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 171
f
i
.x;t/;s.x
0
;t/gD
@s.x
0
;t/
@x
0i
ı.x x
0
/; (11)
f
i
.x;t/;
j
.x
0
;t/gD
i
.x
0
;t/
@
@x
0j
ı.x x
0
/
j
.x;t/
@
@x
i
ı.x x
0
/; (12)
and zero otherwise, where ı.x x
0
/ denotes the standard Dirac delta function in
three-dimensional space. It fulfills
R
d
3
xı.x/ D 1.
In the Newtonian theory, the Hamiltonian of the fluid is given by,
H D
1
2
Z
d
3
x
i
i
%
G
2
Z
d
3
xd
3
x
0
%
.x;t/%
.x
0
;t/
jx x
0
j
C
Z
d
3
x : (13)
For point masses, the total momentum and mass densities read, consistent with
the Eqs. 10 and 12,
i
D
X
a
p
ai
ı.x x
a
/; %
D
X
a
m
a
ı.x x
a
/; (14)
where the position and momentum variables fulfill the standard Poisson bracket
relations,
fx
i
a
;p
aj
gDı
ij
; zero otherwise; (15)
and the Hamiltonian takes the form,
H D
1
2
X
a
p
ai
p
ai
m
a
G
2
X
a¤b
m
a
m
b
jx
a
x
b
j
; (16)
where the self-energy term has been dropped (for regularization techniques, see
Section 3.4).
3 Canonical General Relativity and PN Expansions
In curved spacetime the stress–energy tensor of an ideal fluid takes the form
T
D %.c
2
C h/u
u
C pg
;g
u
u
D1; (17)
where % denotes the proper rest-mass density, h the specific enthalpy, and u
the
four-velocity field of the fluid. Using energy density e D %.c
2
C h/ p (also the
specific internal energy ˘ D e=% c
2
could be used), the equation of state reads
e D e.%;s/ with de D .c
2
C h/d% C %Tds; or dp D %dh %Tds: (18)
172 G. Sch¨afer
The variables of the canonical formalism are chosen to be
%
D
p
gu
0
%; s;
i
D
1
c
p
gT
0
i
: (19)
They fulfill the same (universal) kinematical Lie–Poisson bracket relations as in the
Newtonian theory, see [42], or also [9],
f
i
.x;t/;%
.x
0
;t/gD
@
@x
0i
Œ%
.x
0
;t/ı.x x
0
/; (20)
f
i
.x;t/;s.x
0
;t/gD
@s.x
0
;t/
@x
0i
ı.x x
0
/; (21)
f
i
.x;t/;
j
.x
0
;t/gD
i
.x
0
;t/
@
@x
0j
ı.x x
0
/
j
.x;t/
@
@x
i
ı.x x
0
/: (22)
The evolution equations take the form
@%
@t
D@
i
ıH
ı
i
%
” @
.
p
g%u
/ D 0; (23)
@s
@t
D
ıH
ı
i
@
i
s ” u
@
s D 0; (24)
@
i
@t
D@
j
ıH
ı
j
i
@
i
ıH
ı
j
j
@
i
ıH
ı%
%
C
ıH
ıs
@
i
s; (25)
corresponding to @
p
gT
i
1
2
p
gT
@
i
g
D 0,
v
i
D
ıH
ı
i
; where v
i
D c
u
i
u
0
: (26)
The linear and angular momenta of the fluid read, respectively,
P
i
D
Z
d
3
x
i
;J
i
D
Z
d
3
x
ijk
x
j
k
: (27)
For a system made of point masses simplifications take place,
h D p D s D 0; .dusty matter/; (28)
and further,
%
D
X
a
m
a
ı.x x
a
/;
i
D
X
a
p
ai
ı.x x
a
/; v
i
a
D
dx
i
a
dt
; (29)
Post-Newtonian Methods: Analytic Results on the Binary Problem 173
where p
ai
and x
i
a
, respectively, are the linear momentum and the position vector of
the ath particle. The kinematical Poisson bracket relations are given by
fx
i
a
;p
aj
gDı
ij
; zero otherwise: (30)
Hereof the standard Hamilton equations result,
dp
ai
dt
D
@H
@x
i
a
;
dx
i
a
dt
D
@H
@p
ai
: (31)
Remarkably, the difference to the Newtonian theory comes solely from the Hamilto-
nian, which is thus a dynamical difference and not a kinematical one. This statement
refers to the matter only and not to the gravitational field. The latter is quite different
in general relativity.
