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3.4 A step further: the “waveless” approximation 81

conformal ﬂatness and maximal slicing, appear to give a maximum dimensionless spin of

ADM

∼

0.94 and not unity, as we would obtain for a slice of Kerr.

The ADM mea-

sure of the total angular momentum (J

ADM

) and mass (M

ADM

) quoted here are deﬁned in

Section 3.5.

3.4 A step further: the “waveless” approximation

The conformal thin-sandwich formalism of Section 3.3 seems more suitable for the con-

struction of equilibrium or quasiequilibrium initial data than the transverse-traceless for-

malism of Section 3.2 because we can freely determine some time derivatives of quantities

rather than the quantities themselves. Basically, in the CTT formalism the freely speciﬁable

variables are the conformally related metric and parts of the extrinsic curvature, and in the

CTS formalism the latter are replaced with the time derivative of the conformally related

metric. For equilibrium data it is much easier to make a well-motivated choice for the time

derivative of a quantity – typically zero – than for a quantity itself.

In the CTS formalism, we therefore have a natural choice for the time derivative of

the conformally related metric when constructing equilibrium data, but there still is very

little guidance as to how to choose the conformally related metric itself. Many applications

adopt conformal ﬂatness, so that the conformally related metric is simply a ﬂat metric.

This choice simpliﬁes the equations quite dramatically, but a priori it is not clear whether

or not it leads to a good approximation for quasiequilibrium, for example, for binary black

holes or neutron stars in nearly circular orbit (see Chapters 12 and 15).

To address this problem, the “waveless” approximation

goes one step further and

replaces the conformally related metric as a freely speciﬁable variable with the time deriva-

tive of the extrinsic curvature.

Here we will discuss this approximation only qualitatively

and refer the reader to the literature for details.

Recall that to replace pieces of the extrinsic curvature with the time derivative of the

conformally related metric as freely speciﬁable variables, the CTS formalism employs the

evolution equation for the metric, e.g., equations (2.134)or(3.93). The waveless approxi-

mation is an extension of this proceedure. The key idea is to replace the conformally related

metric with the time derivative of the extrinsic curvature as freely speciﬁable variables, now

employing the evolution equation for the extrinsic curvature, equation (2.135). In addition,

one can choose the time derivatives of both ¯γ

and

to satisfy a helical symmetry in

a near-zone, and to set them to zero in the far zone.

The asymptotic behavior of these

Lovelace et al. (2008).

Measured in terms of the quasilocal spin J

and mass M

of the black hole (see Chapter 7.4 for deﬁnitions), the

maximum dimensionless spin for such a CTS solution is J

∼

0.99; Lovelace et al. (2008).

Shibata et al. (2004).

Several other approaches have been suggested for the construction of equilibrium models; see, e.g., Blackburn and

Detweiler (1992); Andrade et al. (2004); Friedman and Ury

u (2006).

A helical symmetry makes the system appear stationary in the rotating frame of a binary; compare Chapters 12.3.1

and 15.1. Imposing a (nontrivial) helical symmetry globally is generally not compatible with asymptotic ﬂatness.

82 Chapter 3 Constructing initial data

quantities is so chosen to eliminate any standing waves in the far zone, which explains the

name of this approximation.

In the context of Chapter 2 we have considered equation (2.135) as an equation that

determines the time derivative of the extrinsic curvature for a given spatial metric. Instead,

we now consider the time derivative of the extrinsic curvature to be given, and we would

like to solve for the spatial metric, or at least its conformally related part. To do so we

focus on the Ricci tensor R

that appears on the right-hand side of equation (2.135). Using

equation (3.10) we can write this in terms of the Ricci tensor

associated with the

conformally related metric ¯γ

. When expressed in the form (2.143) this tensor contains a

combination of second-order spatial derivatives of the conformally related metric, making

it a rather complicated differential operator. Fortunately we still have a gauge freedom that

we can use to our advantage. As it turns out, we can make a certain gauge choice under

which these second derivatives simplify dramatically, leaving only the last second-order

term in (2.143).

