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Post-Newtonian Theory and the Two-Body Problem 161
It is convenient to perform a change of phase, from the actual orbital phase to
the new phase variable
D
2G M !
c
3
ln
!
!
0
; (116)
where M is the binary’s total mass monopole moment
15
and !
0
D
1
4u
0
expŒ
11
12
Cis
related to the constant u
0
r
0
=c entering the tail integrals in Eq. 60. The logarith-
mic term in corresponds physically to some spreading of the different frequency
components of the wave along the line of sight from the source to the detector, and
expresses the tail effect as a small delay in the arrival time of gravitational waves.
This effect, although of formal 1.5PN order in Eq. 116, represents in fact a very
small modulation of the orbital phase: compared to the dominant phase evolution
whose order is that of the inverse of 2.5PN radiation reaction, this modulation is of
order 4PN and can thus be neglected with the present accuracy.
The spherical harmonic modes of the polarization waveforms can now be
obtained at 3PN order using the angular integration formula (55). We start from the
expressions of the wave polarizations h
C
and h
as functions of the inclination an-
gle i and of the phase . We use the known dependence of the spherical harmonics
(54) on the azimuthal angle. Denoting h h
C
ih
D h.i; / we find that the
latter angular integration becomes
h
`m
D .i/
m
e
i m
Z
2
0
d
0
Z
0
di sin ih.i;
0
/Y
`m
2
.i;
0
/; (117)
exhibiting the azimuthal factor e
i m
appropriate for each mode. Let us introduce a
normalized mode coefficient H
`m
starting by definition with one at the Newtonian
order for the dominant mode having .`; m/ D .2; 2/. This means posing
h
`m
D
2G mx
Rc
2
r
16
5
H
`m
e
im
: (118)
All the modes have been given in [23] up to the 3PN order. The dominant mode
.2; 2/, which is primarily needed for the comparison between post-Newtonian cal-
culations and numerical simulations, reads at 3PN order
H
22
D 1 C
107
42
C
55
42
x C 2 x
3=2
C
2173
1512
1069
216
C
2047
1512
2
x
2
C
107
21
C
34
21
24i
x
5=2
15
The mass monopole M differs from m D m
1
C m
2
as it includes the contribution of the gravi-
tational binding energy. At 1PN order it is given for circular orbits by M D mŒ1
2
CO.c
4
/.
162 L. Blanchet
C
27027409
646800
856
105
C C
2
3
2
428
105
ln.16x/ C
428
105
i
C
278185
33264
C
41
96
2
20261
2772
2
C
114635
99792
3
x
3
C O
1
c
7
:
(119)
3.6 Spin–Orbit Contributions in the Energy and Flux
To successfully detect the gravitational waves emitted by spinning, precessing
binaries, and to estimate the binary parameters, spin effects should be included
in the templates. For maximally spinning compact bodies the spin–orbit coupling
(linear in the spins) appears dominantly at the 1.5PN order, while the spin–spin
one (which is quadratic) appears at 2PN order. The spin effect on the free motion
of a test particle was obtained by Papapetrou [75] in the form of a coupling to
curvature. Seminal later works by Barker and O’Connell [7, 8] obtained the lead-
ing order spin–orbit and spin–spin contributions in the post-Newtonian equations
of motion. Based on these works, the spin–orbit and spin–spin terms were obtained
in the radiation field [59, 63, 65, 72], enabling the derivation of the orbital phase
evolution (the crucial quantity that determines the templates). Finding the 1PN cor-
rections to the leading spin–orbit coupling in both the (translational) equations of
motion and radiation field was begun in [73, 87] and completed in [13, 53]. The
result [53] for the equations of motion was confirmed by an alternative derivation
based on the ADM-Hamiltonian formalism [45].
In Section 3.1 we discussed a covariant formalism for spinning particles [6, 49,
96–98]. We want now to find a convenient three-dimensional variable for the spin.
Restricting ourselves to spin–orbit effects, that is, neglecting O.S
2
/, we can write
the components of the spin tensor S
˛ˇ
A
as
S
0i
A
D
1
c
p
.g/
A
"
ij k
u
A
j
S
A
k
; (120a)
S
ij
A
D
1
c
p
.g/
A
"
ij k
"
u
A
0
S
A
k
C u
A
k
v
l
A
c
S
A
l
#
: (120b)
We have used the momentum–velocity relation (76) and have taken into account
the spin condition (73). A first possibility is to adopt as the basic spin variable the
contravariant components of the spin covector S
A
i
in Eq. 120, which are obtained
by raising the index by means of the contravariant spatial metric, viz
S
A
D .S
i
A
/ with S
i
A
.
