Baumgarte T., Shapiro S. Numerical Relativity. Solving Einstein’s Equations on the Computer
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12.3 The conformal thin-sandwich approach 411
Figure 12.4 Illustration of a helical Killing vector generating a circular binary orbit.
to select models in the initial data set that presumably are close to quasiequilibrium, but
there is no guarantee that the spacetime corresponding to these models will maintain
quasiequilibrium once we evolve off the initial time slice.
Following our discussion in Section 12.1 we seek to construct black hole binaries at
moderately close separations, at which point the orbits have become quasicircular. At these
separations we assume that the individual black holes have settled into a quasiequilibrium
state, whereby no matter and very little gravitational radiation fall into them to disturb this
state. Now a stationary spacetime in strict equilibrium possesses two important Killing
vectors: ∂/∂t, associated with stationary symmetry and ∂/∂φ, associated with rotational
symmetry. The spacetime for a quasiequilibrium binary system has no such Killing vectors,
but it does possess an approximate helical Killing vector ξ
a
hel
given by a linear combination
of these two vectors. In particular, the approximate Killing vector ξ
a
hel
generates a circular
orbit with orbital angular velocity
orb
, as illustrated in Figure 12.4. In the coordinate
system of an inertial observer at infinity we may write ξ
a
hel
as
ξ
a
hel
=
(
∂/∂t
)
a
+
orb
(
∂/∂φ
)
a
. (12.63)
If ξ
a
hel
were an exact helical Killing vector we would have L
ξ
hel
g
ab
= 0. In other words, the
spacetime geometry of the binary system would be invariant if we were to move through an
infinitesimal angle dφ =
orb
dt in an infinitesimal time dt. Consequently, we could choose
a corotating coordinate system in which the time axis t
a
= αn
a
+ β
a
remains aligned with
ξ
a
hel
,
19
so that, in this coordinate system, all partial derivatives of the metric with respect to
time would vanish. In quasiequilibrium, we can expect that, in such a corotating coordinate
system, the corresponding time derivatives will vanish approximately.
The Bowen–York approach provides no mechanism to impose this notion of quasiequi-
librium. We can construct a spatial metric and extrinsic curvature that solve the constraint
19
Note that this choice leads to a spacelike time coordinate at large separations.
412 Chapter 12 Binary black hole initial data
equations, but we have no control over their time evolution. In the conformal thin-sandwich
decomposition of Chapter 3.3, on the other hand, we can explicitly set several of these time
derivatives to zero. This approach therefore seems more promising for the construction of
quasiequilibrium initial data.
In Chapter 3.3 we introduced the time derivative of the conformally related metric
as
¯
u
ij
≡ ∂
t
¯γ
ij
(12.64)
(see equation 3.94) and then expressed the extrinsic curvature
¯
A
ij
in terms of
¯
u
ij
with the
help of the evolution equation (2.134),
¯
A
ij
=
ψ
6
2α
(
¯
Lβ)
ij
−
¯
u
ij
(12.65)
(see equation 3.99). Inserting this into the momentum constraint (12.4) then yields
(
¯
L
β)
i
− (
¯
Lβ)
ij
¯
D
j
ln(ψ
−6
α) = ¯α
¯
D
j
(ψ
6
α
−1
¯
u
ij
) +
4
3
α
¯
D
i
K, (12.66)
while the Hamiltonian constraint (12.3) remains
8
¯
D
2
ψ − ψ
¯
R −
2
3
ψ
5
K
2
+ ψ
−7
¯
A
ij
¯
A
ij
= 0. (12.67)
Here we have again assumed a vacuum spacetime.
For the construction of quasiequilibrium data it now seems natural to choose the require-
ments that
¯
u
ij
= 0 (12.68)
and
∂
t
K = 0 (12.69)
in the corotating coordinate system where t
a
∝ ξ
a
hel
. The latter condition then yields an
equation for the lapse α,
¯
D
2
(αψ) = αψ
7
8
ψ
−8
¯
A
ij
¯
A
ij
+
5
12
ψ
4
K
2
+
1
8
¯
R
+ ψ
5
β
i
¯
D
i
K, (12.70)
while the former reduces the momentum constraint (12.66)to
(
¯
L
β)
i
− (
¯
Lβ)
ij
¯
D
j
ln(ψ
−6
α) =
4
3
α
¯
D
i
K. (12.71)
The extrinsic curvature
¯
A
ij
is given in terms of these unknowns by
¯
A
ij
=
ψ
6
2α
(
¯
Lβ)
ij
. (12.72)
Equations (12.67), (12.70) and (12.71) now form a coupled system of elliptic equations
for the lapse α, the conformal factor ψ and the shift β
i
. Outer boundary conditions follow
12.3 The conformal thin-sandwich approach 413
from asymptotic flatness and equation (12.63),
lim
r→∞
ψ = 1, lim
r→∞
α = 1, lim
r→∞
β
i
=
orb
∂
∂φ
i
= (Ω
orb
× r)
i
. (12.73)
Here we are assuming that the coordinate system is centered on the binary system’s center of
mass-energy at coordinate radius r = 0, where r
2
= x
2
+ y
2
+ z
2
. The above asymptotic
condition on the shift guarantees that the coordinate system corotates with the binary.
