Baumgarte T., Shapiro S. Numerical Relativity. Solving Einstein’s Equations on the Computer

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3.5 Mass, momentum and angular momentum 91

(b) Consider the following form for q,

q =

Aρ

1 + (r/λ)

, (3.167)

where A and λ are constants, r

= ρ

+ z

,andn ≥ 4. Choose n = 5andλ = 1,

integrate either equation (3.146) or equation (3.166) numerically for different A

between 10

−2

≤ A ≤ 16 and make a plot of M

ADM

vs. A. Check your integrations

by showing that for small amplitude A, the mass is proportional to A

and satisﬁes

ADM

≈ 1.4 × 10

−2

On further probing, things can sometimes get a little confusing, as when evaluating the

ADM mass for Schwarzschild spacetime in Painlev

e–Gullstrand coordinates (see exercise

2.32 or Table 2.1). All the above expressions yield an ADM mass of zero in these coordi-

nates. The reason for this is that the shift in Painlev

e–Gullstrand coordinates does not fall

off sufﬁciently fast, violating the condition (3.129). If nothing else, this result provides a

warning to us that we must be careful to check that the metric satisﬁes the correct asymp-

totic conditions in the adopted coordinates when applying the above formulae to calculate

the mass. It also motivates a search for other mass deﬁnitions.

For example, another possibility for determining the mass is the Bondi–Sachs mass.

For stationary spacetimes, the ADM and Bondi–Sachs mass are identical. But they are

different for dynamical spacetimes emitting radiation (gravitational or otherwise). The

ADM mass, evaluated at spatial inﬁnity, remains strictly conserved, while the Bondi–

Sachs mass, evaluated at null inﬁnity, decreases in response to the outgoing radiation.

We refer the reader to other references for a detailed discussion and formulae for the

Bondi–Sachs mass.

One other measure of mass that has proven particularly useful in recent numerical

applications is the Komar mass,

, which can be deﬁned for a spacetime that admits

a timelike Killing vector, say ξ

(t)

. Contracting this Killing vector with the (4-dimentional)

Ricci tensor we can deﬁne a current

(t)

= ξ

(t)

(4)

. (3.168)

The Ricci tensor is related to the stress-energy tensor through Einstein’s equations, suggest-

ing that this current may lead to a reasonable mass deﬁnition. As it turns out, the current

(3.168) is conserved, meaning that its divergence vanishes. To see this, we compute

∇

(t)

= (∇

(t)

)

(4)

+ ξ

(t)

∇

(4)

. (3.169)

Here the ﬁrst term vanishes because of Killing’s equation and the second one can be

rewritten using the contracted Bianchi identity

∇

(t)

∇

(4)

R = 0, (3.170)

See, e.g., Poisson (2004), Sections 4.3, and Section 11.2, and references therein.

See Komar (1959). Our discussion is adapted from Carroll (2004), Section 6.4.

92 Chapter 3 Constructing initial data

where the second term vanishes because the directional derivative of

(4)

R along a Killing

vector has to be zero. Just as the existence of the conserved matter current ρ

gave rise

to a conserved rest-mass (3.127) in the beginning of this section, the conserved current

(3.168) gives rise to the conserved Komar mass,

4π





√

γ n

(t)

. (3.171)

Assuming that the Killing ﬁeld ξ

(t)

is properly normalized in the asymptotically ﬂat region

of spacetime (ξ

(t)

=−1), the above deﬁnition of the Komar mass leads to the expected

result for the value of the mass in familiar cases, as we shall see below. On the other

hand, in a vacuum spacetime we have

(4)

= 0, so that J

(t)

= 0, so we might conclude

from equation (3.171) that the Komar mass vanishes for a black hole. This conclusion is

incorrect, however, since for a black hole the volume integral (3.171) contains a singularity

at which

(4)

is not deﬁned.

However, the current (3.168) can be written as a total

derivative. As a consequence we can convert the volume integral into a surface integral at

inﬁnity that can be evaluated even for black holes.

