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

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13.2 Binary black hole inspiral and coalescence 451

ﬁn

= 3.272

(1 + q)

− 2.075

(1 + q)

, (13.8)

where M

ﬁn

denotes the ﬁnal ADM mass of the remnant black hole.

Exercise 13.2 Evaluate equations (13.7)and(13.8)forq = 1andq = 3and

compare with the values recorded in Table I.4 for the puncture binary black hole

simulations summarized in Appendix I.

Black hole spin

As we discussed above, the linear momentum emitted during one orbital period in the

inspiral phase nearly cancels out for nonspinning black holes, leaving only a relatively

small cumulative effect. Allowing for black hole spin, however, can enhance the effect.

A post-Newtonian analysis shows that, to leading order, the spin-orbit contribution to the

emitted linear momentum is

=−



r v × ∆ − 2v

ˆn × ∆ − (ˆn × v)

(

r ˆn · ∆ + 2 v · ∆

)



. (13.9)

Here and throughout the rest of this chapter boldfaced symbols represent 3-dimensional,

spatial vectors. As before, M = m

+ m

is the total mass, and µ = m

/M is

the reduced mass. We also deﬁne x = x

− x

as the relative position vector, r =|x|

as the coordinate distance, ˆn = x/r as a unit vector pointing from one binary companion

to the other, and v =

x as the relative velocity, where the dot represents a derivative with

respect to time. Finally, ∆ = M(S

− S

) is a measure of the black hole spins.

Note that equation (13.9) provides an expression for the “instantaneous” emission of

linear momentum. The ﬁnal kick speed is again a cumulative quantity that results from

the change in linear momentum integrated throughout the binary inspiral and merger.

However, the rate equation (13.9) does have an immediate consequence for the direction

of the resulting kick. To see this, ﬁrst consider black hole spins S

and S

that are aligned

with the orbital angular momentum and that are therefore perpendicular on v and ˆn, both of

which lie in the orbital plane. For such spins, the last two terms ˆn · ∆ and v · ∆ in equation

(13.9) vanish, and the spin contribution to the kick originates from the ﬁrst two terms.

The resulting kick is therefore in the orbital plane, just like the contribution from unequal

masses, though not necessarily in the same direction. We may call this contribution V

⊥

with the ⊥ symbol denoting a direction perpendicular to the orbital angular momentum.

Then consider spins that are perpendicular to the orbital angular momentum and hence

lie in the orbital plane. For such spins all four terms in (13.9) yield contributions that are

See equation (3.31b) in Kidder (1995); spin-spin effects enter only at higher post-Newtonian order.

452 Chapter 13 Binary black hole evolution

perpendicular on the orbital plane. We will call this contribution V



, with the  symbol

denoting a direction parallel to the orbital angular momentum.

To describe a general kick velocity we need to pick three independent unit vectors.

We choose the ﬁrst unit vector, e

, to be aligned with the direction of the kick velocity

that we would obtain in the absence of any black hole spins, i.e., the (unequal) mass

contribution to the kick velocity. We choose the second unit vector e

to lie in the orbital

plane as well, but orthogonal to e

, and, ﬁnally, the third unit vector e

to be orthogonal to

the orbital plane. We may then write

kick

= V

+ V

⊥

(cos ξ e

+ sin ξ e

) + V



. (13.10)

Here the angle ξ measures the angle between the mass and orbital plane spin contributions

to the kick. The coefﬁcients V

, V

⊥

and V



are the magnitudes of these kicks, which have

to be determined numerically. However, we can already anticipate some of the properties

of these coefﬁcients. The coefﬁcient of the mass contribution V

should depend only on

the mass ratio q; it should vanish for q = 1andq = 0 and should presumably be similar

to equation (13.5). At least to leading order, the spin contribution V

⊥

can depend only on

components of the spins along the orbital angular momentum, S



. Similarly, V



should

depend only on components of the spin in the orbital plane, S

⊥

Soon after accurate binary black hole simulations became feasible, a number of inves-

tigators leaped to explore the effect of black hole spin on black hole recoil.