This last second-order term forms an elliptic operator that we can invert

to ﬁnd the conformally related metric.

In summary, the “waveless” approximation uses the equations of the CTS formalism, as

in Box 3.3, together with equation (2.135). Given a suitable gauge choice, the Ricci tensor

in equation (2.135) turns into an elliptic operator on ¯γ

. The freely speciﬁable quantities are

now K and its time derivative, as well as the time derivatives of both ¯γ

and

. Instead

of making an ad hoc choice for ¯γ

we can now choose the time derivative of

and

obtain ¯γ

as a solution of equation (2.135). Presumably, this approach provides a further

advantage over the CTS formalism of Section 3.3 for the construction of equilibrium and

quasiequilibrium initial data, since we can freely set more time derivatives of quantities

equal to zero, rather than specifying (i.e., guessing) the quantities themselves.

As a concrete example we may return to the example of a rotating black hole. As we

discussed in both Section 3.2 and at the end of Section 3.3, assuming conformal ﬂatness

¯γ

= η

in either the CTT or CTS decomposition never leads to a spatial slice of a

vacuum, stationary, rotating Kerr black hole solution, but instead to a solution that may

be interpreted as a rotating black hole plus some gravitational radiation. In the “waveless”

approximation, by contrast, we would choose the time derivatives of ¯γ

and K

to vanish, and would then obtain the conformally related metric ¯γ

as a result of the

calculation. In this approach we could indeed ﬁnd a slice of the Kerr solution, without any

gravitational wave perturbation.

More generally, the “waveless” approximation allows us to construct stationary slices

of stationary spacetimes exactly, independently of any choice of the conformally related

We will use similar gauges leading to this simpliﬁcation of the Ricci tensors in several places in later chapters; see

Chapters 4.3, 11.3 and 11.5.

One subtlety arises from the fact that the differential operator acting on ¯γ

involves ¯γ

itself. To avoid this problem it

is possible to express this operator – or, in fact, all differential operators in the problem – in terms of a ﬂat reference

metric.

Assuming, of course, that we have imposed suitable boundary conditions.

3.5 Mass, momentum and angular momentum 83

metric. In many parts of this book we will be more interested in solutions that do not

possess exact Killing vectors, for example black hole or neutron star binaries. We will

see, however, that these spacetimes do admit approximate helical Killing vectors. For

these situations it is possible that the “waveless” approximation produces initial data that

represent this symmetry more accurately than initial data that require an ad hoc choice

for the conformally related metric. We will discuss some results for binary neutron stars

obtained with the “waveless” approximation in Chapter 15.3.

3.5 Mass, momentum and angular momentum

There are several important global conserved quantities that characterize an isolated sys-

tem, such as its total mass and angular momentum. Associated with these global parameters,

which are well-deﬁned only for asymptotically ﬂat spacetimes, are conservation laws that

state that the rate of loss of these quantities from an isolated system is equal to the rate

at which matter, ﬁelds and gravitational waves carry them away.

Once we have solved

the constraint equations and constructed a complete set of initial data for a system, we

can then determine the values of the global conserved parameters associated with the

system. During a numerical evolution, monitoring the degree to which these parameters

are conserved provides a very useful check on the accuracy of the numerical integration.

Here we assemble a few useful formulae for evaluating some of these parameters.

Suppose the system contains matter. Then we can derive an expression for its conserved

rest mass M

(sometimes called the baryon mass, if the matter is composed of baryons)

from the continuity equation,

∇

(ρ

) = 0, (3.122)

where ρ

is the rest-mass density. Integrating this expression over a 4-dimensional region

of spacetime  yields





√

−g∇

(ρ

) = 0. (3.123)

Using Gauss’s theorem we can relate the divergence of ρ

inside the region  to the

value of ρ

on the region’s 3-dimensional boundary ∂,





√

−g∇

(ρ

) =



∂



, (3.124)

where d



= N

√

γ d

x and N

is the outward-pointing unit normal vector on ∂.