To show that J

(t)

can be written as a total derivative we start with the deﬁnition of the

Riemann tensor,

∇

(t)

−∇

∇

(t)

(4)

dab

(t)

. (3.172)

Contracting this equation and using the relation ∇

(t)

= 0, which holds for any Killing

vector, we ﬁnd

∇

(t)

(4)

(t)

, (3.173)

so that, by equation (3.168), we have

(t)

=∇

∇

(t)

. (3.174)

Inserting equation (3.174) into equation (3.171) we then have

4π





√

γ n

∇

(t)

. (3.175)

Since ∇

(t)

is antisymmetric, we can use Gauss’s theorem to convert this volume integral

into the surface integral at inﬁnity,

4π



∂

∞

∇

(t)

. (3.176)

Equation (3.176), which can be used even in the case of black holes, could have been the

starting point for our derivation of the Komar mass.

We already faced a similar issue above when we discussed volume integrals for M

ADM

in the case of spacetimes

containing singularities.

3.5 Mass, momentum and angular momentum 93

We now bring this expression into a form that is easier to evaluate in terms of standard

3 + 1 variables. We assume that the coordinates are chosen to reﬂect the symmetry imposed

by the Killing vector ξ

(t)

, meaning that our lapse and shift are the Killing lapse and shift

as deﬁned in equation (2.162), and thus satisfy

(t)

= t

= αn

+ β

. (3.177)

Recalling that dS

= d



∂

∞

, we can rewrite the integrand in equation (3.176)as

∇

(t)

=−n

∇

(t)

=−n

∇

(αn

+ β

)

=−n

∇

α + α∇

+∇

) = σ

∇

α + σ

∇

= σ

∇

α − σ

= σ

α − σ

, (3.178)

where we have used Killing’s equation, and equation (2.52), as well as the rela-

tions n

=−1, n

= 0andn

= 0. Inserting this into equation (3.176) ﬁnally

yields

4π



∂

∞

α − β

). (3.179)

In many cases, the term β

falls off rapidly enough so that it can be neglected in

the integral (3.179). A counterexample is the Schwarzschild spacetime in Painlev

e–

Gullstrand coordinates, for which the shift term carries the entire contribution to the Komar

mass.

Exercise 3.27 Evaluate the Komar mass M

for a Schwarzschild black hole in (1)

Schwarzschild coordinates and (2) Painlev

e–Gullstrand coordinates.

Evidently, exercise 3.27 shows that the Komar mass can handle Painlev

e–Gullstrand

coordinates while the ADM mass cannot. On the other hand, the deﬁnition of the Komar

mass assumes the existence of a timelike Killing vector, which the ADM mass does not.

This property of the Komar mass in fact provides a powerful diagnostic tool for numerical

spacetimes: by searching for solutions for which the ADM mass and Komar mass are both

well-deﬁned and identical, we can identify spacetimes that admit a timelike Killing vector

and are hence in stationary equilibrium. For example, we shall apply this diagnostic in

Chapter 12.3.3 to construct (quasi-)stationary spacetimes containing binary black holes in

circular equilibrium.

Finally, we can turn the surface integral (3.179) back into the volume integral

4π





√

γ (D

α − β

− K

), (3.180)

provided the integrand contains no singularities. Otherwise, we can excise the singu-

larity and restrict the volume integral to the region outside the excision surface just as

94 Chapter 3 Constructing initial data

we did in deriving equation (3.148). With the help of equation (2.137), the momen-

tum constraint (2.133) and the evolution equation (2.134) this integral can be converted

into





√



α(ρ + S) − 2β



, (3.181)

wherewehaveassumedagainthatξ

(t)

= t

is a Killing vector, so that we have ∂

∂

K = 0. For vacuum spacetimes this volume integral vanishes, which is not surprising in

the light of the discussion following equation (3.171). In practice, this realization means

that the Komar mass in equation (3.179) can be evaluated on any surface ∂ enclosing all

matter sources and singularities, and does not need to be evaluated at inﬁnity on ∂

∞

(in

contrast to the ADM mass).

Exercise 3.28 Show that, under the same weak-ﬁeld, slow-velocity assumptions as

in exercises 2.28, 3.7 and 3.23, the difference between the Komar mass M

and the

rest-mass M

becomes

− M

= 3T + 2W + U + 3, (3.182)

in the Newtonian limit. Here we have deﬁned the quantity

 =



x. (3.183)

Equation (3.182) may seem strange at ﬁrst. Recall, however, that the Komar mass is

deﬁned only in the presence of a timelike Killing vector, that is, for stationary spacetimes.

Such spacetimes are in dynamical equilibrium, so that they obey the virial theorem given

2T + W + 3 = 0 (3.184)

in the Newtonian limit.