Most of

the initial calculations focused on black hole spins that are aligned or anti-aligned with

the orbital angular momentum. The resulting kicks are several hundred km/s in mag-

nitude, easily exceeding the maximum kick of approximately 175 km/s found for non-

spinning black holes. Baker et al. (2007c) found that in this case, when the black hole

spins are orthogonal to the orbital plane, the ﬁnal, cumulative kick speed can be esti-

mated from a ﬁtting formula that combines the contributions from both the mass and spin

asymmetries,

kick

=|V

+ V

⊥

|≈V

32q

(1 + q)



(1 − q)

+ 2(1 − q)K cos ξ + K



1/2

. (13.11)

Here the parameter V

determines the overall scaling of the kick speed and the quantity

K is deﬁned as K = k(qS

− S

). In the latter expression k is a dimensionless

number representing the relative importance of the kick contributions from the spin and

mass asymmetries, m

and m

are the irreducible black hole masses, and S

and S

are their

spins. By choosing these parameters to be V

= 276 km/s, ξ = 0.58 rad and k = 0.85,

Baker et al. (2007c) show that equation (13.11) reproduces all the numerical results to

within about 10%.

See, e.g, Herrmann et al. (2007b), Koppitz et al. (2007), Campanelli et al. (2007a) for some early studies.

See also Favata et al. (2004); Koppitz et al. (2007).

13.2 Binary black hole inspiral and coalescence 453

Exercise 13.3 (a) Adopt equation (13.11) to estimate the maximum kick speed for

nonspinning black holes.

(b) Show that equation (13.11) reduces to equation (13.5) for nonspinning black holes

and use this identiﬁcation to determine the terminal radius r

term

in equation (13.5).

Campanelli et al. (2007a) realized that an even more dramatic effect can be achieved

when the black hole spins are not aligned with the orbital angular momentum. As we have

seen above, the kick velocity may then no longer lie in the orbital plane, but instead may

pick up a component orthogonal to this plane that can be much larger than the component

in the plane.

To synthesize the results of different simulations of the effect of black hole spin on the

recoil we return to equation (13.10). This expression predicts the kick velocities rather

well if the coefﬁcients in equation (13.10) are chosen as follows.

Set the magnitude of

the mass contribution V

to be

= V

f (q)

max



1 + B

(1 + q)



, (13.12)

where f (q) is given by equation (13.6), B =−0.93, and V

= 214 km/s. Note that the

second term in this expression does not appear in equation (13.5)or(13.11), which also

explains why the overall coefﬁcient differs. Now set the two spin contributions to be

⊥

= V

⊥

g(q)

max





− q





(13.13)

and



= V



g(q)

max

cos( − 

)



⊥

− q

⊥



. (13.14)

Here we have deﬁned

g(q) =

f (q)

1 − q

(1 + q)

, (13.15)

which assumes a maximum of g

max

= 2

 0.0346 at q = 2/3. The quantity 

measures the angle between the orbital plane component of the vector ∆ and the infall

direction at merger. Finally, set the coefﬁcients appearing above to be V

⊥

= 252 km/s and



= 2, 073 km/s. Note that the coefﬁcients V

, V

⊥

and V



have all the properties that

we anticipated in the discussion below equation (13.10).

ThesizeofthecoefﬁcientV



suggests that binaries can pick up a much larger kick

orthogonal to the orbital plane than in the orbital plane, if the individual black hole

spins have a signiﬁcant component in the plane. As an example, Figure 13.12 shows the

trajectories of the black hole centers for an equal-mass binary.

This simulation starts

Campanelli et al. (2007b).

Gonz

alez et al. (2007).

454 Chapter 13 Binary black hole evolution

−4

−2

0.15

0.1

0.05

−

Figure 13.12 Coordinate trajectories of the black hole centers (punctures) for an equal-mass, spinning black

hole binary. The initial spins are both perpendicular to the orbital angular momentum; each spin vector points

away from the companion along the line joining the centers. Following merger, the remnant receives a kick in the

negative z-axis, perpendicular to the orbital plane. [From Gonz

alez et al. (2007).]

with a binary in an approximately circular orbit. Both black holes have a spin of magnitude

S/m

= 0.8, and both of the spin vectors initially point away from the companion along

the line connecting the centers. As equation (13.10) suggests, this merger should lead

to a kick orthogonal to the orbital plane, which can be seen in Figure 13.12.Giventhe

symmetry of these particular initial data, the plane of the orbit is displaced along the z-axis

during the inspiral and merger without being tilted. Ultimately, the remnant receives a kick

of 2650 km/s in the negative z-direction.

