν)νu2+ + ln2 ln3 ν 83ν2+10ν3 u3 (n′ p′)4
− − 15 45 − 5 − ·
(cid:18)(cid:18)(cid:18) (cid:18)(cid:18) (cid:19) (cid:19) (cid:19)(cid:19)(cid:19)
827 2358912 1399437 390625 27
+ ln2+ ln3+ ln5 ν ν2+6ν3 u2(n′ p′)6+O[νu(n′ p′)8].
− 3 − 25 50 18 − 5 · ·
(cid:18)(cid:18)(cid:18)(cid:18) (cid:19) (cid:19)(cid:19)(cid:19)
(2.7c)
This yields
397 123
ρ(x)=14x2+ π2 x3
2 − 16
(cid:18) (cid:19)
58265 215729 5024 1184 2916 2512
+ π2 + γ + ln2+ ln3+ lnx x4+O(x5lnx). (2.8)
E
1536 − 180 15 15 5 15
(cid:18) (cid:19)
The 4PN-level contribution to the precession function of n′ p′ in Q were recently provided: the powers 8 and
ρ(x) is of the form ρ (x) = (ρc + ρlnlnx)x4, with 10 in·Ref. [32]; and the powers 12 to 20 in Ref. [33].)
4PN 4 4
the rational logarithmic coefficient ρln = 2512 [8], and In addition to analytical confirmations, there are also
4 15
the transcendental nonlogarithmic 4PN coefficient numericalself-forceconfirmationsofthe4PN-levelvalues
of ρ(x) and ρc,DJS based on direct dynamical computa-
58265 215729 4
ρc,DJS = π2 tions of the precession of slightly eccentric orbits.
4 1536 − 180 ThefirstnumericalSFdeterminationoftheprecession
5024 1184 2916 function ρ(u) was made in Ref. [34]. In particular, the
+ 15 γE+ 15 ln2+ 5 ln3. (2.9) latter work confirmed the value ρl4n = 251152 and derived
anestimateofthevalueofthenonlogarithmiccoefficient,
The analytical values of the 4PN-level functions A, D¯,
namely
Q have been recently independently confirmed by two
self-force computations (based on the recently derived ρc,num[34] =69+7. (2.10)
4 −4
eccentric-extension of the first law [28]); see Refs. [31]
and[32]. (Notealsothatthecoefficientsofhigherpowers We wereinformed by Maartenvande Meentthat he has
4
64
veryrecentlyobtainedamuchmoreaccuratedetermina- aln = , (2.15b)
tion of ρc with preliminary results yielding [35] 5 5
4
533 23761π2 1184
ρc,num[35] =64.640566(2), (2.11) d¯c = + γ
4 4 − 45 − 1536 15 E
wherethe number inparenthesesindicates apreliminary 6496 2916
estimate of the uncertainty on the last digit. Note that ln2+ ln3, (2.15c)
− 15 5
bothnumericalestimatesofρc assumetheanalyticalval-
4
592
ues of the 4PN and 5PN logarithmic contributions to d¯ln = , (2.15d)
ρ(u). [There is no doubt about the (1SF) 4PN and 5PN 4 15
logarithmiccontributionstothedynamics: the4PNones 5308 496256 33048
can be straightforwardly deduced from the tail-related q4,3 = + ln2 ln3, (2.15e)
− 15 45 − 5
4PN logarithmic term written in Eq. (6.39) of Ref. [18]
(equivalent to the F[xa,pa]lnrs12 logarithmic contribu- q6,2 = 827 2358912ln2
tion in Eq. (2.3)), while the 5PN ones are straightfor- − 3 − 25
wardly derivable from the higher-tail results of Refs. [7] 1399437 390625
+ ln3+ ln5. (2.15f)
and [17] (Sec. IXA).] Both numerical SF results (2.10)
50 18
and (2.11) confirm, within their respective error bars,
The numerical values of the nonlogarithmic coefficients
the numerical value
are
ρc,DJS =64.6405647571193781901484255 (2.12)
4 ··· ac =23.5033892426034362387576146 , (2.16a)
5 ···
of the result (2.9) predicted by our 4PN dynamics. The
numerical result (2.10) differs from the analytical value d¯c4 =221.5719911921481640232337716···, (2.16b)
(2.9) by
q =28.7110442849559497574969412 , (2.16c)
4,3
···
∆ρc ρc,num[34] ρc,DJS =4+7, (2.13) q = 2.7830076369522324890284545 . (2.16d)
4 ≡ 4 − 4 −4 6,2 − ···
while the morerecentnumericalvalue(2.11)differs from All these 1SF-order coefficients have been indepen-
the analytical value (2.9) only by dently checked, either analytically, or numerically, by
various SF computations (which used the first law of bi-
∆ρc ρc,num[35] ρc,DJS =(1 2) 10−6. (2.14)
4 ≡ 4 − 4 ± × nary dynamics as a bridge). We have already mentioned
above the theoretical consensus on the 4PN logarithmic
Whatisespeciallyimportantinsuchanumericalcheckis
contributions aln and d¯ln. [Note the absence of any log-
thatwearetalkinghereaboutadirectdynamicalcheckof 5 4
arithmic contribution to Q (u,p ).] The nonlogarith-
the 4PNdynamics derivedinRefs.[16]and[17]. Indeed, 4PN r
mic contribution ac to the main EOB radial potential
nouseismadehereofthefirst-law-of-mechanicsbridgeto 5
A(u) is actually not a deep check of the ADM results
go from SF computations of the Detweiler-Barack-Sago
(2.1) and (2.5) because Ref. [16] used the analytical SF
redshift invariant to 4PN dynamical functions. The SF
result of Bini and Damour [13] to calibrate their single
computations done in Refs. [34] and [35] directly esti-
IR ambiguity constant C. Let us, however, note two
mated the effect of the nonlocal gravitational self-force
things. First, all the transcendental contributions to ac
Fµ on slightly eccentric orbits to extract the precession 5
have been reproduced by the local ADM computation
function ρ(u) and, then, its PN expansion coefficients.