When ∂ is spacelike, the factor  =−1andwhen∂ is timelike,  =+1.

Now imagine a “pill-box”-shaped spacetime region that is bounded by two spatial slices



and 

as well as a timelike hypersurface residing entirely outside the source, as

illustrated in Figure 3.5. In this case only the spatial surfaces contribute to the surface

See, e.g., Misner et al. (1973), Chapters 19 and 20.

84 Chapter 3 Constructing initial data

Figure 3.5 A “pill-box”-shaped spacetime region  is bounded by a 3-dimensional surface ∂ consisting of

two spatial slices 

and 

, as well as a timelike hypersurface that lies in the exterior vacuum. An isolated,

gravitating source in the interior is conﬁned within the shaded worldtube. The outward-pointing normal vectors

to δ are also shown.

integral. On 

the normal vector N

points toward the future, and therefore coincides

with the normal vector n

of the spacetime foliation introduced in Section 2.3.From

equation (2.117) we then have

= n

=−αu

. (3.125)

On 

, N

points toward the past, while n

points toward the future, which introduces a

negative sign between them. The conservation law (3.124) can therefore be written as





√

γαu

−





√

γαu

= 0. (3.126)

Equation (3.126) immediately implies that the rest mass, deﬁned as





√

γαu

, (3.127)

is conserved. The rest mass is thus determined by a 3-dimensional volume integral over a

spatial region spanning the matter source.

Deﬁning the total mass-energy of the system is more subtle, since it cannot be deﬁned

locally in general relativity. One useful measure of mass-energy is provided by the ADM

mass, M

ADM

, named after Arnowitt, Deser and Misner.

The ADM mass measures the

total mass-energy of an isolated gravitating system at any instant of time measured within

a spatial surface enclosing the system at inﬁnity. Formally, M

ADM

is deﬁned by an integral

See Arnowitt et al. (1962).

3.5 Mass, momentum and angular momentum 85

over the 2-dimensional surface at inﬁnity, ∂

∞

, of a spatial slice  according to

ADM

16π



∂

∞

√

γγ

(∂

− ∂

)dS

. (3.128)

Here dS

= σ



∂

∞

z is the outward-oriented surface element, where the z

’s a r e

coordinates on ∂

∞

, γ

∂

∞

is the induced metric on ∂

∞

,andσ

is the unit normal

(σ

= 1) to ∂

∞

This deﬁnition requires the spacetime to be asymptotically ﬂat and

the spacetime metric to approach the Minkowski metric sufﬁciently quickly with increasing

distance from the source,

− η

= O(r

−1

). (3.129)

The form of the integrand appearing in equation (3.128) is clearly not covariant and

gives the correct answer only when it is evaluated in asymptotically Cartesian coordinates

(in which case

√

γ = 1on∂

∞

). We will provide a covariant expression shortly (see

equation 3.145).

The existence of such a surface integral to determine the mass-energy

of an isolated system is consistent with the principle that the total mass-energy of such a

system can always by determined by a measurement performed by a distant observer (e.g.,

using Kepler’s third law).

By deﬁnition, the spatial surface ∂

∞

must be taken out to inﬁnity, in which case M

ADM

is rigorously conserved. In numerical applications, the integral is often evaluated on a large

surface at a distant, but ﬁnite, radius from the gravitating source, in the asymptotically ﬂat

region of spacetime. In this case, M

ADM

will change in time whenever there is a ﬂux of

matter or gravitational radiation passing across the surface. However, the rate of change of

ADM

will exactly reﬂect the rate at which mass-energy is carried across the surface by

these ﬂuxes.

Using the identity

∂

− ∂

= 

mnj

− 

jmn

, (3.130)

the integral (3.128) can be rewritten as

ADM

16π



∂

∞

(

mnj

− 

jmn

)dS

16π



∂

∞

(

− γ



)dS

(3.131)

where 

= γ



O Murchadha and York (1974), equation (7). See also Misner et al. (1973), equation (20.7), or Wald (1984), equa-

tion (11.2.14), for equivalent expressions.

