Comparison between self-force and post-Newtonian
dynamics: Beyond circular orbits
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Citation
Akcay, Sarp, Alexandre Le Tiec, Leor Barack, Norichika Sago,
and Niels Warburton. “Comparison Between Self-Force and
Post-Newtonian Dynamics: Beyond Circular Orbits.” Phys. Rev.
D 91, no. 12 (June 2015). © 2015 American Physical Society
As Published
http://dx.doi.org/10.1103/PhysRevD.91.124014
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American Physical Society
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Detailed Terms
PHYSICAL REVIEW D 91, 124014 (2015)
Comparison between self-force and post-Newtonian dynamics:
Beyond circular orbits
Sarp Akcay,1 Alexandre Le Tiec,2 Leor Barack,3 Norichika Sago,4 and Niels Warburton5
1
School of Mathematical Sciences and Complex & Adaptive Systems Laboratory,
University College Dublin, Belfield, Dublin 4, Ireland
2
Laboratoire Univers et Théories, Observatoire de Paris, CNRS, Université Paris Diderot,
92190 Meudon, France
3
School of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom
4
Faculty of Arts and Science, Kyushu University, Fukuoka 819-0395, Japan
5
MIT Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
(Received 5 March 2015; published 3 June 2015)
The gravitational self-force (GSF) and post-Newtonian (PN) schemes are complementary approximation methods for modeling the dynamics of compact binary systems. Comparison of their results in
an overlapping domain of validity provides a crucial test for both methods and can be used to enhance
their accuracy, e.g. via the determination of previously unknown PN parameters. Here, for the first
time, we extend such comparisons to noncircular orbits—specifically, to a system of two nonspinning
objects in a bound (eccentric) orbit. To enable the comparison we use a certain orbital-averaged
quantity hUi that generalizes Detweiler’s redshift invariant. The functional relationship hUiðΩr ; Ωϕ Þ,
where Ωr and Ωϕ are the frequencies of the radial and azimuthal motions, is an invariant characteristic
of the conservative dynamics. We compute hUiðΩr ; Ωϕ Þ numerically through linear order in the mass
ratio q, using a GSF code which is based on a frequency-domain treatment of the linearized Einstein
equations in the Lorenz gauge. We also derive hUiðΩr ; Ωϕ Þ analytically through 3PN order, for an
arbitrary q, using the known near-zone 3PN metric and the generalized quasi-Keplerian representation
of the motion. We demonstrate that the OðqÞ piece of the analytical PN prediction is perfectly
consistent with the numerical GSF results, and we use the latter to estimate yet unknown pieces of the
4PN expression at OðqÞ.
DOI: 10.1103/PhysRevD.91.124014
PACS numbers: 04.70.Bw, 04.25.D-, 04.25.dg, 04.25.Nx
I. INTRODUCTION
With the Advanced LIGO observatories scheduled to
start science runs in 2015 [1], the next few years are likely
to see first direct detections of gravitational waves from
astrophysical sources. Prime targets are inspiraling and
coalescing binary systems of neutron stars and/or black
holes, with predicted rates that may be as high as a few
dozen per observation year [2]. Theoretical templates of the
gravitational waveforms must be developed to enable
detection and interpretation of the weak signals [3]. The
parameter space of these waveforms is too large for
numerical relativity simulations to cover sufficiently well.
Instead, the community has been seeking semianalytical
models that can be informed by a judiciously chosen set of
numerical relativity templates. A leading framework is the
effective one-body (EOB) model, where the two-body
relativistic dynamics is mapped onto a model of (non)
geodesic motion in an effective spacetime [4–7]. EOB
waveforms will play a crucial role in searches based
on matched filtering, and there is an important need to
refine the model, particularly in the strong-field regime
[8–10].
1550-7998=2015=91(12)=124014(32)
One avenue of refinement is provided by the gravitational self-force (GSF) method, a perturbative scheme
based on an expansion in the mass ratio of the binary
[11–13]. The GSF approach is complementary to the postNewtonian (PN) approximation, a weak-field/smallvelocity expansion valid for arbitrary mass ratios [14].
Recently, there has been much activity in attempt to
“synergize” the two schemes. The goal of such crosscultural studies is threefold: to test the two independent
approximation schemes—GSF and PN—and help delineate
their respective domains of validity; to determine yetunknown, high-order expansion terms in both approaches
(hence, improving both approximations); and to help
calibrate the EOB model across the entire inspiral parameter space.
To facilitate such studies requires the identification of
concrete gauge-invariant physical quantities that can be
computed using both approaches. A first such quantity was
identified by Detweiler in 2008 within the GSF framework
[15]: the so-called “redshift” variable, defined for strictly
circular orbits when dissipation is ignored. The functional
relation between the redshift and the orbital frequency is a
gauge-invariant diagnostic of the conservative sector of the
124014-1
© 2015 American Physical Society
AKCAY et al.
PHYSICAL REVIEW D 91, 124014 (2015)
binary dynamics. Detweiler made the first successful
comparison with the PN prediction at 2PN order [15].
This comparison was later extended by Blanchet et al.
to 3PN order and to even higher orders [16–20].
The calculation of the redshift through linear order in
the mass ratio was subsequently confirmed by several other
GSF computations in different gauges [21,22], which
provided an internal consistency check for the GSF
formalism.
Soon after, Barack and Sago considered two more such
“conservative” invariant quantities, namely, the frequency of the innermost stable circular orbit (ISCO)
and the rate of periastron advance [23,24]. These results
led to a plethora of comparisons between PN, GSF and
numerical relativity [25–30] and the subsequent refinement of EOB theory [31–34]. More recently, the geodetic
spin precession along circular orbits was computed by
Dolan et al. through linear order in the mass ratio, and
the numerical results successfully compared to a 3PNaccurate prediction [35]. The results allowed a numerical
prediction of the (hitherto unknown) 4PN expression for
the spin precession. This was later confirmed analytically
by Bini and Damour [9], who also proceeded to obtain all
PN terms up to the 8.5PN order, at linear order in the
mass ratio. Dolan et al. [36] then presented a computation of the leading post-geodesic corrections to certain
tidal invariants defined along the orbit. The PN series for
these tidal invariants were also computed analytically up
to 7.5PN order in Ref. [10], still at linear order in the
mass ratio.
All synergistic work so far has focused on circular
orbits, for simplicity. Here, for the first time, we extend
this program to orbits of arbitrary eccentricity. There are
several reasons to do so. First, eccentricity provides more
“handle” on the strong-field dynamics, giving access to
new degrees of freedom in the EOB formulation. Second,
although most Advanced LIGO binaries would have
completely circularized by the time they enter the observable frequency band, there are scenarios where eccentricity effects could become observable and would give
access to much interesting physics [37–42]. Third,
eccentric inspirals in the extreme-mass-ratio regime will
be key sources for a future mHz-band detector in space
[43–47].
A gauge-invariant quantity for eccentric orbits, suitable
for synergistic studies, was introduced by Barack and Sago
in Ref. [24] (henceforth, BS2011). This quantity is a
straightforward generalization of Detweiler’s redshift,
obtained by averaging the time component of the particle’s
four-velocity with respect to proper time over one epicyclic
period of the motion. In other words, it is the ratio between
the period measured in the coordinate time of a static
observer at infinity and the proper-time period. This
“averaged redshift,” denoted here hUi, is defined with
the dissipative piece of the GSF ignored. The functional
relationship between hUi and the two invariant frequencies
that characterize the motion is a gauge-invariant diagnostic
of the conservative eccentric-orbit dynamics. BS2011
calculated hUi (numerically) through linear order in the
mass ratio for a sample of strong-field orbits, but they
stopped short of attempting a calculation in a weaker-field
regime where a meaningful comparison with PN results
might be possible. The method of BS2011, which is
based on a time-domain numerical integration of the
relevant field equations, was best suited for tackling
strong-field orbits, and its performance deteriorated fast
with increasing orbital radius because of the longer
evolution time required.
Here we extend the range of BS2011’s calculation into
the weaker-field regime, derive a 3PN-accurate formula for
hUi, valid for any mass ratio, and compare between the
numerical GSF results and the analytical PN prediction in
the small mass-ratio limit. This is the first such comparison
for noncircular orbits. It shows a good agreement for large
and medium separations, and allows us to assess the
performance of the PN expansion all the way down to
the innermost stable orbit. Moreover, we are also able,
through fits to the numerical GSF data, to extract some
information about the 4PN approximation.
Our numerical GSF calculation improves on that of
BS2011 in both accuracy and weak-field reach. This
improvement is achieved in two ways. First, our computation is based on the frequency-domain approach of Akcay
et al. [48], in which the field equations are reduced to
ordinary differential equations. This offers significant
computational saving, particularly at lower eccentricities
(e ≲ 0.4). Second, we have found a way to significantly
simplify the expression given in BS2011 for hUi as a
function of the two orbital frequencies. The new form
requires a simpler type of numerical input, which can be
obtained at greater accuracy.
This paper is organized as follows. In Sec. II we review
relevant results for bound motion in Schwarzschild spacetime and for the redshift as defined for circular orbits. We
then extend the definition to eccentric orbits and obtain a
simple expression for the generalized redshift hUi in terms
of calculable perturbative quantities. Section III discusses
the numerics and sources of error, and displays a sample
of numerical results for hUi. In Sec. IV we perform a
detailed derivation of the PN expression for hUi through
3PN order. Our calculations rely crucially on the known
3PN near-zone metric and the 3PN quasi-Keplerian
representation of the motion. The numerical GSF and
analytical PN results are compared in Sec. V. In
Appendix A we establish the equivalence between our
simplified formulation of hUi and that of BS2011.
Appendix B derives some useful PN formulas valid in
the test-mass limit.
Table I summarizes some of our notation, for easy
reference. In the GSF context, we denote the mass of
124014-2
COMPARISON BETWEEN SELF-FORCE AND POST- …
TABLE I.
PHYSICAL REVIEW D 91, 124014 (2015)
Important symbols.
m1
m2
m ¼ m1 þ m2
q ¼ m1 =m2
ν ¼ m1 m2 =m2
Δ ¼ ðm2 − m1 Þ=m
Ωr
Ωϕ
particle’s mass
black hole’s mass
total mass
mass ratio
symmetric mass ratio
reduced mass difference
radial (epicyclic) frequency
average azimuthal frequency
the background Schwarzschild geometry by m2 and the
mass of the orbiting particle by m1, with the assumption
that q ≡ m1 =m2 ≪ 1. In the PN context, the two particles
of masses m1 and m2 have an arbitrary mass ratio q. We will
set G ¼ c ¼ 1, except in Sec. IV where we keep these
constants explicit in PN expressions. We use a metric
signature ð−; þ; þ; þÞ.
where fðrÞ ≡ 1 − 2m2 =r, and V eff ðr; L2 Þ ≡ fðrÞð1 þ L2 =r2 Þ
is an effective potential for thepffiffiradial
motion. Bound
ffi
(eccentric)
geodesics exist for 2 2=3 < E < 1 and L >
pffiffiffi
2 3m2 . For a given bound geodesic, the radial distance
rp ðτ0 Þ is confined to a finite range 2m2 < rmin ≤ rp ðτ0 Þ ≤
rmax < ∞ with rmin , rmax denoting periastron and apastron
radii, respectively. These two turning-point radii can be
mapped bijectively to fE; Lg. Thus the pair frmin ; rmax g
can also parameterize the family of bound geodesics.
Another such pair is given by the dimensionless “semilatus rectum” p and the “eccentricity” e, defined by
p≡
2rmax rmin
;
m2 ðrmax þ rmin Þ
A. Bound geodesic orbits in Schwarzschild spacetime
We first review relevant results for bound geodesic
motion in Schwarzschild spacetime. We consider
a test particle of mass m1 moving on a bound (timelike)
geodesic orbit in the Schwarzschild spacetime of a black
hole of mass m2 . Using Schwarzschild coordinates
ft; r; θ; ϕg, we label the position of the particle by
xαp ðτ0 Þ ¼ ðtp ðτ0 Þ; rp ðτ0 Þ; θp ðτ0 Þ; ϕp ðτ0 ÞÞ, with four-velocity
uα0 ≡ dxαp =dτ0 , where τ0 is a proper-time parameter along
the geodesic, and the label ‘0’ indicates normalization
with respect to the background (Schwarzschild) metric g0αβ ,
i.e., g0αβ uα0 uβ0 ¼ −1. Without loss of generality, we confine
the motion to lie in the equatorial plane, i.e., θp ¼ π=2,
such that uθ0 ¼ 0. We parameterize the geodesics by the two
constants of motion: the specific energy E ≡ −u0t
and specific angular momentum L ≡ u0ϕ , where
u0α ¼ g0αβ uβ0 .
The geodesic equation of motion is given by
β
u0 ∇0β uα0 ¼0, where ∇0β is the covariant derivative compatible with the background metric g0αβ . For the above setup,
this gives
dtp
E
¼
;
dτ0 fðrp Þ
ð2:1aÞ
dϕp L
¼ ;
dτ0 r2p
ð2:1bÞ
2
drp
¼ E 2 − V eff ðrp ; L2 Þ;
dτ0
rmax − rmin
:
rmax þ rmin
ð2:2Þ
These relations can be inverted to yield the Keplerian-like
formulas
rmax ¼
II. GENERALIZED REDSHIFT: FORMULATION
IN SELF-FORCE APPROACH
e≡
pm2
;
1−e
rmin ¼
pm2
:
1þe
ð2:3Þ
We can further express the specific energy and angular
momentum in terms of p and e by solving the equation
E 2 ¼ V eff ðr; L2 Þ at r ¼ frmin ; rmax g. Using Eqs. (2.3), this
yields
ðp − 2 − 2eÞðp − 2 þ 2eÞ 1=2
pm2
E¼
; L ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:
2
pðp − 3 − e Þ
p − 3 − e2
ð2:4Þ
Following Darwin [49], we parameterize the radial
motion using the “relativistic anomaly” χ via
rp ðχÞ ¼
pm2
;
1 þ e cos χ
ð2:5Þ
where χ ¼ 0 and χ ¼ π correspond to periastron and
apastron passages, respectively. Using Eq. (2.1c) with
Eq. (2.5), we obtain
dτ0
m2 p3=2
¼
dχ
ð1 þ e cos χÞ2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p − 3 − e2
;
p − 6 − 2e cos χ
ð2:6Þ
which, with the help of Eqs. (2.1a), (2.1b) and (2.4), also
gives
ð2:1cÞ
124014-3
dtp
m2 p2
¼
dχ ðp − 2 − 2e cos χÞð1 þ e cos χÞ2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðp − 2 − 2eÞðp − 2 þ 2eÞ
;
×
p − 6 − 2e cos χ
dϕp
¼
dχ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
:
p − 6 − 2e cos χ
ð2:7aÞ
ð2:7bÞ
AKCAY et al.
PHYSICAL REVIEW D 91, 124014 (2015)
The functions τ0 ðχÞ, tp ðχÞ and ϕp ðχÞ are all monotonically
increasing along the orbit. The radial periods in coordinate
and proper times are calculated, respectively, via
Z 2π
Z 2π
dtp
dτ0
T r0 ¼
dχ;
T r0 ¼
dχ; ð2:8Þ
dχ
dχ
0
0
and the accumulated azimuthal angle between successive
periastron passages is
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Z 2π
dϕp
p
4e
Φ0 ¼
ellipK
:
dχ ¼ 4
p − 6 þ 2e
p − 6 þ 2e
dχ
0
ð2:9Þ
R
Here ellipKðkÞ ≡ 0π=2 ð1 − ksin2 θÞ−1=2 dθ is the complete
elliptic integral of the first kind and subscripts ‘0’ serve to
distinguish the geodesic values T r0 , T r0 and Φ0 from their
corresponding GSF-perturbed quantities to be introduced
below. For any ðp; eÞ we have Φ0 > 2π; hence, the
periastron advances.
We can now define the radial and (average) azimuthal
frequencies via
Ωr ≡
2π
;
T r0
Ωϕ ≡
Φ0
:
T r0
ð2:10Þ
The pair fΩr ; Ωϕ g provides a gauge-invariant parametrization of eccentric orbits. It should be noted, however, that the
mapping between ðp; eÞ and ðΩr ; Ωϕ Þ is not bijective: there
exist (infinitely many) pairs of physically distinct geodesics
of different fp; eg values but the same set of frequencies.
This degeneracy, first noted in BS2011, was thoroughly
studied in [50]. The phenomenon is a feature of orbits very
close to the innermost stable orbit. Since in this work we
focus on less bound orbits (for the purpose of comparison
with PN theory), the phenomenon of isofrequency pairing
will not be relevant to us.
In the parameter space of eccentric geodesics, stable
orbits are located in the region given by p > 6 þ 2e. The
curve p ¼ 6 þ 2e is called the “separatrix.” Along it both
Φ0 and T r0 diverge, but Ωϕ remains finite. This gives rise to
the so-called “zoom-whirl” behavior [51], where the
orbiting particle zooms in from far away, whirls around
the black hole many times, thus accumulating a large
azimuthal phase, then zooms back out. In the limit
p → 6 þ 2e, the particle sits exactly at the peak of the
effective potential and whirls infinitely on an unstable
circular geodesic.
B. The redshift invariant for circular orbits
Now let the particle’s mass m1 be finite but small, i.e.,
q ≡ m1 =m2 ≪ 1;
ð2:11Þ
theory, Detweiler and Whiting [52] showed that such a
particle follows a geodesic motion in a certain smooth,
effective, locally-defined spacetime with metric
gαβ ¼ g0αβ þ hRαβ :
ð2:12Þ
Here, hRαβ is a certain smooth piece of the physical
(retarded) metric perturbation produced by the particle.
