Blanchet L., Spallicci A., Whiting B. (Eds.) Mass and Motion in General Relativity
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Post-Newtonian Methods: Analytic Results on the Binary Problem 201
where r
denotes the four-dimensional covariant derivative. The Landau–Lifshitz
form of the Einstein field equations fits very well into Lorentz-covariant schemes.
It takes the form [51],
@
@
U
.g
˛ˇ
/ D
16G
c
4
LL
.g
˛ˇ
;@
g
˛ˇ
/; (186)
with U
Dg
g
g
g
and g
D
p
gg
,whereg denotes the
determinant of the metric tensor. t
LL
is known as the Landau–Lifshitz stress–
energy (or energy-momentum) pseudo-tensor of the gravitational field. It is unique
in the sense of symmetry and dependence on the metric coefficients and its first
derivatives only. The equations of motion now read
@
@
@
U
0 ! @
LL
D 0 (EOM); (187)
with
LL
gT
C
c
4
16G
t
LL
.g
˛ˇ
;@
g
˛ˇ
/: (188)
Applying the condition of harmonic coordinates
@
g
D 0 or;@
H
D 0 with H
D
p
gg
; (189)
where
denotes the Minkowski metric, the Einstein field equations in Landau–
Lifshitz form read (so-called relaxed field equations because they do not imply
the equations of motion; rather the condition of harmonic coordinates implies the
equations of motion),
˛ˇ
@
˛
@
ˇ
H
D
16G
c
4
; (190)
where
DgT
C
c
4
16G
(191)
and
DH
˛ˇ
@
˛
@
ˇ
H
C @
˛
H
ˇ
@
ˇ
H
˛
C
1
2
g
g
˛ˇ
@
H
˛
@
H
ˇ
C
1
8
.2g
˛
g
ˇ
g
g
˛ˇ
/.2g
g
g
g
/@
˛
H
@
ˇ
H
g
˛
g
ˇ
@
H
@
˛
H
ˇ
g
˛
g
ˇ
@
H
@
˛
H
ˇ
C g
˛ˇ
g
@
H
˛
@
H
ˇ
: (192)
The
object starts with quadratic nonlinearities of the gravitational field. It is
another stress–energy pseudo-tensor of the gravitational field.
202 G. Sch¨afer
6.1 PM and PN Expansions
The formal retarded solution (resulting under the condition of no incoming radia-
tion) of the inhomogeneous wave equation (190) reads
H
.x;t/ D
4G
c
4
Z
d
3
x
0
.x
0
;t
jx x
0
j
c
Ic
2
/jx x
0
j
1
: (193)
A PM expansion in powers of G can now be introduced in the form,
H
.x;t/ D
1
X
nD1
G
n
H
Œn
.x;tIc/: (194)
If additionally the virial theorem holds,
GM
Rc
2
V
2
c
2
; (195)
where respectively M; R; and V are typical masses, radii, and velocities of the
system in question, the PM expansion may be further expanded into a PN series
in powers of 1=c. Yet, because of the retardation structure of the solution, a PN
expansion is achievable only in the near and far zones and this only in a general-
ized form with log-c terms showing up at higher orders starting from 4 PN, that is,
.1=c
2
/
4
D 1=c
8
, on.
Let us assume now that the matter source is bounded by a sphere with radius R
centered in the origin of the coordinate system and that for the typical gravitational
wave length the relation R holds. The near zone is then defined by jxj.
The formal PN expansion (near-zone PN expansion) is defined by
H
nz
.x;t/ D
4G
c
4
1
X
nD0
.1/
n
nŠ
Z
d
3
x
0
@
n
c
n
@t
n
.x
0
;tIc
2
/jx x
0
j
n1
: (196)
Additionally, the expansion
.x
0
;tIc
2
/ D
1
X
nD1
1
c
2n
.n/
.x
0
;t/ (197)
applies. In the far zone, where r Djxj holds, a formal PN expansion (far-zone
PN expansion) yields
H
fz
.x;t/ D
4G
c
4
r
1
X
nD0
1
nŠ
Z
d
3
x
0
@
n
c
n
@t
n
.x
0
;t
r
c
Ic
2
/.x
0
n/
n
: (198)
Post-Newtonian Methods: Analytic Results on the Binary Problem 203
Here, the expansion
x
0
;t
r
c
Ic
2
D
1
X
nD1
1
c
2n
Œn
x
0
;t
r
c
(199)
applies. The Eqs. 196 and 197 on the one side and (198)and(199) on the other are
somewhat simplified in the sense that they do not show up log-c terms at higher
orders in the expansions that result from the badly defined integrals of non-compact
support. On the other side, as the expressions stand, they are mathematically not
defined at all. Details can be found in the works by Blanchet, Damour, Will, and
collaborators; particularly see the contribution by L. Blanchet in this volume; for
tail terms, also see our Section 6.3.
