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

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Appendix H Moving puncture representions of Schwarzschild 641

with f (α) = 2/α. Retrace our derivation above for the more general form f (α) =

n/α,wheren is some arbitrary constant.

Hint: Try α

= 1 − 2M/r

+ C

2α/n

as the generalization of (H.17). You

should ﬁnd C

= (3n +

√

4 + 9n

)

/(128n

) e

−2α

(b) In Chapter 4.3 we argued that maximal slicing can be recovered in the limit

n →∞. Demonstrate that the solution of part (a) does indeed reduce to the maxi-

mal slicing solution of the previous section (with C = 3

√

/4) in this limit.

The above steps complete the solution in terms of the areal radius r

. For numerical purposes

it would again be desirable to express this solution in terms of an isotropic radius; while that does

not seem to be possible analytically, the transformation can be carried out numerically. We refer

the reader to Hannam et al. (2008) for a more detailed discussion of these solutions, as well as

their geometrical properties.

Binary black hole puncture

simulations as test problems

Simulating the late inspiral, merger and ringdown of a binary black hole was for many years

the “holy grail” of numerical relativity. Dozens of researchers have spent many years attempting

to formulate a stable algorithm capable of solving this important problem. As discussed in

Chapter 13, several different techniques and corresponding codes now exist that can evolve

coalescing binary black holes successfully. In this appendix we summarize two simulations in

sufﬁcient detail to enable the reader to assess the requirements and caliber of a typical computation.

We employ one widely adopted method for the simulations, based on the BSSN evolution scheme

and binary puncture initial data. The examples chosen here, and the quoted results, can be used

as test problems for the construction of new algorithms and new modules.

We emphasize that performing a simulation of binary black hole coalescence provides a highly

nontrivial and comprehensive laboratory for testing a 3 + 1 code capable of solving Einstein’s

ﬁeld equations in vacuum in the strong-ﬁeld regime of general relativity. It tests the caliber of not

only the basic evolution scheme, but also the all-important diagnostic routines. These routines

measure globally conserved quantities like the total mass and angular momentum of the system,

the masses and spins of the individual black holes, the location and area of the black hole horizons,

the asymptotic gravitational waveforms, the recoil kick imparted to the black hole remnant, and

other “vital statistics” characterizing a dynamical spacetime containing black holes. Appreciable

effort invariably goes into implementing and testing these diagnostic modules in order to extract

meaningful and reliable physical information from the numerical output.

We consider two cases of merging, nonspinning black holes: one is an equal-mass system,

and the other has a 3:1 mass ratio. Both cases employ initial data constructed using the binary

puncture technique to obtain quasiequilibrium binaries in circular orbit, as discussed in Chap-

ter 12.2. The initial data for the two cases are speciﬁed by the puncture parameters listed in

Table I.1. The code employs AMR with equatorial symmetry; the AMR grid setup is described in

Table I.2. The basic evolution scheme is BSSN with moving puncture gauge conditions. Details

concerning the implementation of the evolution scheme are summarized in Table I.3. A number

of the global diagnostic parameters, and several key physical results, are listed in Table I.4.

In reading through the summaries of these simulations as provided by the tables and ﬁgures

below it should become apparent that performing such calculations requires an understanding of

much of the formalism and computational machinery reviewed in this book. This point is implicit

in the many references cited below to earlier equations and discussions throughout the main text

where many of the quantities that appear in tables and ﬁgures are introduced. The simulation of

the inspiral and merger of a binary black hole thus represents the culmination of our training

and ability to evolve a pure vacuum, dynamical spacetime in 3 + 1 dimensions. It is also the logical

642

Appendix I Binary black hole puncture simulations 643

Table I.1 Binary black hole puncture initial data. The freely chosen parameters are each puncture’s

coordinate position x,“bare”massM, momentum P and spin S. The resulting initial puncture geometry

is speciﬁed by the 3-metric γ

and extrinsic curvature K

, as constructed in Chapter 12.2 for nonspinning

black holes in quasiequilibrium circular orbit. In the last ﬁve lines we list diagnostic parameters computed

from these initial data, including the total ADM mass M (equation 3.140), the total angular momentum J

(equation 3.191), and the irreducible mass M

irr

of each black hole (equation 7.2). Quantities like x, P, etc.

are listed above in “code units”, (where 1 code unit =1.011M) and can be scaled to arbitrary M. [After

Etienne et al. (2009).]