The physical perturbation itself is a solution of the
linearized Einstein equation, sourced by the particle’s
energy-momentum, with suitable “retarded” boundary
conditions. How hRαβ may be computed in practice, on a
Schwarzcshild background, is discussed, for example,
in Ref. [23].
Within linear perturbation theory, hRαβ may be split into a
dissipative piece and a conservative (time-symmetric)
piece, and the effects of the two pieces may be considered
separately. The conservative part of the perturbation is
R;adv
R;ret
R
defined as hR;cons
¼ 12 ðhR;ret
αβ
αβ þ hαβ Þ, where hαβ ≡ hαβ
R;adv
and hαβ is a smooth perturbation constructed just like
hR;ret
but starting with the particle’s “advanced” metric
αβ
R;cons
perturbation. Replacing hR;ret
in the effective
αβ → hαβ
metric (2.12) amounts to “turning off” the dissipation.
The resulting equations of motion capture only
conservative aspects of the dynamics.
In Ref. [15] Detweiler considered a particle in circular
geodesic motion in the “conservative” effective spacetime
R;cons
0
gcons
:
αβ ¼ gαβ þ hαβ
ð2:13Þ
In the absence of dissipation the orbit remains circular, and
the spacetime possesses a helical Killing vector field,
which, on the orbit, is proportional to the 4-velocity
uα ¼ dxα =dτ. We introduce here τ as a proper-time paramα
eter along the geodesic in the effective metric gcons
αβ , with u
α β
normalized with respect to that metric, i.e., gcons
αβ u u ¼ −1.
Thanks to the helical symmetry, all components of the
particle’s 4-velocity are invariant under gauge transformations that respect the helical symmetry [21]. Detweiler
proposed to use the functional relationship between ut and
Ω ≡ uϕ =ut as a gauge-invariant benchmark for the
conservative self-force effect beyond the geodesic approximation. The frequency Ω is the circular-orbit reduction of
the frequency Ωϕ defined earlier for eccentric orbits. The
quantity ut (or rather, its inverse) may be assigned a
heuristic meaning of “redshift” (as measured in the smooth
metric hR;cons
by a static asymptotic observer located along
αβ
the helical symmetry axis), but it should be remembered
that the true redshift, as measured in the physical metric of
the particle, is, of course, divergent.
Detweiler obtained [15]
and consider the effect of self-interaction on the motion
through OðqÞ. Within the context of linear perturbation
124014-4
ut ðΩÞ ¼ ut0 ðΩÞ þ qutgsf ðΩÞ;
ð2:14Þ
COMPARISON BETWEEN SELF-FORCE AND POST- …
where
and
ut0
2=3 −1=2
¼ ½1 − 3ðm2 ΩÞ
PHYSICAL REVIEW D 91, 124014 (2015)
is the geodesic limit,
1
qutgsf ¼ ut0 uα uβ hR;cons
αβ
2
ð2:15Þ
is the OðqÞ correction arising from self-interaction.
Note that the correction utgsf is defined for a fixed value
of Ω at the background, which ensures its gauge invariance.
In Ref. [15] and subsequent work [16,17] (see also
[18–20]), Detweiler and collaborators calculated numerically the post-geodesic correction utgsf ðΩÞ, and showed that
it is consistent with corresponding PN expressions in an
overlapping domain of validity.
Detweiler’s numerical results were derived using the
Regge-Wheeler gauge. An independent calculation using a
direct numerical integration of the Lorenz-gauge form of
the perturbation equations later recovered the same invariant relation utgsf ðΩÞ [21]. This comparison highlighted a
subtlety in the notion of invariance as applied to utgsf ðΩÞ:
the gauge transformation between the Lorenz-gauge metric
perturbation and the Regge-Wheeler one does not leave
utgsf ðΩÞ invariant, due to a certain minor gauge irregularity
of the Lorenz-gauge metric (that was first identified in
Ref. [53] and further discussed in [21]). Specifically, the
physical metric perturbation does not vanish at infinity
when expressed in the Lorenz gauge; see Eq. (2.23)
below. While the perturbation remains helically symmetric,
the transformation to an “asymptotically flat” gauge like
Regge-Wheeler’s (or the harmonic gauge of PN theory), in
which Eq. (2.15) applies, has a generator that itself
does not have a helical symmetry. As a result, the
transformation introduces a correction to utgsf ðΩÞ.
Denoting by ĥαβ the Lorenz- gauge metric perturbation,
one finds [21]
1
qutgsf ¼ ut0 uα uβ ĥR;cons
þ αEðut0 Þ2 :
αβ
2
ð2:16Þ
The parameter α is extracted from the Lorenz-gauge
perturbation as prescribed in Eq. (2.23) below; for a circular
orbit it reads α ¼ qðm2 ΩÞ2=3 ut0 . One must be mindful,
when working in the Lorenz gauge (as we do here), to
take proper account of this gauge irregularity. We shall
return to this point in more detail when discussing
eccentric orbits.
subscripts ‘0’ dropped. The functional relation between
these invariant frequencies and any gauge-dependent
set of parameters can be written as the sum of a
“geodesic” term and a GSF correction; such relations were
derived in explicit form in BS2011 but will not be
needed here.
The GSF-perturbed orbit is a geodesic in the effective
R;cons
0
metric gcons
, with tangent four-velocity uα
αβ ¼ gαβ þ hαβ
t
normalized in gcons
αβ . It is easily checked that u is no longer
gauge-invariant in a pointwise sense when the orbit is
noncircular. Instead, BS2011 suggested to consider the
orbital average
Z T
r
1
T
hUi ≡ hut i ≡
ut dτ ¼ r ;
ð2:17Þ
Tr 0
Tr
where T r is the radial period measured in proper time τ.
BS2011 argued that hUi is invariant under gauge transformations that respect the periodicity of the orbit and are
well behaved (in a certain sense) at infinity. We may split
hUi in the form
hUiðΩi Þ ¼ hUi0 ðΩi Þ þ qhUigsf ðΩi Þ;
where Ωi ≡ fΩr ; Ωϕ g, hUi0 is the geodesic limit of hUi
taken with fixed Ωi , and qhUigsf is the GSF correction,
defined for fixed Ωi . The functional relation hUigsf ðΩi Þ is
an invariant measure of the GSF effect on the eccentric
orbit, and it is the quantity that we will use for our GSF–PN
comparison in this paper.
The geodesic limit of hUi is given by
hUi0 ¼
T r0
;
T r0
ð2:19Þ
where the periods T r0 and T r0 may be calculated via (2.8)
given the parameters p; e of the geodesic orbit. BS2011
describes a practical method for (numerically) inverting the
relations Ωi ðp; eÞ in order to obtain pðΩi Þ and eðΩi Þ. This
method may be used in conjunction with Eqs. (2.8) and
(2.19) in order to compute hUi0 for given frequencies Ωi .
Our goal now is to express hUigsf ðΩi Þ explicitly in terms
of calculable perturbative quantities (the metric perturbation and/or the GSF). Since fixing Ωi fixes T r , the only
contribution to hUigsf ðΩi Þ comes from the OðqÞ difference
T r − T r0 . From the normalizations g0αβ uα0 uβ0 ¼ −1 and
Þuα uβ ¼ −1 one obtains
ðg0αβ þ hR;cons
αβ
C. The redshift invariant generalized to eccentric orbits
Now consider an eccentric orbit subject to the
conservative effect of the GSF. In absence of dissipation,
the orbit remains bound and has a constant radial
period T r and a constant accumulated azimuthal phase Φ
per radial period. Hence, it possesses a well defined pair of
frequencies fΩr ; Ωϕ g, defined via Eq. (2.10) with the
ð2:18Þ
dτ0
1
1
¼ 1 þ uα0 uβ0 hR;cons
≡ 1 þ hRuu ;
αβ
2
2
dτ
ð2:20Þ
where terms of Oðq2 Þ and higher are omitted. Here we have
used uα0 ¼ dxα =dτ0 and uα ¼ dxα =dτ, assigning to each
given physical point on the perturbed orbit the same
coordinate value it has on the background orbit [this choice
124014-5
AKCAY et al.
PHYSICAL REVIEW D 91, 124014 (2015)
is consistent with the “fixed frequency” mapping introduced in Eq. (2.18)]. Since the contraction hRuu automatically picks out the conservative piece of hRαβ , the label
‘cons’ becomes redundant and we have dropped it.
Neglecting subleading terms in the mass ratio q, we
now obtain
Z T r
dτ
1
T r − T r0 ¼
1 − 0 dτ ¼ − T r0 hhRuu i;
ð2:21Þ
2
dτ
0
Eq. (2.25) reduces to (2.16) in the circular-orbit limit.
As in the circular case, the expression for hUigsf involves
only the R field along the orbit (and the parameter α), and
not the GSF itself. Our result (2.25) is much simpler than
the one derived in BS2011 using a different procedure. In
that work, certain simplifications that reduce the expression
for hUigsf to the form (2.25) have been overlooked. In
Appendix A we establish the equivalence between the two
results.
where the average is taken with respect to τ (or τ0 ) over a
radial period. The OðqÞ perturbation of hUi ¼ T r =T r at
fixed Ωi therefore reads
III. NUMERICAL CALCULATION OF THE
GENERALIZED REDSHIFT
qhUigsf
T
1
¼ − r0 2 ðT r − T r0 Þ ¼ hUi0 hhRuu i:
2
ðT r0 Þ
ð2:22Þ
This would be our final result for hUigsf if hRuu were to be
calculated in a suitable “asymptotically flat” gauge. Our
calculation, however, will be performed in the Lorenz
gauge, which suffers from the aforementioned irregularity
at infinity. Let us now describe this irregularity more
specifically. For either circular or noncircular orbits, the
Lorenz-gauge metric component ĥtt tends to a finite nonzero value at r → ∞ (other components are regular). This
behavior is due entirely to the static piece of the mass
monopole perturbation, and therefore the asymptotic value
of ĥtt does not depend on the angular direction even for
eccentric orbits; it depends only on the orbital parameters.
To remove this gauge artifact, following BS2011 we
introduce the normalized time coordinate t ¼ ð1 þ αÞt̂,
where t̂ denotes the original Lorenz-gauge time coordinate,
and α ¼ αðΩi Þ is given by
1
α ¼ − ĥtt ðr → ∞Þ:
2
A. Details of numerics and sources of error
ð2:23Þ
This normalization, which amounts to an OðqÞ gauge
transformation away from the Lorenz gauge, corrects the
asymptotic behavior. Under t̂ → t we have, at leading
order,
ĥRuu → hRuu þ 2αg0tt ðut0 Þ2 ¼ hRuu − 2αEut0 :
ð2:24Þ
Thus, to re-express hUigsf in Eq. (2.22) in terms of the
Lorenz-gauge perturbation, we need simply replace
hhRuu i → hĥRuu i þ 2αEhUi0 . We finally get
1
qhUigsf ¼ hUi0 hĥRuu i þ αEhUi20 :
2
We have used the frequency-domain computational
framework of Ref. [48] in order to compute hUigsf for a
large sample of orbits, focusing primarily on obtaining
weak-field data for PN comparisons. Our calculation is
based on Eq. (2.25), which takes as input the regularized
Lorenz-gauge metric perturbation evaluated along the orbit
(as well as the asymptotic value α, also to be read off the
Lorenz-gauge perturbation). Since the GSF correction
qhUigsf (defined at fixed frequencies Ωi ) is of OðqÞ, it
is sufficient to use as input the metric perturbation
calculated along geodesic orbits. For convenience we shall
use p; e (as defined in Sec. II A), rather than Ωi , to
parameterize these geodesics, and will thus express our
results in the form hUigsf ¼ hUigsf ðp; eÞ. It is important to
emphasize that our results refer to the GSF correction to the
functional relation hUigsf ðΩi Þ defined for fixed invariant
frequencies Ωi , even though we use the geodesic parameters p and e as independent variables. These two facts
should not be confused.
ð2:25Þ
Equation (2.25) is one of our main results, giving hUigsf
in terms of quantities directly calculable using existing GSF
codes: the R field ĥRαβ in the Lorenz gauge, and the
corresponding asymptotic parameter α. It is clear that
We use the eccentric-orbit GSF code of Ref. [48] to
obtain the metric perturbation ĥRαβ ðχÞ along the geodesic
orbit. This code employs a frequency-domain approach,
coupled with the method of extended homogeneous solutions of Ref. [54], to compute the regularized metric
perturbation ĥRαβ . It then outputs ĥRuu ðχÞ at 2400 evenly
spaced points along the orbit, and interpolates the numerical data using Mathematica’s Interpolation function.
In its default setting, Interpolation fits cubic polynomials between successive data points. Since ĥRuu ðχÞ is
very smooth this level of interpolation is sufficient for our
purposes. We subsequently calculate the orbital average
hĥRuu i using NIntegrate with the appropriate numerical
integration options/controls offered by MATHEMATICA. The
coefficient α is extracted, using (2.23), from the static
monopole piece of the metric perturbation, whose construction is prescribed in App. B of [48]. Since this piece is
essentially known analytically (its computation involves
the evaluation of a certain orbital integral, easily done with
MATHEMATICA at extremely high accuracy), numerical
124014-6
COMPARISON BETWEEN SELF-FORCE AND POST- …
PHYSICAL REVIEW D 91, 124014 (2015)
error in our calculation of hUigsf comes entirely from the
numerical evaluation of ĥRuu ðχÞ. Reference [48] contains a
thorough analysis of error sources for ĥRuu ðχÞ and the GSF.
Here, we briefly review two dominant sources.
Each Fourier mode of our computation has associated
~ r þ mΩ
~ ϕ , where n~ and m
~ are
with it a frequency, ω ¼ nΩ
integer harmonic numbers. The dominating source of
numerical error depends on the value of ω. For modes
of sufficiently large frequency (m2 ω ≳ 10−4 ), the dominant
error comes from the estimation of the contribution from
the tail of uncomputed multipoles of large l values.
Typically, we compute the contributions to all the l-modes
up to and including l ¼ 15, and estimate the remaining
contribution to the mode sum by fitting numerical
data to suitable power-law models of the large-l behavior
[23,48,55]. This is a relatively well-modeled and
well-controlled source of error, and it can be reduced in
a straightforward manner using additional computational
resources.
For modes with small frequencies, m2 ω ≲ 10−4 , a
second source of numerical error takes over. This comes
from rounding errors introduced when inverting the
matrix of amplitude coefficients as part of the procedure
for computing inhomogeneous solutions to the Lorenzgauge field equations [48]. When ω is very small, the
matrix becomes nearly singular, and its inversion using
machine-precision arithmetic introduces large errors. The
problematic “nearly-static” modes occur generically in our
TABLE II. Numerical data for the GSF contribution hUigsf to the generalized redshift (defined with fixed invariant frequencies Ωi ), for
various eccentric geodesic orbits in a Schwarzschild background. The orbital parameters e (eccentricity) and p (semilatus rectum) are
defined in Sec. II A. Parenthetical figures indicate estimated error bars on the last displayed decimals.
e
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
e
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
e
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
e
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
p ¼ 10
−0.12878023ð4Þ
−0.1277540ð3Þ
−0.1260434ð2Þ
−0.123648ð3Þ
−0.120567ð2Þ
−0.1168020ð6Þ
−0.112352ð2Þ
−0.107221ð2Þ
p ¼ 15
−0.07751154ð5Þ
−0.0768706ð2Þ
−0.07580395ð6Þ
−0.07431376ð9Þ
−0.07240329ð5Þ
−0.0700768ð5Þ
−0.0673398ð5Þ
−0.064199ð1Þ
p ¼ 20
p ¼ 25
−0.0556761252ð8Þ
−0.05522166ð7Þ
−0.05446527ð1Þ
−0.05340854ð9Þ
−0.0520537ð3Þ
−0.05040377ð4Þ
−0.0484623ð4Þ
−0.0462337ð9Þ
−0.0434829334ð1Þ
−0.043132423ð1Þ
−0.042548963ð6Þ
−0.041733648ð3Þ
−0.04068802ð6Þ
−0.0394141ð2Þ
−0.0379143ð1Þ
−0.0361916ð2Þ
p ¼ 30
p ¼ 35
p ¼ 40
p ¼ 50
−0.0356833158ð1Þ
−0.035398479ð5Þ
−0.03492427ð6Þ
−0.034261480ð8Þ
−0.03341121ð1Þ
−0.0323749ð2Þ
−0.0311543ð2Þ
−0.0297515ð6Þ
−0.0302606957ð1Þ
−0.0300209627ð8Þ
−0.029621799ð3Þ
−0.029063791ð5Þ
−0.028347763ð7Þ
−0.0274748ð4Þ
−0.0264462ð2Þ
−0.0252634ð2Þ
−0.0262706836ð5Þ
−0.0260637929ð8Þ
−0.025719277ð2Þ
−0.025237591ð2Þ
−0.024619373ð3Þ
−0.02386545ð8Þ
−0.0229768ð4Þ
−0.0219547ð5Þ
−0.0207905297ð5Þ
−0.0206282073ð6Þ
−0.0203578673ð9Þ
−0.019979802ð2Þ
−0.01949443ð6Þ
−0.01890227ð5Þ
−0.0182040ð1Þ
−0.0174004ð4Þ
p ¼ 60
p ¼ 70
p ¼ 80
p ¼ 90
−0.0172030750ð2Þ
−0.0170695612ð3Þ
−0.016847175ð1Þ
−0.016536122ð4Þ
−0.01613669ð1Þ
−0.01564925ð2Þ
−0.01507427ð8Þ
−0.0144123ð3Þ
−0.0146718447ð3Þ
−0.0145584700ð2Þ
−0.0143696130ð3Þ
−0.0141054255ð4Þ
−0.0137661204ð6Þ
−0.013351971ð9Þ
−0.0128633ð1Þ
−0.0123002ð5Þ
−0.0127901170ð3Þ
−0.0126916092ð2Þ
−0.0125275072ð7Þ
−0.0122979274ð7Þ
−0.012003033ð2Þ
−0.011643033ð7Þ
−0.01121819ð3Þ
−0.0107288ð2Þ
−0.0113362814ð8Þ
−0.0112491974ð8Þ
−0.0111041186ð3Þ
−0.0109011371ð5Þ
−0.010640382ð2Þ
−0.010322020ð3Þ
−0.00994625ð3Þ
−0.0095133ð2Þ
p ¼ 100
p ¼ 110
p ¼ 120
p ¼ 130
−0.0101792669ð2Þ
−0.0101012344ð10Þ
−0.0099712296ð7Þ
−0.0097893274ð4Þ
−0.0095556341ð8Þ
−0.009270281ð4Þ
−0.00893344ð2Þ
−0.0085453ð2Þ
−0.0092365822ð2Þ
−0.0091658975ð1Þ
−0.0090481316ð3Þ
−0.0088833462ð1Þ
−0.0086716263ð10Þ
−0.008413085ð2Þ
−0.00810786ð1Þ
−0.0077561ð2Þ
−0.0084537150ð1Þ
−0.0083891149ð2Þ
−0.0082814820ð1Þ
−0.0081308692ð1Þ
−0.0079373488ð7Þ
−0.007701017ð5Þ
−0.00742198ð3Þ
−0.0071004ð2Þ
−0.0077931956ð9Þ
−0.0077337155ð1Þ
−0.0076346120ð1Þ
−0.0074959280ð1Þ
−0.0073177296ð5Þ
−0.007100089ð2Þ
−0.00684311ð3Þ
−0.0065469ð2Þ
124014-7
AKCAY et al.