This scenario has been explored further by Campanelli et al. (2007b), who evolve six

different sets of initial data. All binaries have equal-mass black holes, and all have black

hole spins of magnitude S/m

= 0.5 that lie in the orbital plane and are anti-aligned with

each other. The different sets of initial data differ only in the direction of these black hole

spins. It turns out that in many respects these particular initial data lead to a surprisingly

similar evolution. The total radiated energy and angular momentum, for example, are all

identical to within 3%. The projection of the black holes’ trajectory into the orbital xy

13.2 Binary black hole inspiral and coalescence 455

−3.5 −1.5 0.5 2.5

x/M

−3.5

−1.5

0.5

2.5

y/M

SP2

SPA

SPB

SPC

SPD

SPE

0 100 200 300

t/M

−0.5

−0.3

−0.1

0.1

0.3

0.5

z/M

SP2

SPA

SPB

SPC

SPD

SPE

Figure 13.13 The left panel shows the projection into the xy (orbital) plane of the punture trajectories for six

different sets of initial data (with only one puncture shown per binary). All binaries have equal masses and black

hole spins of magnitude S/m

= 0.5 that reside in the orbital plane and are anti-aligned with each other;

however, the direction of the spins within the orbital plane are different for each set of initial data. In this case the

spin hardly affects the trajectory of the black holes within the xy plane. The right panel shows the z-location of

the orbital plane as a function of time. During the inspiral, the orbital plane oscillates along the z-axis, but does

not tilt. Following merger, the remnant recoils along the z-axis. [From Campanelli et al. (2007b).]

plane is also hardly affected by the black hole spins; this is demonstrated in the left panel

of Figure 13.13. What is affected, however, is the z location of the orbital plane. During the

inspiral this plane containing the two punctures oscillates along the z-axis, i.e., orthogonal

to the orbital plane. The two punctures always have identical z-coordinates throughout

the inspiral, meaning that the orbital plane oscillates without tilting. This behavior can be

understood in terms of the symmetry of the initial data: the two anti-aligned black hole

spins “cancel” each other out and cannot affect the orbital angular momentum, which

therefore has to keep its original orientation. The location of the orbital plane along the

z-axis as a function of time for these different sets of initial data is shown in the right panel

of Figure 13.13. Following merger, the emission of linear momentum again ceases, and

the remnant recoils with the linear momentum it has at the time of merger. Campanelli

et al. (2007b) ﬁnd that the maximum kick that binary black holes can receive during

merger is approximately 4000 km/s – signiﬁcantly larger than the values for nonspinning,

unequal-mass binaries.

Clearly these ﬁndings raise some very interesting astrophysical questions. Kick speeds

of several thousand km/s easily exceed the escape speeds from the centers of even giant

elliptical galaxies. However, we also have strong observational evidence that supermassive

black holes reside at the centers of these galaxies, implying that these black holes have not

undergone mergers resulting in such large kicks. Early analysis of these questions suggests

456 Chapter 13 Binary black hole evolution

that the likelihood of the black hole spins being aligned in such a way that merger leads to

ejection is rather small.

So far we have focused our discussion on the effect of black hole spin on the recoil of

the binary remnant. Clearly, however, spin affects the inspiral and merger in other ways as

well. For example, black hole spins that are aligned with the orbital angular momentum

increase the binary’s total angular momentum. If this total angular momentum exceeds the

maximum angular momentum of a Kerr black hole, then the binary cannot merge until

a sufﬁcient amount of angular momentum has been radiated away. Quite generally, we

expect binaries with black hole spins aligned with the orbital angular momentum to merge

more slowly than binaries with spins that are anti-aligned. This effect, sometimes referred

to as “orbital hang-up”, has been explored with numerical simulations.

Another very important effect is spin ﬂip. In the discussion above, and in Figures 13.12

and 13.13, we have focused on binaries for which the two black hole spins are in the

orbital plane and anti-aligned with each other. For these binaries we have found that the

orbital plane gets shifted without being tilted, which we explained in terms of the two

black hole spins canceling each other out. Clearly this situation changes if the two black

hole spins are not anti-aligned. In this case a spin-orbit interaction affects both the black

hole spins and the orbital angular momentum during the inspiral and may change their

orientation.

As an example, we show results of a simulation by Campanelli et al. (2007)inFig-

ure 13.14. Here the initial black hole spins, both of magnitude S/m

= 0.5013, again lie

in the orbital plane, but they are now aligned with each other. At the initial time, they both

point in the positive y-direction, perpendicular on the x-axis that initially connects the two

black hole punctures. During the inspiral, the spins rotate by approximately 90

◦

within the

orbital plane and also pick up a nonzero z-component orthogonal to the orbital plane.