3.1 Canonical Variables of the Gravitational Field
Within the ADM canonical formalism of general relativity, in generalized isotropic
coordinates, the independent gravitational field variables h
TT
ij
and
ij
TT
enter in the
form
g
ij
D
1 C
1
8
4
ı
ij
C h
TT
ij
; (32)
ij
DQ
ij
C
ij
TT
; (33)
where g
ij
D g
ji
ij
is the metric of the curved three-dimensional hypersurfaces
t Dconst,
ij
c
3
=16G is the canonical conjugate to
ij
,thatis,
ij
D
1=2
.K
ij
ij
K
k
k
/,whereK
ij
DK
ji
is the extrinsic curvature of the t D const slices, and
D det.
ij
/,
il
lj
D ı
ij
,andfor Q
ij
holds
Q
ij
D @
i
j
C @
j
i
2
3
ı
ij
@
k
k
: (34)
Obviously,
ii
D0,or
i
i
D
ij
h
TT
ij
. The canonical conjugate to h
TT
ij
reads
ij
TT
c
3
=16G. The index TT means tranverse-traceless, that is, h
TT
ii
D
ii
TT
D0,
@
j
h
TT
ij
D@
j
ij
TT
D0.
Using those variables, the Einstein field equation
p
gG
00
D
8G
c
4
p
gT
00
can
be put into the form, employing point masses for the source,
1=2
R D
1
1=2
i
j
j
i
1
2
i
i
j
j
C
16G
c
3
X
a
m
2
a
c
2
C
ij
p
ai
p
aj
1=2
ı
a
; (35)
174 G. Sch¨afer
and the field equations
p
gG
0
i
D
8G
c
4
p
gT
0
i
read
2@
j
j
i
C
kl
@
i
kl
D
16G
c
3
X
a
p
ai
ı
a
; (36)
where ı
a
D ı.x x
a
/.TheEqs.35 and 36 are the famous four constraint equations
of general relativity.
In the gauge Eqs. 32–34 the ADM Hamiltonian can be written, [1],
H
h
x
i
a
;p
ai
;h
TT
ij
;
ij
TT
i
D
c
4
16G
Z
d
3
x
h
x
i
a
;p
ai
;h
TT
ij
;
ij
TT
i
; (37)
resulting from the solution of the four (elliptic-type) constraint equations. The addi-
tional Hamilton equations of motion for the gravitational field are given by
@
ij
TT
@t
D
16G
c
3
ıH
ıh
TT
ij
;
@h
TT
ij
@t
D
16G
c
3
ıH
ı
ij
TT
: (38)
The transition to a Routh functional simplifies a lot the construction of the dy-
namics of the matter and of the gravitational field. The Routh functional is chosen
in the form [46],
R
x
i
a
;p
ai
;h
TT
ij
;@
t
h
TT
ij
D H
c
3
16G
Z
d
3
x
ij
TT
@
t
h
TT
ij
: (39)
The evolution equations for the matter and the gravitational field now read
ı
R
R.t
0
/dt
0
ıh
TT
ij
.x
k
;t/
D 0; Pp
ai
D
@R
@x
i
a
; Px
i
a
D
@R
@p
ai
: (40)
The conservative dynamics results from the on-field-shell Routh functional
R
shell
.t/ D R
h
x
i
a
;p
ai
;h
TT
ij
Œx
k
a
;p
ak
; @
t
h
TT
ij
Œx
k
a
;p
ak
i
; (41)
with solved field equations, in the form
Pp
ai
.t/ D
ı
R
R
shell
.t
0
/dt
0
ıx
i
a
.t/
; Px
i
a
.t/ D
ı
R
R
shell
.t
0
/dt
0
ıp
ai
.t/
; (42)
where
ı
R
R
shell
.t
0
/dt
0
ız.t/
D
@R
shell
@z.t/
d
dt
@R
shell
@Pz.t/
C; z D .x
i
a
;p
ai
/: (43)
Post-Newtonian Methods: Analytic Results on the Binary Problem 175
Using the matter equations of motion in the Routhian R
shell
the Routhian can be
brought into the form R.x
i
a
;p
ai
/. Herein, however, the meaning of the variables x
i
a
and p
ai
has changed, see [19,27,64].