PHYSICAL REVIEW D 91, 124014 (2015)
calculation, since, given any orbital parameters, there will
~ and n~ in the Fourier sum for which m2 ω is
exist values of m
~ is truncated
very small. In practice, the sum over n~ and m
once our results reach a desired accuracy. Consequently, the
problem is less severe for low-eccentricity orbits, where the
effective frequency band is narrow, and more severe at high
eccentricity, where the broad frequency band implies a
higher chance of encountering nearly static modes.
Ultimately, this restricts our calculation to orbits with
eccentricities of e ≲ 0.4. Low-ω modes are encountered
also when the fundamental frequencies themselves are
small, as with weak-field orbits—the main focus of the
present work. Our code incorporates several methods
for mitigating this small-frequency problem (see
Ref. [48] for details), but even with these techniques
employed, our current calculation appears limited to orbits
with p ≲ 130; at larger p we observe a rapid reduction
in accuracy.
The issue of nearly static modes has been addressed in a
very recent paper by Osburn et al. [56], who proposed
additional mitigation methods. These may be used to
improve the performance of weak-field calculations in
future work.
B. Numerical results
Table II displays a sample of our numerical results
for hUigsf . Parenthetical figures indicate estimated error
bars on the last displayed decimals; for instance,
−0.0556761ð1Þ stands for −0.0556761 1 × 10−7.
Additional data may be made available to interested readers
upon request from the authors. Some of the data shown in
the table will be plotted in Sec. V, where we discuss the
comparison with PN results.
As a check of our frequency-domain computation, we
compare our results for hUigsf with those obtained in
FIG. 1 (color online). Numerical GSF output for hUigsf (black data points) versus analytical PN approximations (solid curves). Each
panel shows hUigsf as a function of semilatus rectum p for a fixed eccentricity e. Insets display, on a log-log scale, the relative differences
ΔnPN
rel ≡ j1 − U nPN =hUigsf j, where U nPN is the PN approximation through nPN order. In both the main plots and the insets, the three
curves correspond, top to bottom, to the 1PN, 2PN and 3PN approximations. Solid curves in the insets are the analytical PN residues
1 − U 1PN =U 3PN (upper curve) and 1 − U 2PN =U 3PN (middle curve); for the lower curve we have fitted the simple model
1 − U 3PN =hUigsf ¼ p−4 ðα1 þ α2 ln p þ α3 =pÞ.
124014-8
COMPARISON BETWEEN SELF-FORCE AND POST- …
PHYSICAL REVIEW D 91, 124014 (2015)
TABLE III. Our frequency-domain numerical results for hUigsf
and the corresponding time-domain values from BS2011. Each
cell shows our result (top) in comparison to BS2011’s (bottom).
The relative disagreement between the two data sets is ∼10−6 ,
roughly consistent with the magnitude of error bars. As discussed
in the text, evidence suggests that our results are accurate to
within the error bars given, whereas the magnitude of error in the
time-domain data is slightly underestimated in some cases. No
time-domain data exist for p > 20 to allow comparison in the
weak-field domain.
e
p ¼ 10
p ¼ 15
p ¼ 20
0.1 Here
−0.1277540ð3Þ −0.0768706ð2Þ −0.05522166ð7Þ
BS2011 −0.1277554ð7Þ −0.0768709ð1Þ −0.05522177ð4Þ
0.2
−0.123648ð3Þ −0.07431376ð9Þ −0.05340854ð9Þ
−0.1236493ð7Þ −0.0743140ð1Þ −0.05340866ð4Þ
0.3
−0.1168020ð6Þ −0.0700768ð5Þ −0.05040377ð4Þ
−0.1168034ð6Þ −0.0700771ð1Þ −0.05040388ð4Þ
0.4
−0.107221ð2Þ −0.064199ð1Þ −0.0462337ð9Þ
−0.1072221ð5Þ −0.0641991ð1Þ −0.04623383ð4Þ
BS2011 using a time-domain method. BS2011 provided a
small sample of numerical results in the range p ≤ 20 and
e ≤ 0.5. The comparison is shown in Table III. There is
evidently a good agreement between the two sets of
numerical results, although in some of the entries the
values appear not fully consistent given the stated error bars
(in all these cases the BS2011 values are smaller than ours).
We have strong evidence to suggest that the source of
disagreement is a slight underestimation of the magnitude
of systematic error in the time-domain analysis of BS2011:
We have tested the output of our frequency-domain code
against accurate GSF data published in Ref. [56], and
against yet unpublished redshift data calculated by
van de Meent [57] (using a very different frequencydomain method based on a semianalytical treatment of
Teukolsky’s equation [58]). These comparisons strongly
favor the frequency-domain data in the table. We have also
gone back to the time-domain code of BS2011 and
implemented more robust error analysis methods, again
suggesting that the error reported in BS2011 is slightly
underestimated in some cases.
Also evident from the table is the fact that our code’s
accuracy starts to degrade for e ¼ 0.4; however, it still
matches BS2011’s results to five or six significant digits.
No published numerical data exist to allow comparison
beyond p ¼ 20. (Reference [56] gives results for e ≤ 0.7
and p ≤ 90, but these are for the GSF components, not
for hUigsf .)
except that we will consider the orbital average of a
quantity that is related to the orbital dynamics of a binary
of nonspinning compact objects, modeled as point particles, while Refs. [59,60] calculated the orbital-averaged
fluxes of energy and angular momentum radiated at
infinity. Furthermore, while these fluxes are invariant under
the exchange 1↔2 of the bodies’ labels, and requires
knowledge of the gravitational field in the wave zone, the
generalized redshift hUi is a property of one particle, whose
evaluation involves the near-zone metric.
A. Redshift variable in standard harmonic
coordinates
1. The regularized 3PN metric
Throughout Sec. IV we assume that m1 < m2 , and
treat m1 as the “particle” orbiting the “black hole” of
mass m2 . The redshift of the particle can be computed
from the knowledge of the regularized PN metric
gαβ ðy1 Þ ≡ gαβ ðt; y 1 Þ, generated by the two bodies and
evaluated at the coordinate location y 1 ðtÞ of the particle,
as [16]
U≡
We shall now derive the invariant relation hUiðΩr ; Ωϕ Þ
within the PN approximation. Our calculations will be
similar in spirit to those performed by Arun et al. [59,60],
vα1 vβ1 −1=2
¼ −gαβ ðy1 Þ 2
;
c
ð4:1Þ
where vα1 ¼ ðc; v1 Þ, with v1 ¼ dy 1 =dt the coordinate
velocity of the particle. The generalized redshift will be
given by the proper-time average of Eq. (4.1) over one
radial period.
The regularized PN metric gαβ ðy1 Þ was itself computed
up to 2.5PN order, in harmonic coordinates, in Ref. [61].
This calculation was then extended to 3PN order in
Ref. [16], partly based on existing computations of the
3PN equations of motion using Hadamard regularization
[62] and dimensional regularization [63]. Reference [16]
performed two calculations of the 3PN regularized metric,
using both Hadamard and dimensional regularizations,
obtaining the same metric but expressed in two different
harmonic coordinate systems. The two metrics were found
to differ by an infinitesimal 3PN coordinate transformation
in the “bulk,” i.e., outside the particle’s worldlines, and also
by an intrinsic shift of these worldlines. Combining
Eqs. (4.2) and (A15) of Ref. [16], the 3PN-accurate
expression of the regularized metric reads, in the standard
harmonic coordinates corresponding to the use of
Hadamard regularization,1
As usual we denote by r12 ¼ jy 1 − y 2 j the coordinate separation, by n12 ¼ ðy 1 − y 2 Þ=r12 the unit direction from particle 2
to particle 1, and by v12 ¼ v1 − v2 the relative velocity, where
va ¼ dy a =dt is the 3-velocity of particle a. The Euclidean scalar
product between two 3-vectors A and B is denoted ðABÞ.
Parentheses around indices are used to indicate symmetrization,
i.e., Aði BjÞ ¼ Ai Bj þ Aj Bi .
1
IV. GENERALIZED REDSHIFT:
POST-NEWTONIAN CALCULATION
ut1
124014-9
AKCAY et al.
PHYSICAL REVIEW D 91, 124014 (2015)
2Gm2 Gm2
Gm1
Gm2
Gm2 3
2
2
ðn12 v2 Þ4 − 3ðn12 v2 Þ2 v22 þ 4v42
4v2 − ðn12 v2 Þ − 3
þ 6
g00 ðy1 Þ ¼ −1 þ 2
þ
−2
r12
r12
c r12 c4 r12
c r12 4
Gm2
Gm2
Gm1
87
47
55
39
23
þ
3ðn12 v2 Þ2 − v22 þ 2
þ
− ðn12 v1 Þ2 þ ðn12 v1 Þðn12 v2 Þ − ðn12 v2 Þ2 − ðv1 v2 Þ þ v21
4
2
4
2
4
r12
r12
r12
47
Gm1 17 Gm2
Gm
5
Gm2
þ 8 2 − ðn12 v2 Þ6 − 5ðn12 v2 Þ2 v42 þ 3ðn12 v2 Þ4 v22 þ 4v62 þ
þ v22 −
þ
ð−4ðn12 v2 Þ4
4
2 r12
r12
r12
c r12 8
Gm1
617
491
225
41
ðn v Þ4 þ
ðn12 v1 Þ3 ðn12 v2 Þ −
ðn12 v1 Þ2 ðn12 v2 Þ2 þ ðn12 v1 Þðn12 v2 Þ3
−
þ 5ðn12 v2 Þ2 v22 − v42 Þ þ
24 12 1
6
4
2
r12
53
79
101
49
27
ðn12 v2 Þ2 ðv1 v2 Þ þ ðv1 v2 Þ2 − ðn12 v1 Þ2 v21
þ ðn12 v2 Þ4 − ðn12 v1 Þ2 ðv1 v2 Þ þ 42ðn12 v1 Þðn12 v2 Þðv1 v2 Þ þ
8
4
4
4
8
23
273
39
305
139
ðn12 v2 Þ2 v21 − 25ðv1 v2 Þv21 þ v41 −
ðn12 v1 Þ2 v22 þ
ðn12 v1 Þðn12 v2 Þv22
þ ðn12 v1 Þðn12 v2 Þv21 −
4
8
8
8
4
291
77
235 4
G2 m 2
r
32
ðn12 v2 Þ2 v22 − 62ðv1 v2 Þv22 þ v21 v22 þ
v2 þ 2 1 ln 12
32ðn12 v12 Þ2 − v212
−
8
2
8
3
r0
r12
r12
6603 584
r12
1881
r
2 2789 364
2
þ ðn12 v1 Þ
−
ln 0
þ
ln 0
− 44 ln 12
þ ðn12 v1 Þðn12 v2 Þ
þ ðn12 v2 Þ −
100
5
50
5
20
r1
r1
r01
3053 584
r
1613 364
r
1421 44
r
þ ðv1 v2 Þ −
−
ln 12
þ
ln 12
þ ln 12
þ v21 −
þ v22
0
0
150 15
100
15
60
3
r1
r1
r01
Gm1
1091 44
r
32
r12
G2 m1 m2
r12
2 − 32 v2
− ln 12
ln
þ
−
þ
þ
ln
32ðn
v
Þ
12 12
60
3
3
3 12
r12
r01
r0
r0
r212
109 141 2
197 177 2
2
2 391 213 2
þ ðn12 v1 Þ −
−
π þ ðn12 v1 Þðn12 v2 Þ −
þ
π þ ðn12 v2 Þ
−
π
12 16
6
8
6
16
812 59 2
299 47 2
1097 71 2
Gm1
3571 15 2 44
r
2
2
þ ðv1 v2 Þ
− π þ v1 −
þ π þ v2 −
þ π þ
− π − ln 12
−
9
8
18 16
18
16
60
8
3
r12
r01
64
r
Gm2
28141 15 2 44
r
32
r12
G2 m22
Gm2
2
2
− π þ ln 12
ln
þ ln 12
þ
−
þ
þ
−ðn
v
Þ
þ
2v
−
2
12 2
2
3
630
8
3
3
r0
r12
r02
r0
r12
r212
þ oðc−8 Þ;
ð4:2aÞ
4Gm2 i Gm2 i
Gm1 Gm2
Gm1 i
Gm1
2
2
i
v
2ðn
þ
4
v
þ
v
Þ
−
4v
−
2
þ
v
þ
n
f10ðn12 v1 Þ þ 2ðn12 v2 Þg
12 2
2
12
r12
r12
r12 1
r12
c3 r12 2 c5 r12 2
Gm2
Gm2 i
3
Gm1
4
2 2
4
−
v − ðn12 v2 Þ þ 4ðn12 v2 Þ v2 − 4v2 þ
48ðn12 v1 Þ2 − 44ðn12 v1 Þðn12 v2 Þ
ðn v Þ þ 7
2
r12 12 2
r12
c r12 2
Gm2
Gm2
G2 m1 m2 95 3 2
2
2
2
2
2
þ 10ðn12 v2 Þ þ 40ðv1 v2 Þ − 16v2 − 26v1 þ
− π
−2ðn12 v2 Þ þ v2 − 2
þ
6 4
r12
r12
r212
G2 m2
1013 24
r
17
43
i Gm1
þ ln 12
þ 2 1 −
þ
v
− ðn12 v1 Þ2 − 15ðn12 v1 Þðn12 v2 Þ þ ðn12 v2 Þ2 þ 3ðv1 v2 Þ
1
75
5
2
2
r01
r12
r12
2
2
2
17 2 15 2
G m1 m2
57 3 2
G m1 1973 24
r12
Gm1 21
i
− ln 0
ðn v Þ3
− þ π þ 2
þ n12
þ v1 − v2 þ
2
2
2 4
75
5
2 12 1
r1
r12
r212
r12
43
29
3
1
− ðn12 v1 Þ2 ðn12 v2 Þ − ðn12 v1 Þðn12 v2 Þ2 þ ðn12 v2 Þ3 − 11ðn12 v1 Þðv1 v2 Þ − 19ðn12 v2 Þðv1 v2 Þ þ ðn12 v1 Þv21
2
2
2
2
2
39
41
3
Gm2
Gm2
G m1 m2 51
2
2
2
2
2
þ
ðn v Þ
þ ðn12 v2 Þv1 þ ðn12 v1 Þv2 þ ðn12 v2 Þv2 þ
ðn v Þ 2ðn12 v2 Þ − v2 − 2
2
2
2
2 12 1
r12 12 2
r12
r212
97
9
G2 m2
2398
1088
72
r
þ oðc−7 Þ;
ðn12 v1 Þ þ
ðn12 v2 Þ þ ðn12 v12 Þ ln 12
− ðn12 v2 Þ − π 2 ðn12 v12 Þ þ 2 1 −
2
4
25
25
5
r01
r12
g0i ðy1 Þ ¼ −
ð4:2bÞ
124014-10
COMPARISON BETWEEN SELF-FORCE AND POST- …
PHYSICAL REVIEW D 91, 124014 (2015)
2Gm
Gm
Gm1 Gm2
Gm1 Gm2
þ 4vi2 vj2 þ ni12 nj12 −8
gij ðy1 Þ ¼ δij þ 2 2 δij þ 4 2 δij −ðn12 v2 Þ2 þ
þ
þ
r12
r12
r12
r12
c r12
c r12
Gm2 ij 3
Gm2
Gm1
71
47 2 59 Gm1 25 Gm2
4
2
2
2
2
þ 6
δ
− ðn12 v12 Þ þ v12 −
ðn v Þ − v2 ðn12 v2 Þ −
ðn v Þ þ
þ
4 12 2
4
4
5 r12
6 r12
r12 12 2
r12
c r12
Gm1 Gm2
Gm1 ði jÞ
Gm1 i j
Gm1
þ vi2 vj2 4v22 − 2ðn12 v2 Þ2 − 10
þ 24
−16ðn12 v1 Þ2
−
v v − 16
v v þ ni12 nj12
r12
r12
r12 1 2
r12 1 1
r12
182 Gm1
Gm2
Gm1
Gm2
Gm1
ði jÞ
2
2
þ
−2ðn12 v2 Þ þ 3
þ 40
þ 32ðn12 v1 Þðn12 v2 Þ − 12v12 þ
þ2
ðn12 v12 Þn12 v1
5 r12
r12
r12
r12
r12
Gm2
ði jÞ Gm1
þ n12 v2
f−60ðn12 v1 Þ þ 36ðn12 v2 Þg þ 2
ðn v Þ þ oðc−6 Þ:
ð4:2cÞ
r12
r12 12 2
Since we are considering only the conservative part of
the binary dynamics, we did not include in (4.2) the
dissipative 2.5PN radiation-reaction terms; these can be
found in Eqs. (7.6) of Ref. [61]. Notice the occurrence at
3PN order of some logarithmic terms, containing two
constants r01 and r02 (one for each body) that have the
dimension of a length. These ultraviolet (UV) regularization parameters come from regularizing the self-field of
point particles using the Hadamard regularization of
Ref. [62]. The constants r01 and r02 are gauge-dependent,
as they can be arbitrarily changed by a coordinate transformation of the bulk metric [16,62] or by some shifts of the
worldlines of the particles [63]. The metric coefficient
g00 ðy1 Þ also involves a constant r0 that originates from the
infrared (IR) regularization of the metric at spatial infinity,
as discussed in Ref. [16]. This arbitrary IR scale should also
disappear from the final gauge-invariant results.