Simultaneously the orbital plane tilts, leaving a remnant with an angular momentum that

has nonzero components in both the y and z directions. This is not too surprising, of

course, since the total angular momentum of the initial data also had nonzero components

in these two directions – the y-components from the black hole spins, and the z-component

from the orbital angular momentum. Some of this angular momentum is radiated away

during the inspiral, but part of it remains with the merger remnant. The individual black

holes therefore experience a spin ﬂip: originally, both of their spins were aligned with the

y-axis, but after merger the remnant has a signiﬁcant component in the z-direction.

These spin-ﬂips may explain the so-called X-shaped radio jets. We show some examples

of observations of such objects in Figure 13.15. While the details of jet emission are still

Schnittman and Buonanno (2007); Bogdanovic et al. (2007).

See, e.g., Campanelli et al. (2006b).

The masses and spins of the individual black holes can be determined with the help of the isolated or dynamical

horizon formalism introduced in Chapter 7.4; the spins, in particular, are given by (7.74), where φ

is an approximate

Killing vector on the horizon. Some subtleties in the determination of the direction of these spins are discussed in

Campanelli et al. (2007).

13.2 Binary black hole inspiral and coalescence 457

−3

−2

−1

−3

−2

−1

−0.3

−0.2

−0.1

0.1

0.2

0.3

Figure 13.14 Puncture trajectories together with black hole spin directions, shown at time intervals of 4M.The

initial orbital plane is in the xy (horizontal) plane, and the initial black hole spins reside in this plane, both

pointing in the positive y-direction (to the right). During the course of the inspiral the spins rotate by

approximately 90

◦

within the orbital plane, and also pick up a nonzero z-component orthogonal to the orbital

plane. Simultaneously the orbital plane tilts, leaving a remnant with an angular momentum that has nonzero

components in both the y and z directions. Note that the z-scale is 1/10th the x and y-scale. [From Campanelli

et al. (2007).]

3C52

3C223.1

NGC326

3C403

Figure 13.15 Examples of X-shaped radio jets. The VLA radio observations of 3C52 are from Leahy and

Williams (1984), of 3C223.1 and 3C403 from Dennett-Thorpe et al. (1999), and of NGC326 from Murgia et al.

(2001). [From Merritt and Ekers (2002).]

458 Chapter 13 Binary black hole evolution

uncertain, it is believed that jets are emitted along the rotation axes of accreting Kerr

black holes. While most jets point along well-deﬁned directions, some jets appear as if

the direction of the emission has changed very abruptly some time in the past: the more

distant part of the jet points in one direction, while the closer part of the jet, emitted

more recently, points along another direction. Together, these two axes form an X-shaped

ﬁgure. The numerical simulations of binary black hole mergers described above provide a

very plausibe explanation for this apparent spin ﬂip of the spinning black hole, namely its

merger with another black hole.

Rotating stars

Rotating stars in general relativity are of long-standing theoretical interest. With the discov-

ery of radio pulsars in 1967

and their identiﬁcation as magnetized, rotating neutron stars,

rotating relativistic stars have also become objects of intense observational scrutiny. Radio

pulsars have become particularly useful for relativity, since they provide excellent cosmic

clocks. The original idea of Baade and Zwicky (1934) that neutron stars could be formed

during supernovae events, coupled with the discovery of radio pulsars in some supernova

remnants like the Crab, ﬁrmly linked rotating neutron stars with stellar collapse and super-

novae explosions. Following the detection of radio pulsars, X-ray pulsars were discovered

in 1971.

X-ray pulsars proved the existence of rotating neutron stars that are accreting gas

supplied by a companion star in a binary system. The discovery of the ﬁrst binary pulsar

by Hulse and Taylor

in 1974 proved that rotating, relativistic stars can reside in binary

systems, making neutron stars even more interesting to relativists. General relativity is

required to describe the gravitational ﬁeld of neutron stars, since their compaction, M/R,

where M is the mass and R is the characteristic radius of the star, is large (∼0.1–0.2).