6.2 PN Expansion in the Near Zone
Up to the 2 PN order the metric coefficients read
g
00
D1 C
2
c
2
V
2
c
4
V
2
C
8
c
6
O
X C V
i
V
i
C
V
3
6
; (200)
g
0i
D
4
c
3
V
i
8
c
5
O
R
i
; (201)
g
ij
D ı
ij
1 C
2
c
2
V C
2
c
4
V
2
C
4
c
4
O
W
ij
: (202)
With the following choice of the matter variables, respectively, mass, mass-current,
and stress density,
D
T
00
C T
ii
c
2
;
i
D
T
0i
c
;
ij
D T
ij
; (203)
the 2 PN potentials can be put into the form
V.x;t/ D G
ret
f
4G
g
G
Z
d
3
z
jx zj
.z;t jx zj=c/; (204)
V
i
D G
ret
f
4G
i
g
; (205)
O
W
ij
D G
ret
˚
4G.
ij
ı
ij
kk
/ @
i
V@
j
V
; (206)
O
R
i
D G
ret
4G.V
i
V
i
/ 2@
k
V@
i
V
k
3
2
@
t
V@
i
V
; (207)
204 G. Sch¨afer
O
X D G
ret
4GV
ii
C 2V
i
@
t
@
i
V C V@
2
t
V
C
3
2
.@
t
V/
2
2@
i
V
j
@
j
V
i
C
O
W
ij
@
2
ij
V
: (208)
The potentials of the orders 2.5 PN and 3.5 PN are radiation–reaction potentials.
They are most compactly given under Burke–Thorne coordinate conditions, reading
U
reac
.x;t/ D
G
5c
5
x
ij
O
M
Œ5
ij
.t/C
G
c
7
1
189
x
ij k
O
M
Œ7
ij k
.t/
1
70
x
kk
x
ij
O
M
Œ7
ij
.t/
; (209)
U
reac
i
.x;t/ D
G
c
5
1
21
Ox
ij k
O
M
Œ6
jk
.t/
4
45
ij k
x
jm
O
S
Œ5
km
.t/
; (210)
where the source multipole moments are given by
O
M
ij
D
Z
d
3
y
Oy
ij
C
1
14c
2
y
kk
Oy
ij
@
2
t
20
21c
2
Oy
ij k
@
t
k
; (211)
O
M
ij k
D
Z
d
3
y Oy
ij k
; (212)
O
S
ij
D
Z
d
3
y
km<i
Oy
j>k
m
: (213)
The used definitions read y
ij
y
i
y
i
, y
<ij >
Oy
ij
D STF.y
ij
/, and, for example,
O
M
Œ7
ij k
indicates the seventh time derivative of M
ij k
. Explicitly, the 1 PN metric
including the gravitational radiation reaction through 3.5 PN order is given by
g
00
D1 C
2
c
2
.U C U
reac
/ C
1
c
4
@
2
t
2U
2
4U U
reac
; (214)
g
0i
D
4
c
3
.U
i
C U
reac
i
/; (215)
g
ij
D ı
ij
1 C
2
c
2
.U C U
reac
/
; (216)
where the potentials have the integral representations
U.x;t/D G
Z
d
3
y
jx yj
.y;t/; (217)
U
i
.x;t/ D G
Z
d
3
y
jx yj
i
.y;t/; (218)
Post-Newtonian Methods: Analytic Results on the Binary Problem 205
.x;t/D G
Z
d
3
y jx yj.y;t/: (219)
These integrals are well defined. Evidently, multipole expansion and PN expansion
nicely fit together; see also [59].