We introduce the expression (4.2) of the regularized 3PN
metric into the definition (4.1) of the redshift, and expand in
powers of 1=c, keeping all terms up to Oðc−8 Þ. This gives
an expression for U as a function of the two masses m1 and
m2 , the coordinate separation r12 , and the scalar products
ðn12 v1 Þ, ðn12 v2 Þ, ðv1 v1 Þ, ðv1 v2 Þ and ðv2 v2 Þ, as well as the
regularization constants r0 , r01 and r02 , in an arbitrary
reference frame. The resulting expression is too lengthy
to be displayed here.
2. Reduction to the center-of-mass frame
We wish to specialize the previous expression to the
center-of-mass (c.m.) frame, which is consistently defined at
3PN order by the vanishing of the center-of-mass integral
deduced from the 3PN binary equations of motion [64]. This
condition yields expressions for the individual positions y a
and velocities va relatively to the c.m. in terms of the relative
position y ≡ y 1 − y 2 and relative velocity v ≡ v1 − v2 [65].
Since these results play an important role in our algebraic
manipulations, we recall here the expressions for the functional relationships y a ½y; v in the harmonic gauge that was
used to derive the regularized metric (4.2). Thus,
y 1 ¼ ½X2 þ νðX1 − X2 ÞPy þ νðX1 − X2 ÞQv þ oðc−6 Þ;
ð4:3aÞ
y 2 ¼ ½−X1 þ νðX1 − X2 ÞPy þ νðX1 − X2 ÞQv þ oðc−6 Þ;
ð4:3bÞ
where ν ≡ m1 m2 =m2 is the symmetric mass ratio and
Xa ≡ ma =m, with m ≡ m1 þ m2 the total mass of the binary.
The coefficients P and Q depend on the parameters m and ν,
the separation r ≡ jyj, the relative velocity squared
v2 ≡ ðvvÞ, and the radial velocity r_ ≡ ðnvÞ. They explicitly
read [65]
1 v2 Gm
1
Gm 2 19 3
Gm 2
1 3
G2 m 2 7 ν
4 3 3
_
þ
−
ν
þ
v
þ
ν
þ
r
þ
ν
þ
−
v
−
−
2r
8 2
r
8 2
r
8 4
4 2
c2 2
c4
r2
1
5 11
Gm 4 53
15 2
Gm 4 1 5
21 2
Gm 2 2
5 21
11 2
6
2
þ 6 v
− ν þ 6ν þ
v
− 7ν − ν þ
r_
− νþ ν þ
v r_ − þ ν − ν
16 4
r
16
2
r
16 8
16
r
16 16
2
c
2
2
2
2
3
3
2
Gm
101 33
Gm
7 73
Gm
14351 22
r
ν ν
þ 2 v2
−
− ν þ 3ν2 þ 2 r_ 2 − þ ν þ 4ν2 þ 3
þ ln 00 þ −
;
ð4:4aÞ
12
8
3 8
1260
3
r0
8 2
r
r
r
P¼
Q¼−
7 Gm_r Gm_r 2
15 21
19
Gm 235 21
2 5
_
þ
ν
þ
r
−
ν
−
þ
ν
:
v
−
þ
4 c4
8
4
12 24
r
24
4
c6
ð4:4bÞ
Again, we did not include the radiation-reaction contributions at 2.5PN order. A logarithmic term contributes at 3PN order
in (4.4a); it involves a particular combination r000 of the gauge constants r01 and r02 , that is defined by
124014-11
AKCAY et al.
PHYSICAL REVIEW D 91, 124014 (2015)
ðX1 − X2 Þ ln r000 ¼ X21 ln r01 − X22 ln r02 :
ð4:5Þ
By computing the time derivatives of Eqs. (4.3)–(4.4)
and by applying, where necessary, an iterative orderreduction of all accelerations by means of the c.m.
equations of motion given in Eqs. (3.9)–(3.10) of
Ref. [65], we obtain expressions analogous to (4.3)–(4.4)
for the particle’s individual velocities va as functions of the
relative variables y and v. Replacing the positions and
velocities by their c.m. expressions y a ½y; v and va ½y; v
yields 3PN-accurate expressions for the scalar products
ðn12 v1 Þ, ðn12 v2 Þ, ðv1 v1 Þ, ðv1 v2 Þ, ðv2 v2 Þ as functions of r, r_
and v2 . Finally, the c.m. expression for the redshift
U½r; r_ ; v2 in standard harmonic coordinates takes the form
U ¼1þ
1
1
1
1
U þ 4 U 1PN þ 6 U 2PN þ 8 U 3PN þ oðc−8 Þ;
2 N
c
c
c
c
ð4:6Þ
where the various PN contributions read
1 Δ ν 2
1 Δ Gm
UN ¼
þ − v þ
þ
;
4 4 2
2 2
r
2 2
3
3
5
11 2 4
5 5
ν 2 Gm 2
Δ
Gm 2
1 Δ
Gm
þ Δ − ν − Δν þ ν v þ þ Δ − − ν
v þ
−ν ν
r_ þ þ − 2ν
;
U1PN ¼
16 16
8
8
4 4
2
r
2
r
4 4
r2
ð4:7aÞ
ð4:7bÞ
5 5
3 19
157 2 75 2 85 3 6
27 27
37 13
5
19
11
Gm 4
þ Δ − ν − Δν þ
ν þ Δν − ν v þ
þ Δ − ν − Δν þ ν2 − Δν2 þ ν3
v
32 32
2 16
32
32
16
16 16
8
4
16
16
2
r
5 7
11 23
Gm 2 2
23 23
17
59 2 1 2 3 3 G2 m2 2
2
r_ v þ
þ Δ − 2ν − Δν þ ν − Δν − ν
þ − þ Δ − ν − Δν þ 6ν ν
v
8 8
8
8
r
8 8
8
8
4
2
r2
1 Δ
19
55 2 5 2 9 3 G2 m2 2
3
9
9
3 2 Gm 4
r_
þ þ − 5ν þ Δν − ν þ Δν − ν
r_ þ − Δ þ ν þ Δν − ν ν
4 4
8
8
4
2
8
16
16
2
r
r2
3 3
1 Δ 9 1
G m
þ þ − ν − Δν
;
ð4:7cÞ
4 4 8 8
r3
U2PN ¼
35 35
221
705 2 939 2 855 3 665 3 3059 4 8
65 65
49 163
þ
Δ − 2ν −
Δν þ
ν þ
Δν −
ν −
Δν þ
ν v þ
þ Δ− ν−
Δν
U3PN ¼
256 256
128
64
128
32
64
128
32 32
4
16
169 2 45 2 57 3 145 3 255 4 Gm 6
61 19
131 229
251 2 67 2 289 3
þ
ν þ Δν þ ν þ
Δν −
ν
v þ − þ Δþ
ν−
Δν þ
ν þ Δν −
ν
8
4
8
16
8
r
32 32
32
32
16
4
8
2 2
Gm 2 4
231 231
19 39
297 2 129 2 597 3
55
Gm 4
15 33
9
×ν
r_ v þ
þ
Δ − ν − Δν þ
ν þ
Δν −
ν − Δν3 þ ν4
− Δ þ Δν
v
þ
r
32 32
2
4
16
16
16
4
32 32
2
r2
87
111 3 Gm 4 2
3 3
323
283
667 2 617 2 725 3 27 3 161 4
− 9ν2 − Δν2 þ
ν ν
r_ v þ þ Δ −
νþ
Δν −
ν −
Δν þ
ν − Δν þ
ν
16
8
r
8 8
32
32
32
32
16
2
4
G2 m 2 2 2
35 35
39731 71 2
23761 71 2
64933 2 35 2
1
× 2 r_ v þ
þ Δþ −
þ π νþ −
þ π Δν −
ν þ Δν þ12ν3 − Δν3 − 2ν4
8 8
2520 64
1260 64
5040
16
2
r
3 3
16
r
11
r
11
r
G m 2
5
15 35
þ ð1 − Δ þ ν − ΔνÞνln 0 þ ð−1 − Δ þ 3ν þ ΔνÞνln 0
− νln
v þ
Δ − ν − Δν
3
r0
3
r1
3
r2
16
32 32
r3
2 2
35
5
15
Gm 6
425 73
177
Δν 2
3
Gm
þ ν2 þ Δν2 − ν3 ν
r_ þ −
− Δþ
ν þ − ν2 þ Δν2 − 9ν3 ν 2 r_ 4
16
8
8
r
48 6
16
8 3
2
r
5 5
17917 213 2
5767 213 2
122333 2 7
r
2
3 5
3
4
þ
π νþ
−
π Δν þ
ν þ Δν −16ν þ Δν − 14ν þ 16νln
þ þ Δ−
4 4
420
64
840 64
1680
16
2
r0
3 3
r
r
Gm 2
3 3
67853 15 2
_
þ11ð−1 þ Δ − ν þ ΔνÞνln 0 þ11ð1 þ Δ − 3ν − ΔνÞνln 0
þ
Δ
−
þ
π
ν
r
þ
r1
r2
16 16
5040 32
r3
4 4
20141 15 2
656 2 16
r
11
r
11
r
G m
þ π Δν −
ν þ νln
−
þ ð−1 þ ΔÞνln 0 þ ð1 þ Δ − 2νÞνln 0
:
ð4:7dÞ
5040 32
315
3
r0
3
r1
3
r2
r4
124014-12
COMPARISON BETWEEN SELF-FORCE AND POST- …
pffiffiffiffiffiffiffiffiffiffiffiffiffi
Here, Δ ≡ ðm2 − m1 Þ=m ¼ X2 − X1 ¼ 1 − 4ν denotes
the reduced mass difference, so that the test-mass limit of
particle 1 corresponds to ν → 0. Since the redshift (4.1) is a
property of particle 1, the expressions (4.7) are not symmetric
by exchange 1↔2 of the bodies’ labels. The redshift of
particle 2 is simply obtained by setting Δ → −Δ in Eqs. (4.7).
As expected, the regularization constants r0 , r01 and r02 that
enter the expression (4.2) of the regularized 3PN metric
appear in the c.m. expression (4.6)–(4.7) for the redshift. In
Sec. IV D we will check that the orbital averaging cancels out
the dependance on these arbitrary length scales.
PHYSICAL REVIEW D 91, 124014 (2015)
U½r; r_ ; v2 in SH coordinates by means of the functional
equality U0 ¼ U þ δξ U, where
δξ U ¼ −
∂U
∂U
∂U
δξ r −
δξ r_ − 2 δξ v2 þ Oðξ2 Þ;
∂r
∂ r_
∂v
with
δξ r ¼ ðnξÞ þ Oðξ2 Þ;
_ þ ðvξÞ − r_ ðnξÞ þ Oðξ2 Þ;
δξ r_ ¼ ðnξÞ
r
r
B. Redshift variable in alternative coordinates
In the previous section we obtained an expression for the
redshift variable in the standard harmonic (SH) coordinate
system, namely, the coordinate system in which the 3PN
equations of motion were originally derived [62,65]. These
coordinates are such that the equations of motion involve
some gauge-dependent logarithmic terms at 3PN order.
Importantly, these logarithms prevent the use of the 3PN
quasi-Keplerian representation of the binary motion
(reviewed in Sec. IV C below), thus impeding the averaging
of the redshift over an orbit. Therefore, it is useful to have
the expression for the redshift in a modified harmonic (MH)
coordinate system, without logarithmic terms in the equations of motion, such as the one used in Refs. [59,60].
Alternatively, we shall use ADM-type coordinates, which
are also free of such logarithms at 3PN order in the
equations of motion. Both the MH coordinates and the
ADM coordinates are suitable for a 3PN quasi-Keplerian
parametrization of the motion [66]. This will require us to
re-express the redshift in terms of the variables r, r_ and v2
in these alternative coordinate systems.
1. Modified harmonic coordinates
The trajectories y 0a ðtÞ of the particles in MH coordinates
are related to their counterparts y a ðtÞ in SH coordinates by
some 3PN shifts ξa ðtÞ of the worldlines induced by a
coordinate transformation in the “bulk,” namely, y 0a ¼ y a þ
ξa [62]. Therefore, in the c.m. frame, the MH coordinate
separation y 0 is related to the SH coordinate separation y
through y 0 ¼ y þ ξ, where the relative shift ξ ≡ ξ1 − ξ2 is
given by [59]
22 G3 m3 ν
r
ξðSH→MHÞ ¼ −
ln 0 n þ oðc−6 Þ;
ð4:8Þ
6
2
3 cr
r0
with n ≡ y=r the unit direction pointing from particle 2 to
particle 1. Following [59], we introduced the “logarithmic
barycenter” r00 of the constants r01 and r02 , (not to be
confused with the IR constant r0 ):
ln r00 ≡ X1 ln r01 þ X2 ln r02 :
ð4:9Þ
The expression U 0 ½r; r_ ; v2 for the redshift in MH
coordinates can then be deduced from the formula for
ð4:10Þ
_ þ Oðξ2 Þ:
δξ v2 ¼ 2ðvξÞ
ð4:11aÞ
ð4:11bÞ
ð4:11cÞ
Since the relative shift (4.8) comes at 3PN order, the
nonlinear terms Oðξ2 Þ in Eqs. (4.10) and (4.11) contribute
at leading 6PN order, and can thus be neglected. Plugging
the expression (4.8) into Eqs. (4.11), we find the explicit
expressions
22 G3 m3 ν
r
ðδξ rÞðSH→MHÞ ¼ −
ð4:12aÞ
ln 0 ;
6
2
3 cr
r0
22 G3 m3 ν
r
_
ðδξ r_ ÞðSH→MHÞ ¼ −
r
−
2_
r
ln
; ð4:12bÞ
6
3
3 cr
r00
ðδξ v2 ÞðSH→MHÞ
44 G3 m3 ν 2
r
2
2
r_ þ ðv − 3_r Þ ln 0
:
¼−
6
3
3 cr
r0
ð4:12cÞ
In order to compute the change δξ U in the redshift induced
by the relative shift (4.8), we only require the Newtonian
expression for U½r; r_ ; v2 , which is given by Eq. (4.7).
Combined with (4.10) and (4.12), this gives
11 G3 m3 ν
ð1 þ Δ − 2νÞ_r2
ðδξ UÞðSH→MHÞ ¼
3 c 8 r3
2Gm
r
2
2
ν ln 0
þ ð1 þ ΔÞðv − 3_r Þ −
r
r1
þ ð1 þ Δ − 3ν − ΔνÞðv2 − 3_r2 Þ
Gm
r
−
þ oðc−8 Þ:
ð1 þ Δ − 2νÞ ln 0
r
r2
ð4:13Þ
Adding the above shift to the formula (4.6)–(4.7) for the
redshift in SH coordinates yields the expression for the
redshift in MH coordinates. Since U 0 ¼ U þ δξ U is a
functional equality, the resulting MH redshift is expressed
as a function of the “dummy” variables r, r_ and v2 .
Adding together Eqs. (4.7d) and (4.13), we find that the
UV regularization constant r02 disappears from the
124014-13
AKCAY et al.
PHYSICAL REVIEW D 91, 124014 (2015)
expression for U0 ½r; r_ ; v2 in MH coordinates. However the
UV and IR constants r01 and r0 remain and enter the result
through the logarithmic contributions
11 G3 m3 ν 2
2 Gm
v
−
3_
r
−
½U0 log ¼
3 c 8 r3
r
r
16
r
× ð1 − Δ þ 2νÞ ln 0 − ln
:
ð4:14Þ
r1
11
r0
For circular orbits, such that r_ ¼ 0 and v2 ¼ Gm=rþ
Oðc−2 Þ, these logarithmic contributions cancel out. We will
see that the factor ðv2 − 3_r2 − Gm=rÞ=r3 vanishes when
ξðSH→ADMÞ ¼
averaged over one radial period, such that the constants r0
and r01 will cancel out from the final, gauge-invariant result
for the orbital-averaged redshift, as expected.