The surface element dS

can be written alternatively as dS



ilm



∧



(= 

ilm

in an integral), where



ilm

√

γ [ilm] is the 3-dimensional Levi-Civita symbol and [ilm] is the total antisymmetrization symbol. See, e.g.,

Lightman et al. (1975), Problem 8.10; see also Appendix C and Poisson (2004), Section 3.2.1 for discussion of

alternative expressions for surface elements and Section 3.3 for a derivation of Gauss’s theorem.

A “gauge-invariant” form of equation (3.128) for asymptotic Minkowski spacetimes is provided by equation (85) in

Yo r k , J r . (1979): M

ADM

16π



∂

∞

− δ

tr h)dS

,whereγ

= η

+ h

,andwhereh

= O(r

−1

86 Chapter 3 Constructing initial data

To get comfortable with the above formulation of M

ADM

, we will evaluate this mass for

a Schwarzschild spacetime as described by the Kerr–Schild metric (see exercise 2.33 and

Table 2.1). Taking derivatives of the spatial metric

= η

+ 2Hl

, (3.132)

where H = M/r and l

= x

/r with r

= x

+ y

+ z

,weﬁnd



(3 + 8H )l

and γ



=−

. (3.133)

The metric evaluated on a 2-sphere S

at large radius r  M is

∂

∞

= r

(dθ

+ sin

θdφ

), (3.134)

so that the oriented surface element becomes

= σ



∂

∞

z = l

sin θdθdφ. (3.135)

Inserting these expressions into equation (3.131) we obtain

ADM

= lim

r→∞

16π



(4 + 8H )l

sin θdθdφ = M. (3.136)

While this result is not surprising, the identiﬁcation of M

ADM

with the parameter M

appearing in the Kerr–Schild metric is reassuring.

To bring the expression for M

ADM

into a more familiar and useful form for numerical

applications, we employ the conformal transformation of the spatial metric according to

equation (3.5). With equation (3.7) the integrand in equation (3.131) can then be expressed



− γ



= ψ

−4

(



− ¯γ



) − 8ψ

−5

ψ. (3.137)

Substituting this relation into the integrand in equation (3.131) and assuming that the

asymptotic behavior of the conformal factor satisﬁes

ψ = 1 +

O(r

−1

), (3.138)

we ﬁnd

ADM

16π



∂

∞

(



− ¯γ



−

2π



∂

∞

ψd

. (3.139)

Note that in writing the above expression we used the fact that dS

is conformally invariant

on ∂

∞

, i.e., d

= dS

Exercise 3.18 Show that in Cartesian coordinates, when the conformal metric

satisﬁes ¯γ = 1, then



= 0, so that equation (3.139) simplies to

ADM

16π



∂

∞



−

2π



∂

∞

ψd

, (3.140)

where



=−∂

¯γ

3.5 Mass, momentum and angular momentum 87

Exercise 3.19 Consider again a Schwarzschild spacetime described in Kerr–Schild

coordinates (see exercise 2.33 and Table 2.1). Now assume a conformal decomposi-

tion of the spatial metric as in equation (3.6), so that the conformal factor becomes

ψ = γ

1/12



1 +



1/12

. (3.141)

(a) Verify that the conformally related metric satisﬁes

¯γ



1 +



−1/3





= δ



−



+ O





(3.142)

and that its inverse satisﬁes

¯γ

= δ

−



−



+ O





. (3.143)

(b) Apply equation (3.140) to determine M

ADM

. In particular, show that the ﬁrst

integral in that expression contributes 2M/3 and the second contributes M/3to

ADM

If the conformal metric falls off sufﬁciently fast,

¯γ

= η

+ O(r

−1+a

), (3.144)

with a > 0, then the ﬁrst integral vanishes and the ADM mass reduces to

ADM

=−

2π



∂

∞

ψd

=−

2π



∂

∞

ψdS

. (3.145)

In the writing the last expression we used the fact that we are evaluating the surface

integral at inﬁnity, whereby we can drop the bar over the variables in the integrand.