6.3 PN Expansion in the Far Zone
In the far zone, the multipole expansion of the transverse-traceless (TT) part of
the gravitational field, obtained by algebraic projection with P
ij km
.n/, reads, for
example, [70],
H
ij
fzTT
.x;t/ D
G
c
4
P
ij km
.n/
r
1
X
lD2
(
1
c
2
l2
2
4
lŠ
M
Œl
kmi
3
i
l
t
r
c
N
i
3
i
l
C
1
c
2
l1
2
8l
.l C 1/Š
pq.k
S
Œl
m/pi
3
i
l
.t
r
c
/n
q
N
i
3
i
l
)
; (220)
where the leading mass-quadrupole tensor takes the form, for example, [15],
M
ij
t
r
c
D
b
M
ij
t
r
c
C
2Gm
c
3
Z
1
0
d v
ln
v
2b
C
11
12
b
M
Œ2
ij
t
r
c
v
C O
1
c
4
(221)
with
r
D r C
2Gm
c
2
ln
r
cb
C O
1
c
3
showing a leading-order tail term. Notice the modification of the standard PN
expansion through tail terms. The Eq. 220 nicely shows that multipole expansions
in the far zone also do induce PN expansions.
The gravitational luminosity is generally given by (H
ij
fzTT
Dh
TTfz
ij
),
L.t/ D
c
3
32G
I
fz
@
t
H
ij
fzTT
2
r
2
d˝: (222)
206 G. Sch¨afer
Through 1.5 PN order, the luminosity explicitly reads,
L.t/ D
G
5c
5
1
X
nD0
1
c
2
n
O
L
n
.t/
D
G
5c
5
M
Œ3
ij
M
Œ3
ij
C
1
c
2
5
189
M
Œ4
ij k
M
Œ4
ij k
C
16
9
S
Œ3
ij
S
Œ3
ij
: (223)
On reasons of energy balance, for any representation of the Einstein theory, the
time-averaged energy loss has to fulfill a relation of the form
<
dE.t
r
c
/
dt
> D < L.t/ >; (224)
where the time averaging procedure takes into account typical periods of the system.
The derivation of this equation in Section 3.6 is known to be valid for the first two
radiation emissions and reaction levels.
7 Energy Loss and Gravitational Wave Emission
The energy flux to n PN order in the far zone, denoted n PN(fz), implies energy loss
to (n C 5/2) PN order in the near zone, denoted (n C 5/2) PN(nz). Hereof it follows
that energy-losscalculations are quite efficient via energy-fluxcalculations. Because
of this we will apply the balance property between emitted and lost energies to some
PN orders to easily derive the energy loss from the energy flux. In general, only after
averaging over orbital periods will both expressions coincide (see Eq. 224). In the
case of circular orbits, however, this averaging procedure is not needed.
7.1 Orbital Decay to 4 PN Order
The binding energy of our binary system on circular orbits is given by E
circ
.There-
fore, for the energy loss to 4 PN order, we get
dE
circ
dt
D L D
32c
5
5G
2
x
5
1
1247
336
C
35
12
x C 4x
3=2
; (225)
where the 1.5 PN(fz) energy flux is taken from Ref. [8] where the 2 PN(fz) energy
flux also can be found; for the 3.5 PN(fz) energy flux see [11,13].
Taking into account the Eq. 129 we obtain a differential equation for x which is
easily solved with accuracy 1=c
8
. In terms of the dimensionless time variable
Post-Newtonian Methods: Analytic Results on the Binary Problem 207
D
c
3
5Gm
.t
c
t/; (226)
where t
c
denotes the coalescence time, the solution reads [8],
x D
1
4
1=4
1 C
743
4032
C
11
48
1=4
1
5
3=8
: (227)
Taking into account the relation between phase and frequency
d
dt
D !, respec-
tively,
d
d
D
5
x
3=2
, the phase evolution results in
D
c
1
5=8
1 C
3715
8064
C
55
96
1=4
3
4
3=8
: (228)
7.2 Gravitational Waveform to 1.5 PN Order
The radiation field can be decomposed into two orthogonal polarization states. The
polarization states h
C
and h
are defined by
h
C
D
1
2
.u
i
u
j
v
i
v
j
/h
TT
ij
; (229)
h
D
1
2
.u
i
v
j
C v
i
u
j
/h
TT
ij
; (230)
where u and v denote two vectors in the polarization plane forming an orthogonal
right-handed triad with the direction n from the source to the detector. The detector
is directly sensitive to a linear combination of the polarization waveforms h
C
and
h
, namely,
h.t/ D F
C
h
C
.t/ C F
h
.t/; (231)
where F
C
and F
are the so-called beam-pattern functions of the detector depending
on two angles giving the direction n of the source as seen from the detector and
a polarization angle specifying the orientation of the vectors u and v around that
direction.