3.2 Brill–Lindquist Initial-Value Solution for Binary Black Holes
The Brill–Lindquist solution for multiple black holes is a pure vacuum solution of
the constraint equations at initial time t under the conditions of time symmetry, that
is, p
ai
D 0 D
ij
, and of conformal flatness, that is, h
TT
ij
D 0 [16]. A related vacuum
solution is the Misner–Lindquist solution where an additional isometry condition is
imposed [53, 55]. Under those conditions, the only remaining constraint equation
reads, not using vacuum but (point-mass) sources,
1 C
1
8
D
16G
c
2
X
a
m
a
ı
a
;.h
TT
ij
D 0 D p
ai
D
ij
/: (44)
In the case of two black holes, its solution is given by, see [47],
D
4G
c
2
˛
1
r
1
C
˛
2
r
2
(45)
with (a; b D 1; 2 and b ¤ a)
˛
a
D
m
a
m
b
2
C
c
2
r
ab
G
0
@
s
1 C
m
a
C m
b
c
2
r
ab
=G
C
m
a
m
b
2c
2
r
ab
=G
2
1
1
A
; (46)
resulting into the Brill–Lindquist solution for binary black holes. Obviously, each
Brill–Lindquist black hole is represented by a Dirac delta function (fictitious image
mass-point; see Section 3.4). In the Misner–Lindquist case, infinite many fictitious
image mass-points are needed for each black hole [47,53, 55].
The energy of the Brill–Lindquist solution simply reads
H
BL
D .˛
1
C ˛
2
/c
2
D .m
1
C m
2
/c
2
G
˛
1
˛
2
r
12
: (47)
The methods that have been used for the obtention of the Brill–Lindquist solution
from sources (notice, in the original work of Brill and Lindquist this solution has
been obtained without any regularization as a purely vacuum solution) are analyti-
cal Hadamard regularization and mass renormalization [47], as well as dimensional
regularization based on the d-dimensional metric
ij
D
1 C
1
4
d 2
d 1
4
d 2
ı
ij
(48)
176 G. Sch¨afer
with solution ( denotes the Euler gamma function)
D
4G
c
2
.
d 2
2
/
d 2
2
˛
1
r
d 2
1
C
˛
2
r
d 2
2
!
(49)
(for more details see Section 3.4). The PN expansion of the Brill–Lindquist initial
energy expression is straightforward to all orders of 1=c
2
. Once it had fixed the static
ambiguity parameter !
static
(see [19]) in nondimensional-regularization calculations
to the correct value of zero [47]. At that time, however, it was not quite clear that the
Brill–Lindquist solution delivers the correct boundary conditions for the point-mass
model.
The truncation of the constraint equations in the form h
TT
ij
0 as well as
dropping an additional term in the Hamiltonian constraint connected with the
energy density of the field momentum results in a remarkable, fully explicitly solv-
able conservative so-called skeleton dynamics that allows a PN expansion of the
Hamiltonian, and of all the metric coefficients too, to all orders, see next section.