[12,14],theneededcalibrationofC usingonlyarational
One has therefore here a direct confirmation of the way
shift. Second, the analytical determination of ac in Ref.
Refs.[16,17]computed(viawell-establishedEOBresults 5
[13] (which was preceded by accurate numerical estima-
[6]) the precession effect of the nonlocal 4PN dynamics.
tions [7, 9]) was later analyticallyconfirmedin Refs. [36]
LetusnowsummarizethenumericalSFconfirmations
and [37], as well as numerically confirmed by extremely
of the other 1SF predictions one candraw from the 4PN
high-accuracy self-force results [38].
results(2.7). The4PN-levelcoefficientsofthethreeEOB
potentials A, D¯ and Q [the latter being considered at The nonlogarithmic 1SF 4PN contribution d¯c4 to the
second EOB potential D¯(u) has been fully confirmed
O(p4), i.e. at the level of the fourth power of the eccen-
r (again using the first law of mechanics) by recent SF
tricity]areoftheformA (u)=ν(ac+alnlnu+νa′)u5,
4PN 5 5 5 works, both analytically and numerically. As already
D¯ (u) = ν(d¯c +d¯lnlnu+νd¯′)u4, and Qˆ (u,p ) =
4PN 4 4 4 4PN r mentioned, direct analytical checks have been obtained
νq (ν)u3p4+νq (ν)u2p6+O(p8),withq (ν)=q +
4,3 r 6,2 r r 4,3 4,3 in Refs. [31] and [32]. In addition, Ref. [31] has pointed
νq′ +ν2q′′ and q (ν)=q +νq′ +ν2q′′ . Among
4,3 4,3 6,2 6,2 6,2 6,2 outthat the recentnumericalSF results of Ref. [39]pro-
the 4PN coefficients, the 1SF contributions are the ones vide a numerical check on the value d¯c at the accuracy
at order O(ν), i.e. the unprimed ones in our notation, 4
level 0.05. (The latter error level takes into account
namely ±
the fact that the a terms have been fully confirmed.)
5
ac5 = 2257152π2 − 426307 + 1528γE+ 2556ln2, (2.15a) recTehntely1ScFon4fiPrNmecdo,ntbriobtuhtiaonnaslytoticQa(lrly,p[r3)2h]aavnedalnsuombeerein-
5
cally, with an uncertainty δq = 4 [39] (see also the with the same nonlocal (or “tail”) action6, but with a
4,3
±
recentnumericaldeterminationofthe coefficientq (u)of different local Hamiltonian,
4
νp4 in the EOB Q potential [40]).
r
Letusemphasizethatthetermsproportionaltohigher HBlo3cFM(xa,pa;s)=HBlo3cFM(xa,pa;r12)
powers of ν in Eq. (2.6) are much less sensitive than the r
termsoforderν tosubtleregularizationambiguities. Ac- +F[xa,pa]ln 12. (3.2)
s
tually, as shown in Refs. [10, 12] the regularization sub-
tleties are in direct correspondence with the power of ν. (Reference[21]doesnotexplicitlydisplaythelnsdepen-
The terms of order ν3 and ν4 are not ambiguous at all dence of the local Hamiltonian but agrees with Ref. [16]
(and have been independently derived, in the effective on the cancellation of that dependence.) For simplicity,
field theory approach, for the relation between energy we do not discuss here the issue of the order reduction
and orbital frequency for circular orbits, in Ref. [11]), of the derivatives of x and p entering (via I(3)) both
a a ij
whilethetermsoforderν2aredelicate,butcanbeunam- thenonlocalaction(2.2)andthecoefficientF ofthelog-
biguously derivedwhenusing dimensionalregularization arithm, Eq. (2.4). Indeed, there is complete agreement,
for treating the UV divergences[12]. Actually, as explic- at the level of the action, between Refs. [16] and [21] for
itly shown in Ref. [21] (see also the next section), the what concerns the nonlocal piece of the action. [As said
recent harmonic-coordinates Fokker-action computation in Ref. [16], the order reduction of I(3) (i.e. its on-shell
ofRef.[21]agrees(modulosomecontacttransformation) ij
replacement by a local function of x (t) and p (t)) en-
with the action (2.1) for all powers of ν, except for the a a
tails a suitable nonlocal shift of the dynamical variables
firstpower. We think thatthis is relatedto the factthat
(whichwasexplicatedinEqs.(5.14),(5.15)ofRef.[21]).]
the most delicate IR effects are linked with the nonlocal
Theimportantissueisthe differencebetweenthetwolo-
contribution(2.2),whichiseasilyseentobepurelyofor-
cal Hamiltonians which, according to Eq. (5.19) of Ref.