Combining equation (3.184) with (3.182) shows that, just like

the ADM mass, the Komar mass may be interpreted as the total mass-energy (cf. exercise

3.23). As we have discussed above, this argument can be turned around to identify systems

in dynamical equilibrium by searching for spacetimes for which the ADM and Komar

mass are equal. In fact, a relativistic formulation of the virial theorem can be derived by

imposing the equality of the ADM and Komar mass.

We now turn to the angular momentum of an isolated system. First we shall treat the case

of an axisymmetric spacetime, for which there exists is a global rotational Killing vector,

(φ)

. Retracing the steps that lead to the Komar mass integral (3.176), we can construct a

surface integral for a conserved angular momentum according to

=−

8π



∂

∞

∇

(φ)

. (3.185)

See, e.g., Shapiro and Teukolsky (1983), equation (7.1.21).

See Gourgoulhon and Bonazzola (1994).

3.5 Mass, momentum and angular momentum 95

Equation (3.185)istheKomar angular momentum expression for an asymptotically ﬂat,

axisymmetric spacetime. This quantity measures the angular momentum about the symme-

try axis and gives the expected magnitude in familiar examples, as we will see in exercise

3.29 below. The fact that the surface can be chosen arbitrarily can be used to prove that in

an axisymmetric spacetime gravitational radiation carries no angular momentum.

The

integrand can be rewritten as

∇

(φ)

=−σ

∇

(φ)

= σ

(φ)

∇

=−σ

(φ)

, (3.186)

where we have used Killing’s equation, the relation ξ

(φ)

= 0, equation (2.52) and, in

the last step, the fact that both ξ

(φ)

and σ

are spatial. Inserting the above expression into

equation (3.185) yields

8π



∂

∞

(φ)

. (3.187)

Adopting Cartesian coordinates, we can assign the symmetry axis to be along x

,in

which case the spatial rotational Killing vector about this axis becomes

(φ)

= ξ

(i)

= 

kml

(i)

, (3.188)

where e

(i)

= δ

is the basis vector along x

and 

kml

is the 3-dimensional Levi-Civita

tensor, both evaluated in the asymptotically ﬂat region. The angular momentum about the

axis may then be calculated from



ijk

8π



∂

∞

. (3.189)

If desired, this integral can also be converted into a volume integral.

Exercise 3.29 Show that the angular momentum of solution (3.66)isJ

(which, by

virtue of exercises 3.11 and 2.34 also shows that the angular momentum of a Kerr

black hole is J = aM).

In the absence of axisymmetry, it is often useful to evaluate the ADM angular momentum,

whichisdeﬁnedas

ADM

8π



∂

∞

− δ

K )ξ

(i)

, (3.190)

or, substituting equation (3.188),

ADM

8π



ijn



∂

∞

− δ

K ). (3.191)

Lightman et al. (1975), Problem 18.9

See, e.g., Lightman et al. (1975), Problem 10.9.

96 Chapter 3 Constructing initial data

This expression requires some stronger asymptotic gauge conditions beyond asymptotic

ﬂatness, but these conditions are met by a conformally ﬂat metric, ¯γ

= η

Exercise 3.30 Prove that J

ADM

reduces to J

in axisymmetry.

Hint: First choose the surface ∂

∞

to be a 2-sphere and show that the piece of

the integral involving K in the expressions for J

ADM

vanishes. Now generalize to

arbitrary surfaces.

Exercise 3.31 Show that in a stationary, axisymmetric and nonsingular spacetime

the mass-energy can be calculated from

ADM

= M





√

γ n

(2T

− δ

T )ξ

(t)

, (3.192)

and the angular momentum can be calculated from

ADM

= J

=−





√

γ n

(φ)

. (3.193)

Here we assume that the interior volume  covers the entire matter distribution and

contains no singularities.

Finally, if we replace the rotational Killing vector in equation (3.190) by the spatial

translational Killing vector, we obtain an expression for the ADM linear momentum.A

spatial translational Killing vector along x

in the asymptotically ﬂat region takes the form

(i)

= e

(i)

. (3.194)

Inserting this relation into (3.190) yields the linear momentum along x

ADM

8π



∂

∞

− δ

K ). (3.195)

Once again, the surface integrals given above for J

ADM

and P

ADM

can be converted into

volume integrals, if desired.