We have yet to choose a conformal background metric ¯γ
ij
and the extrinsic curvature K .
The above equations simplify dramatically for conformally flat spaces ¯γ
ij
= η
ij
and maxi-
mal slicing K = 0.
20
As an alternative, the background can be chosen to be a superposition
of Kerr–Schild geometries.
21
Once we have chosen a background solution we are almost ready to solve equations
(12.67), (12.70) and (12.71)forα, ψ and β
i
. The last issue that we need to address before
attempting a numerical integration are the singularities that lurk in the interior of the black
holes. In the context of the Bowen–York approach we discussed two different methods
of eliminating these singularities. The first approach was the conformal-imaging method,
which imposes an isometry and enables us to excise the black hole interior; the second
approach was the puncture method which absorbs the singularity in analytical terms.
Unfortunately, neither one of these two methods works well in the context of the con-
formal thin-sandwich approach. The conformal-imaging approach leads to inconsistencies
on the black hole throats. In a nutshell, the isometry requires both α = 0andβ
i
= 0
on the throats, which, according to (12.72), means that the extrinsic curvature
¯
A
ij
can
remain finite only if (
¯
Lβ)
ij
= 0. This requirement leads to boundary conditions on both
the shift and its derivatives, making the system overdetermined.
22
While it would be
desirable to avoid boundary conditions altogether with the puncture approach, it seems
impossible to construct even single equilibrium black holes with a combination of the
conformal thin-sandwich and puncture methods.
23
These considerations lead us to con-
struct new black-hole boundary conditions that are based explicitly on quasiequilibrium
requirements.
12.3.2 Quasiequilibrium black hole boundary conditions
Before discussing the quasiequilibrium black hole conditions
24
it is useful to review briefly
some of the formalism that we introduced to describe apparent horizons in Chapter 7.3.
In particular, we considered a closed 2-dimensional hypersurface of and called it S.
20
See Gourgoulhon et al. (2002); Grandcl
´
ement et al. (2002)andCook and Pfeiffer (2004).
21
See Marronetti and Matzner (2000); Yo et al. (2004); Cook and Pfeiffer (2004).
22
See Cook (2002) for a discussion. In Gourgoulhon et al. (2002)andGrandcl
´
ement et al. (2002), who implemented
this approach, the problem is avoided by adding a small and otherwise insignificant artificial term.
23
At least if the background solution is based on an expression of the form (12.49); see Hannam et al. (2003), but
compare with Hannam (2005).
24
In this section we closely follow Cook and Pfeiffer (2004), but see Jaramillo et al. (2004) for an alternative approach.
414 Chapter 12 Binary black hole initial data
We called its outward pointing unit normal s
a
, and found that s
a
induces a 2-dimensional
metric,
m
ab
= γ
ab
− s
a
s
b
, (12.74)
on S. We constructed a future-pointing, outgoing null vector k
a
from
k
a
=
1
√
2
(n
a
+ s
a
). (12.75)
The geometry of all of these objects is illustrated in Figure 7.3. As before, we define the
projection of the gradient of k
a
into S as
ab
= m
c
a
m
d
b
∇
c
k
d
. (12.76)
(see equation 7.78). We now have all the tools in hand to formulate the required conditions.