Equal mass (q = 1) Unequal mass (q = 3)

Location of punctures x

= (0, 4.891, 0), x

= (5.25, 0, 0),

−

= (0, −4.891, 0) x

−

= (−1.75, 0, 0)

“Bare” mass

= M

−

= 0.4856 M

= 0.234, M

−

= 0.735

Spin S

= S

−

= 0 S

= S

−

= 0

Momentum P

= (0.0969, 0, 0), P

= (0, 0.09407, 0),

−

= (−0.0969, 0, 0) P

−

= (0, −0.09407, 0)

ADM mass M = 0.9894 M = 0.9895

Angular momentum J = 0.9479 J = 0.6587

Irreducible mass M

irr

= M

−

irr

= 0.5000 M

irr

= 0.2498, M

−

irr

= 0.7494

Mean coord. radius of BHs’ horizon

= R

−

= 0.2359 R

= 0.1097, R

−

= 0.3605

Binary coord. separation (|x

− x

−

|)9.887M 7.074M

Table I.2 Grid setup. The parameter L is the location of the outer boundary, i.e.,

max

= y

max

= z

max

= L, x

min

= y

min

=−L, z

min

= 0 (equatorial symmetry). N

−

) is the number of

AMR reﬁnement levels (Chapter 6.2.5) centered at the “+”(“−”) puncture. R

−

) are the radii of the

reﬁnement levels centered at the “+”(“−”) puncture in code units (where 1 code unit = 1.011M).



(

−

) is the grid spacing in each reﬁnement level centered at the “+”(“−”) puncture (in code units).

Note that the grid spacing doubles in each successive reﬁnement level. The coarest grid spacing is twice

that of the outermost reﬁnement level, which is 8 for the equal-mass case and 10.24 for the unequal-mass

case. N

−

) is the number of grid points across the mean diameter of the apparent horizon of the “+”

(“−”) black hole at t = 0. [After Etienne et al. (2009).]

Equal-mass (q = 1) Unequal mass (q = 3)

L 320 (= 323.4M) 409.6 (= 413.9M)

810

−

(160, 80, 140/3, 20, 5, 2.5, 1.25, 0.625) (256, 128, 64, 24, 12, 6, 3, 1.5, 0.75, 0.375)



(4, 2, 1, . . . , 0.0625, 0.03125) (5.12, 2.56, 1.28, . . . , 0.02, 0.01)

−

(160, 80, 140/3, 20, 5, 2.5, 1.25, 0.625) (256, 128, 64, 32, 16, 8, 4, 2, 1)



−

(4, 2, 1, . . . , 0.0625, 0.03125) (5.12, 2.56, 1.28, . . . , 0.04, 0.02)

≈ 15 ≈ 22

−

≈ 15 ≈ 36

644 Appendix I Binary black hole puncture simulations

Table I.3 Evolution scheme. The basic BSSN scheme is summarized in Chapter 11.5 (see Box 11.1).

“Moving puncture” gauge conditions are adopted for the lapse and shift (see equation 4.51 and discussion

following equation 4.83). To enforce the auxiliary constraints in (1) the technique described in Etienne

et al. (2008) is implemented; to enforce the auxiliary constraint in (2) the technique suggested by

Marronetti et al. (2007) is adopted (see also equations 14.41–14.43 and related discussion). Kreiss and

Oliger (1973) dissipation (Chapter 6.2.3) is employed here to reduce any high frequency noise arising at

the moving reﬁnement boundaries. Carpet is a Cactus-based module that implements AMR (see

http://www.carpetcode.org for full documentation). Upwind differencing vs. centered spatial

differencing, as well as the method of lines (MOL) algorithm for time-integrations, are discussed in

Chapter 6.2.3. The Courant factor is deﬁned below equation (6.72). Prolongation refers to interpolations

in space and time at the AMR reﬁnement boundaries. A range of extraction radii is chosen to conﬁrm

convergence of waveforms. [After Etienne et al. (2009).]

Formalism BSSN

Lapse 1+log slicing: ∂

α =−2αK ,where∂

≡ ∂

− β

∂

Shift -freezing: ∂

= (3/4)B

, ∂

= ∂



− ηB

, η = 0.25 = 0.2474/M

Conformal variable Evolve e

−2φ

(instead of φ), where det γ

≡ e

12φ

Symmetry Equatorial

Other Numerical (1) Enforce auxiliary constraints det( ˜γ

) = 1andTr(

) = 0

techniques (2) Enforce auxiliary constraint



=−∂

˜γ

(3) Add 5th order Kreiss–Oliger dissipation of the form

(/64)(x

∂

+ y

∂

+ z

∂

) f

in all lapse, shift and BSSN evolution equations, ∂

f .

Here x, y and z are grid spacings and  = 0.1.