Similarly, we can replace the operator D by its ﬂat-space counterpart. We note, however,

that not every conformal metric satisﬁes the fall-off condition (3.144); we have already

seen a counterexample in exercise 3.19.

Exercise 3.20 Show that the constant M appearing in the isotropic form of the

metric for a Schwarzschild spacetime, equation (2.35), is the ADM mass M

ADM

Exercise 3.21 Show that the ADM mass of the wormhole solution (3.26)isgiven

by equation (3.27).

With the help of Gauss’s theorem the 2-dimensional surface integral (3.145) can be

converted into a 3-dimensional volume integral,

ADM

=−

2π



∂

∞

ψd

=−

2π







¯γ

ψ. (3.146)

An identical integral can be written in terms of the unbarred (physical) metric: M

ADM

=−

2π





√

γ D

ψ.In

fact, the interior metric can be replaced by a ﬂat spatial metric, in which case M

ADM

=−

2π





x∇

ψ,where∇

the ﬂat-space Laplacian operator. The choice is usually made depending on which form of the integrand is particularly

useful numerically.

88 Chapter 3 Constructing initial data

Substituting the Hamiltonian constraint (3.37) then yields

ADM

16π







¯γ



16πψ

ρ + ψ

−7

− ψ

R −



. (3.147)

Of course, equations (3.146) and (3.147) only apply if the integrand is regular everywhere

inside the volume . They therefore do not apply, for example, if  contains a singularity

arising from a black hole. In this case equation (3.146) must be modiﬁed so that  does

not contain the singularity:

ADM

=−

2π



∂

∞

ψd

=−

2π







¯γ

ψ −

2π



∂

ψd

, (3.148)

where ∂

is some suitable inner surface of  that surrounds the black hole singularity,

excising it from the interior of . Generalization of equation (3.148) for multiple black

holes is obvious, as is the substitution of the Hamiltonian constraint (3.37)for

ψ in the

integrand of the ﬁrst term on the right hand side.

Exercise 3.22 Use equation (3.140) with ¯γ = 1, together with Hamiltonian con-

straint (3.37), to derive the following general expression for M

ADM

, which allows

for a black hole singularity inside ∂

ADM

16π







16πψ

ρ + ψ

−7

− ψ

R −

+ ∂





16π





(



− 8

ψ)d

(3.149)

This formula for evaluating M

ADM

has proven very useful as a diagnostic in numerical

simulations involving black holes.

Exercise 3.23 Assume a perfect gas, so that the stress–energy tensor is given by

= (ρ

+ ρ

+ P)u

+ Pg

(3.150)

where  is the speciﬁc internal energy density and P the pressure (see Section 5.2.1).

Show that under the assumptions of exercise 2.28 the difference between the ADM

mass M

ADM

and the rest-mass M

is given by

ADM

− M

= T + W + U (3.151)

in the Newtonian limit, where T is the kinetic energy,

T =



x, (3.152)

Note that in Chapter 11.5 we adopt a different convention for the conformal rescaling of A

via equation (11.34),

whereby ψ = e

, the second term in equation (3.147) becomes ψ

,and ¯γ = 1.

Cao et al. (2008); see footnote .

3.5 Mass, momentum and angular momentum 89

W the gravitational potential energy,

W =



d

x, (3.153)

and U is the internal energy,

U =



d

x. (3.154)

Exercise 3.23 reinforces our interpretation of the ADM mass as the total mass-energy.

Given this interpretation, the ADM mass for the boosted black hole solution of exercise

3.13 may seem quite confusing at ﬁrst sight.