To begin with, we would like the surface S to be an apparent horizon. According to
equations (7.15) and (7.17), this means that the trace of
ab
, i.e., the expansion of the k
a
,
has to vanish,
= m
ab
ab
= m
ab
∇
a
k
b
= 0. (12.77)
In the context of the isolated horizon formalism in Chapter 7.4 we have also seen that for
the black hole to be in equilibrium, the shear of
ab
also has to vanish,
σ
ab
=
(ab)
−
1
2
m
ab
= 0 (12.78)
(see equation 7.84). Finally, we would like to construct a coordinate system that tracks
the horizon, i.e., we want to require that in quasiequilibrium the coordinate location of
the apparent horizon not change as the initial data begin to evolve. Consider a sequence
of apparent horizons S residing in neighboring spatial slices , as we did in Chapter 7.4.
For isolated black holes, the resulting worldtube H is smooth and is generated by k
a
.To
obtain a coordinate system that tracks the horizon, the time vector of the evolution, t
a
,
must reside in the null surface generated by k
a
. This requirement implies that t
a
,onS,
must be tangent to H. For that to be the case, t
a
must be a linear combination of k
a
,which
generates H , and some spatial vector h
a
tangent to S,
t
a
= ak
a
+ bh
a
, (12.79)
where a and b are two arbitrary coefficients. Since h
a
is spatial we have n
a
h
a
= 0, and
since it is tangent to S we also have s
a
h
a
= 0. Combining equations (12.75) and (12.79),
and using the fact that k
a
is a null vector, it is then easy to see that we must have
t
a
k
a
= 0. (12.80)
Conditions (12.77), (12.78) and (12.80) represent the geometric conditions that guar-
antee that the surface S is an apparent horizon, in quasiequilibrium, and will appear in
12.3 The conformal thin-sandwich approach 415
quasiequilibrium in the coordinate system defined by the lapse and shift that we are con-
structing. Our next task is to translate these geometric conditions into horizon boundary
conditions for the conformal factor ψ, the lapse α and the shift β
i
.
We start with condition (12.80). Inserting both t
a
= αn
a
+ β
a
andequation(12.75)we
immediately find
β
i
s
i
= α. (12.81)
The left-hand side of equation (12.81) suggests that we should split the shift into compo-
nents that are normal and tangential to the surface S,
β
i
= β
i
+ β
⊥
s
i
, (12.82)
where
β
i
≡ m
i
j
β
j
and β
⊥
≡ β
i
s
i
. (12.83)
With this notation the condition (12.81) becomes a condition on the normal component of
the shift,
β
⊥
= α. (12.84)
Exercise 12.8 Show that condition (12.84) holds for a Schwarzschild black hole in
Kerr–Schild coordinates (see Table 2.1).
We next consider the apparent horizon condition (12.77). From equation (7.29)wemay
write the expansion as
=
1
√
2
m
ij
(D
i
s
j
− K
ij
). (12.85)
Since we would like to apply this condition to the conformally decomposed constraint
equations, we also apply a conformal transformation to m
ij
and s
i
. Given the rescaling
(12.1)forγ
ij
, and given the relation (12.74) between γ
ij
, m
ij
and s
i
, it is natural to choose
m
ij
= ψ
4
¯
m
ij
and s
i
= ψ
2
¯
s
i
. (12.86)
Note that the normalization γ
ij
s
j
s
i
= 1 then implies that the conformally rescaled normal
vectors
¯
s
i
are normalized with respect to the conformally related metric, ¯γ
ij
¯
s
j
¯
s
i
= 1.
Exercise 12.9 Show that
D
i
s
j
= ψ
2
¯
D
i
¯
s
j
− 2
¯
s
i
¯
D
j
ln ψ + 2¯γ
ij
¯
s
k
¯
D
k
ln ψ
. (12.87)
Inserting equation (12.87) into the expansion (12.85), and using the abbreviation
J ≡ m
ij
K
ij
, (12.88)
416 Chapter 12 Binary black hole initial data
we find
=
ψ
−2
√
2
¯
m
ij
¯
D
i
¯
s
j
+ 4
¯
s
k
¯
D
k
ln ψ − ψ
2
J
. (12.89)
According to equation (12.77) the expansion hastovanishforS to be an apparent
horizon, which yields the condition
¯
s
k
¯
D
k
ln ψ =
1
4
ψ
2
J −
¯
m
ij
¯
D
i
¯
s
j
. (12.90)
Equation (12.90) imposes a Neumann-type boundary condition on the normal gradient of
the conformal factor ψ.