3.3 Skeleton Hamiltonian
In Ref. [32] the skeleton dynamics has been developed. The skeleton approach to
general relativity requires the conformal flat condition for the spatial three-metric
for all times (not only initially as for the Brill–Lindquist solution)
ij
D
1 C
1
8
4
ı
ij
: (50)
Hereof, in our coordinate system, maximal slicing follows,
ij
ij
D 2
p
ij
K
ij
D 0: (51)
Under the conformal flat condition for the spatial three-metric, the momentum con-
straint equations become
j
i;j
D
8G
c
3
X
a
p
ai
ı
a
: (52)
The solution of these equations is constructed under the condition that
j
i
(and not
ij
,seeEqs.33 and 34) is purely longitudinal, that is,
j
i
D @
i
V
j
C @
j
V
i
2
3
ı
ij
@
l
V
l
: (53)
This condition is part of the definition of the skeleton model. At spacelike infinity,
the surface-area integrals of
j
i
or
ij
are proportional to the total linear momentum
of the binary system.
Post-Newtonian Methods: Analytic Results on the Binary Problem 177
Furthermore, in the Hamilton constraint equation, which in our case reads
D
j
i
i
j
.1 C
1
8
/
7
16G
c
2
X
a
m
a
ı
a
.1 C
1
8
/
1 C
p
2
a
.1 C
1
8
/
4
m
2
a
c
2
1=2
; (54)
we perform a truncation of the numerator of the first term in the following way
j
i
i
j
2V
j
@
i
i
j
C @
i
2V
j
i
j
!2V
j
@
i
i
j
D
16G
c
3
X
a
p
aj
V
j
ı
a
; (55)
that is, we drop from
j
i
i
j
the term @
i
2V
j
i
j
. This is the second crucial trunca-
tion condition additional to the conformal flat one. Without this truncation neither an
explicit solution can be achieved nor a PN expansion is feasible. From [46] we know
that at the 3 PN level the h
TT
ij
-field is needed to make the sum of the corresponding
terms from
j
i
i
j
analytic in 1=c.
With the aid of the ansatz
D
4G
c
2
X
a
˛
a
r
a
(56)
and by making use of dimensional regularization, the energy and momentum con-
straint equations result in an algebraic equation of the form, [32],
˛
a
D
m
a
1 CA
˛
b
r
ab
1 C
p
2
a
=.m
2
a
c
2
/
.1 C A˛
b
=r
ab
/
4
1
2
C
p
ai
V
ai
=c
.1 C A˛
b
=r
ab
/
7
; (57)
where A G=.2 c
2
/ and b ¤ a.
With these inputs the skeleton Hamiltonian for binary black holes becomes
(at least initially, for p
a
D 0, the solution is consistent with general relativity)
H
sk
c
4
16G
Z
d
3
x D c
2
X
a
˛
a
: (58)
The Hamilton equations of motion read
P
x
a
D
@H
@p
a
;
P
p
a
D
@H
@x
a
: (59)
In the center-of-mass frame of the binary system, we define
p p
1
Dp
2
; r x
1
x
2
;r
2
D .x
1
x
2
/ .x
1
x
2
/: (60)
178 G. Sch¨afer
Further, we will employ the following convenient dimensionless quantities
O
t D
tc
3
Gm
; Or D
rc
2
Gm
;
O
p D
p
c
;
O
H
sk
D
H
sk
c
2
; (61)
O
j D
Jc
Gm
; Op
r
D
p
r
c
;
O
p
2
DOp
2
r
C
O
j
2
=Or
2
; (62)
where J D r p is the orbital angular momentum in the center-of-mass frame and
p
r
D pr=r the radial momentum. The total rest-mass is denoted by m D m
1
Cm
2
and the reduced mass by D m
1
m
2
=m. The binary skeleton Hamiltonian
O
H
Sk
can
be put into the following form [37],
O
H
sk
D 2 Or
1
C
2
2
with (63)
1
D 1 C
4 Or
2
1 C
4
2
Op
2
r
C
O
j
2
=Or
2
2
4
2
!