Exercise 3.24 Show that the ADM mass of the boosted black hole solution of

exercise 3.13 is

ADM

= M +

+ O(P

). (3.155)

This result seems to suggest that the kinetic energy of a black hole is 5P

/(8M), and

not P

/(2M), as we might have expected. However, the blame for this result does not go

to the ADM mass, but instead to our incorrect interpretation of the parameter

M as the

black hole’s mass. In Chapter 7.1 we will ﬁnd that the irreducible mass M

irr

, based on the

black hole’s horizon area (see equations 1.70 and 7.2), is a suitable deﬁnition for the black

hole’s quasilocal mass. This mass differs from

M for nonzero momentum P.Inexercise

7.20 we will ﬁnd that if the ADM mass (3.155) is expressed in terms of M

irr

instead of M,

we indeed ﬁnd the expected Newtonian expression P

/(2M

irr

) for the kinetic energy.

The boosted black hole solution treated in exercises 3.13 and 3.24 is constructed by

specifying initial data on a conformally ﬂat, t = constant slice for a nonspinning black

hole endowed with some linear momentum P

. As we discussed in Section 3.2,the

analogous approach for constructing a rotating black hole did not lead to a slice of a

rotating Kerr black hole (recall that this approach generates a solution that has a perturbation

containing gravitational radiation). Likewise, there is no reason to expect that the same

approach to constructing nonspinning “boosted” black hole initial data will lead to a time

slice of a Schwarzschild black hole spacetime that has been “boosted” by applying a

conventional Lorentz boost to the static Schwarzschild solution (it does not). We follow

the later approach to constructing a boosted black hole in exercise 3.25 and show that

it leads to the expected result for the ADM mass on t = constant slices in a boosted

coordinate system. Speciﬁcally, if the black hole mass is M in a static frame, then we ﬁnd

ADM

= γ M in an asymptotic frame in which the black hole is observed to be moving

with velocity v,whereγ = 1/

√

1 − v

. This result is reassuring.

Exercise 3.25 Consider the Schwarzschild metric (2.35), expressed in Cartesian

isotropic coordinates, in the limit of r = (x

+ y

+ z

)

1/2

 M,

=−



1 −





1 +



(dx

+ dy

+ dz

). (3.156)

90 Chapter 3 Constructing initial data

Now boost this metric in the z-direction by applying a Lorentz transformation

x =

x, y =

y, z = γ (

z − v

t), t = γ (

t − v

z) (3.157)

where γ = (1 − v

)

−1/2

(a) Find the new spacetime metric in the boosted (barred) coordinate system and

show that the spatial metric on

t = constant slices is given asymptotically by





1 +





+ 2γ



, (3.158)

where r is now given in terms of the boosted coordinates as r = (

+ γ

(

z −

)

1/2

,andwherewehavedeﬁnedz



= diag(0, 0, 1).

(b) The easiest approach for evaluating the ADM mass may be to use expression

(3.128), keeping only leading-order terms when evaluating the integral on a sphere of

constant

r = (

)

1/2

with

r  M and

r  v

t. Without loss of generality,

we may set

t = 0. Show that

∂



=−



(η

+ 2γ

), (3.159)

where m



= ∂r/∂ x



= (

y,γ

z)/r , and note that the surface element is dS

sin θdθdφ,wherel

= x

r = (

z)/

r/r = (1 + γ

)

−1/2

,whereµ =

r = cos θ ,insertall

the above expressions into (3.128), carry out the trivial integration over φ,andshow

that the integral reduces to

ADM

= γ



−1

dµ

(1 + γ

)

3/2

. (3.160)

(d) Finally, evaluate the integral (3.160)toﬁnd

ADM

= γ M. (3.161)

Exercise 3.26 Consider the Brill wave deﬁned in exercise 3.5, subject only to the

following regularity and asymptotic restrictions on q:

q = 0forρ = 0, (3.162)

∂

q = 0forρ = 0, (3.163)

∂

q = 0for z = 0, (3.164)

q ∼ r

−a

, a ≥ 2. (3.165)

(a) Show that M

ADM

can be evaluated as

ADM

2π







x, (3.166)

and thereby prove that the mass-energy is positive deﬁnite for any q = 0.

In the

context of a gravitational wave at a moment of time symmetry, the mass-energy is

sometimes referred to as the Brill mass.

Brill (1959); Eppley (1977).