For our binary system, the two polarizations h
C
and h
are chosen such that
the polarization vectors u and v lie, respectively, along the major and minor axis
of the projection onto the plane of the sky of the circular orbit, with u oriented
toward the ascending node, the point at which black hole 1 crosses the plane of the
sky moving toward the observer. The result, to 1.5 PN(fz) order, reads [14](the2
PN(fz) wave form is given therein too)
h
C;
D
2Gx
c
2
r
h
H
Œ0
C;
C x
1=2
H
Œ1=2
C;
C xH
Œ1
C;
C x
3=2
H
Œ3=2
C;
i
; (232)
208 G. Sch¨afer
where the plus polarization is given by
H
Œ0
C
D.1 C c
2
i
/cos2 ; (233)
H
Œ1=2
C
D
s
i
8
ım
m
Œ.5 C c
2
i
/cos 9.1 C c
2
i
/cos3 ; (234)
H
Œ1
C
D
1
6
Œ19 C 19c
2
i
2c
4
i
.19 11c
2
i
6c
4
i
/cos2
4
3
s
2
i
.1 C c
2
i
/.1 3/cos4 ; (235)
H
Œ3=2
C
D
s
i
192
ım
m
fŒ57 C 60c
2
i
c
4
i
2.49 12c
2
i
c
4
i
/cos
27
2
Œ73 C 40c
2
i
9c
4
i
2.25 8c
2
i
9c
4
i
/cos3
C
625
2
.1 2/s
2
i
.1 C c
2
i
/cos5 g2.1 C c
2
i
/cos2 ; (236)
and the cross polarization by
H
Œ0
D2c
i
sin2 ; (237)
H
Œ1=2
D
3
4
s
i
c
i
ım
m
Œsin 3sin3 ; (238)
H
Œ1
D
c
i
3
Œ17 4c
2
i
.13 12c
2
i
/sin2
8
3
c
i
s
2
i
.1 3/sin4 ; (239)
H
Œ3=2
D
s
i
c
i
96
ım
m
fŒ63 5c
2
i
2.23 c
2
i
/sin
27
2
Œ67 15c
2
i
2.19 15c
2
i
/sin3
C
625
2
.1 2/s
2
i
sin5 g4c
i
sin2 ; (240)
where c
i
Dcosi and s
i
Dsini and i denotes the inclination angle between the di-
rection of the detector, as seen from the binary’s center-of-mass, and the normal to
the orbital plane which is assumed to be right-handed with respect to the sense of
motion so that 0 i . ım D m
1
m
2
, and the phase variable is given by
D 3x
3=2
ln
x
x
0
; (241)
where is the actual orbital phase of the binary, namely, the angle oriented in
the sense of motion between the ascending node and the direction of black hole
Post-Newtonian Methods: Analytic Results on the Binary Problem 209
1( D0 mod 2 when the two black holes lie along u, with black hole 1 at the
ascending node). The logarithmic phase modulation originates from the propaga-
tion of tails in the wave zone. The constant scale x
0
can be chosen arbitrarily; it
relates to the arbitrary constant b in the Eq. 221. For details on higher order PN
levels, see, for example, [4].
Acknowledgements I thank the organizers of the Orl´eans School on Mass for their kind invitation
and Luc Blanchet for helpful remarks on the manuscript.