Before proceeding it is useful to illustrate this formalism by applying it to a Schwarz-
schild black hole in isotropic coordinates, for which
¯γ
ij
= η
ij
(12.91)
and
ψ = 1 +
M
2r
. (12.92)
We a lso have K
ij
= 0, which immediately gives us J = 0. To apply the boundary condition
(12.90) on the horizon at r = M/2 we first have to construct the normal
¯
s
i
. They have to
be radial, and they have to be normalized with respect to the conformally related metric,
and therefore
¯
s
i
= (1, 0, 0) (12.93)
in spherical polar coordinates. We then have
¯
m
ij
¯
D
i
¯
s
j
=−
¯
m
ij
¯
r
ij
=
2
r
. (12.94)
Evaluating the right hand side of equation (12.90)gives
¯
s
k
¯
D
k
ln ψ = ψ
−1
∂
r
ψ =−ψ
−1
M
2r
2
. (12.95)
Inserting the last two equations into equation (12.90)wefind
M
2r
2
=
1
4
1 +
M
2r
2
r
, (12.96)
which holds on the horizon at r = M/2. This result is not surprising, of course, but it is
reassuring.
Exercise 12.10 Show that condition (12.90) holds for a Schwarzschild black hole
in Kerr–Schild coordinates (see Table 2.1).
We now return to our derivation of the quasiequilibrium boundary conditions and exam-
ine condition (12.78). For simplicity we start with the first term,
(ab)
, and substitute equa-
tion (12.75)fork
a
. Since in the end we are interested in an equation for the contravariant
12.3 The conformal thin-sandwich approach 417
components of the shift vector β
i
, it is also easier to consider the contravariant components
of
ab
, obtaining
(ab)
= m
ac
m
bd
∇
(c
k
d)
=
1
√
2
m
ac
m
bd
∇
(c
n
d)
+∇
(c
s
d)
. (12.97)
We now introduce the abbreviation
H
ab
= m
ac
m
bd
∇
(c
s
d)
= D
(a
s
b)
, (12.98)
where we have also introduced
D
i
as the covariant derivative operator compatible with
m
ij
. In complete analogy with our construction of the 3-dimensional covariant derivative
D
i
from the 4-dimensional covariant derivative ∇
a
by projecting all indices into (see
equation 2.40 and the following equations), we construct
D
i
by projecting all indices of the
4-dimensional covariant derivative into S. With the help of H
ab
, as well as the definition
of the extrinsic curvature (2.49), we may now rewrite equation (12.97)as
(ij)
=
1
√
2
H
ij
− m
i
l
m
j
m
K
lm
, (12.99)
where, without loss of generality, we have also switched to spatial indices. It turns out to
be convenient to write the extrinsic curvature as
K
ij
= A
ij
+
1
3
γ
ij
K =
1
2α
(Lβ)
ij
− u
ij
+
1
3
γ
ij
K, (12.100)
where we have used equations (3.31) and (3.93). We also remind the reader that the
longitudinal operator, i.e., the vector gradient or conformal Killing operator, is defined as
(Lβ)
ij
= D
i
β
j
+ D
j
β
i
−
2
3
γ
ij
D
k
β
k
(12.101)
(see equation 3.50). Before proceeding we decompose the shift term according to equa-
tion (12.82) and write
1
2α
m
i
l
m
j
m
(Lβ)
lm
=
1
2α
m
i
l
m
j
m
(Lβ
)
lm
+
β
⊥
α
H
ij
−
1
3α
m
ij
D
k
(β
⊥
s
k
), (12.102)
wherewehaveusedm
ij
s
j
= 0 as well as the definition (12.98). Substituting the expressions
(12.100) and (12.102) into equation (12.99)wenowfind
√
2
(ij)
=
1
2α
−m
i
l
m
j
m
(Lβ
)
lm
+
1
3α
m
ij
D
k
(β
⊥
s
k
) + m
i
l
m
j
m
u
lm
−
1
3
m
ij
K + H
ij
1 −
β
⊥
α
. (12.103)
Applying a conformal transformation to the first three terms casts this relation as
√
2
(ij)
=
ψ
−4
2α
−
¯
m
i
l
¯
m
j
m
(
¯
Lβ
)
lm
+
1
3α
¯
m
ij
D
k
(β
⊥
s
k
) +
¯
m
i
l
¯
m
j
m
¯
u
lm
−
1
3
m
ij
K + H
ij
1 −
β
⊥
α
, (12.104)
418 Chapter 12 Binary black hole initial data
wherewehaveusedtherescalingrule(3.98) derived in exercise 3.15 for (Lβ
)
ij
. According
to the condition (12.78) we can now compute the shear σ
ij
by taking the trace-free part,
with respect to m
ij
,of
(ij)
. In the process all terms in (12.104) that are proportional to
m
ij
drop out, and we find
σ
ij
=−
1
√
2
ψ
−4
α
¯
D
(i
β
j)
−
1
2
¯
m
ij
¯
D
c
β
c
−
1
2
¯
m
i
l
¯
m
j
m
¯
u
lm
−
1
2
¯
m
ij
¯
m
lm
¯
u
lm
+
1
√
2
H
ij
−
1
2
m
ij
H
1 −
β
⊥
α
. (12.105)
So far this derivation has been completely general, and it is now time to specialize to
the situation at hand. To begin with, we can insert the boundary condition (12.84), which
immediately eliminates the last term. For the construction of quasiequilibrium data we
also assume
¯
u
ij
= 0, which eliminates the middle term. For our purposes, the geometrical
condition of vanishing shear (12.78) then reduces to
¯
D
(i
β
j)
−
1
2
¯
m
ij
¯
D
c
β
c
= 0. (12.106)
The trained eye will recognize this as the conformal Killing’s equation (A.27) for a 2-
dimensional surface S. This means that the shear σ
ab
vanishes if the tangential shift β
i
is a conformal Killing vector of
¯
m
ij
on the black hole horizon S. This condition has a
surprisingly transparent physical interpretation, as we discuss below.