1=2
8 Op
2
r
C 7
O
j
2
=Or
2
2
8 Or
2
7
2
; (64)
2
D 1 C
C
4 Or
1
1 C
4
2
Op
2
r
C
O
j
2
=Or
2
2
C
4
1
!
1=2
8 Op
2
r
C 7
O
j
2
=Or
2
2
8 Or
2
7
1
; (65)
where
D
1
p
1 4
and
C
D
1 C
p
1 4
with D =m.
The conservative skeleton Hamiltonian has the following nice properties. It is
exact in the test-body limit where it describes the motion of a test particle in
the Schwarzschild spacetime. It is identical to the 1 PN accurate Hamiltonian for
the binary dynamics in general relativity. Further, as explained earlier, when point
particles are at rest, the Brill–Lindquist initial value solution is reproduced. It is re-
markable that the skeleton Hamiltonian allows a PN expansion in powers of 1=c
2
to
arbitrary orders. The skeleton Hamiltonian thus describes the evolution of a kind of
black hole under both conformal flat conditions for the three-metric and analyticity
conditions in 1=c
2
for the Hamiltonian. Of course, gravitational radiation emission
is not included. It can, however, be added to some reasonable extent, see [37].
Restricting to circular orbits and defining x D .Gm!=c
3
/
2=3
,where! is the
orbital angular frequency, the skeleton Hamiltonian reads explicitly to 3 PN order,
O
H
sk
D
x
2
C
3
8
C
24
x
2
C
27
16
C
29
16
17
48
2
x
3
C
675
128
C
8585
384
7985
192
2
C
1115
10368
3
x
4
C O.x
5
/: (66)
In Ref. [32] the coefficients are given to the order x
11
inclusively.
Post-Newtonian Methods: Analytic Results on the Binary Problem 179
In the Isenberg–Wilson–Mathews approach to general relativity only the
conformal flat condition is employed. Thus the energy stops being analytic in
1=c at 3 PN. Through 2 PN order, the Isenberg–Wilson–Mathews energy of a binary
is given by
O
H
IWM
D
x
2
C
3
8
C
24
x
2
C
27
16
39
16
17
48
2
x
3
: (67)
We already quote here the 3 PN result of general relativity. It reads, see Eq. 129,
O
H
3PN
D
x
2
C
3
8
C
24
x
2
C
27
16
19
16
C
1
48
2
x
3
C
675
128
C
205
192
2
34445
1152
C
155
192
2
C
35
10368
3
!
x
4
: (68)
The difference between
O
H
IWM
and
O
H
sk
through 2 PN order shows the effect of trun-
cation in the field-momentum part of
O
H
sk
and the difference between
O
H
IWM
and
O
H
3PN
reveals the effect of conformal flat truncation. In the test-body limit, D 0,
all the Hamiltonians coincide and for the equal-mass case, D 1=4, their differ-
ences are largest.
3.4 Functional Representation of Compact Objects
Before going on with the presentation of more dynamical expressions we will dis-
cuss in more detail the ı-function-source model we are employing. Although we are
interested in both neutron stars and black holes, our matter model will be based
on black holes because these are the simplest objects in general relativity and
neutron stars resemble them very much as seen from outside. The simplest black
holes are the isolated nonrotating ones. Their solution is the Schwarzschild metric
which solves the Einstein field equations for all time. In isotropic coordinates, the
Schwarzschild metric reads
ds
2
D
1
MG
2rc
2
1 C
MG
2rc
2
!