References
1. R. Arnowitt, S. Deser, C.W. Misner, in Gravitation: an Introduction to Current Research,ed.
by L. Witten (Wiley, New York, 1962), pp. 227–265
2. B.M. Barker, R.F. O’Connell, Gen. Rel. Grav. 11, 149 (1979)
3. L. Blanchet, Phys. Rev. D 65, 124009 (2002)
4. L. Blanchet, Living Rev. Rel. 9,URL:http://www.livingreviews.org/lrr-2006-4
5. L. Blanchet, A. Buonanno, G. Faye, Phys. Rev. D 74, 104034 (2006); Erratum, ibidem 75,
049903(E) (2007); Erratum, ibidem 81, 089901(E) (2010)
6. L. Blanchet, T. Damour, G. Esposito-Far`ese, Phys. Rev. D 69, 124007 (2004)
7. L. Blanchet, T. Damour, G. Esposito-Far`ese, B.R. Iyer, Phys. Rev. Lett. 93, 091101 (2004)
8. L. Blanchet, T. Damour, B.R. Iyer, Phys. Rev. D 51, 5360 (1995)
9. L. Blanchet, T. Damour, G. Sch¨afer, Mon. Not. R. Astron. Soc. 242, 289 (1990)
10. L. Blanchet, G. Faye, J. Math. Phys. 42, 4391 (2001)
11. L. Blanchet, G. Faye, B.R. Iyer, B. Joguet, Phys. Rev. D 65, 061501(R) (2002); Erratum, ibidem
71, 129902(E) (2005)
12. L. Blanchet, B.R. Iyer, Class. Q. Grav. 20, 755 (2003)
13. L. Blanchet, B.R. Iyer, B. Joguet, Phys. Rev. D 65, 064005 (2002); Erratum, ibidem 71,
129903(E) (2005)
14. L. Blanchet, B.R. Iyer, C.M. Will, A.G. Wiseman, Class. Q. Grav. 13 (1996)
15. L. Blanchet, G. Sch¨afer, Class. Q. Grav. 10, 2699 (1993)
16. D.R. Brill, R.W. Lindquist, Phys. Rev. 131, 471 (1963)
17. A. Buonanno, T. Damour, Phys. Rev. D 59, 084006 (1999)
18. T. Damour, Phys. Rev. D 64, 124013 (2001)
19. T. Damour, P. Jaranowski, G. Sch¨afer, Phys. Rev. D 62, 044024 (2000)
20. T. Damour, P. Jaranowski, G. Sch¨afer, Phys. Rev. D 62, 084011 (2000)
21. T. Damour, P. Jaranowski, G. Sch¨afer, Phys. Rev. D, 62, 021501(R) (2000); Erratum, ibidem
63, 029903(E) (2001)
22. T. Damour, P. Jaranowski, G. Sch¨afer, Phys. Lett. B 513, 147 (2001)
23. T. Damour, P. Jaranowski, G. Sch¨afer, Phys. Rev. D 63, 044021 (2001); Erratum, ibidem 66,
029901(E) (2002)
24. T. Damour, P. Jaranowski, G. Sch¨afer, in Proc.11th Marcel Grossmann Mtg, Berlin 23–29
July 2006, ed. by H. Kleinert, R.T. Jantzen, R. Ruffini (World Scientific, Singapore, 2008),
pp. 2490–2492
25. T. Damour, P. Jaranowski, G. Sch¨afer, Phys. Rev. D 77, 064032 (2008)
26. T. Damour, P. Jaranowski, G. Sch¨afer, Phys. Rev. D 78, 024009 (2008)
27. T. Damour, G. Sch¨afer, J. Math. Phys. 32, 127 (1991)
28. T. Damour, G. Sch¨afer, Nuovo Cimento B 101, 127 (1988)
29. P.D. D’Eath, Phys. Rev. D 12, 2183 (1975)
30. F. Estabrook, H. Wahlquist, S. Christensen, B. DeWitt, L. Smarr, E. Tsiang, Phys. Rev. D 7,
2814 (1973)
210 G. Sch¨afer
31. G. Faye, L. Blanchet, A. Buonanno, Phys. Rev. D 74, 104033 (2006)
32. G. Faye, P. Jaranowski, G. Sch¨afer, Phys. Rev. D 69, 124029 (2004)
33. V. Fock, The Theory of Space Time and Gravitation, 2nd rev. edn. (Pergamon Press, New York,