Numerical relativity is important for treating many dynamical processes involving neutron

stars accurately. It provides a valuable tool for probing neutron star formation from stellar

core collapse in a supernova and for tracking the dynamical evolution and assessing the

ﬁnal fate of neutron stars subject to instabilities. It also furnishes gravitational radiation

waveforms generated by these events, as well as by binary inspirals and mergers. Numeri-

cal relativity is absolutely essential for following the catastrophic collapse of an unstable

star to a black hole.

There are other types of rotating, relativistic stars besides neutron stars that are of

astrophysical interest. These stars also require general relativity for an accurate description

and numerical relativity for a reliable analysis of their dynamical evolution and ﬁnal fate.

For example, rotating supermassive stars, although they have not been observed as yet,

Hewish et al. (1968). For their roles in the discovery of radio pulsars (with J. Bell), A. Hewish and M. Ryle shared the

Nobel Prize in Physics in 1974.

Schreier et al. (1972); Tananbaum et al. (1972). For his role in the discovery and interpretation of cosmic X-ray sources,

R. Giacconi shared the Nobel Prize in Physics in 2002.

Hulse and Taylor (1975). For their discovery of this binary, and for showing that its orbit decays at the rate predicted

by general relativity via the emission of gravitational waves, R. Hulse and J. Taylor were awarded the Nobel Prize in

Physics in 1993.

For a detailed discussion of the physics of neutron stars, see, e.g., the texts by Shapiro and Teukolsky (1983), Glendenning

(1996) and Haensel et al. (2007), as well as references therein.

459

460 Chapter 14 Rotating stars

might form in the early Universe and slowly evolve to the point of onset of collapse to black

holes. Such stars might be the progenitors of supermassive black holes observed in the

centers of most galaxies. Rotating white dwarfs are abundant in nature. While even massive

white dwarfs in equilibrium are only marginally relativistic, they will undergo collapse

once they accrete too much mass and exceed the Chandrasekhar limit.

Numerical relativity

simulations are necessary to follow such collapse and determine the ﬁnal outcome. Rotating

boson stars, constructed from complex scalar ﬁelds, are also relativistic in nature and

numerical relativity is again essential for analyzing their dynamical behavior.

Simulating rotating, relativistic stars using numerical relativity is a new enterprise. Its

development is due partly to the construction of new formulations of numerical relativity

that are able to integrate the multidimensional Einstein ﬁeld equations in a stable fashion

(see Chapter 11), and partly to the advent of parallel computer hardware with sufﬁcient

size and power to handle large-scale, multidimensional problems.

In this chapter we ﬁrst will discuss the numerical construction of rotating, relativistic

ﬂuid stars in stationary equilibrium. Such models, interesting in their own right, comprise

the initial data for many evolution calculations. We will then discuss some representative

simulations of ﬂuid stars in numerical relativity that probe the stability of these conﬁgu-

rations. These simulations are capable of tracking the rapid dynamical and, in some cases,

the slow secular evolution of rotating ﬂuid stars, both in axisymmetry (2 + 1)andinthree

spatial dimensions (3 + 1).

14.1 Initial data: equilibrium models

Up until the 1970s, models of rotating stars in general relativity were restricted to slow

rotation. Slowly rotating relativistic stars can be constructed by perturbing nonrotating,

spherically symmetric models.

Such a treatment involves the integration of ordinary

differential equations, which is straightforward. Since the 1970s, rapidly rotating relativistic

stars have been constructed numerically by many authors.

We will focus on the method

of Komatsu et al. (1989a,b, hereafter KEH), which allows for the construction of both

uniformly and differentially rotating stellar conﬁgurations.

14.1.1 Field equations

We consider rotating equilibrium models that are stationary and axisymmetric. The metric

can then be written, following KEH, in the form

=−e

γ +ρ

+ e

2σ

(dr

dθ

) + e

γ −ρ

sin

θ(dφ − ωdt)

, (14.1)

The limit is 1.4M



, the maximum mass of a nonrotating star supported by cold, degenerate electrons. For his derivation

of this limit in 1931, Chandrasekhar shared the Nobel Prize in 1983.

Hartle (1967); Hartle and Thorne (1968).

See, e.g., Wilson (1972a); Bonazzola and Schneider (1974); Butterworth and Ipser (1975, 1976); Friedman et al. (1986),

Komatsu et al. (1989a,b), Cook et al. (1992, 1994a,b), and Salgado et al. (1994a,b). For a comparison of methods, see

Stergioulas and Friedman (1995); Nozawa et al. (1998).