25
As it turns out, it is remarkably simple to construct a tangential shift β
i
so that it is
a conformal Killing vector of
¯
m
ij
on S. We first observe that any closed, 2-dimensional
surface S is conformally equivalent to the unit 2-sphere. By this we mean that we can
construct any metric
¯
m
ij
on S from the flat metric η
ij
on the unit sphere through a
combination of conformal and coordinate transformations.
26
In exercise A.7 of Appendix A
we also show that a Killing vector of a metric g
ab
is automatically a conformal Killing
vector of the conformally related metric
¯
g
ab
. This means that a Killing vector of the flat
metric η
ij
on the unit sphere is automatically a conformal Killing vector of
¯
m
ij
on S,and
hence a solution to the boundary condition (12.106).
We have therefore reduced the problem to finding Killing vectors of the unit sphere.
This is very simple – in spherical coordinates θ and φ, for example, we have ξ
i
= e
i
(φ)
(see
equation A.22 in Appendix A). On the unit sphere, all azimuthal vectors associated with
any rotation axis passing through its center are Killing vectors. Viewing the unit sphere
as embedded in a flat 3-dimensional space we can express these vectors ξ
i
in Cartesian
coordinates as ξ
i
=
ijk
ˆ
z
j
ˆ
n
k
,where
ˆ
z
i
is a unit vector aligned with the axis of rotation and
where
ˆ
n
i
is the unit normal on the sphere. Since the product of a Killing vector ξ
i
with a
25
Cook and Pfeiffer (2004).
26
The metric
¯
m
ij
has at most three independent components, which we can account for by a conformal transformation
and a transformation of the two coordinates.
12.3 The conformal thin-sandwich approach 419
constant is still a Killing vector, we see that
β
i
=
spin
ξ
i
(12.107)
is a Killing vector of η
ij
on the unit sphere, hence a conformal Killing vector of
¯
m
ij
on the
black hole horizon S, and thus the desired solution to (12.106).
It is remarkable that condition (12.107) leaves us exactly the freedom that we need to
specify an arbitrary spin on the black hole. The parameter
spin
determines the angular
speed of the black hole, and ξ
i
the axis of rotation. Given that we implement this boundary
condition in the corotating coordinate system associated with the helical Killing vector ξ
a
hel
defined by equation (12.63), we can construct corotating black holes by setting
spin
= 0.
As seen by an inertial observer, these black holes are rotating at an angular velocity
orb
.
To construct black holes that are not rotating as seen by an inertial observer it seems
reasonable to set the magnitude of
spin
equal to
orb
and choose ξ
i
perpendicular to
the orbital plane. We will refer to this approach as the “leading-order” approximation.
Evaluating the resulting black holes’ quasi-local spin J
S
with the help of equation (7.74)
reveals that generally this approach does not lead to exactly zero spin. This suggests that
we must vary
spin
until the J
S
does indeed vanish.
27
To summarize, we have now expressed the geometrical quasiequilibrium boundary
conditions (12.77), (12.78) and (12.80) in terms of the boundary conditions (12.84)for
β
⊥
,(12.90)forψ, and (12.107)forβ
i
. Interestingly the geometric conditions do not
provide a boundary condition for the lapse α, which apparently we can choose freely.