2. ADM-type coordinates
Similarly, the individual trajectories y 0a ðtÞ of the particles
in ADM coordinates are related to the trajectories y a ðtÞ in SH
coordinates by some shifts ξa ðtÞ of the worldlines: y 0a ¼
y a þ ξa [64,67]. In the c.m. frame, the ADM coordinate
separation y 0 is related to the SH coordinate separation y
through y 0 ¼ y þ ξ, where the relative shift ξ ¼ ξ1 − ξ2 reads
[59,65]
Gm
5 2 ν 2 Gm 1
9
Gmν
1 11
15
4
2 2 5
_
_
−
v
−
νv
r
þ
3ν
n
þ
ν_
r
v
þ
þ
ν
þ
r
−
ν
þ
−
v
8
8
r 4
4
2 8
16 16
c4
c6
2 2
1
5
Gm 2 451 3
Gm 2 161 5
G m 2773 21 2 22
r
4
þ r_ − þ ν −
v
þ ν þ
r_
− ν þ 2
þ π − ln 0
n
16 16
r
48 8
r
48 2
280 32
3
r0
r
21
5 29
Gm 43
2 17
2
þ v
− ν þ r_ − þ ν þ
þ 5ν r_ v þ oðc−6 Þ;
ð4:15Þ
8
4
12 24
r
3
from which the authors of Ref. [59] deduced, using Eqs. (4.11), the transformation of variables that we need to compute the
redshift in ADM coordinates2:
ðδξ rÞðSH→ADMÞ ¼
ðδξ r_ ÞðSH→ADMÞ
Gm 5 2 19 2 Gm 1
Gmν 4 1 11
39 99
2 2
4 23 73
_
_
νv
ν_
r
þ
3ν
þ
−
ν
þ
r
þ
ν
þ
r
−
ν
v
−
−
þ
v
8
r 4
2 8
16 16
48 48
c4 8
c6
2 2
Gm 2 451 3
Gm 2 283 5
Gm
2773 22
r
21
þ
−
;
ð4:16aÞ
v
þ ν −
r_
þ ν þ 2
þ ln 0 − π 2
r
48 8
r
16 2
280
3
r0
32
r
Gm
19 2 19 2 Gm
1 ν
Gmν
39 99
163 443
4
2
2
− þ
þ 6 r_ v − þ ν þ r_ v
−
ν
¼ 4 r_ − νv þ ν_r þ
4
4
r
4 2
8
8
24
24
cr
cr
23 73
Gm 2 1603 17
Gm 2 1777 131
4
þ r_ − þ ν −
v
þ ν þ
r_
þ
ν
12 12
r
48
4
r
48
24
G2 m2 3121 44
r
21
11
þ 2
− ln 0 þ π 2 − ν ;
ð4:16bÞ
105
3
r0
16
4
r
ðδξ v2 ÞðSH→ADMÞ ¼
Gm
13 4 5 2 2 3 4 Gm 2 1 21
Gm 2
19
_
1
þ
νv
ν_
r
ν_
r
v
þ
ν
−
r
ν
þ
v
þ
þ
−
4
2
4
r
2 2
r
2
c4 r
Gmν 6
13 31
31 75
3
5 25
Gm 4 9 25
2
4
4
2
6
þ 6
− ν þ r_ v − ν þ r_ − þ ν −
v
þ ν
v − þ ν þ r_ v
4
4
8
8
2
8 8
r
8 4
cr
Gm 2 2
131 121
Gm 4 99 259
G2 m2
3839 44
r
21
þ
r_ v −
þ
ν þ
r_
−
ν þ 2 v2 −
þ ln 0 − π 2 þ ν
r
8
4
r
4
12
420
3
r0
16
r
2 2
Gm
28807
r
63
13
− 44 ln 0 þ π 2 − ν :
ð4:16cÞ
þ 2 r_ 2
420
r0
16
2
r
The expression U 0 ½r; r_ ; v2 for the redshift in ADM coordinates can then be deduced from the result (4.6)–(4.7) in SH
coordinates via the functional equality U0 ¼ U þ δξ U. Using the expressions (4.10) and (4.16), the SH redshift is found to be
modified by 2PN and 3PN corrections that read
2
The remainder Oðξ2 Þ in Eqs. (4.10) and (4.11) is of order 4PN, which is still negligible in the transformation to ADM coordinates.
124014-14
COMPARISON BETWEEN SELF-FORCE AND POST- …
PHYSICAL REVIEW D 91, 124014 (2015)
Gm
13 4 5 2 2
3 4
¼ 6 ð1 þ Δ − 2νÞν − v þ v r_ þ r_
16
8
16
cr
Gm 2 1 Δ 33
37
21
þ
v
þ þ ν þ Δν − ν2
r
8 8 16
16
4
Gm 2 1 Δ 11
19
19
G2 m2 1 Δ 3
3
−
r_
þ þ ν þ Δν − ν2 − 2
þ þ ν þ Δν
r
4 4 16
16
4
8 8 2
2
r
Gmν
65 65
161
205 2
þ 8
þ Δ−
ν − 6Δν þ
ν
−v6
32 32
16
16
cr
61 61
297
175
185 2
þ v4 r_ 2
þ Δ−
ν−
Δν þ
ν
32 32
32
32
16
9
15
21
45 2
2 _4 9
þv r
þ Δ − ν − Δν þ ν
32 32
8
16
16
5
5
35
25
25
− r_ 6
þ Δ − ν − Δν þ ν2
32 32
32
32
16
Gm 4 661 661
1669
125
11 2
þ
r_
þ
Δ−
νþ
Δν − ν
r
96
96
96
96
6
2 2
Gm
3
39
33
287 3
4 3
2
2
þ 8 2 v
þ Δ − ν − Δν − 18ν − 14Δν þ
ν
16 16
16
16
8
cr
3 3
53
29
847 2 361 2
þ v2 r_ 2 − − Δ þ ν þ Δν þ
ν þ
Δν − 36ν3
8 8
32
32
32
32
Gm 2 5
5
2189 21 2
2329 21 2
5263 21 2 2
þ
v
þ Δþ
− π νþ
− π Δν þ
þ π ν
r
16 16
1120 64
1120 64
1680 32
Δν2
11
r
11
r
þ
− 8ν3 þ ð1 þ ΔÞν2 ln 0 þ ð1 þ Δ − 3ν − ΔνÞν ln 0
3
r1
3
r2
16
Gm 2 5 5
53099 63 2
50159 63 2
15821 63 2 2
−
r_
þ Δ−
þ π ν−
þ π Δν þ
þ π ν
r
4 4
3360 64
3360 64
420
32
11
59
r
r
þ Δν2 − ν3 þ 11ð1 þ ΔÞν2 ln 0 þ 11ð1 þ Δ − 3ν − ΔνÞν ln 0
8
4
r1
r2
G2 m2
1 Δ
2493 21 2
1933 21 2
þ 2
þ π νþ
þ π Δν þ 12ν2
− − þ
8 8
560 64
560 64
r
22
r
11
r
þ oðc−8 Þ:
− ν2 ln 0 − ð1 þ Δ − 2νÞν ln 0
3
r1
3
r2
ðδξ UÞðSH→ADMÞ
Although the additional contribution (4.17) in ADM coordinates is more involved than its counterpart (4.13) in MH
coordinates, they share the same logarithmic terms. Thus,
adding Eqs. (4.7d) and (4.17) we find that the UV regularization constant r02 disappears from the expression for
U0 ½r; r_ ; v2 in ADM coordinates, while the constants r0
and r01 remain and enter the final result through the
logarithmic terms (4.14).
C. The generalized quasi-Keplerian representation
Before we discuss the orbital averaging of the redshift
in Sec. IV D, we must summarize the 3PN generalized
ð4:17Þ
quasi-Keplerian (QK) representation of the motion of
Memmesheimer et al. [66]. Indeed, since averaging over
one radial period is most conveniently performed using an
explicit solution of the equations of motion, the generalized
QK representation is an essential input for our 3PN
calculation. The QK representation was originally introduced by Damour and Deruelle [68] to account for the
leading-order 1PN general relativistic effects in the timing
formula of the Hulse-Taylor binary pulsar. It was later
extended at 2PN order in Refs. [69–71], in ADM coordinates, and more recently at 3PN order [66] in both ADM
and harmonic coordinates.
124014-15
AKCAY et al.
PHYSICAL REVIEW D 91, 124014 (2015)
We first introduce the mean anomaly
l ≡ Ωr ðt − tper Þ;
ð4:18Þ
where tper is the coordinate time at a periastron passage and
Ωr ¼ 2π=T r is the radial frequency (also known as the
mean motion n), i.e., the frequency associated with the
periodicity T r of the radial motion. The mean anomaly
simply maps one radial period t ∈ ½tper ; tper þ T r Þ to the
trigonometric interval l ∈ ½0; 2πÞ. We then adopt a parametric description of the binary’s motion in polar coordinates, in the c.m. frame, in terms of the eccentric anomaly
u ∈ ½0; 2πÞ. At 3PN order, this parametrization reads
rðuÞ ¼ ar ð1 − er cos uÞ;
ð4:19aÞ
lðuÞ ¼ u − et sin u þ f t sin V þ gt ðV − uÞ
þ it sin 2V þ ht sin 3V;
with βϕ ≡ ½1 − ð1 − e2ϕ Þ1=2 =eϕ . Equations (4.18)–(4.20)
provide a 3PN-accurate generalization of the usual
Keplerian representation of the Newtonian motion.3
The previous generalized QK representation is complete
only once the orbital elements Ωr , K, ar , et , er , eϕ , f t , gt , it ,
ht , f ϕ , gϕ , iϕ and hϕ have been related to the first integrals
of the motion, namely, the binding energy E and the orbital
angular momentum J, both per reduced mass μ ¼ m1 m2 =m.
Following Ref. [59], we shall instead make use of the
convenient, dimensionless, coordinate-invariant quantities4
ε≡−
ð4:19bÞ
ϕðuÞ ¼ ϕper þ KðV þ f ϕ sin 2V þ gϕ sin 3V
þ iϕ sin 4V þ hϕ sin 5VÞ;
angle of return to periastron is given by Φ ¼ 2πK (equivalent
to ΔΦ ¼ 2πk), and the true anomaly V is defined by
βϕ sin u
VðuÞ ¼ u þ 2 arctan
;
ð4:20Þ
1 − βϕ cos u
ð4:19cÞ
where ϕper is the value of the orbital phase when t ¼ tper , at a
periastron passage, K ≡ 1 þ k is the fractional angle of
advance of the periastron per orbital revolution, such that the
2E
;
c2
j≡−
2EJ2
;
ðGmÞ2
ð4:21Þ
such that ε > 0 and j > 0 for a generic bound eccentric orbit
(since E < 0 for such orbits). Notice the PN scalings ε ¼
Oðc−2 Þ and j ¼ Oðc0 Þ. Therefore, we shall consider expansions in powers of the PN parameter ε, with coefficients
depending on j and ν. In ADM coordinates, the 3PNaccurate expressions for the orbital elements read [59,66]
ε3=2 c3
ε
ε2
192
1 þ ½−15 þ ν þ
555 þ 30ν þ 11ν2 þ 1=2 ð−5 þ 2νÞ
8
Gm
128
j
3
ε
5760
þ
−29385 − 4995ν − 315ν2 þ 135ν3 þ 1=2 ð17 − 9ν þ 2ν2 Þ
3072
j
16
2
2
3
− 3=2 ð10080 − 13952ν þ 123π ν þ 1440ν Þ þ oðε Þ ;
j
3ε ε2 3
15
ADM
ð−5 þ 2νÞ þ 2 ð7 − 2νÞ
¼1þ þ
K
j
4 j
j
3
ε 24
1
þ
ð5 − 5ν þ 4ν2 Þ − 2 ð10080 − 13952ν þ 123π 2 ν þ 1440ν2 Þ
128 j
j
5
2
2
þ 3 ð7392 − 8000ν þ 123π ν þ 336ν Þ þ oðε3 Þ;
j
Gm
ε
ε2
1
2
aADM
½−7
þ
ν
þ
ð−68
þ
44νÞ
¼
þ
1
þ
1
þ
10ν
þ
ν
r
4
j
16
c2 ε
3
ε
1
þ
3 − 9ν − 6ν2 þ 3ν3 þ ð864 − 2212ν − 3π 2 ν þ 432ν2 Þ
j
192
1
2
2
3
− 2 ð6432 − 13488ν þ 240π ν þ 768ν Þ þ oðε Þ ;
j
ΩADM
¼
r
ð4:22aÞ
ð4:22bÞ
ð4:22cÞ
In the Newtonian limit, ar is the semimajor axis, the three eccentricities coincide (et ¼ er ¼ eϕ ≡ e), an eccentric orbit does not
precess (K ¼ 1), and f t ¼ gt ¼ it ¼ ht ¼ f ϕ ¼ gϕ ¼ iϕ ¼ hϕ ¼ 0.
4
For circular orbits, we have the well-known Newtonian limits ε ∼ v2 =c2 ∼ Gm=ðrc2 Þ and j ∼ 1.
3
124014-16
COMPARISON BETWEEN SELF-FORCE AND POST- …
PHYSICAL REVIEW D 91, 124014 (2015)
ε
1 − j þ ½−8 þ 8ν þ jð17 − 7νÞ
4
2
ε
24
þ
8 þ 4ν þ 20ν2 − 1=2 ð5 − 2νÞ þ 24j1=2 ð5 − 2νÞ
8
j
4
− jð112 − 47ν þ 16ν2 Þ þ ð17 − 11νÞ
j
3
ε
þ
24ð−2 þ 5νÞð−23 þ 10ν þ 4ν2 Þ þ 15jð528 − 200ν þ 77ν2 − 24ν3 Þ
192
2
− 72j1=2 ð265 − 193ν þ 46ν2 Þ − ð6732 − 12508ν þ 117π 2 ν þ 2004ν2 Þ
j
2
þ 1=2 ð16380 − 19964ν þ 123π 2 ν þ 3240ν2 Þ
j
2
− 3=2 ð10080 − 13952ν þ 123π 2 ν þ 1440ν2 Þ
j
1=2
96
2
2
3
þ 2 ð134 − 281ν þ 5π ν þ 16ν Þ þ oðε Þ
;
j
ε
eADM
¼
1 − j þ ½24 − 4ν þ 5jð−3 þ νÞ
r
4
ε2
8
2
2
þ
52 þ 2ν þ 2ν − jð80 − 55ν þ 4ν Þ þ ð17 − 11νÞ
j
8
3
ε
þ
−768 − 344ν − 6π 2 ν − 216ν2 þ 3jð−1488 þ 1556ν − 319ν2 þ 4ν3 Þ
192
4
− ð588 − 8212ν þ 177π 2 ν þ 480ν2 Þ
j
1=2
192
þ 2 ð134 − 281ν þ 5π 2 ν þ 16ν2 Þ þ oðε3 Þ
;
j
ε
ADM
eϕ
¼ 1 − j þ ½24 þ jð−15 þ νÞ
4
ε2
1
þ
−32 þ 176ν þ 18ν2 − jð160 − 30ν þ 3ν2 Þ þ ð408 − 232ν − 15ν2 Þ
j
16
3
ε
þ
−16032 þ 2764ν þ 3π 2 ν þ 4536ν2 þ 234ν3 − 36jð248 − 80ν þ 13ν2 þ ν3 Þ
384
6
− ð2456 − 26860ν þ 581π 2 ν þ 2689ν2 þ 10ν3 Þ
j
1=2
3
2
2
3
3
þ 2 ð27776 − 65436ν þ 1325π ν þ 3440ν − 70ν Þ þ oðε Þ
;
j
eADM
¼
t
f ADM
t
pffiffiffiffiffiffiffiffiffiffi
ε2 1 − j
¼−
νð4 þ νÞ
8 j1=2
ε3 j1=2
1
4148
2
2
2
3
p
ffiffiffiffiffiffiffiffiffiffi
ν þ π ν þ 200ν þ 11ν
þ
νð−64 − 4ν þ 23ν Þ þ 2 576 −
3
64 1 − j
j
1
4232
þ
þ oðε3 Þ;
−576 þ
ν − π 2 ν − 209ν2 − 35ν3
j
3
124014-17
ð4:22dÞ
ð4:22eÞ
ð4:22fÞ
ð4:22gÞ
AKCAY et al.
PHYSICAL REVIEW D 91, 124014 (2015)
gADM
t
3ε2 5 − 2ν
ε3
1
þ
¼
ð10080 − 13952ν þ 123π 2 ν þ 1440ν2 Þ
1=2
3=2
2
192 j
j
1
2
þ 1=2 ð−3420 þ 1980ν − 648ν Þ þ oðε3 Þ;
j
ε3 1 − j
νð23 þ 12ν þ 6ν2 Þ þ oðε3 Þ;
32 j3=2
13ε3 1 − j 3=2 3
hADM
¼
ν þ oðε3 Þ;
t
j
192
iADM
¼
t
f ADM
¼
ϕ
ε2 1 − j
νð1 − 3νÞ
8 j2
ε3 4ν
1
þ
ð−11 − 40ν þ 24ν2 Þ þ 2 ð−256 þ 1192ν − 49π 2 ν þ 336ν2 − 80ν3 Þ
256 j
j
1
2
2
3
þ 3 ð256 − 1076ν þ 49π ν − 384ν − 40ν Þ þ oðε3 Þ;
j
3ε2 ν2
ð1 − jÞ3=2
32 j2
pffiffiffiffiffiffiffiffiffiffi ε3 1 − j
1 220
2
2
ν νð9 − 26νÞ þ
þ π þ 104ν þ 50ν
−
j 3
256 j
1 220
2
2
− 2
þ oðε3 Þ;
þ π þ 32ν þ 15ν
3
j
ð4:22hÞ
ð4:22iÞ
ð4:22jÞ
ð4:22kÞ
gADM
¼−
ϕ
iADM
¼
ϕ
ð4:22lÞ
ε3 ð1 − jÞ2
νð5 þ 28ν þ 10ν2 Þ þ oðε3 Þ;
128 j3
ð4:22mÞ
5ε3 ν3
ð1 − jÞ5=2 þ oðε3 Þ:
256 j3
ð4:22nÞ
hADM
¼
ϕ
The eccentricities eADM
, eADM
and eADM
are all such that
t
r
ϕ
2
−2
j ¼ 1 − e þ Oðc Þ at Newtonian order; they start differing from each other at leading 1PN order.