ij
/
A
S
A
j
; (121)
where
ij
is the inverse of the covariant spatial metric
ij
g
ij
that is, satisfies
ik
kj
D ı
i
j
. The choice of spin variable (121) has been adopted in [73, 87]. How-
ever, to express final results (to be used in gravitational-wave templates) it is better
Post-Newtonian Theory and the Two-Body Problem 163
to use a different set of spin variables characterized by having some conserved
Euclidean lengths. Such spins with constant Euclidean magnitude will be denoted
by S
c
A
. They can be computed in a straightforward way at a given post-Newtonian
order in terms of the previous variables (121). For instance we find up to 1PN order,
S
c
A
D
1 C
.U /
A
c
2
S
A
1
2c
2
.v
A
S
A
/ v
A
C O
1
c
4
: (122)
The (regularized) gravitational potential .U /
A
is defined by Eq. 81. The constant-
magnitude spin variable S
c
A
obeys a spin precession equation that is necessarily of
the form
dS
c
A
dt
D ˝
A
S
c
A
: (123)
Indeed this equation implies that jS
c
A
jDconst. The precession angular frequency
vector ˝
A
for two-body systems has been computed for the leading spin–orbit and
spin–spin contributions [63] and for the 1PN correction to the spin–orbit [13,45,53].
For two bodies we conveniently use the following combinations (introduced in [63])
of the two spins:
S
c
S
c
1
C S
c
2
; (124a)
˙
c
m
S
c
2
m
2
S
c
1
m
1
: (124b)
Furthermore, recalling the orthonormal triad fn; ;
O
zg used in Section 3.4,where
O
z
is the unit vector in the direction perpendicular to the orbital plane, we denote by
S
c
z
S
c
O
z and ˙
c
z
˙
c
O
z the projections along that perpendicular direction.
The spin–orbit terms have been computed at 1PN order both in the equations of
motion and in the radiation field. In the equations of motion they correct the orbital
frequency and invariant conserved energy with terms at orders 1.5PN and 2.5PN.
In the presence of spins the energy gets modified to E D E
mono
C E
spin
where the
monopole part has been obtained in Eq. 110 and where the spin terms read
E
spin
D
c
2
2G m
x
x
3=2
14
3
S
c
z
C 2 ˙
c
z
Cx
5=2
11
61
9
S
c
z
C
3
10
3
˙
c
z
CO
1
c
6
:
(125)
We recall that .m
1
m
2
/=m; see footnote 13. This expression is valid for quasi-
circular orbits, and we neglect the spin–spin terms. Similarly the gravitational-wave
flux will be modified at the same 1.5PN and 2.5PN orders. Posing F D F
mono
C
F
spin
where F
mono
is given by Eq. 111,wefind
164 L. Blanchet
F
spin
D
32c
5
5G
2
m
2
2
x
5
x
3=2
4S
c
z
5
4
˙
c
z
Cx
5=2
9
2
C
272
9
S
c
z
C
13
16
C
43
4
˙
c
z
C O
1
c
6
:
(126)
Having in hand the spin contributions in E and F , we can deduce the evolution
of the orbital phase from the energy balance equation (101). In absence of preces-
sion of the orbital plane, for example, for spins aligned or anti-aligned with the
orbital angular momentum, the gravitational-wave phase will reduce to the “carrier”
phase
GW
2 (keeping only the dominant harmonics), where is the orbital
phase that is obtained by integrating the orbital frequency. However, in the general
case of non-aligned spins, we must take into account the effect of precession of
the orbital plane induced by spin modulations. Then the gravitational-wave phase
is given by ˚
GW
D
GW
C ı
GW
, where the precessional correction ı
GW
arises
from the changing orientation of the orbital plane, and can be computed by standard
methods using numerical integration [2]. Thus, the carrier phase
GW
constitutes the
main theoretical output to be provided for the gravitational-wave templates, and can
directly be obtained numerically from using the integration formula (115).
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Post-Newtonian Methods: Analytic Results
on the Binary Problem
Gerhard Sch
¨
afer
Abstract A detailed account is given on approximation schemes to the Einstein
theory of general relativity where the iteration starts from the Newton theory of
gravity. Two different coordinate conditions are used to represent the Einstein field
equations, the generalized isotropic ones of the canonical formalism of Arnowitt,
Deser, and Misner and the harmonic ones of the Lorentz-covariant Fock-de Donder
approach. Conserved quantities of isolated systems are identified and the Poincar´e
algebra is introduced. Post-Newtonian (PN) expansions are performed in the near
and far (radiation) zones. The natural fitting of multipole expansions to PN schemes
is emphasized. The treated matter models are ideal fluids, pure point masses, and
point masses with spin and mass-quadrupole moments modeling rotating black
holes. Various Hamiltonians of spinning binaries are presented in explicit forms
to higher PN orders. The delicate use of black holes in PN expansion calculations
and of the Dirac delta function in general relativity find discussions.
