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

Подождите немного. Документ загружается.

10.2 Head-on collision of two black holes 361

Topology of the event horizon

The most interesting case is scenario two, the head-on collison of well-separated, boosted

spheres in which the particle velocity dispersion in each sphere is initially zero. In this case

each cluster collapses on itself to form a black hole prior to merger. Hence this scenario

simulates the head-on collision of two nonrotating black holes. The spheres each have an

initial coordinate radius a/M = 0.8 and their centers are located at z

=±1.4M along the

collision axis. Searches for apparent horizons are performed by solving equation (7.51)on

each time slice. There are no apparent horizons initially. The spheres are given an inward

boost velocity of v = 0.15 as measured by a normal observer.

Figure 10.4 shows spatial snapshots of the collision at several different instants of time.

Common as well as disjoint horizons appear, as shown in the ﬁgures. The event horizon

is found

by integrating null geodesics backwards in time in the curved background, as

discussed in Chapter 7.2. These integrations can only be performed after the evolution has

terminated and the global spacetime has been fully determined. Disjoint event horizons

appear within the spheres of matter by t/M ≈ 0.13, much earlier than the formation of the

common apparent horizon at t/M ≈ 6.5 or the disjoint apparent horizons at t/M ≈ 7.1.

Note how the tidal ﬁeld of each companion causes a distortion in the shape of the nascent

black holes and their disjoint event horizons, giving rise to the “hour glass” appearance

of the merging holes when they ﬁrst come into contact. The disjoint event horizons

grow towards each other until they coalesce at t/M ≈ 2.18. By the time the integrations

terminate at t/M ≈ 11.7, all of the matter is encompassed by a single black hole and the

exterior spacetime is close to Schwarzschild. The common event and apparent horizons

coincide and have an area

A/16π M

≈ 1.1, a polar circumference C

pole

/4π M ≈ 1.1and

an equatorial circumference

/4π M ≈ 1.0. The deviations of these ratios from unity,

their Schwarzschild values, is a measure of the numerical error in the evolution. The total

energy in gravitational waves radiated in the collision is E/M ≈ 3 × 10

−4

Figure 10.5 is a spactime diagram of the collision and merger. The time axis is along

the vertical direction, while spacelike hypersurfaces (with one of the spatial directions

suppressed) are horizontal planes at any instant of time. The collision axis goes from left to

right and the black hole horizon is shown as the dark shaded surface. Some of the light rays

which generate the horizon (“null generators of the horizon”) are shown. Their trajectories

were traced by numerically propagating light rays in the background spacetime. The inset

shows a closeup view of the formation and merger of the two horizons and how the rays

enter the horizons at those early events.

Figure 10.5 is the famous “pair of pants” picture of the event horizon for coalescing

black holes that was sketched in general relativity textbooks in the 1970s.

The ﬁgure

shown here, produced over 20 years later, was the ﬁrst real calculation of such a diagram.

Many things were known about the the topology of the merging horizons and the null

Hughes et al. (1994).

Hawking and Ellis (1973); Misner et al. (1973).

362 Chapter 10 Collapse of collisionless clusters in axisymmetry

Figure 10.4 Spatial snapshots of the head-on collision of boosted clusters. Top panel: Initial data. Each sphere

has an initial radius a/M = 0.8, an inward boost velocity v = 0.15, and their centers are at z

/M =±1.4.

Sample particles from the collisionless matter distribution are represented by dots. A clock displaying the elapsed

fraction of the total evolution time appears in the corner. Middle panel: Horizon coalescence. At time

t/M = 2.18 the two event horizons coalesce (shaded hour glass). The cluster’s centers are at z/M =±1.07. The

scale of the image has been enlarged by about a factor of 2 over the top panel. Bottom panel: Final state. By time

t/M ≈ 11.7 shown here, the event and common apparent horizons have settled down to a quasistationary state

and coincide. [From Matzner et al. (1995).]

10.2 Head-on collision of two black holes 363

Figure 10.5 Spacetime diagram for the collision depicted in Figure 10.4. The diagram shows some of the null

generators of the horizon. The time axis is vertical, the collision axis goes from left to right; one of the spatial

dimensions is suppressed. The inset on the left zooms in on the caustic and crossover structure at the birth of the

event horizon. [From Matzner et al. (1995).]

generators prior to these numerical simulations, but a few important details were not. It

was well known that the black hole is spanned by null generators, which can intersect or

cross each other only at those points at which they enter the horizon. Once on the horizon,

a null generator can never propagate off, nor can it ever cross another null generator.

These properties were all understood and nicely corroborated by the simulation. But in

addition, the simulation revealed for the ﬁrst time a line of crossover points for the null

generators that extends from the “crotch” on the “pair of pants” down along each inside

trouser seam, around each bottom, and continuing a small distance up each outside seam

(see Figure 10.5). The points on the outside seam at which the line of crossover terminates

are caustics, where the intensity of the intersecting light rays becomes inﬁnite. For a single,

isolated cluster undergoing spherical collapse to a black hole, the caustic would arise at

the base of the spacetime diagram at the point at which the event horizon ﬁrst forms, and

there would be no crossovers. Here, the gravitational tidal ﬁeld of the colliding black holes

shifts the location of the caustic and produces the line of crossovers. An analysis of the

simulation

shows that the line of crossovers is spacelike, which means that they cannot be

traced by light rays or particles moving slower than c. As this line approaches the caustics

on the sides, it becomes asymptotically null.

The topology of the event horizon for merging black holes is somewhat different in the

case of vacuum black holes, or “eternal” black holes not formed from matter collapse.