The expressions (4.22) are specific to the ADM coordinates. Before we give the corresponding expressions in
MH coordinates, let us recall an important point related to
the use of gauge-invariant variables. As shown in Ref. [69],
the functional forms of Ωr ¼ 2π=T r and K ¼ Φ=ð2πÞ as
functions of gauge-invariant variables like ε and j are
identical in different coordinate systems. In particular we
have the exact same relations in MH coordinates as in
ADM coordinates:
ΩMH
¼ ΩADM
≡ Ωr ;
r
r
ð4:23aÞ
K MH ¼ K ADM ≡ K:
ð4:23bÞ
We may therefore use any combination of Ωr and K
instead of the constants of the motion ε and j to
parameterize in a physically meaningful way a given
eccentric orbit (assuming a one-to-one relation).
Following Ref. [59], we introduce the frequency
Ωϕ ≡ KΩr , which is a natural generalization of the
circular-orbit frequency Ω,5 and we define the dimensionless coordinate-invariant parameters (remember that
k ¼ K − 1)
GmΩϕ 2=3
3x
x≡
ð4:24Þ
;
ι≡ :
k
c3
The PN parameter x is Oðc−2 Þ, while ι is merely
Newtonian at leading order (the relativistic periastron
advance first appears at 1PN order). The choice of
variables (4.24) is the obvious generalization of the
gauge-invariant variable x that is commonly used for
Note that Ωϕ coincides with
R the average angular frequency of
_ ≡ 1 T r ϕðtÞdt.
_
the motion: Ωϕ ¼ hϕi
Tr 0
124014-18
5
COMPARISON BETWEEN SELF-FORCE AND POST- …
PHYSICAL REVIEW D 91, 124014 (2015)
circular orbits. It will thus facilitate checking the circularorbit limit. In Sec. IV E, we shall express our final results
in terms of either of the two sets of gauge-invariant
parameters ðε; jÞ or ðx; ιÞ.
To compute the invariant relationships hUiðε; jÞ and
hUiðx; ιÞ from the expressions (4.6)–(4.7) and (4.10) for
the redshift in MH coordinates, we shall also need
expressions for the orbital elements ar , et , er , eϕ , f t , gt ,
it , ht , f ϕ , gϕ , iϕ and hϕ in these coordinates. They are given
by Eqs. (4.22c)–(4.22l), to which we must add the
differences [66]
Gmε
5
1 1 17
þ ν
−
¼ 2 − νþ
8
j 4 4
c
Gmε2 ν
ν2 1
1
11499 21 2
19 2
þ 2
þ þ
− þ −
þ π νþ ν
32 32 j
2
560
32
4
c
1 3
14501 21 2
þ 2
þ
− π ν − 5ν2
þ oðε2 Þ;
420
16
j 2
ε2
1 17
1
MH
ADM
p
ffiffiffiffiffiffiffiffiffiffi
et − et
þ ν 1−
¼
j
1−j 4 4
ε3
19 52
225 2 1 29
79039 21 2
201 2
þ pffiffiffiffiffiffiffiffiffiffi − − ν þ
ν þ
þ
− π ν−
ν
32 3
32
j 16
1680 16
16
1−j
1
3
14501 21 2
2
þ 2 − þ −
þ oðε3 Þ;
þ π ν þ 5ν
2
420
16
j
ε2
1 73
5
1 1 17
MH
ADM
er − er
þ ν−j ν−
þ ν
¼ pffiffiffiffiffiffiffiffiffiffi
8
j 2 2
1−j 2 8
ε3
13
5237 21 2
19 2
143
37 2
þ pffiffiffiffiffiffiffiffiffiffi
þ −
þ π νþ ν þj −
νþ ν
1680 32
16
64
64
1 − j 16
1 13
3667 105 2
51
þ
þ
−
π ν − ν2
j 8
56
32
4
1
14501 21 2
2
þ 2 −3 þ −
þ oðε3 Þ;
þ π ν þ 10ν
210
8
j
ε2
1 71
ν 1 1 141
ADM ¼ pffiffiffiffiffiffiffiffiffiffi
eMH
−
ν
þ
j
þ
þ
ν
−
e
−
ϕ
ϕ
4 16
32 j 4 32
1−j
3
ε
13
36511 21 2
1723 2
17
33 2
þ pffiffiffiffiffiffiffiffiffiffi − þ
−
π ν−
ν þj
νþ
ν
32
8960 128
256
256
256
1−j
1
13
21817 147 2
169 2
þ
− þ −
þ
π νþ
ν
j
16
480
64
8
1 3
621787 273 2
1789 2
þ 2
þ oðε3 Þ;
þ
−
π ν−
ν
13440 128
128
j 2
19ε2 1 − j 1=2
ADM
f MH
−
f
¼
ν
t
t
j
8
ε3
296083 21 2
989 2
361
171 2
þ π νþ
ν þj
ν−
ν
þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −1 þ −
6720
32
64
64
64
jð1 − jÞ
1
276133 21 2
799 2
1þ
− π ν−
ν
þ oðε3 Þ;
þ
j
6720
32
64
aMH
r
aADM
r
− gADM
¼ oðε3 Þ;
gMH
t
t
124014-19
ð4:25aÞ
ð4:25bÞ
ð4:25cÞ
ð4:25dÞ
ð4:25eÞ
ð4:25fÞ
AKCAY et al.
PHYSICAL REVIEW D 91, 124014 (2015)
11ε3 1 − j
νð19 − 10νÞ þ oðε3 Þ;
ð4:25gÞ
32 j3=2
ε3 1 − j 3=2
MH
ADM
ht − ht
¼
νð23 − 73νÞ þ oðε3 Þ;
ð4:25hÞ
j
192
ε2 1 1
ε3 1 1045
99 2 1
5
139633 21 2
117 2
−
þ
ν− ν þ
− þ −
þ π νþ
ν
ð1 þ 18νÞ þ
¼−
32
j
4
3360
16
8
8 j j2
j 32 192
1 3
92307 21 2
351 2
þ 2
þ oðε3 Þ;
þ
− π ν−
ν
ð4:25iÞ
2240 16
32
j 2
iMH
− iADM
¼
t
t
ADM
f MH
ϕ − fϕ
gMH
ϕ
iMH
ϕ
−
iADM
ϕ
−
gADM
ϕ
pffiffiffiffiffiffiffiffiffiffi ε2 ð1 − jÞ3=2
7
5
1
49709 21 2 445
3 1−j
ν
− νþ
−
þ
π þ
ν
¼
νþε
j
128 32
j
13440 128
128
32
j2
1 100783 21 2 847
þ 2
−
π −
ν
þ oðε3 Þ;
26880 128
256
j
ð4:25jÞ
where l0 ≡ dl=du can be computed from Eqs. (4.19b) and
(4.20). We first perform the orbit averaging in MH
coordinates.
ε3 ð1 − jÞ2
¼
νð149 − 198νÞ þ oðε3 Þ;
384 j3
ð4:25kÞ
1. Orbital average in MH coordinates
ADM
hMH
¼
ϕ − hϕ
3
5=2
ε ð1 − jÞ
256
j3
νð1 − 5νÞ þ oðε3 Þ: ð4:25lÞ
Notice, in agreement with the comment made earlier in
Sec. IV B 2, that the MH coordinates differ from the ADM
coordinates at leading 2PN order.
D. Orbital average of the redshift
We are finally in a position to compute the generalized
redshift
1
hUi ≡
Tr
Z
Tr
0
hUi
1
¼
Tr
Z
0
Tr
dt
1
¼
UðtÞ 2π
Z
0
2π
ð4:26Þ
UðτÞdτ;
dl
1
¼
UðlÞ 2π
Z
0
2π
l0 ðuÞ
du;
UðuÞ
ð4:27Þ
6
6
4
X
X
l0
αN ðet ; εÞ
ln ð1 − et cos uÞ
þ
βN ðet ; εÞ
:
¼
N
U N¼−1 ð1 − et cos uÞ
ð1 − et cos uÞN
N¼2
ð4:28Þ
which coincides with the ratio T r =T r of the coordinate time
period T r and the proper time period T r of the radial
motion.6 The averaged redshift (4.26) can be written in the
convenient alternative forms7
−1
Using the generalized QK representation (4.18)–(4.20),
(4.22)–(4.23) and (4.25), the variables r, r_ and v2 ¼ r_ 2 þ
r2 ϕ_ 2 that enter the expression (4.6)–(4.7) and (4.13) for the
redshift in MH coordinates can be expressed as functions of
the binding energy ε, the time eccentricity et ≡ eMH
t , and
the eccentric anomaly u. The integrand in Eq. (4.27) then
reads
Beware that, although we are using the same symbol to denote
the generalized redshift in Eqs. (2.17) and (4.26), the former
definition is restricted to linear order in the mass ratio, while the
latter holds for any q.
7
Notice the simple relation hUiτ h1=Uit ¼ 1, where h·iτ (resp.
h·it ) denotes an averaging over one radial period with respect to
the proper time τ (resp. the coordinate time t).
The computation of the coefficients αN and βN is straightforward, but the resulting expressions are too cumbersome
to be reported here. The integral in (4.27) is readily
performed thanks to the following formulas, which are
valid for all integers N ≥ 1:
Z
1 2π
du
I N ðeÞ ≡
2π 0 ð1 − e cos uÞN
ð−ÞN−1 dN−1
1
;
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
¼
ð4:29aÞ
N−1
ðN − 1Þ! dy
y2 − e2 y¼1
Z
1 2π ln ð1 − e cos uÞ
log
du
I N ðeÞ ≡
2π 0 ð1 − e cos uÞN
ð−ÞN−1 dN−1 Yðy; eÞ
¼
;
ð4:29bÞ
ðN − 1Þ! dyN−1 y¼1
where
124014-20
COMPARISON BETWEEN SELF-FORCE AND POST- …
pffiffiffiffiffiffiffiffiffiffiffiffi2ffi
1
1−e þ1
Yðy; eÞ ≡ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln
2
y2 − e2
p
ffiffiffiffiffiffiffiffiffiffiffiffi
ffi
1 − e2 − 1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
þ 2 ln 1 þ
y þ y2 − e2
PHYSICAL REVIEW D 91, 124014 (2015)
ð4:30Þ
MH 2
MH 3
MH 4
4
hUi ¼ 1 þ U MH
N ε þ U 1PN ε þ U 2PN ε þ U 3PN ε þ oðε Þ;
We note that the logarithmic contributions in (4.28) arise
at 3PN order from those terms proportional to ln ðr=r0 Þ and
ln ðr=r01 Þ in Eq. (4.14). Indeed, combining Eqs. (4.19a),
(4.22d), (4.22e), (4.25b) and (4.25c), one finds
r
a
ln
¼ ln r þ ln ð1 − et cos uÞ þ Oðc−2 Þ; ð4:31aÞ
r0
r0
r
a
ln 0 ¼ ln 0r þ ln ð1 − et cos uÞ þ Oðc−2 Þ: ð4:31bÞ
r1
r1
Hence, some coefficients αN in (4.28) depend on the
regularization constants r0 and r01 through ln ðar =r0 Þ
and ln ðar =r01 Þ. However, when averaged over one
radial period, these terms cancel out from the final
expression, because they appear only through the vanishing
combination
2I 2 ðeÞ − 5I 3 ðeÞ þ 3ð1 − e2 ÞI 4 ðeÞ ¼ 0:
Implementing all the above integrations, the expression
(4.6)–(4.7) and (4.13) for the redshift in MH coordinates
can be averaged over an orbit. Up to 3PN order, the
generalized redshift (4.26) then takes the form
ð4:33Þ
where the PN coefficients depend on thepsymmetric
ffiffiffiffiffiffiffiffiffiffiffiffiffi mass
ratio ν, the reduced mass difference Δ ¼ 1 − 4ν, and the
time eccentricity et in MH coordinates (hence, et ≡ eMH
t ).
They read
3 3
ν
U MH
N ¼ þ Δ− ;
4 4
2
3 3
45
9
3 þ 3Δ
ν − Δν þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ;
U MH
1PN ¼ − − Δ −
4 4
16
16
1 − e2t
U MH
2PN ¼
ð4:32Þ
The final expression for hUiðε; eMH
t Þ is thus free of the
regularization constants r0 and r01 .
ð4:34aÞ
ð4:34bÞ
1 Δ 111
15
75
3
þ þ
ν − Δν þ ν2 þ Δν2
8 8
16
16
32
32
21 21
57
15
1
þ Δ þ ν − Δν þ 3ν2 pffiffiffiffiffiffiffiffiffiffiffiffiffi
−
2
2
4
4
1 − e2t
41 41
37
37
1
þ Δ − ν − Δν þ 5ν2
;
þ
2
2
4
4
ð1 − e2t Þ3=2
43
119
93
15
45
νþ
Δν − ν2 þ Δν2 − ν3
4
16
32
8
64
405 405
1419
525
3 2 15 3
1
2
þ
Δþ
ν−
Δν − 3ν − Δν − ν pffiffiffiffiffiffiffiffiffiffiffiffiffi
þ
16
16
16
16
4
4
1 − e2t
1
45 45
1
þ Δ − 9ν − 9Δν
þ
− ð27 þ 27Δ þ 18ν − 18ΔνÞ
2
2
2
1 − et
ð1 − e2t Þ2
1467 1467
3281 287 2
9185 287 2
−
Δþ
−
π νþ
−
π Δν
þ −
8
8
48
256
48
256
4193 41 2 2 415 2 107 3
1
þ −
þ π ν −
Δν þ
ν
48
64
16
4
ð1 − e2t Þ3=2
873 873
861 2
861 2
þ
Δ þ −278 þ
π ν þ −278 þ
π Δν
þ
4
4
256
256
695 123 2 2 135 2 51 3
1
þ
−
π ν þ
Δν − ν
:
4
64
4
2
ð1 − e2t Þ5=2
ð4:34cÞ
U MH
3PN ¼ −
ð4:34dÞ
For notational simplicity we did not add a label on et to indicate that it is the time eccentricity in MH coordinates. (No such
label is required over ε, which is gauge invariant.) This point should be remembered when comparing expressions derived in
different gauges, as we shall do next.
2. Orbital average in ADM coordinates
We shall now perform an independent calculation in ADM coordinates. We start from the expression for the redshift in
ADM coordinates, as given by (4.6)–(4.7) and (4.17), employ the appropriate QK parametrization and perform the orbital
124014-21
AKCAY et al.
PHYSICAL REVIEW D 91, 124014 (2015)
averaging as outlined above. We find that the form (4.28) is obtained also in the ADM case, with the same coefficients βN
but different coefficients αN in general. The result for the generalized redshift in ADM coordinates is of the form
2
ADM 3
ADM 4
4
ε þ U ADM
hUi ¼ 1 þ U ADM
N
1PN ε þ U 2PN ε þ U 3PN ε þ oðε Þ;
ð4:35Þ
), and read
where the various coefficients depend on ν, Δ, and the time eccentricity in ADM coordinates (hence, et ≡ eADM
t
3 3
ν
U ADM
¼ þ Δ− ;
N
4 4
2
ð4:36aÞ
3 3
45
9
3 þ 3Δ
ν − Δν þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ;
U ADM
1PN ¼ − − Δ −
4 4
16
16
1 − e2t
ð4:36bÞ
1 Δ 111
15
75
3
U ADM
þ
ν − Δν þ ν2 þ Δν2
2PN ¼ þ
8 8
16
16
32
32
21 21
57
15
1
þ Δ þ ν − Δν þ 3ν2 pffiffiffiffiffiffiffiffiffiffiffiffiffi
−
2
2
4
4
1 − e2t
41 41
37
37
1
þ Δ − ν − Δν þ 5ν2
;
þ
2
2
4
4
ð1 − e2t Þ3=2
43
119
93
15
45
νþ
Δν − ν2 þ Δν2 − ν3
4
16
32
8
64
405 405
1419
525
3 2 15 3
1
2
þ
Δþ
ν−
Δν − 3ν − Δν − ν pffiffiffiffiffiffiffiffiffiffiffiffiffi
þ
16
16
16
16
4
4
1 − e2t
1
45 45
1
þ Δ − 9ν − 9Δν
þ
− ð27 þ 27Δ þ 18ν − 18ΔνÞ
2
2
1 − e2t
ð1 − e2t Þ2
1461 1461
3893 287 2
9797 287 2
−
Δþ
−
π νþ
−
π Δν
þ −
8
8
48
256
48
256
4193 41 2 2 415 2 107 3
1
þ −
þ π ν −
Δν þ
ν
48
64
16
4
ð1 − e2t Þ3=2
435 435
1163 861 2
1163 861 2
þ
Δþ −
þ
π νþ −
þ
π Δν
þ
2
2
4
256
4
256
695 123 2 2 135 2 51 3
1
þ
−
π ν þ
Δν − ν
:
4
64
4
2
ð1 − e2t Þ5=2
ð4:36cÞ
U ADM
3PN ¼ −
Although the coefficients (4.34) and (4.36) coincide
ADM
through 2PN order, the 3PN coefficients U MH
3PN and U 3PN
are different. A useful internal check of the PN
calculations of the generalized redshift in MH and
ADM coordinates is the verification that the equality
of Eqs. (4.33)–(4.34) and (4.35)–(4.36) holds if and
only if the time eccentricities eMH
and eADM
are related
t
t
by
eMH
t
¼
eADM
t
1−
1 þ 17ν ε2
3
þ Oðε Þ :
1 − ðeADM
Þ2 4
t
ð4:37Þ
This relation is in perfect agreement with what is
predicted from using different QK representations of
the motion, namely, Eq. (4.22d) together with (4.25b).