1 Introduction
In the weak-field slow-motion limit of any theory of gravity the Newtonian theory of
gravity comes into play because it describes the motion and structure of gravitating
objects very well in that regime. According to current knowledge from experiments
and observations, the most reliable theory of gravity is the Einstein theory of general
relativity [72].
Most of the objects in the Universe seem to have velocities v, which are small
compared with the speed of light (in the following denoted by c), that is, v=c . 1=3.
In those cases it is to be expected that the Newtonian theory is a good starting point
for an iteration scheme toward full general relativity thinking in terms of expansion
G. Sch¨afer (
)
Friedrich-Schiller-Universit¨at Jena, Theoretisch-Physikalisches Institut,
Max-Wien-Platz 1, D-07743 Jena, Germany
e-mail: G.Schaefer@tpi.uni-jena.de
L. Blanchet, A. Spallicci, and B. Whiting (eds.), Mass and Motion in General Relativity,
Fundamental Theories of Physics 162, DOI 10.1007/978-90-481-3015-3
6,
c
Springer Science+Business Media B.V. 2011
167
168 G. Sch¨afer
of general relativity in powers of dimensionless v=c. The conservative nature of
most of the phenomena, that is, no measurable gravitational radiation damping, the
power series has to be one with ordering parameter .v=c/
2
(recall v
2
=c
2
. 1=10 in
the regime of our interest) because of motion-inversion symmetry in those cases. We
shall see later that at the order .v=c/
5
the gravitational radiation damping enters for
the first time. Calling the order .v=c/
2n
the nth PN order (n PN), the gravitational
dissipation (because of gravitational radiation damping from radiation emission)
enters at the 2.5 PN order so the exponent n will also include half-integer numbers.
In case of bound systems the virial theorem holds, which tells us that v
2
GM=r ,
where G is the Newtonian gravitational constant, r a typical distance of two bod-
ies or a typical radius of one body, and M the total mass of the system. So an
expansion of general relativity in powers of .v=c/
2
is at the same time an expansion
in powers of GM=c
2
r if bound systems are considered. From the dynamics of bound
systems the dynamics of low-velocity scattering is straightforwardly obtained.
Depending on the mathematical representation of the Einstein theory (choice of
coordinates, choice of variables, etc.) and on the physical aspects under investiga-
tion (physics in the near zone, physics in the far zone, etc.), there exist many of those
expansions that all are called PN expansions. The crucial ordering parameter in all
PN expansions is always 1=c
2
. The Einstein field equations as well as the equa-
tions of motion that follow from them by integrability conditions can be formally
expanded in powers of 1=c
2
. The solutions of these equations, however, cannot be
expanded in this way, in general. Only under conditions where no gravitational radi-
ation is present an all-over-in-space expansion in powers of 1=c
2
is feasible. This is
the case for stationary systems only. Yet for those parts of a radiating system where
radiation plays no role, that is, in the conservative parts, an expansion in powers of
1=c
2
can be carried out too. In general, PN expansions in powers of 1=c are feasible
in the near zone (r with a typical wavelength of the gravitational radiation
and r the radius of a sphere enclosing the matter source) and in the far or radiation
zone (r ) of a radiating system, but they are not analytic in 1=c because c is
showing up in log-terms. In the near zone, up to 3.5 PN order, which means order
1=c
7
D .1=c
2
/
3:5
, a PN expansion in powers of 1=c is valid, where the even powers
are connected with the conservative dynamics and the odd powers with the dissi-
pative radiation reaction dynamics. From the 4 PN order on (corresponding to 1.5
PN order in the far zone), log-c terms enter via tail effects which emerge from back
scattering [4]. Post-Minkowskian (PM) series are global series in space where only
the weak-field limit is assumed without restrictions to the velocities of the bodies.
Here, the ordering parameter is G or, dimensionless GM=c
2
r, which is equivalent
to a nonlinearity expansion of the Einstein field equations [4]. If the virial theorem
holds, namely, .GM=c
2
r/ .v=c/
2
, PM series can be further expanded into PN
series. Having much less information on their respective orders, the PN series are
much easier to be worked out analytically, compare, for example, the 1 PN point-
mass Hamiltonian, see [51], with the 1 PM point-mass Hamiltonian, only recently
achieved in [52].