Exercise 3.32 Show that the linear momentum of solution (3.80)isP

ADM

. (Recall

that this solution assumes K = 0.)

Exercise 3.33 Show that the linear momentum of the boosted black hole in exercise

3.25 is given by P

ADM

= γ Mv

,whereγ = 1/

√

1 − v

The importance of monitoring the conserved quantities assembled here when perform-

ing a numerical simulation of an evolving system cannot be overstated. As we mentioned

Yo r k , J r . (1979); Gourgoulhon et al. (2002).

3.5 Mass, momentum and angular momentum 97

earlier, the degree to which they are conserved helps calibrate the accuracy of the simula-

tion. Often these quantities are evaluated on a large, but ﬁnite, surface in the asymptotically

ﬂat region near the outer boundary of the computational grid. In this case, any changes

in the calculated mass, angular momentum or linear momentum of the system must be

accounted for by the integrated ﬂux of these quantities carried off by any matter or gravi-

tational waves crossing the surface. We will see how well this principle works in action in

some representative simulations described in later chapters.

Choosing coordinates:

the lapse and shift

In Chapter 2 we performed a 3 + 1 decomposition of Einstein’s ﬁeld equations and have

seen that these can be split into two distinct sets: constraint equations and evolution

equations. The constraint equations contain no time derivatives and relate ﬁeld quantities

on a given t =constant spacelike hypersurface. The evolution equations contain ﬁrst-order

time derivatives that tell us how the ﬁeld quantities change from one hypersurface to the

next. In Chapter 3 we have brought the constraint equations into a form that is suitable

for numerical implementation, that is, we cast the equations in terms of spatial differential

operators that can be inverted with standard numerical techniques. We will provide a brief

introduction to some common numerical algorithms for solving these (elliptic) equations

in Chapter 6. The 3 + 1 evolution equations that we derived, e.g., equation (2.134)forγ

and equation (2.135)forK

, are not quite ready for numerical integration. For one thing,

we have yet to impose coordinate conditions by specifying the lapse function α and the

shift vector β

that appear in these equations. The lapse and shift are freely speciﬁable

gauge variables that need to be chosen in order to advance the ﬁeld data from one time

slice to the next. As it turns out, ﬁnding kinematical conditions for the coordinates that

allow for a well-behaved, long time evolution is nontrivial in general. However, geometric

insight and numerical experimentation can be combined to produce good gauge choices

for treating many of the most important physical and astrophysical problems requiring

numerical relativity for solution, as we shall see.

What constitutes a “good” coordinate system? Clearly, the adopted coordinates must

not allow the appearance of any singularities, which could have dire consequences for

a numerical simulation. Such a singularity, which is often associated with a black hole,

could be either a coordinate singularity or a physical singularity. Recall, for example, the

case of a Schwarzschild black hole in Schwarzschild coordinates. Singularities associated

with the Schwarzschild radius r

= 2M (the horizon) are coordinate in origin and are

removable by a coordinate transformation. Singularities at r

= 0, however, are physical

and invariant, resulting from the inﬁnite curvature at the origin, and are not removable

by a coordinate transformation. We know that encountering such a physical singularity

during a space voyage would be disastrous for any traveler; be assured that encountering

one in a numerical simulation would be equally disastrous for a numerical relativist!

Such a singularity will result in one or more of the ﬁeld variables blowing up to inﬁnity,

Unless one’s code is speciﬁcally designed to explore the nature of singularities; see, e.g., Berger (2002).

Chapter 4 Choosing coordinates: the lapse and shift 99

leading to underﬂows and overﬂows in the output and eventually causing the code to

crash.

To avoid coordinate singularities associated with horizons, like the one at r

= 2M

that we discussed above, black hole simulations have sometimes been carried out using

“horizon penetrating” coordinates in which light cones do not pinch-off at the horizon

as they do in Schwarzschild coordinates. Kerr–Schild coordinates provide one example

of a “horizon penetrating” coordinate system that is well-behaved at r

= 2M.Many

simulations also have relied on “singularity-avoiding” gauge conditions to prevent or

postpone the appearance of physical singularities in the computational domain. These

gauges have been especially important in treating stellar collapse to black holes, where

physical singularities are not present in the initial spacetime, but inevitably arise following

the formation of a black hole. More recently, black hole “excision” techniques to prevent

the appearance of singularities have been developed whereby the black hole interior and

its curvature singularity are excised from the computational domain altogether. In codes

that solve the partial differential equations on a discrete spacetime coordinate lattice, it

has proven adequate in some cases to retain the black hole interior and its singularity

within the computational domain by simply avoiding placing the singularity on any lattice

point on which the variables are evaluated. This is the trick commonly used to evolve

“puncture” black holes, which contain interior coordinate singularities, rather than physical

singularities, as we discussed in Chapter 3.1.2.