der ν1 [in S/(Mµ)]. [This is also explicitly displayed in
[21], is
Eqs. (7.5)–(7.7) of Ref. [17].] As we shall further discuss
below, physical effects mixing local and nonlocal effects, G4Mm2m2
and thereby being sensitive to IR divergences, are very HBlo3cFM−HDloJcS = c8r41 2
delicate to determine unambiguously. We can, however, 12
conclude that the non-IR-sensitive part of the results of
(n p ) (n p ) 2
12 1 12 2
Rteerfm. s[2i1n]tphreoavcidtieonan(2i.n1d)ewpheincdheanrtecoofnofirdrmeraνtinonwiothf anll th2e. ×"aB3FM m·1 − m·2
≥ (cid:16) (cid:17)
To summarize this section, many independent SF re-
p p 2 GM
sults have confirmed all the (IR-sensitive) terms linear 1 2
in ν, while the other PN calculations at the 4PN level +bB3FM m1 − m2 +cB3FM r12 #,
(cid:16) (cid:17)
(Ref. [11] and, especially, Ref. [21]), have confirmed all (3.3)
the terms nonlinear in ν (i.e. ν2, ν3, and ν4). We
∝
conclude that all the results of Refs. [16, 17] have been with
(piecewise) confirmed.
1429 826 902
(a,b,c)B3FM = 315 ,315,315 . (3.4)
(cid:18) (cid:19)
III. INCOMPATIBILITIES BETWEEN THE 4PN
Before discussing further the originof the discrepancy
HARMONIC FOKKER ACTION RESULTS OF
REF. [21] AND SELF-FORCE RESULTS [i.e.thefactthat(a,b,c)B3FM =(0,0,0)],wewishtoem-
6
phasize two things: (i) the discrepant terms in Eq. (3.3)
representonlythree(Galileo-invariant)termsamongthe
The simplest way to compare the results of the re-
hundreds (exactly 219) of contributions to the (non-
cent harmonic Fokker action computation [21] to self-
center-of-mass) two-body Hamiltonian, as displayed in
force data is to compute the SF effects induced by the
the Appendix of Ref. [16], or in Sec. VIII E of Ref. [14];
difference between the dynamics of Ref. [21], and that
(ii) the three discrepant terms (3.3) [when expressed in
of Refs. [16, 17]. This difference has been worked out in
the center of mass, and in reduced variables, i.e. in the
Ref. [21].
senseofEq.(2.6)]arelinear inthe symmetricmassratio
Reference [21] obtained, after applying a suitable con-
ν, i.e. they are of 1SF order.
tact transformation (alluded to, though not explicitly
The discrepancy Eq. (3.3) therefore implies 1SF-
presented, at the beginning of their Sec. V B) to their
detectable effects away from all the 1SF checks of the
original harmonic-coordinates result an action of the
ADM dynamics (2.1) reviewed in the previous section.
same form as the ADM results (2.1), namely
SB3FM = paidxia−HBlo3cFM(xa,pa;s)dt
Z hXa i 6 Weuseherethefactthat, atthe4PNlevel,MB3FM=MDJS+
+Snonloc(s), (3.1) O(1/c2)withMDJS≡Mhere=m1+m2.
6
Let us now quantify the corresponding 1SF differences. changesaboveentailacorrespondingchangeinthe4PN-
To do that it is convenient to transcribe the 4PN-level levelprecessioncoefficientρc givenbyδρc =10δac+δd¯c,
4 4 5 4
Hamiltonian difference (3.3) in terms of corresponding i.e.
differences in the three EOB potentials A, D¯, Q. (We
recall in passing that the EOB parametrization of the δB3FMρc4 =(2a+28b+20c)B3FM, (3.11)
dynamics in terms of A, D¯, Q is completely gauge fixed,
i.e.
and therefore directly linked to gauge-invariant quanti-
ties.) δB3FMρc = 44026 139.7650794. (3.12)
Considering, for more generality, a 4PN-level Hamil- 4 315 ≈
tonian difference with general coefficients (a,b,c) in Eq.
(3.3), the corresponding additional contributions to A, As we recalled above, SF computations [34, 35] of the
D¯, Q are easily found to be precession function ρ(u) [6], and of its PN coefficients,
do not rely on the first law of binary mechanics, but
δa,b,cA=(2b+2c)νu5, (3.5a) involve direct computations of the additional precession
induced by the nonlocal self-force. The difference (3.12)
δa,b,cD¯ =(2a+8b)νu4, (3.5b) is therefore (assuming the correctness of the Delaunay
reductionofRef.[17])adirectdynamicalconsequenceof
δa,b,cQ=0. (3.5c)
the Fokker-action result of Ref. [21]. It would be inter-
esting to confirm this result by a purely dynamical com-
Note in passing that those contributions are invariant
putationofthe (nonlocal)4PNprecession. We note here
under the “gauge transformation”
thatthedifferenceisalreadyexcludedbythe“old”result
δg(a,b,c)=g(4, 1,1), (3.6) (2.13), and even more so (by 108 standard deviations!)