1966)
34. T. Futamase, Y. Itoh, Living Rev. Rel. 10,URL:http://www.livingreviews.org/lrr-2007-2
35. J.B. Gilmore, A. Ross, Phys. Rev. D 78, 124021 (2008)
36. W. Goldberger, I.Z. Rothstein, Phys. Rev. D 73, 104029 (2006)
37. A. Gopakumar, G. Sch¨afer, Phys. Rev. D 77, 104023 (2008)
38. M. Hannam, S. Husa, D. Pollney, B. Br¨ugmann, N. O’Murchadha, Phys. Rev. Lett. 99, 241102
(2007)
39. A. Hanson, T. Regge, Ann. Phys. (N.Y.) 87, 498 (1974)
40. S. Hergt, G. Sch¨afer, Phys. Rev. D 77, 104001 (2008)
41. S. Hergt, G. Sch¨afer, Phys. Rev. D 78, 124004 (2008)
42. D.D. Holm, Physica D 17, 1 (1985)
43. L. Infeld, J. Pleba´nski, Motion and Relativity (Pergamon, New York, 1960)
44. Y. Itoh, Phys. Rev. D 69, 064018 (2004)
45. P. Jaranowski, G. Sch¨afer, Phys. Rev. D 55, 4712 (1997); C. K¨onigsd¨orffer, G. Faye, G. Sch¨afer,
Phys. Rev. D 68, 044004 (2003)
46. P. Jaranowski, G. Sch¨afer, Phys. Rev. D 57, 7274 (1998); Erratum, ibidem 63, 029902(E)
(2001)
47. P. Jaranowski, G. Sch¨afer, Phys. Rev. D 60, 124003 (1999) and 61, 064008 (2000)
48. P. Jaranowski, G. Sch¨afer, Phys. Rev. D 65, 127501 (2002)
49. L. Kidder, Phys. Rev. D 52, 821 (1995)
50. K.D. Kokkotas, G. Sch¨afer, Mon. Not. R. Astron. Soc. 275, 301 (1995)
51. L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields (Pergamon Press, Oxford, 1985)
52. T. Ledvinka, G. Sch¨afer, J. Biˇc´ak,Phys.Rev.Lett.100, 251101 (2008)
53. R.W. Lindquist, J. Math. Phys. 4, 938 (1963)
54. R.-M. Memmesheimer, A. Gopakumar, G. Sch¨
afer, Phys. Rev. D 71, 044021 (2005)
55. C.W. Misner, Ann. Phys. (N.Y.) 24, 102 (1963)
56. M.E. Pati, C.M. Will, Phys. Rev. D 65, 104008 (2002)
57. R.A. Porto, I.Z. Rothstein, Phys. Rev. D 78, 044012 (2008)
58. R.A. Porto, I.Z. Rothstein, Phys. Rev. D 78, 044013 (2008)
59. O. Poujade, L. Blanchet, Phys. Rev. D 65, 124020 (2002)
60. E. Racine, A. Buonanno, L. Kidder, Phys. Rev. D 80, 044010 (2009)
61. T. Regge, T. Teitelboim, Ann. Phys. (N.Y.) 88, 286 (1974)
62. M. Riesz, Acta Math. 81, 1 (1949)
63. M. Ruffert, H.-Th. Janka, G. Sch¨afer, Astron. Astrophys. 311, 532 (1996)
64. G. Sch¨afer, Phys. Lett. A 100, 128 (1984)
65. G. Sch¨afer, in Symposia Gaussiana, Conf. A: Mathematical and Theoretical Physics, ed. by
M. Behara, R. Fritsch, R.G. Lintz (Walter de Gruyter, Berlin, 1995), pp. 667–679
66. G. Sch¨afer, Gen. Rel. Grav. 36, 2223 (2004)
67. J. Steinhoff, S. Hergt, G. Sch¨afer, Phys. Rev. D 78, 101503(R) (2008)
68. J. Steinhoff, G. Sch¨afer, Phys. Rev. D 80, 088501 (2009)
69. J. Steinhoff, G. Sch¨afer, S. Hergt, Phys. Rev. D 77, 104018 (2008)
70. K.S. Thorne, Rev. Mod. Phys. 52, 299 (1980)
71. K.S. Thorne, J.B. Hartle, Phys. Rev. D 31, 1815 (1985)
72. C.M. Will, Living Rev. Rel. 9,URL:http://www.livingreviews.org/lrr-2006-3