ð4:36dÞ
E. Gauge-invariant formulations
To compare the analytical PN predictions with the
numerical results of the GSF calculation (Sec. III), it is
best to use a coordinate-invariant relationship. We shall
thus replace the coordinate-dependant time eccentricity et
in favor of the coordinate-invariant angular momentum
variable j. Substituting the PN expansion (4.22d) into
Eq. (4.36), or alternatively Eqs. (4.22d) and (4.25b) into
(4.34), we get
hUi ¼ 1 þ U N ε þ U 1PN ε2 þ U 2PN ε3 þ U 3PN ε4 þ oðε4 Þ;
ð4:38Þ
where
124014-22
COMPARISON BETWEEN SELF-FORCE AND POST- …
PHYSICAL REVIEW D 91, 124014 (2015)
UN ¼
3 3
ν
þ Δ− ;
4 4
2
3 3
45
9
3 þ 3Δ
U 1PN ¼ − − Δ − ν − Δν þ pffiffi ;
4 4
16
16
j
1 Δ 111
15
75
3
U 2PN ¼ þ þ
ν − Δν þ ν2 þ Δν2
8 8
16
16
32
32
33 33
135
9
1
2 p
ffiffi
þ Δþ
ν − Δν þ 3ν
−
8
8
8
8
j
35 35
25
25
1
þ
þ Δ − ν − Δν þ 5ν2 3=2 ;
2
2
4
4
j
43
119
93
15
45
νþ
Δν − ν2 þ Δν2 − ν3
4
16
32
8
64
297 297
3819
1509
453 2 459 2 9 3 1
þ
Δþ
ν−
Δν þ
ν −
Δν − ν pffiffi
þ
128 128
64
64
128
128
8
j
9 9
1
−
þ Δ þ 27ν − 9Δν
2 2
j
945 945
17 287 2
10063 287 2
−
þ
Δþ
þ
π νþ −
þ
π Δν
16
16
96 256
96
256
4471 41 2 2 65 2 85 3 1
þ
− π ν þ Δν − ν
96
64
32
8
j3=2
693 693
875 861 2
875 861 2
þ
Δþ −
þ
π νþ −
þ
π Δν
þ
4
4
4
256
4
256
271 123 2 2 21 2 21 3 1
þ
:
−
π ν þ Δν − ν
2
64
2
2
j5=2
ð4:39aÞ
ð4:39bÞ
ð4:39cÞ
U 3PN ¼ −
ð4:39dÞ
Since the relationship hUiðε; jÞ is coordinate-invariant, it is physically meaningful. However, the binding energy E and
angular momentum J are not easily accessible to perturbative GSF calculations, so a direct comparison is not obvious.
Thankfully, Eq. (4.38) can also be expressed using the invariant parameters (4.24) defined with respect to the fundamental
frequencies Ωr and Ωϕ . Indeed, inverting the PN expansions (4.22a) and (4.22b) yields
5 ν 2
5 5
ν2 5 − 2ν 5 − 3ν 5 2
− − xþ
− ν − þ pffi −
þ 2 x
ε¼x 1þ
4 12 ι
8 8
24
ι
ι
ι
185 75
25 2
35 3
105 35
7
1
þ −
− ν−
ν −
ν þ
− ν − ν2 pffi
192 64
288
5184
8
6
6
ι
2
15 15
ν 1
95
211 41 2
5 2 1
3
3
þ − þ νþ
−
þ
− π ν − ν 3=2 x þ oðx Þ ;
4
4
8
9
96
2
4 ι
ι
pffi
27 5
5
35
373 41 2
55
− ν þ νι x þ − þ
−
π ν − ν2 þ ½5 − 2ν ι
j¼ιþ
4 2
12
8
16 128
24
2
35 25
ν
115
665 205 2
15
1 2
þ − þ νþ
−
−
π ν − ν2
x þ oðx2 Þ:
ιþ
16 48
16
12 128
8
ι
8
ð4:40aÞ
ð4:40bÞ
We thus have the leading-order relationships x ¼ ε þ Oðc−2 Þ and ι ¼ j þ Oðc−2 Þ. Introducing the expansions (4.40) into
Eq. (4.38)–(4.39), our main PN result reads
hUi ¼ 1 þ V N x þ V 1PN x2 þ V 2PN x3 þ V 3PN x4 þ oðx4 Þ;
124014-23
ð4:41Þ
AKCAY et al.
PHYSICAL REVIEW D 91, 124014 (2015)
where the various PN coefficients, which depend on the variable ι as well as on the particle’s masses, read up to 3PN order
VN ¼
V 1PN ¼
3 3
ν
þ Δ− ;
4 4
2
3
3
7
5
ν2 3 þ 3Δ
3 3
1
þ Δ − ν − Δν þ þ pffi −
þ Δ−ν
;
16 16
2
8
ι
2 2
ι
24
41 41
3
43
99
5
ν3
− Δ − ν − Δν þ ν2 þ Δν2 þ
32 32
4
16
32
32
48
57 57
3
1
3
3
5
ν2 1
þ
þ Δ − 22ν − Δν − 2ν2 pffi −
þ Δ − 14ν − Δν þ
8
8
2
ι
4 4
2
6 ι
37 37
5
5
1
15 15
5
1
−
þ Δ þ ν þ Δν − 5ν2 3=2 þ
þ Δ− ν 2;
8
8
2
2
4
4
2
ι
ι
ð4:42aÞ
ð4:42bÞ
V 2PN ¼ −
605 605
385
323
565 2 419 2 895 3
25
35 4
−
Δþ
νþ
Δν þ
ν þ
Δν −
ν þ
Δν3 þ
ν
256 256
48
128
64
128
864
1728
10368
117 117
89
499
1267 2 91 2 5 3 1
þ
Δ− ν−
Δν þ
ν − Δν þ ν pffi
þ
128 128
2
16
96
32
6
ι
3
411 411
63
129
297 2 15 2 ν 1
þ
þ
Δ− νþ
Δν −
ν − Δν −
16
16
2
8
16
16
8 ι
1755 1755
4339 41 2
497 41 2
þ −
−
Δþ
−
π νþ
−
π Δν
64
64
48
128
24 128
181 41 2 2 115 2 5 3 1
þ −
þ π ν þ
Δν þ ν
144 96
16
4
ι3=2
1797 1797
355 123 2
355 123 2
þ
Δþ −
þ
π νþ −
þ
π Δν
þ
128
128
16 128
16 128
1321 123 2 2 99 2 33 3 1
þ
−
π ν − Δν þ ν
:
32
64
32
4
ι5=2
ð4:42cÞ
V 3PN ¼ −
ð4:42dÞ
The noncircular nature of the motion only explicitly enters the result at leading 1PN order via the invariant parameter ι.
Since we have the qualitative behavior ι ∼ 1 − e2, this suggests that the effect of the eccentricity on hUi will be moderate (at
least in the weak-field regime).
1. Circular-orbit limit
Another key check of the results (4.39) and (4.42) is provided by the circular-orbit limit. For such orbits, the two
constants of the motion are no longer independent variables. Indeed, the angular momentum variable, say j⊙, is related to
the energy ε by the 3PN gauge-invariant expansion [72]
9 ν
81
ν2 2
j⊙ ¼ 1 þ
þ εþ
− 2ν þ
ε
4 4
16
16
945
7699 41 2
ν2 ν3 3
þ −
þ π νþ þ
þ
ε þ oðε3 Þ:
64
192 32
2 64
ð4:43Þ
It can be checked that the eccentricities et , er , eϕ all vanish when j is replaced by (4.43) in Eqs. (4.22d)–(4.22f) and
(4.25b)–(4.25d). The invariant result (4.38)–(4.39) then reduces to
124014-24
COMPARISON BETWEEN SELF-FORCE AND POST- …
PHYSICAL REVIEW D 91, 124014 (2015)
3 3
ν
9 9
45
9
þ Δ− εþ
þ Δ − ν − Δν ε2
U⊙ ¼ 1 þ
4 4
2
4 4
16
16
81 81
265
103
139 2 3
2
þ Δ−
ν−
Δν þ
ν þ Δν ε3
þ
8
8
16
16
32
32
891 891
3809 287 2
4945 287 2
þ
Δ−
−
π ν−
−
π Δν
þ
16
16
24 128
48 128
7727 41 2 2 143 2 205 3 4
þ
− π ν þ
Δν −
ν ε þ oðε4 Þ:
96 32
16
64
ð4:44Þ
Setting et → 0 in Eq. (4.34) or (4.36) yields the same expression.
We then replace the constant of the motion ε in favor of the frequency-related parameter x [recall Eq. (4.24)], using the
well-known 3PN-accurate expression for the binding energy as a function of the circular-orbit frequency, namely, [see, e.g.,
Eq. (232) of Ref. [14]]
3 ν
27 19
ν2 2
ε⊙ ¼ x 1 þ − −
xþ − þ ν−
x
4 12
8
8
24
675
34445 205 2
155 2
35 3 3
3
þ
−
π ν−
ν −
ν x þ oðx Þ :
þ −
64
576
96
96
5184
ð4:45Þ
Finally, replacing ε in (4.44) using (4.45), we recover the known 3PN result for the circular-orbit redshift (see Eq. (4.10) of
Ref. [16]):
3 3
ν
27 27
5
5
ν2 2
þ Δ− xþ
þ Δ − ν − Δν þ
U⊙ ¼ 1 þ
x
4 4
2
16 16
2
8
24
135 135
37
67
115 2
5
ν3 3
2
þ
Δ − ν − Δν þ
ν þ Δν þ
þ
x
32
32
4
16
32
32
48
2835 2835
2183 41 2
12199 41 2
þ
þ
Δ−
− π ν−
− π Δν
256
256
48
64
384
64
17201 41 2 2 795 2 2827 3
25
35 4 4
3
þ
−
π ν þ
Δν −
ν þ
Δν þ
ν x
576
192
128
864
1728
10368
þ oðx4 Þ:
ð4:46Þ
Interestingly, at Newtonian order, the averaged redshift hUi along an eccentric orbit has the same functional form as U ⊙ in
the case of a circular orbit. This shows that the effect of the eccentricity cancels out at Newtonian order, because of the
orbital averaging.
Alternatively, we can also combine Eqs. (4.40b), (4.43), (4.45) to obtain the PN expansion of the invariant relation ι⊙ ðxÞ
in the circular-orbit limit, namely,
9 7
9
397 41 2
28 2 2
ι⊙ ¼ 1 þ − þ ν x þ − þ
− π ν þ ν x þ oðx2 Þ;
2 3
4
12 32
9
ð4:47Þ
and introduce this expression into (4.41)–(4.42) to recover (4.46).
Our third and last check of the correctness of the formula (4.42) will be to recover the known result in the testparticle limit.
2. Extreme mass-ratio limit
The 3PN result (4.41)–(4.42) is valid for any mass ratio q ¼ m1 =m2 . To extract from this result the contribution due to the
conservative piece of the GSF, we introduce an alternative set of dimensionless coordinate-invariant parameters, better
suited than ðx; ιÞ to the extreme mass-ratio limit q ≪ 1:
124014-25
AKCAY et al.
Gm2 Ωϕ 2=3
y≡
;
c3
PHYSICAL REVIEW D 91, 124014 (2015)
λ≡
3y
:
k
ð4:48Þ
We substitute the relations x ¼ yð1 þ qÞ2=3 and ι ¼ λð1 þ
qÞ2=3 in (4.41)–(4.42), and expand in powers of the mass
ratio q, neglecting terms of Oðq3 Þ or higher. The 3PN result
for the sum of the test mass, GSF and post-GSF contributions reads
hUi ¼ hUi0 þ qhUigsf þ q2 hUip-gsf þ Oðq3 Þ;
ð4:49Þ
where
3
3
6
3 2
41
57
3
37
15 3
þ pffiffiffi − y þ − þ pffiffiffi − − 3=2 þ 2 y
hUi0 ¼ 1 þ y þ
2
8
16 4 λ 2λ 4λ
2λ
λ λ
605
117
411 1755
21
1797 20 4
þ pffiffiffi þ
−
y þ oðy4 Þ;
þ −
þ
þ
−
128 64 λ 8λ 32λ3=2 4λ2 64λ5=2 λ3
2 2
14 16
5
5 3
8 293 36
hUigsf ¼ −y − 4 − y − 6 þ pffiffiffi − þ 3=2 þ 2 y þ
− pffiffiffi þ
λ
λ
3 4 λ λ
λ
λ
λ
1789 41 2 1
56
355 123 2 1
40 4
þ
− π
þ
π
y þ oðy4 Þ;
− þ −
þ
24
64
8
64
λ3=2 λ2
λ5=2 3λ3
2
69
29 16 15
5
275 1533 207
hUip-gsf ¼ y þ 4 − y2 þ
þ pffiffiffi − þ 3=2 þ 2 y3 þ
þ pffiffiffi −
λ
8 4 λ λ λ
24 32 λ 4λ
λ
6377 41 2 1
56
2031 369 2 1
40 4
þ −
y þ oðy4 Þ:
þ π
−
π
þ þ
−
96
32
16
64
λ3=2 λ2
λ5=2 3λ3
ð4:50aÞ
ð4:50bÞ
ð4:50cÞ
In the test-particle limit q ¼ 0, we recover the 3PN expansion of the fully relativistic result (2.19) for a geodesic orbit,
as derived in App. B. The 3PN prediction (4.50c) could be compared with future calculations of the second-order
GSF [73–78].
Finally, we may express the result (4.50b) for the 3PN expansion of the GSF contribution to the generalized redshift by
means of the usual parametrization of bound timelike geodesic orbits in Schwarzschild in terms of the semilatus rectum p
and eccentricity e (see Sec. II A). Substituting for y and λ from Eqs. (B1) into (4.50b), we find
pffiffiffiffi
je
2je
123 2 pffiffiffiffi
3=2
2 1
hUigsf ¼ −
je
π
1þ
þ ð5 je − 4je þ 9je − 5je Þ 2 þ 95 −
64
p
p
p
79 41 2 3=2
27 5=2
1
2
3
−3
þ oðp Þ ;
ð4:51Þ
− 16je þ − þ π je − 16je − je þ 4je
6 64
2
p3
where je ≡ 1 − e2 . For small eccentricities, we may write
hUigsf ¼ a þ be2 þ ce4 þ de6 þ Oðe8 Þ;
ð4:52Þ
where the weak-field expansions of the coefficients aðpÞ,
bðpÞ, cðpÞ and dðpÞ read
1 2
5
121 41 2 1
a¼− − 2− 3−
− π
þ oðp−4 Þ;
p p
3
32
p
p4
ð4:53aÞ
1
4
7
5 41 2 1
þ π
þ oðp−4 Þ; ð4:53bÞ
b¼ þ 2þ 3−
p p
3 32
p
p4
2
1
705 123 2 1
c¼− 2þ 3þ
þ oðp−4 Þ; ð4:53cÞ
−
π
8 256
p 4p
p4
5
475 41 2 1
d¼− 3þ −
þ
π
þ oðp−4 Þ;
12 128
2p
p4
ð4:53dÞ
and higher-order terms in the eccentricity all contribute at
leading 2PN order.
V. COMPARISON OF POST-NEWTONIAN AND
SELF-FORCE RESULTS
In Fig. 1 we plot our data for hUigsf as a function of p for
a sample e ¼ f0.1; 0.2; 0.3; 0.4g of eccentricities. We
show, superposed, the corresponding 1PN, 2PN and 3PN
predictions from Eq. (4.51). The insets display the relative
differences between the GSF data and the successive PN
approximations. We make the following observations:
(i) There is an excellent agreement between the numerical GSF results and the analytical PN prediction at
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COMPARISON BETWEEN SELF-FORCE AND POST- …
PHYSICAL REVIEW D 91, 124014 (2015)
“large” p, in what should be considered a very
strong test of both calculations. This is a first
demonstration of such an agreement for noncircular
orbits.
(ii) The PN series appears to converge uniformly to the
GSF result at any p for any fixed e in our survey, at
least through 3PN order.
(iii) The 3PN formula reproduces the GSF results
extremely well even in what might be considered
a “strong-field” regime: at p ¼ 10 it does so to
within ∼1% for e ¼ 0.1 and to within a few percent
for e ¼ 0.4; at p ¼ 20 the agreement is already at
the level of one part in a thousand.
We can make the comparison more quantitative by
attempting to extract the large-p behavior of the numerically computed function hUigsf ðp; eÞ. Our strategy will be
to fit the numerical data against the PN model (4.51),
leaving the numerical coefficients as unknown fitting
parameters, later to be compared with the analytically
known values. Given the relative sparseness of data
available, we shall not attempt a simultaneous fit over p
and e, but rather fit over each of the two dimensions
separately, as described below. We will follow a “marginalization” procedure, whereby each of the PN orders is
fitted for in turn, assuming the analytic values of all terms at
lower PN order. Since the circular limit of hUigsf has been
computed previously at great accuracy [17,18,22,34], we are
able to accurately “remove” the circular (e-independent) part
of hUigsf from the data, fitting only for the e-dependent
residue. This should allow to fit the eccentricity-related
terms of interest here with greater accuracy.