− by the recent one (2.14).
which corresponds to the most general canonical trans-
formationrespectingthestructure(3.3)(withgenerating
function gνp /r3). Inserting the values (3.4) in Eqs. IV. LOGICAL BASIS OF THE ACTION-ANGLE
r
(3.5a)–(3.∝5b) leads to changes in A and D¯ (with respect DELAUNAY-LIKE METHOD OF REF. [17]
to the values computed in Ref. [17])
Independently of the above SF confirmations of the
δB3FMA= 384νu5, (3.7) 4PN results of Refs. [16, 17], let us reassess the logi-
4PN 35 cal basis and the consistency of the methodology used
δ4BP3NFMD¯ = 9341656νu4. (3.8) itnraostu,rwDheelareu,nainy-EouOrBopdienriiovna,tioline,tahnedflianwdiscaotfe,Rbeyf.c[o2n1-]
that have led to the nonzero values (3.4), which are in-
Inotherwords,theonly1SF4PN-levelcoefficientsthat compatible with many SF results. Our Ref. framework
are different are ac and d¯c with is the action-angle formulation of (planar) Hamiltonian
5 4
dynamics. In [17] we used the standard Delaunay no-
δB3FMac = 384 10.97143, (3.9) tation (modulo the use of calligraphic letters L ,
5 35 ≈ G ). Namely, ( ,ℓ; ,g) are the two planar ac→tioLn-
→ G L G
δB3FMd¯c = 9466 30.05079. (3.10) angle canonical pairs: L = √a is conjugate to the mean
4 315 ≈ anomaly angle ℓ, while = a(1 e2) is conjugate to
G −
the argumentof the periastrong =ω. In orderto better
Suchlarge,4PN-leveldeviationsawayfromthe results p
exhibit the meaning (and consistency) of our approach
of Ref. [17] are in violent contradiction with the many
inthecircularlimit,weshalluseherethecombinationof
SF confirmations of the ADM 4PN dynamics reviewed
Delaunay variables introduced by Poincar´e, namely, the
in the previous section. They have been obtained here
twoaction-anglepairs(Λ,λ;I ,̟)where(using suitably
r
by using the Delaunay-like reductionused in Ref. [17] to
scaled variables as in Ref. [17])
convert the nonlocal dynamics (2.1) into the (formally)
local one (2.5). The authors of Ref. [21] express doubts Λ= =I +I , (4.1a)
r ϕ
aboutsomeaspectsoftheresultsof[17]. Weshalladdress L
theirconcernsinthefollowingsectionsandconcludethat λ=ℓ+g, (4.1b)
their concerns are unsubstantiated. Therefore we con-
I = = I , (4.1c)
clude that taking the results of Ref. [21] at face value r L−G L− ϕ
does lead to the large changes (3.9) and (3.10) above, ̟ = g = ω. (4.1d)
− −
and are therefore strictly incompatible with extant SF
knowledge. Here, Iϕ = is the angular momentum (Iϕ =
We wish to go further and point out an even more 21π pϕdϕ = pGϕ) and Ir is the radial action (Ir =
blatant contradiction with SF tests of periapsis preces- 1 p dr). The sum Λ = I + I is conjugate to
2π H r r ϕ
sion. First,wenote(using,e.g.Eq.(8.3)in[17])thatthe the mean longitude λ = ℓ + g, while I is conjugate
r
H
7
to (minus) the argument of the periastron ̟ = ω. subtleties linked to the Pf prescription and to the slow
−
The latter (surprising) minus sign is necessary to have decay of 1/τ for τ + are only of a technical na-
| | | | → ∞
dL dℓ + dG dg = dΛ dλ + dI d̟ with I = ture. Forinstance,theoscillatorynatureoftheintegrand
r r
∧ ∧ ∧ ∧
= √a a(1 e2) > 0. In the circular limit, I′(3)I(3) [mimicked as x′x /(r′3r3) in our toy example]
L−G − − ij ij i i
λ becomes the usual polar angle ϕ in the orbital plane ensures the large τ convergence8 of the nonlocal inte-
p
and I 0, so that the action variable Λ becomes equal | |
r → gral (2.2) without having to assume that I(3) tends to
to the angular momentum I = p . In this limit, the ij
general Hamilton equation λ˙ϕ= ∂Hϕ/∂Λ gives back the zero when |τ| → +∞. [Indeed, the fact that −xi/r3 is
theon-shellvalueofthesecondtimederivativeofx ,and
usual circular link between the orbital frequency Ω = ϕ˙ i
I(3) is the third time derivative of I , ensures that their
and the derivative of the energy with respect to the an- ij ij
large-time averages both vanish for conservative bound
gular momentum p . However, when dealing with non-
ϕ
motions. Note that our toy model is the “electromag-
circular orbits, and tackling the nonlocal action (2.2), it
netic”(dipolar)analogofthegravitational(quadrupolar)
is important to work with the clearly defined canonical
tailaction.] For simplicity, aswe wish hereto emphasize
action-angle pairs (Λ,λ;I ,̟). In principle, it would be
r
issues of principle without getting bogged down by sec-
better, when discussing the circular limit, to replace the
ondary technical issues, we will not specify the weight
second pair (I ,̟) by the associated Poincar´e variables
r
ξ =√2I cos̟, η =√2I sin̟,becausetheyarecanon- µ(τ) used in our toy model, but proceed as if it were a
r r
smooth, integrable even function of τ.
ical(dξ dη =dI d̟)andregularinthecircularlimit
r
∧ ∧ Thefirsttwosteps,(a)and(b), ofourprocedureyield
(while̟becomesilldefinedasI 0). Keepinginmind
r
→ (when considered, for illustration, at linear order in the
suchanadditionalchangeofvariables,weshall,however,
eccentricity e)
find simpler to express our methodology in terms of I
r
and ̟.