28
The lapse determines the rate at which proper time advances compared to coordinate time,
and in fact it seems quite natural that we can freely choose this rate, even on the horizon
of a black hole in quasiequilibrium.
We can now construct a black hole binary by solving the equations (12.67), (12.70)
and (12.71) subject to the inner boundary conditions (12.84), (12.90), and (12.107). We
impose these boundary conditions on two coordinate spheres chosen to represent the
apparent horizons of the two black holes. The coordinate separation of these two spheres
controls the binary separation, while the coordinate radius of the two spheres controls the
black hole masses.
12.3.3 Identifying circular orbits
We now have almost all pieces in place to construct quasiequilibrium binary black holes
in the conformal thin-sandwich approach. The one last missing piece is the value of the
orbital angular velocity
orb
, which enters through the outer boundary for the shift β
i
in
27
Caudill et al. (2006).
28
Cook (2002) derives a boundary condition for the lapse from an additional geometric condition on the expansion of
the ingoing null vectors. However, Cook and Pfeiffer (2004) find the resulting system of boundary conditions to be
degenerate, leading to an ill-posed elliptic system, and conclude that the boundary value of the lapse can be chosen
freely; cf. Jaramillo et al. (2004); Matera et al. (2008).
420 Chapter 12 Binary black hole initial data
equation (12.73). Finding the correct
orb
is equivalent to identifying a circular orbit. In
the Bowen–York effective potential approach of Section 12.2 we identified circular orbits
by locating extrema of the binding energy along curves of constant angular momentum
(see exercises 12.1 and 12.4). Grandcl
´
ement et al. (2002) suggest an alternative way of
identifying circular orbits, namely by varying
orb
until the ADM mass M
ADM
given by
equation (3.128) equals the Komar mass M
K
givenbyequation(3.179),
M
ADM
= M
K
. (12.108)
For metrics that are sufficiently well-behaved asymptotically the ADM mass is always well
defined, while the Komar mass is meaningful only for stationary spacetimes. Equating
the two masses therefore singles out quasiequilibrium spacetimes, which, for binaries,
identifies circular orbits.
We can also justify the condition (12.108) in terms of the virial relation. In exercises 3.23
and 3.28 we showed that, to first order, equation (12.108) imposes the virial theorem (3.184)
in Newtonian gravity. Gourgoulhon and Bonazzola (1994) show that equation (12.108)in
fact leads to a relativistic virial theorem, which we would expect to hold for circular orbits in
binaries, at least approximately. In exercise 12.1 we have seen that finding energy extrema
to identify circular orbits is equivalent to imposing the virial theorem for Newtonian
point masses, and in Section 12.4 we will present evidence that a similar result holds for
relativistic black hole binaries.
29
Notice that computing the Komar mass (3.179) requires knowledge of a timelike Killing
vector, which in a numerical relativity context is usually expressed in terms of the lapse α
and the shift β
i
. The conformal thin-sandwich approach allows us to impose the existence
of at least an approximate timelike Killing vector, and provides us with the corresponding
Killing lapse and shift. By contrast, the transverse-traceless approach of Section 12.2 only
involves data intrinsic to one time slice (as opposed to the slice and its neighborhood).
It does not provide a true notion of stationary equilibrium, which requires vanishing time
derivatives, it does not allow us to impose the existence of a timelike Killing vector, and
it does not provide a lapse or shift. We therefore cannot compute the Komar mass from
Bowen–York data alone.
30
Before proceeding we point out one subtlety. The Komar mass (3.179) contains both
lapse and shift terms. In many situations the latter fall off sufficiently rapidly so that they
do not contribute to the mass – in fact, several authors define the Komar mass from the
lapse term alone. In general, however, the shift term has to be included,
31
even though it
leads to a complication in the current context.
32
A corotating observer moving along with
29
Skoge and Baumgarte (2002) demonstrate this equivalence analytically for a relativistic spherical shell of collisionless
particles.
30
Given Bowen–York data, a lapse condition can be constructed that imposes some of the conditions for the existence
of an approximate helical Killing vector and the computation of the Komar mass; see Tichy et al. (2003), Tichy and
Br
¨
ugmann (2004) and footnote .
31
As exercise 3.27 demonstrates, the shift term carries the entire contribution to the Komar mass in some coordinate
systems.
32
See the discussion in Caudill et al. (2006).