2
c
2
dt
2
C
1 C
MG
2rc
2
4
d x
2
; (69)
where M is the gravitating mass of the black hole and r
2
D.x
1
/
2
C.x
2
/
2
C.x
3
/
2
,
d x
2
D.dx
1
/
2
C .dx
2
/
2
C .dx
3
/
2
. It should be pointed out that the origin of
the coordinate system r D0 is not located where the Schwarzschild singularity
R D0 (radial Schwarzschild coordinate R) in Schwarzschild coordinates is located,
rather it is located on the other side of the Einstein–Rosen bridge, at infinity.
The relation between isotropic coordinates and Schwarzschild coordinates reads
R Dr.1 C
MG
2rc
2
/
2
if MG=2c
2
r and R
0
Dr.1 C
MG
2rc
2
/
2
if 0 r MG=2c
2
,
180 G. Sch¨afer
where R
0
is another Schwarzschild radial coordinate appropriate for the geometry
of the other side of the Einstein–Rosen bridge. The regimes 0 R<2GM=c
2
and
0 R
0
<2GM=c
2
are not accessible to isotropic coordinates. The harmonic ra-
dial coordinate, say here, %, relates to the Schwarzschild radial coordinate through
R D % C MG=c
2
. Evidently, the origin of the harmonic coordinates is located at
R D MG=c
2
which is a spacelike curve in the region between event horizon and
Schwarzschild singularity.
For two black holes, the metric for maximally sliced Brill–Lindquist initial time-
symmetric data reads
ds
2
D
0
@
1
ˇ
1
G
2r
1
c
2
ˇ
2
G
2r
2
c
2
1 C
˛
1
G
2r
1
c
2
C
˛
2
G
2r
2
c
2
1
A
2
c
2
dt
2
C
1 C
˛
1
G
2r
1
c
2
C
˛
2
G
2r
2
c
2
4
d x
2
; (70)
where the ˛
a
coefficients are given in Eq. 46 and where the ˇ
a
coefficients can be
found in [48] (notice @
t
r
a
D 0, initially).
The total energy results from the ADM mass-energy expression
E
ADM
D
c
4
2G
I
i
0
ds
i
@
i
D
c
4
2G
Z
d
3
x D .˛
1
C ˛
2
/c
2
; (71)
where D 1C
˛
1
G
2r
1
c
2
C
˛
2
G
2r
2
c
2
and ds
i
D n
i
r
2
d˝ is a two-dimensional surface-area
element with unit radial vector n
i
D x
i
=r and solid angle element d˝.
Introducing the inversion map r
0
1
D ˛
2
1
G
2
=4c
4
r
1
or, r
0
1
D r
1
˛
2
1
G
2
=4c
4
r
2
1
and
r
1
D r
0
1
˛
2
1
G
2
=4c
4
r
02
1
,wherer
1
D xx
1
, r
1
Djxx
1
j, r
0
1
D x
0
x
1
, r
0
1
Djx
0
x
1
j,
the three-metric at the throat of black hole 1, dl
2
D
4
d x
2
, transforms into
dl
2
D
04
d x
0
2
D
1 C
˛
1
G
2r
0
1
c
2
C
˛
1
˛
2
G
2
4r
2
r
0
1
c
4
4
d x
0
2
; (72)
where r
2
D
˛
2
1
G
2
4c
4
r
0
1
r
02
1
C r
12
with r
12
D r
1
r
2
. From the new metric function
0
D 1 C
˛
1
G
2r
0
1
c
2
C
˛
1
˛
2
G
2
4r
2
r
0
1
c
4
the proper mass of the throat 1 results in, taking into
account r
2
D r
0
1
˛
2
1
G
2
=4c
4
r
02
1
C r
12
,
m
1
D
c
2
2G
I
i
0
ds
0
i
@
0
i
0
D
c
2
2G
Z
d
3
x
0
0
0
(73)
D ˛
1
C
˛
1
˛
2
G
2r
12
c
2
:
This construction as performed in Ref. [16] is a purely geometrical or vacuum one
without touching singularities. Thus having the two individual masses m
1
and m
2
the gravitational interaction energy is obtained as E D E
ADM
.m
1
C m
2
/c
2
.