1 +∞ 1
The basic methodology we used in Ref. [17] for deal- H (t)= +ε dτµ(τ) cos(λ′ λ)
toy −2Λ2 Λ4Λ′4 −
ing with the nonlocal 4PN dynamics, Eqs. (2.1) and Z−∞ h
(2.2), consists of four steps: (a) we reexpress the action +2ecos(2λ λ′+̟)+2e′cos(2λ′ λ+̟′)
(2.1) as a nonlocal action in the action-angle variables − −
(Λ,λ;I ,̟); (b) we expand it (formally to infinite or-
r +O(e2) . (4.4)
der) in powers of the eccentricity, i.e. in powers of √I ;
r
(c) we “order reduce” the nonlocal dependence on the i
Here and below f′ denotes the variable f taken at time
action-angle variables by using the on-shell equations of
t′ =t+τ, while f denotes its value at time t.
motion7; and (d) we eliminate, `a la Delaunay, the peri-
Step(c)consistsinwriting,foranyvariablef,itsvalue
odictermsintheorder-reducedHamiltonianbyacanon-
f′ attheshiftedtimet′ intermsofcanonicalvariablesat
ical transformation of the action-angle variables. After
time t and of an integral over intermediate times involv-
thesefoursteps,weendupwithanHamiltonianwhichis
ingtherhsofthe(Hamiltonian)equationsofmotion. For
an ordinary (local) function of the (transformed) action
a nonlocal action, the latter read S (t) = 0, S (t) = 0,
variables alone. λ Λ
etc., with
In order not to get distracted by irrelevant technicali-
ties,letusillustratethismethodology[whichwasapplied δS δH
S (t) Λ˙(t) , (4.5a)
inRef.[17]tothefull4PNaction(2.1)–(2.2)]onasimpler λ
≡ δλ(t) ≡− − δλ(t)
toy example which contains some of the key ingredients
of the action (2.1)–(2.2), namely the action S (t) δS λ˙(t) δH , etc., (4.5b)
Λ
≡ δΛ(t) ≡ − δΛ(t)
S = [p(t) dr(t) H (t)dt], (4.2)
toy · − toy where δ/δf(t) denotes a functional derivative [acting on
Z the action S or on dtHnonlocal(t)]. In the normal case
1 1
H (t)= p(t)2 of local actions [41, 42], the use of the identities (4.5a)
toy 2 − r(t) and(4.5b)isthemaRintoolallowingonetoshowhowthe
replacement of the equations of motion within an action
+∞ r(t′) r(t)
+ε dt′µ(t′ t) · , (4.3) isequivalenttoasuitableshiftofthedynamicalvariables
− r(t′)3 r(t)3
Z−∞ | | | | (a “field redefinition”). In the case of nonlocal actions,
onemustintegratethe identities (4.5a)and(4.5b)before
where µ(τ) is an even function of the time difference
replacing λ′ λ(t +τ),Λ′ Λ(t+τ), in the non-
τ t′ t. In the real case (2.1)–(2.2), µ(τ) = 1/τ ≡ ≡ ···
≡ − | | local piece of the action. In this integration, the terms
with an additional partie finie (Pf) prescription. The
8 We have convergence both at τ → +∞ and τ → −∞, without
7 Asdiscussedindetail below,thisisequivalent toapplyingsuit- having absolute convergence. See Appendix A for more discus-
ablenonlocalshiftsofthephase-spacevariables. sionofconvergence issuesinreducedactions.
8
S (t),S (t), are treated as additional terms, which S (t ),S (t ), . Inotherwords,theuseoftheseshifts
λ Λ λ 1 Λ 1
would be zer·o··on shell, but which are now considered as justifies the na·iv··e replacement of λ′,Λ′, by the solu-
···
“source terms” on the rhs of the usual Hamilton equa- tion of the (Newtonian) equations of motion, namely
tions of motion, λ˙(t) δH/δΛ(t) = S , . In view
− Λ ··· τ
of the well-known possibility of neglecting “double-zero λ′ =λ+ , (4.8a)
Λ3
terms,” it is enough to work linearly in these (simple-
zero) source terms. Moreover, as we are working within Λ′ =Λ. (4.8b)
a PN-expanded scheme (and as the nonlocality enters
only at order ε = 1/c8), it is actually enough [mod- Note that the very simple time structure of the unper-
ulo terms of order O(c−2ε)], for the purpose of replacing turbedsolutioninaction-anglevariablesplaysaveryuse-
λ′,Λ′, inthe nonlocalpiece ofthe action,to compute ful role in our Delaunay-based reduction procedure.
···
them assolutionsofthe (sourced)Newtonian-level equa- With the replacements (4.8a) and (4.8b) [i.e. after
tionsofmotion,i.e.solutionsofthedifferentialequations steps (a), (b), and (c)] we have a Hamiltonian of the
(4.5a) and (4.5b) with δH/δΛ, δH /δΛ, , where form
0
···→ ···
H = 1/(2Λ2). Theexplicitformofthesesourcedequa-
0 − 1 +∞ 1 τ
tions of motion are H′ (t)= +ε dτµ(τ) cos
toy −2Λ2 Λ8 Λ3
Λ˙(t)= S (t), (4.6a) Z−∞ (cid:26) (cid:16) (cid:17)
λ
− τ
+2e cos λ ̟
1 Λ3 − −
λ˙(t)= Λ3(t) +SΛ(t), etc. (4.6b) (cid:20) (cid:16) (cid:17)
2τ
+cos +λ+̟ +O(e2) , (4.9)
Viewingtheseequationsasdifferentialequationswithre- Λ3
(cid:18) (cid:19)(cid:21) (cid:27)
specttot′ =t+τ (orτ)fortheunknownsΛ′ Λ(t′)and
λ′ λ(t′), and imposing the initial condition≡s at t′ = t where we recall that e is the function of Ir/Λ defined so
(i.e≡. τ = 0), Λ′(t′ = t) = Λ, λ′(t′ = t) = λ, yields as a that 1 √1 e2 =Ir/Λ, i.e.