The most extensively performed PN calculations in the literature can be
divided into two classes, the class applying harmonic coordinate conditions,
Post-Newtonian Methods: Analytic Results on the Binary Problem 169
see, for example, [33], and the other class using Arnowitt–Deser–Misner (ADM)
generalized isotropic coordinate conditions [1]. Whenever detailed comparisons be-
tween the two classes have been made, agreement of the results could be achieved,
see, for example, [12, 23]. This is quite an important aspect in view of the use
of Dirac delta functions for modeling black holes in explicit calculations. These
functions are a-priori problematic in a nonlinear theory like general relativity so
sophisticated regularization methods have to be employed. The only regularization
method that turned out to be successful to the highest orders of the explicit calcula-
tions is the dimensional one [6,22]. In Section 3.4 a detailed account of black holes
together with their representing Dirac delta functions will be given because they
blatantly violate the weak-field condition. In this case, only external velocities of
the black holes can be small and their relative gravitational interaction weak, and
only in this sense black holes and Dirac delta functions can fit into PN schemes.
The binary point-mass dynamics at 3 PN order and the gravitational waves emit-
ted hereof (3 PN wave generation) could have been managed successfully through
only dimensional regularization [6, 7, 22]. Extended body calculations have been
performed too but they are much more complicated and still not fully under control
at 3 PN order [56]. Only a surface-integral-based method of the Einstein–Infeld–
Hoffmann type did succeed too [34,44].
The many-body systems treated most successfully with PN approximation meth-
ods are point-mass systems composed from two or more objects and spinning binary
systems with even rotationally deformed components; for the most recent results
with spin, see [5, 25, 31,40,41,67,69].
Recently, an effective field theory approach has been advocated for PN cal-
culations [36]. Applications to point-mass systems and to systems with spinning
components have already been performed through higher PN orders [35, 57, 58].
An obvious difference in the spin dynamics between [67]and[58] found clarifica-
tionin[68] showing the correctness of the result in [67].
In this article, Latin indices from the mid–alphabet are mostly running from
1to3,v D .v
i
/; v
2
D v
i
v
i
, @
t
and @
i
denote the partial derivatives with respect
to the time and space coordinates t and x
i
, respectively, and the functional deriva-
tive (Fr´echet derivative) of a functional, say FŒf, with respect to a function f
takes the form ıF Œf =ıf . Greek indices are mostly running from 0 to 3, whereby
x
0
D ct. The signature of the four-dimensional metric g
is C2. Particles are
numbered with indices from the beginning of the Latin alphabet.
2 Systems in Newtonian Gravity in Canonical Form
In the Newtonian theory, the equations governing the motion of gravitating ideal
fluids are (i) the equation for the conservation of mass,
@
t
%
C div.%
v/ D 0; (1)
170 G. Sch¨afer
where %
is the mass density and v the velocity filed of the fluid, (ii) the equations
of motion,
%
@
t
v C
%
2
grad v
2
%
v curl v Dgrad p C %
grad U; (2)
where p is the pressure in the fluid and U the gravitational potential, and (iii) the
equation of state, using internal energy density and specific enthalpy h,
D .%
;s/ with d D hd%
C %
T ds; or dp D %
dh %
T ds; (3)
with (iv) the conservation law for the specific entropy s along the flow lines,
@
t
s C v grad s D 0: (4)
The gravitational potential reads,
U.x;t/D G
Z
d
3
x
0
%
.x
0
;t/
jx x
0
j
: (5)
It results from (v) the Newtonian field equation,
U D4G%
: (6)
is the Laplacian and jjmeans the standard Euclidean distance.
Written in form of Hamilton equations of motion, that is, @
t
A.x;t/DfA.x;t/;Hg;
the equations above take the forms, (i) the mass conservation equation
@%
@t
D@
i
ıH
ı
i
%
; (7)
notice v
i
D
ıH
ı
i
, (ii) the equations of motion
@
i
@t
D@
j
ıH
ı
j
i
@
i
ıH
ı
j
j
@
i
ıH
ı%
%
C
ıH
ıs
@
i
s; (8)
and (iv) the entropy conservation law
@s
@t
D
ıH
ı
i
@
i
s; (9)
where the Hamiltonian is given by H D H.%
;
i
;s/with
i
the linear momentum
density of the fluid, see [42]. Here, use has been made of the kinematical Lie–
Poisson bracket relations between the fundamental variables
f
i
.x;t/;%
.x
0
;t/gD
@
@x
0i
Œ%
.x
0
;t/ı.x x
0
/; (10)