Grid-driver code Carpet

Spatial differencing 4th order upwind on shift advection terms,

4th order centered otherwise

Temporal differencing Method of lines (MOL) with 4th order Runge–Kutta (RK4)

Courant (CFL) factor 0.25

Prolongation 5th order spatial prolongation, 2nd order temporal prolongation

Wave extraction radii 10.11M–54.58M for equal-mass case,

30.32M–70.75M for unequal-mass case

starting point of many nonvacuum dynamical scenarios involving strong gravitational ﬁelds and

requiring the tools of numerical relativity for solution.

Of special signﬁcance is the degree to which total energy and angular momentum are conserved

when properly accounting for losses due to gravitational wave emission. The fractional errors are

seen in Table I.4 to be a few times 10

−4

for the energy and 10

−3

for the angular momentum.

Also, different measures of the mass and spin of the ﬁnal (stationary) remnant Kerr black hole

agree closely, given the adopted computational resources.

Appendix I Binary black hole puncture simulations 645

Table I.4 Simulation results: the black hole remnant and emitted radiation. M

and J

are the ﬁnal ADM mass and angular momentum measured at radius 43.90M in the

equal-mass case, and at 50.35M in the unequal-mass case. Here M is the initial ADM

mass as listed in Table I.1. The value of J

ﬂuctuates slightly; the quoted error is the

ﬂuctuation amplitude at late times. M

is the mass of the ﬁnal black hole as determined

by its irreducible mass (equations 7.2 and 7.4); J

is the spin parameter of the ﬁnal

hole as determined by the ratio of polar and equatorial circumferences, assuming the Kerr

relation. E

and J

are the energy and angular momentum carried off by

gravitational radiation (equations 9.130 and 9.131.) The waves are extracted at radius

50.53M, where the Weyl scalar ψ

is evaluated (equation 9.121). The parameter v

kick

the kick velocity imparted to the black hole in the unequal-mass case (equation 9.132).

(Wavedatafort − r < 50M, arising from gravitational radiation present in the initial

data, are removed before computing the kick velocity.) The parameter

δE ≡ (M − M

− E

)/M measures the fractional violation of energy conservation;

δ J ≡ (J − J

−  J

)/J measures the fractional violation of angular momentum

conservation. [After Etienne et al. (2009).]

Equal mass (q = 1) Unequal mass (q = 3)

/M 0.9615 0.9808

0.636 ± 0.008 0.52 ± 0.04

/M 0.9617 0.9809

0.6852 0.5413

E

/M 0.03794 0.01934

J

0.3306 0.1513

kick

— 174 km s

−1

δE 4 × 10

−4

−2 × 10

−4

δ J 4 × 10

−3

9 × 10

−4

−2

−4

−2

−4

y / M

x/M

24 0

−2

−4

Figure I.1 Trajectory of the coordinate centroid of the “+” (black solid line) and “−” (blue dash line) black hole

during nonspinning binary black hole inspiral and merger. The left-hand panel is the equal-mass case, the

right-hand panel in the unequal mass case. [From Etienne et al. (2009).]

646 Appendix I Binary black hole puncture simulations

00300

100 200 300

600 900

(t − r)/M

(r/M) h

(t − r)/M

0.4

0.3

0.15

−0.15

−0.3

0.2

−0.2

−0.4

Figure I.2 Gravitational waveforms for the inspiral and merger of nonspinning binary black holes. Shown here

are the dominant l = 2, m = 2 wave amplitudtes (r/M )h

(black solid line) and (r/M)h

(blue dash line) as

functions of retarded time t − r . The left-hand panel is the equal-mass case while the right-hand panel is the

unequal-mass case. The waveforms are extracted at radius r = 50.53M and computed by integrating the s =−2

spin-weighted spherical harmonics of the Weyl scalar ψ

twice with time (equations 9.126 and 9.134). [From

Etienne et al. (2009).]

The coordinate trajectories of the two black holes are plotted in Figure I.1 for the two cases;

the emitted gravitational waveforms are shown in Figure I.2.

The cases analyzed here have been considered by many different investigators. The actual

numerical results reported here were obtained by Etienne et al. (2009), who performed these

simulations to test the vacuum sector of their relativistic hydrodynamics AMR code. Usually,

vacuum codes that can simulate these two cases successfully can automatically evolve more

complicated scenarios, such as spinning black hole binaries. Such codes can also treat black hole

binaries for which approaches other than the puncture method are adopted to construct the initial

data (e.g., the conformal thin-sandwich approach is one such alternative; see Chapter 12.3).

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