− −
unique solution (to linear order in the source terms)
2
I
t′ e= 1 1 r . (4.10)
Λ′ =Λ dt S (t ), (4.7a) s − − Λ
− 1 λ 1 (cid:18) (cid:19)
Zt
t′ t 3 t′ At this stage, Eq. (4.9) defines an ordinary Hamil-
λ′ =λ+ − + dt (t′ t )S (t ) tonian, expressed in terms of action-angle variables
Λ3 Λ4 Zt 1 − 1 λ 1 (Λ,λ;Ir,̟). [Strictly speaking, the variables Λ, λ, etc.,
t′ at the stage of Eq. (4.9) differ from the ones entering
+ dt1SΛ(t1). (4.7b) the original action by the shifts ξΛ, ξλ, etc., mentioned
Zt above. They should be denoted as Λshifted, λshifted, etc.,
but, for simplicity, we do not indicate the shifted nature
(We do not assume here t < t′ but use the convention
t′ t of Λ, λ, ... .]
that t = − t′.) When replacing the identities (4.7) As we are now dealing with an ordinary, local Hamil-
in a nonlocal action S = dtdt′µ(t′ t) (Λ,Λ′,λ′ tonian,ourfourthstep(d), i.e.the Delaunayelimination
R R − S −
λ, ) (keeping only the terms linear in the source of periodic terms by suitable canonical transformations
··· R
terms, which is allowed modulo “double-zero” terms), is standard. For instance, a generating function propor-
the extra terms involving the source contributions will tional to
have the form dt [ξ (t )S (t )+ξ (t )S (t )+ ],
1 Λ 1 Λ 1 λ 1 λ 1
where the quantities ξ (t ),ξ (t ), are given by i·n·t·e- +∞ 1 τ
grals over t anRd t′ of aΛfu1nctλion1of·d··ynamical variables: ε dτµ(τ)Λ5 sin Λ3 −λ−̟ (4.11)
ξ (t ) = dtdt′µ(t′ t) (t′ t,t t,Λ(t),λ(t)), . Z−∞ (cid:16) (cid:17)
Λ 1 1
(Note that, because o−f thAe in−equalit−ies t < t < t′··o·r willeliminate thecontribution+2ecos(τ/Λ3 λ ̟)on
t′ < t <RRt, for given values of t and τ =1t′ t, t the rhs of Eq. (4.9). − −
1 1
−
and t′ both range over a bounded interval.) The ex- Aftercompletingstep(d)(formallytoallordersinthe
tra terms involving the source contributions can then be eccentricity) we end up with an ordinary (local) Hamil-
“field-redefined away”by correspondingshifts of the dy- tonian that depends only on (shifted) action variables.
namical variables: δΛ(t ) = ξ (t ),δλ(t ) = ξ (t ), . E.g., for our toy model
1 Λ 1 1 λ 1
···
[These shifts have, in general, nonlocal structures of
1 +∞ 1
the type δΛ(t1) = dtKΛ(t t1,Λ(t),λ(t)), δλ(t1) = H′′ (Λ,I )= +ε dτµ(τ)
dtK (t t ,Λ(t),λ(t)), etc.]− toy r −2Λ2 Λ8
λ − 1 R Z−∞
After performing these (nonlocal) shifts, we are left
wRith an action obtained by replacing λ′,Λ′, by the cos τ +O(e2) . (4.12)
rhs of Eq. (4.7) in which one has set to zero·t·h·e terms × Λ3
(cid:20) (cid:16) (cid:17) (cid:21)
9
Letusemphasizethatourprocedureallowsone,inpar- 1 ∂ +∞ 1 τ
= +ε dτµ(τ) cos .
ticular, to construct (formally to all orders in eccentric- Λ3 ∂Λ Λ8 Λ3
ity)astrictly conserved energythatconsistentlyincludes (cid:18)Z−∞ (cid:16) (cid:17)((cid:19)4.16)
the nonlocaltails at the 4PN level. [The nonlocaleffects
are recognizable through their integral nature, e.g. the One sees that the computation of the rhs will involve
integral ε +∞dτµ(τ)( ) in Eq. (4.12).] Namely, our two types of contributions coming from the tail term:
final, Delau−n∞ay Hamilto·n··ian is strictly conserved. This a “normal” contribution where the Λ differentiation of
shows thaRt the contrary statement below Eq. (5.17) in the second term acts on the prefactor 1/Λ8 of the tail
Ref. [21] is incorrect (at least when allowing for an ex- integral, and a “new” contribution where ∂/∂Λ acts on
pansion in powers of I /Λ e2). the argument of cos(τ/Λ3).