Let us now describe this procedure in more detail. We
assume the e-expanded form (4.52) of the full PN expression (4.51). The term aðpÞ is the circular-orbit limit of
hUigsf , which has been computed to at least ten significant
figures in Refs. [17,18,22,34]. By subtracting off these
numerical data from ours, we construct a new data set for
the difference
ðeÞ
hUigsf ≡ hUigsf − aðpÞ ¼ bðpÞe2 þ cðpÞe4 þ :
ð5:1Þ
We assume that the functions bðpÞ; cðpÞ; … admit expansions in p−1 as in Eqs. (4.53), but pretend that the PN
coefficients are unknown:
b ¼ p−1 þ b1 p−2 þ b2 p−3 þ ;
c ¼ c1 p−2 þ c2 p−3 þ ;
ð5:2Þ
where subscripts are mnemonics for the PN order at which
coefficients occur, and we have fixed the “Newtonian,” 1=p
term of bðpÞ at its known value of unity. Our goal is to
determine the coefficient bn ; cn ; … from the numerical data
ðeÞ
for hUigsf . To this end, we first prepare subsets of data
where in each subset p is fixed and e varies. We fit each
TABLE IV. Best-fit values for the PN coefficients bn and cn
[Eqs. (5.1) and (5.2)] as extracted from the numerical data,
compared to their known exact values. Parenthetical figures are
estimated fitting uncertainties in the last displayed decimals,
obtained by varying the value of the truncation index N in the
fitting model [e.g., Eq. (5.3) for bðpÞ]. The exact value of b3 is
−ð5=3 þ 41π 2 =32Þ.
Coefficient
b1
b2
b3
c1
Estimate
Exact result
þ4.0002ð8Þ
þ7.02ð3Þ
−14.5ð4Þ
−2.00ð1Þ
þ4
þ7
−14.312…
−2
subset with respect to e using the model (5.1), including
terms through Oðe6 Þ. This yields three one-dimensional
data sets, representing bðpÞ, cðpÞ and dðpÞ.
Focusing first on the data set for bðpÞ, we fit it against
the PN model
bðpÞ ¼ p−1 þ
N
X
p−ðnþ1Þ ðbn þ blog
n ln pÞ;
ð5:3Þ
n¼1
log
log
in which blog
1 ¼ b2 ¼ b3 ¼ 0, since logarithmic terms
are known not to occur before the 4PN order [79,80].8
The truncation order N is left as a control parameter; by
varying it we obtain a rough estimate of the numerical
uncertainty in the fitted values of the parameters. We
apply a marginalization procedure, whereby to determine
bn we set all bn0 <n at their known analytic values. We use
this procedure to estimate the values of b1 , b2 and b3 ,
and we later similarly determine c1 . Our results are
shown in Table IV, alongside the known analytic values
for these parameters. We see a good agreement through
3PN order in the Oðe2 Þ term, and at 1PN order in the
Oðe4 Þ term.
Unfortunately, the accuracy of our current code (and its
limited utility at e ≳ 0.4) does not seem to allow us an
accurate extraction of bn≥4 , cn≥2 , or any of the blog
n ’s. The
reason for this can be appreciated from Fig. 2, where we
compare the amplitudes of the 3PN and 4PN terms with the
ðeÞ
amplitude of numerical noise in our hUigsf data. Note that,
while the “signal” from the b3 term lies well above the
noise, the c2 signal is buried deep inside it. Since our data is
limited to relatively small eccentricities, it is clear why we
have less “handle” on the cn [Oðe4 Þ] terms than on the bn
[Oðe2 Þ] terms.
8
The form of the circular-orbit limit, in which the PN
expansion of hUigsf is known analytically up to a very high
order [17–20,81], suggests that the function bðpÞ could also
involve powers of ln p. However, those would contribute at even
higher orders than we consider here, so we do not include them in
the PN model (5.3).
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AKCAY et al.
10
2
10
4
10
10
10
PHYSICAL REVIEW D 91, 124014 (2015)
0.100
Error
Error
b3 e2 p
4
c2 e p
4
(b4 blog
4 ln
6
b4 e2 p
blog
4
8
b3 e2 p
0.001
3
2
c2 e p
10
5
5
b4 e2 p
blog
4
5
10
7
10
9
4
3
2
(b4 blog
4 ln p) e p
5
p) e p
ln p e2 p
4
5
5
ln p e2 p
5
10
20
40
60
80
100
120
140
20
p
40
60
80
100
120
140
p
FIG. 2 (color online). Absolute magnitude of various PN terms (“signal”) compared to the magnitude of numerical error in the data
(“noise”), shown as a function of p for e ¼ 0.1 (left panel) and e ¼ 0.3 (right panel). The red (upper solid) curve shows the 3PN term of
the Oðe2 Þ piece of hUigsf , and the blue (lower solid) curve shows the 2PN term of the Oðe4 Þ piece. Comparison with the magnitude of
numerical noise (black dots) suggests that b3 should be easily discernible while c2 might not. This is confirmed by attempting to fit the
data against PN models, as detailed in the text. The dashed curve estimates the amplitude of the 4PN term of the Oðe2 Þ piece of hUigsf ,
which is not known analytically. We used here the values b4 ¼ −1500 and blog
4 ¼ 250 chosen from the middle of the estimated range
shown in Eqs. (5.4). This 4PN signal appears to lie just over the noise and is detectable. However, as it can be seen from the near overlap
of the densely dashed (green) and sparsely dashed (brown) curves, the jb4 j and blog
4 terms become almost equal in magnitude as p
.
increases, making it very difficult to extract the individual values of b4 and blog
4
Figure 2 also suggests that we might have just enough
“signal” coming from the Oðe2 Þ terms at 4PN to allow a
rough estimation of the coefficients b4 and blog
4 , which
are not known analytically. We have experimented
fitting to a large number of models of the form (5.3),
where all the analytically known coefficients are prespecified, and varying both the cutoff N and the number
of nonzero logarithmic terms. We find that fitting
uncertainties are almost as large as the fitted values
themselves. However, we are able to confidently constrain the values of b4 and blog
to lie within the
4
following ranges:
−2000 ≲ b4 ≲ −1000;
ð5:4aÞ
þ150 ≲ blog
4 ≲ þ350:
ð5:4bÞ
Future analytic calculations of the 4PN terms may be
checked against these predictions.
Our current code does not allow the determination of
unknown PN coefficients related to eccentricity with any
greater accuracy. To improve on our predictions would
require (i) to push the reach of the computation to higher
eccentricities and larger p, and at the same time (ii) to
reduce the numerical error in the calculation of hUigsf .
Some improvement may be achieved using the method of
Ref. [56], which is a slightly more advanced variant of our
method. More significant improvements may have to await
the development of eccentric-orbit GSF codes based on the
Teukolsky equation [58,82]. We expect such codes to start
delivering accurate numerical results in the very near
future.
ACKNOWLEDGMENTS
We are grateful to Maarten van de Meent for
providing us with unpublished comparison data generated by a new GSF code now being developed by him.
S. A. also thanks Maarten van de Meent and Haris
Markakis for many useful discussions. The research
leading to these results received funding from the
European Research Council under the European
Union’s Seventh Framework Programme (FP7/20072013)/ERC Grant Agreement No. 304978. S. A. and
L. B. acknowledge additional support from STFC
through Grant No. PP/E001025/1. S. A.’s work is further
supported by the Irish Research Council, funded under
the National Development Plan for Ireland. A. L. T.
acknowledges support from a Marie Curie FP7
Integration Grant under the 7th European Union
Framework Programme (PCIG13-GA-2013-630210).
N. S. acknowledges the support of the Grand-in-Aid
for Scientific Research (No. 25800154). N. W. gratefully
acknowledges support from a Marie Curie International
Outgoing Fellowship (PIOF-GA-2012-627781) and the
Irish Research Council, which is funded under the
National Development Plan for Ireland.
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COMPARISON BETWEEN SELF-FORCE AND POST- …
PHYSICAL REVIEW D 91, 124014 (2015)
APPENDIX A: EQUIVALENCE BETWEEN OUR
EXPRESSION FOR hUigsf AND BS2011’S
BS2011 give their formula for hUigsf in their Eq. (84).
Adjusting notation and rearranging the terms, their expression reads
ΔT r
qhUigsf ¼ α þ
ðhUi0 þ Cr Ωr þ Cϕ Ωϕ Þ
T r0
ΔΦ
ΔT r
− Cϕ Ωϕ
− hUi0
;
ðA1Þ
Φ0
T r0
where
Cr ≡
∂hUi0
;
∂Ωr
Cϕ ≡
∂hUi0
;
∂Ωϕ
Cϕ ¼
hUi0
Cr ¼
½ðE − LΩϕ ÞhUi0 − 1;
Ωr
ðA3aÞ
ðA3bÞ
which shall be derived below. Substituting these Cr and Cϕ
into Eq. (A1) and using T r0 ¼ T r0 =hUi0 , we obtain
qhUigsf ¼ αEhUi20 þ
hUi0
ðEΔT r − LΔΦ − ΔT r Þ:
T r0
ðA4Þ
Now, writing ðdτ=dχÞ2 ¼ −gαβ ðdxα =dχÞðdxβ =dχÞ and perturbing linearly with Δ, holding fixed p, e and χ (hence,
also, rp and drp =dχ), as in BS2011, we find
1
dτ
Δðdτ=dχÞ ¼ EΔðdt=dχÞ − LΔðdϕ=dχÞ − ĥRuu 0 ; ðA5Þ
2
dχ
which, upon integrating over a radial period, gives
1
ΔT r ¼ EΔT r − LΔΦ − T r0 hĥRuu i:
2
hUi−1
0 ¼ E − Ωϕ L − Ωr J r ;
ðA6Þ
Substituting in Eq. (A4) now reproduces our Eq. (2.25)
for hUigsf .
It remains to establish the relations (A3a) and (A3b).
This can be achieved by manipulating the explicit ellipticintegral representations of Ωϕ ; Ωr and hUi0 , given in
BS2011, but this approach involves much ungainly algebra
and will not be presented here. A much neater derivation
ðA7Þ
H
R
where Jr ≡ ð2πÞ−1 u0r dr ¼ ð2πE 0 Þ−1 0T r ður0 Þ2 dt is the
invariant action variable (per mass m1 ) associated with the
radial motion [84]. This relation is the Schwarzschild
reduction of Eq. (3.4) of Ref. [83]. In addition, we require
a relation between the partial derivatives of E, L and J r with
respect to Ωi . The necessary relation follows most directly
from the general variational formula (“first law”),
ðA2Þ
and ΔX denotes the GSF correction to a quantity X, holding
fixed p and e, rather than the invariant frequencies Ωi .
BS2011 give explicit expressions for ΔT r, ΔΦ and ΔT r in
terms of GSF quantities, but these will not be needed here.
We observe that the expression (A1) is much more
complicated than our result, Eq. (2.25). Our goal here is
to show that the two expressions are, in fact, identical.
To this end, we note the two key relations,
LhUi20 ;
uses general results derived from the Hamiltonian formulation of geodesic motion in Kerr spacetime [83]. Start by
averaging uα0 u0α ¼ −1 with respect to t over a radial period
of the geodesic orbit, to obtain
δE ¼ Ωϕ δL þ Ωr δJ r ;
ðA8Þ
established in [83] [this form is the reduction of Eq. (3.5)
therein to Schwarzschild spacetime, with a fixed black-hole
mass m2 , and with suitable notational adjustments]. Here
δE, δL and δJr correspond to an arbitrary variation of a
geodesic with frequencies Ωi onto a nearby geodesic. If we
regard E, L and Jr as functions of Ωi , we obtain
∂E
∂L
∂J
− Ωϕ
− Ωr r ¼ 0:
∂Ωi
∂Ωi
∂Ωi
ðA9Þ
Taking the partial derivative of (A7) with respect to Ωϕ and
using (A9) immediately leads to (A3a). Equation (A3b), in
turn, is obtained by taking the derivative of (A7) with
respect to Ωr , then using Eq. (A9), and finally substituting
for J r from (A7).
The above establishes the equivalence of our simple
expression (2.25) and the BS2011 result (A1). The simplification obtained here owes itself primarily to the two
key relations (A3a) and (A3b), which have unfortunately
gone unnoticed (by two of us) in BS2011.
APPENDIX B: POST-NEWTONIAN
EXPANSION OF hUi0
Here we consider a test mass on a bound geodesic orbit
around a nonspinning black hole of mass m2 and obtain the
PN expansion of the relationship hUi0 ðΩr ; Ωϕ Þ. This
calculation provides a powerful check of our PN result
(4.41)–(4.42), because it is based on a different formalism,
it makes use of an alternative parametrization of the motion,
and it is performed using a different coordinate system.
Since the relationships (2.7)–(2.10) cannot be inverted
analytically to yield the expressions for the parameters p
and e as functions of the frequencies Ωr ¼ 2π=T r0 and
Ωϕ ¼ Φ0 =T r0 , we shall work perturbatively, expanding all
quantities in powers of the small parameter 1=p. From T r0
and Φ0 we define the invariant parameters y ≡ ðm2 Ωϕ Þ2=3
and λ ≡ 3yðΦ0 =2π − 1Þ−1 [recall Eq. (4.48)]. Expanding
the formulas (2.7)–(2.10) up to 3PN order, we obtain
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AKCAY et al.
PHYSICAL REVIEW D 91, 124014 (2015)
pffiffiffiffi 1
je
2ð1 − je Þ
17
þ
½1 − je þ 5je ½je − je y¼
1þ
p
2
p
p2
133
40 3 1
3=2
5=2
2
−3
− 48je − 35je þ 27je þ 25je − je
þ
þ oðp Þ ;
3
3
p3
11 7
1
75 23
73 2 1
3=2
λ ¼ je 1 −
þ j
þ − þ je − 5je þ je
4 4 e p
16 8
16
p2
1849 849
45 3=2 17 2 95 5=2 2341 3 1
−3
þ
j − je − je þ je −
j
þ −
þ oðp Þ ;
96
64 e 4
16
4
192 e p3
ðB1aÞ
ðB1bÞ
where we introduced the notation je ≡ 1 − e2 . In the limit of vanishing eccentricity, e → 0, we have the simple relation
y ¼ p−1 þ oðp−4 Þ. Actually, we know that for circular orbits the relation y ¼ 1=p holds exactly, such that in Schwarzschild
coordinates the semimajor axis coincides with an invariant measure of the orbital radius. [This, however, is no longer true at
OðqÞ in the GSF approximation.] For circular orbits, the 3PN-accurate relationship between the invariants y and λ then
reads
9
9
27
λ ¼ 1 − y − y2 − y3 þ oðy3 Þ:
2
4
4
ðB2Þ
Inverting the relations (B1) yields expressions for the semilatus rectum p and eccentricity e (or equivalently je ¼ 1 − e2 ) as
functions of the invariant parameters y and λ. Up to 3PN order, we find
1 y
1 19
9
5
151 2
¼
1þ
−
yþ
−
þ
y
p λ
4 4λ
16 16λ 8λ2
65
5
25
1
2255 3
3
p
ffiffi
ffi
y þ oðy Þ ;
þ
−
þ
þ
−
ðB3aÞ
64 4 λ 64λ 4λ2 32λ3
7 11
5
63
13
je ¼ λ 1 þ
þ
y þ 2 þ pffiffiffi þ
−
y2
4 4λ
λ 16λ 16λ2
5 145 221
95
289
263
3
3
y þ oðy Þ :
þ
þ þ pffiffiffi þ
−
þ
ðB3bÞ
6 8 λ 32λ 8λ3=2 64λ2 192λ3
We now have all the pieces required to compute the relation hUi0 ðy; λÞ up to the required PN order. The generalized
redshift is defined as
Z T
Z 2π
r0
1
1
dt0 dχ −1
hUi0 ≡
ut0 ðτ0 Þdτ0 ¼
;
ðB4Þ
T r0 0
T r0 0 dχ ut0 ðχÞ
where T r0 is the proper time period of the radial motion. From the expressions (2.4), (2.5), (2.7a) and (2.8), we find
pffiffiffiffi 21
pffiffiffiffi
je 3
1
55 2 1
3=2
hUi0 ¼ 1 þ
þ 6 je − je
þ 23 je − 6je − 12je þ je
8
p
16
p 2
p2
249 pffiffiffiffi
75 5=2 525 3 1
3=2
2
−3
je − 24je − 105je þ 12je þ je −
j
þ
þ oðp Þ :
ðB5Þ
2
4
128 e p3
Finally, substituting for ðp; eÞ in terms of ðy; λÞ in Eq. (B5), using (B3), we obtain the 3PN-accurate coordinate-invariant
relation,
3
3
6
3 2
41
57
3
37
15 3
p
ffiffi
ffi
p
ffiffi
ffi
hUi0 ¼ 1 þ y þ
þ
− y þ − þ
− −
y
þ
2
8
16 4 λ 2λ 4λ3=2 2λ2
λ λ
605
117
411 1755
21
1797 20 4
y þ oðy4 Þ:
þ −
þ 2þ
−
ðB6Þ
þ pffiffiffi þ
−
3=2
128 64 λ 8λ 32λ
4λ
64λ5=2 λ3
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COMPARISON BETWEEN SELF-FORCE AND POST- …
PHYSICAL REVIEW D 91, 124014 (2015)
In the circular-orbit limit, we may introduce the
PN expansion (B2) for λðyÞ in (B6), expand in powers
of y up to the appropriate PN order, and recover the
3PN expansion of the fully relativistic result U ⊙ ¼
ð1 − 3yÞ−1=2 . Although the result (B6) can in principle
be extended up to an arbitrarily high PN order, we only
need here the 3PN approximation to the exact result.
Comparing with the formula (4.50a) derived from our
3PN calculation valid for any mass ratio, we find perfect
agreement.
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