In these simulations it is again the particular

choice of coordinates that prevents numerical lattice points from reaching the physical

spacetime singularity.

In fact, it has even been possible in some simulations to replace the black hole interior

and its spacetime singularity with smooth, but otherwise arbitrary, initial data in order

to evolve the spacetime numerically. The interior data in such cases is “junk” in that it

typically fails to satisfy the constraint equations. However, these “junk-ﬁlling” initial data

are completely adequate to permit a reliable evolution of the exterior ﬁeld, as long as

suitable gauge conditions are implemented to insure that the computational scheme allows

no information to leak out from the black hole interior to the exterior during the evolution.

Given the causal nature of the black hole event horizon, it is not surprising that such a

scheme can be implemented. Examples of all of these approaches will be discussed later on.

But now we are jumping ahead ourselves. First we must step back and note that the

problem of picking an appropriate coordinate system typically is split into two parts:

choosing a time slicing (i.e., a time coordinate), and picking a spatial gauge (i.e., spatial

coordinates). The time slicing determines what shape the spatial slices  take in the

enveloping spacetime. The lapse α determines how the shape of the slices  changes in

time, since it relates the advance of proper time to coordinate time along the normal vector

connecting one spatial slice to the next, as illustrated in Figure 2.4. Picking a time

Recall the discussion following equation (3.19).

See Section 4.5 and Chapter 13.1.3.

100 Chapter 4 Choosing coordinates: the lapse and shift

slicing or a time coordinate therefore amounts to making a choice for the lapse function.

Letting the lapse vary with position across the spatial slice takes advantage of the freedom

that proper time can advance at different rates at different points on a given slice (“the

many-ﬁngered nature of time”). The shift β

, on the other hand, determines how spatial

points at rest with respect to a normal observer n

are relabeled on neighboring slices. The

spatial gauge is therefore imposed by a choice for the shift vector.

In the rest of this chapter we will discuss a few different gauge choices that are commonly

used in numerical simulations. We will focus here on some of the simpler conditions that

lend themselves to straightforward geometric interpretation and provide us with valuable

intuition. In later chapters we will observe these and other choices in action when we

study actual simulations that construct numerical spacetimes. There we will see how well

different gauge conditions perform in different physical situations. A brief summary of a

few common gauge choices is provided in Box 4.1 for the reader eager and willing to order

from a limited, but representative, menu.

4.1 Geodesic slicing

Since the lapse α and the shift β

can be chosen freely, let us ﬁrst consider the simplest

possible choice,

α = 1,β

= 0. (4.1)

In the context of numerical relativity this gauge choice is often called geodesic slicing;the

resulting coordinates are also known as Gaussian-normal coordinates.

Recall that coordinate observers move with 4-velocities u

= t

= e

(0)

(i.e., spatial

velocities u

= 0). Thus with β

= 0, coordinate observers coincide with normal observers

= n

). With α = 1, the proper time intervals that they measure agree with coordinate

time intervals. Their acceleration is given by equation (2.51),

= D

ln α = 0. (4.2)

Evidently, since their acceleration vanishes, normal observers are freely-falling and there-

fore follow geodesics, hence the name of this slicing condition. Clearly, the evolution

equations (2.134) and (2.135) simplify signiﬁcantly when geodesic slicing is adopted.

Exercise 4.1 Consider the Robertson–Walker metric for a ﬂat Friedmann cosmol-

ogy,

=−dt

+ a

(t) η

, (4.3)

where the expansion factor a(t) is a function of time only. We can immediately

read off the lapse and shift to be α = 1andβ

= 0, showing that this spacetime is

geodesically sliced.

(a) Identify the spatial metric γ

and ﬁnd the extrinsic curvature K

Darmois (1927) used these coordinates in his very early development of the 3+1 decomposition.