r
In addition, our construc∼tion also gives two conserved We recall that, in the full 4PN case, the analog of our
action variables9: Λ and I . These conserved quanti- final result (4.12) [say, for simplicity, at the O(e0) level]
r
ties are such that the difference, Λ I is conserved is the one discussedin Sec. V ofRef. [16], whichleads to
r
and reduces to the usual orbital an−gular momentum (with J now denoting Iϕ Λ Ir)
≡ −
I = 1 p dϕ = p in the local case. We can then
cϕonside2rπ thaϕt I := Λϕ I defines a strictly conserved Ecirc(J)=Hloc (r,p;s)+F(r,p)lnr
H ϕ − r 4PN DJS s
angular momentum that consistently includes the 4PN
nonlocal dynamical effects. (Indeed, one easily checks 1G2M +∞ dτ
Pf f(τ), (4.17)
that the rotational symmetry ϕ′ = ϕ+const remains a − 5 c8 2s/c τ
symmetry at all stages of our construction so that I is Z−∞ | |
ϕ
indeed the Noether conserved quantity associated with wheref(τ):= I(3)(t+τ)I(3)(t) circ,asinSec.VofRef.
ij ij
this symmetry.)
[16]. [In the circular limit f(τ) does not depend on t.]
Let us also emphasize another consistency feature of Theexplicitva(cid:2)lueofthelast,tail(cid:3)term10 isgiven,inview
ourfinal(reduced,local,Delaunay)Hamiltonian. Asthe
of Eqs. (5.7)–(5.9) in Ref. [16] by
above detailed derivation clearly shows, our final Hamil-
tonianaction(intermsofΛshifted,λshifted,...butwithout 64G2M Ω(J)s
[Htail(s)]circ(J)= (µΩ3r2 )2ln 4eγE
explicitly indicating the shifts) is simply 5 c8 12 c
(cid:18) (cid:19)
S′′ = Λdλ+Ird̟ H′′(Λ,Ir)dt , (4.13) 64 ν 4eγEsˆ
− = µc2 ln , (4.18)
Z 5 j10 j3
and the corresp(cid:2)ondingequations ofmotion(cid:3)are the usual (cid:18) (cid:19)
ones: where j cJ/(GMµ), and sˆ s/m with m GM/c2.
≡ ≡ ≡
∂H′′(Λ,I ) ∂H′′(Λ,I ) Ifwealternativelydecidetodefinethetailcontribution
λ˙ = r , ̟˙ = r , Λ˙ =0, I˙r =0. to the reduced Hamiltonian by incorporating the term
∂Λ ∂I
r F ln(r/s) in it, we must simply replace the scale s by
(4.14)
r = r , so that (using the fact that along Newtonian
We said above that, in the circular limit, the mean 12
circular orbits rcirc =mj2)
anomaly λ = ℓ+g = ℓ+ω reduces to a usual polar an-
gle ϕ. (Indeed, this is true in the Newtonian case, and,
64 ν 4eγE
as all the Delaunay shifts are periodic in the unique fast [Htail(s=r12)]circ(J)= µc2 ln . (4.19)
5 j10 j
angular variable λ, we have that ϕ=λ+ terms periodic (cid:18) (cid:19)
inλthatvanishwithe.) Thereforeourderivationshows,
Evidently, we must keep in mind that the meaning of
among other things, that the orbital frequency Ω (= λ˙) the two quantities (4.18) and (4.19) is different and that
of a circular (Ir = 0) binary (with conserved angular their J derivative will differ by a term involving the J
momentum Λ=Iϕ) is given by the usual formula derivative of rcirc =mj2. But the latter difference has a
conceptually trivial origin [in view of the local nature of
∂H′′(Λ,I ) ∂H′′(I +I ,I )
Ω=λ˙ = r = r ϕ r . the Fln(r/s) term in the Hamiltonian] and does not in-
∂Λ ∂I
(cid:12)Ir=0 ϕ (cid:12)Ir=0 terferewiththe issuewe wishto emphasize here,namely
(cid:12)(cid:12) (cid:12)(cid:12) (4.15) theexistence,intheorbitalfrequency,Ωcirc =dHcirc/dJ,
In particular, this mea(cid:12)ns, in our toy example, t(cid:12)hat the
of a contribution coming from differentiating the J de-
circular angular frequency is of the form pendence of the argument4eγEΩ(J)s/c of the logarithm
∂H′′ (Λ,0) in the first (or second) line of Eq. (4.18).
toy
Ω =
toy
∂Λ
10 Here,wefinditconvenienttoworkwithatailtermdefinedwith
9 It would beinteresting to study whether our (4PN-order) con- some given, constant scale s. Such a scale is not a dynamical
structionofconservedactionvariables,andconservedenergy, is variable and is not affected by the Λ differentiation above, i.e.
relatedtothe(1SF-order)workofRef.[43]. thef differentiationinthepresentcircularcase.
10
AsclearlypointedoutalreadyinRef.[16],the compu- where Ωcirc(J)=dHcirc(J)/dJ is its unperturbed value.
0 0
tationofthetailcontributiontothe(circular,Delaunay) From the two results (4.21) and (4.22), it is easy to
function H(J) = E(J) “involves the evaluation of the derive the perturbed value of the function EΩ(Ω)
nonlocal piece Htail(s) along circular motion (without Hcoirrdc(Jcirc(Ω)), where Jcirc(Ω) is the inverseof the func≡-
4PN tion Ωcirc(J). One gets
any differentiathion)”. Tihis lack of any differentiation in
theevaluationof Htail circ(J)distinguishesitamongthe EΩ(Ω)=EΩ(Ω)+εEΩ(Ω)+O(ε2), (4.23)
0 1
other gauge-invariant functions one can associate with
the sequenceofci(cid:2)rcular(cid:3)orbits,andrenderedveryclear11 with
that this necessarily led to the correct value of the func- dEΩdHcirc(J)
tion Htail circ(J). Our discussion above has confirmed E1Ω(Ω)= H1circ(J)− dΩ0 1dJ ,
theconsistencyofthisresultandhasshownthat,indeed, (cid:20) (cid:21)J=J0circ(Ω)
(cid:2) (cid:3) (4.24)
when computing the orbital frequency it is correct to J-
where Jcirc(Ω) is the inverse of the function Ωcirc(J).
differentiate the argument of the tail log. 0 0
Let us apply the convenient result (4.24) to the r -
Letusnowderivethetail-logcontributiontothefunc- 12
scale tail perturbation written as
tional relation between energy and orbital frequency,
E(Ω), corresponding to Eq. (4.19). 64 ν 4eγEΩrcirc(J)
We start by deriving a useful general result about the εHtail(r12) = µc2 ln 0 , (4.25)
1 5 j10 c
function E(Ω), or equivalently the function E(x) [where (cid:18) (cid:19)
x (GMΩ/c3)2/3]. Let us first consider the case of an where we leave open for the moment the functional de-
≡
ordinary Hamiltonian dynamics of the type (with J pendence of Ω within the tail logarithm. In our case
≡
pϕ) EΩ = 1µc2Ω2/3 and jcirc(Ω) = Ω−1/3 (where Ω
0 −2 0 ≡
mΩ), so that the general result (4.24) yields the simple
Hord(r,p ,J)=H (r,p ,J)+εH (r,p ,J). (4.20)
r 0 r 1 r expression b b b
When considering the sequence of circular orbits, it is 1 d
well known12 that, modulo O(ε2), one can replace from E1Ω(Ω)= 1+ 3j dj H1circ(j), (4.26)
the start the radius r on the rhs by its unperturbed cir- (cid:18) (cid:19)
cular value as a function of J, rcirc(J), defined as the which involves a crucial j derivative [by contrast to the
0
solution of ∂H0(r,pr = 0,J)/∂r = 0 for a given J. (We derivationofthefunctionH1circ(j)=[Htail(s=r12)]circ(j)
are assuming here that the pr dependence of Hord starts which involved no differentiation].
at order p2r so that one can consistently set pr = 0 be- Applying Eq. (4.26) to Eq. (4.25) yields
fore considering the J dependence.) This yields for the
circular-reduced dependence of Hord as a function of J 448 4eγEΩrcirc(j)
EΩ (Ω)= µc2νx5 ln 0
r12-tail − 15 c
Hord(J)=Hcirc(J)+εHcirc(J)+O(ε2), (4.21) (cid:20) (cid:18) (cid:19)
circ 0 1
1dlnΩ 1dlnrcirc(j)
where Hcirc(J) = H (rcirc(J),p = 0,J) and Hcirc(J) = 0 , (4.27)
0 0 0 r 1 −7 dlnj − 7 dlnj
H (rcirc(J),p = 0,J). Note that the O(ε) contribution (cid:21)
1 0 r
to Hcoirrdc(J) is directly equal to the zeroth-order-circular where (on shell) ln 4eγEΩr0circ(j) = ln(4eγEx21). As
value of the original O(ε) contribution to the Hamilto- c
nian(4.20)[withoutanycontributionfromεr1circ(J),and r0circ(j) = mj2, the(cid:16)contribution(cid:17)of the last term13 in
without any differentiation]. Furthermore, the on-shell the bracket is 2/7. Moreover, as our Delaunay-based
−
vanishing (along exact circular orbits) of ∂Hord/∂r also method led us to have Ω = Ωcirc(j) = j−3 in Eq. (4.25)
0
ensures that the circular-reduced J dependence of the [seenotablyEq.(4.11)inRef.[17]],thepenultimateterm
orbitalfrequency Ω=∂Hord(r,pr,J)/∂J is simply given in the bracket yields thebaddbitional term
by J differentiation of Hord(J):
circ 1dlnΩcirc(j) 3
0 =+ , (4.28)
dHord(J) dHcirc(J) − 7 dlnj 7
Ωcirc(J)= circ =Ωcirc(J)+ε 1 +O(ε2),
dJ 0 dJ
(4.22) so that the bracket in Eq. (4.27) becomes
4eγEΩr 3 2 4eγEΩr 1
0 0
ln + =ln + . (4.29)
c 7 − 7 c 7
(cid:18) (cid:19) (cid:18) (cid:19)
11 In view, e.g., of the result of Ref. [44] about the consistency of
the order reduction of higher-order Hamiltonians, see also Sec.
IIIinRef.[17]andbelow.
12 Theon-shellvanishingof∂Hord/∂rensuresthatwhenreplacing 13 Note, in passing, that if we were considering the s-
theexact solutionrcirc (J)=rcirc(J)+εrcirc(J), theεrcirc(J) scaled tail (with a fixed scale s) the latter term would be
exact 0 1 1
perturbationwillonlycontribute atorderO(ε2). −1dlns/dlnj=0.
7