Figure 7.2 shows the results of a head-on collision of two nonrotating, vacuum black holes

Matzner et al. (1995).

364 Chapter 10 Collapse of collisionless clusters in axisymmetry

that are initially at rest at ﬁnite separation at a moment of time symmetry (K

= 0).

The initial data are the analytic Misner initial data (3.26) discussed in Chapter 3.1.2.The

initial separation between horizons L (see equation 3.28) is determined by the parameter

µ appearing in the three-metric; for the case shown, µ = 2.2, we have L/M = 8.92. The

simulation proceeds using maximal slicing for a time t ≈ 150M, at which point the

merger is complete (note the spacetime diagram in Figure 7.2 goes only up to t/M = 25).

At late times, the horizon oscillates with decaying amplitude, emitting gravitational wave

quasinormal “ringdown” radiation of a vibrating Schwarzschild black hole. The ﬁgure

shows the event horizon structure from t/M = 5tot/M = 20. The event horizon is found

by integrating equation (7.10). Figure 7.2 differs from Figure 10.5 in two ways. First, as

discussed above, the black holes in Figure 10.5 are “born” from the collapse of matter,

while the black holes in Figure 7.2 are eternal. Second, Figure 10.5 plots the coordinate

circumference of the event horizon, while Figure 7.2 plots its proper polar circumference.

In the latter case, the monotonic increase of the area at early times and its constancy at late

times are evident, in compliance with black hole area theorems.

In contrast to black holes ﬁlled with matter, the legs in Figure 7.2 do not exhibit cusps,

but continue into the past from the times shown in the diagram. As in the case with matter,

there does exist a crossover line along the inside seam of the “pair of pants”. Note that

while the spacetime is symmetric about the initial time, the horizon is not, as it expands

monotonically into the future. This is evident in the expansion of the null generators seen

in the ﬁgure at the initial time.

Exercise 10.3 Can one conclude from the time-symmetry of the spacetime that

its past evolution describes a time-reversed collision: ingoing radiation propagates

from inﬁnity onto a single, Schwarzschild black hole, bifurcating it into two holes

that move apart until coming to a momentary pause, followed by outgoing radiation

as the two holes merge?

10.3 Disk collapse

Many astrophysical systems are best represented by inﬁnitesimally thin disks of collision-

less matter. Not surprisingly, numerous studies of the dynamical behavior of collisionless

disk systems have been performed over the years in Newtonian gravitation.

As it hap-

pens, thin disks provide particularly simple, but amazingly useful, sources for simulations

in numerical relativity. The collapse of an axisymmetric disk of collisionless matter pro-

vides the simplest example of matter collapse exhibiting the two most signiﬁcant and

challenging aspects of relativistic gravitation: black hole formation and gravitational radi-

ation generation. Since the matter source resides entirely in the equatorial plane, the matter

evolution equations in axisymmetry are 1-dimensional (i.e., they depend on r alone). The

Anninos et al. (1995) and references therein; Matzner et al. (1995).

See, e.g., Hockney and Eastwood (1981); Fridman and Polyachenko (1984); Binney and Tremaine (1987).

10.3 Disk collapse 365

source inﬂuences the gravitational ﬁeld, which is 2-dimensional (i.e., a function of r and

θ), via “jump conditions” across the equator. When the disk matter is initially at rest,

the ensuing collapse provides an interesting analogy to Oppenheimer–Snyder collapse

to a black hole in spherical symmetry, but with the added important feature of gravita-

tional radiation production. Since the gravitational ﬁeld is dynamical in disk collapse, the

full machinery of numerical relativity is required to follow the evolution, while spherical

Oppenheimer–Snyder collapse is analytic.

The same equations discussed earlier in this chapter to evolve the axisymmetric collapse

of isolated clusters and the head-on collision of binary clusters are readily adapted to han-

dle inﬁnitesimally thin disks of collisionless particles.

These equations, together with

the relevant jump conditions, are summarized in Appendix F. Dynamical simulations that

solve these equations have explored different relativistic systems and dynamical effects.

The growth of ring instabilities in “cold” equilibrium disks, and their suppression in disks

with sufﬁcient velocity dispersion (i.e., “hot” disks), has been studied. Gravitational radia-

tion from oscillating disks, as well as damping of the oscillations by radiation reaction, has

been determined. The calculation of gravitational waveforms from disk collapse to black

holes has been particularly useful. It has provided another arena for advancing computa-

tional machinery – in this case perturbation methods – that can be used in conjunction with

numerical simulation data to determine late-time gravitational waveforms when the ﬁnal

state of a dynamical system approaches a Schwarzschild black hole.

We will discuss this

application below.

We focus on the collapse of a disk of nonrotating, cold matter to a black hole. In this

case all the disk matter is instantaneoulsy at rest at t = 0 and the spacetime evolution

begins at a moment of time symmetry. The construction of appropriate initial data can be

accomplished in much the same way that initial data for spacetimes containing relativistic

prolate spheroids can be built from their Newtonian counterparts, as in exercise 10.1.

For thin disks, however, we need to construct oblate spheroids, taking the limit as their

eccentricity goes to unity to get ﬂat conﬁgurations. Such a construction is summarized in

exercise 10.4.

Exercise 10.4 Consider as in exercise 10.1 a homogeneous spheroid in Newtonian

gravitation. The potential obeys Poisson’s equation, ∇



= 4πρ

, where the den-

sity ρ

is again given by equation (10.2). Now treat an oblate spheroid, for which

the eccentricity is deﬁned to be e = (1 −c

), and the Newtonian gravitational

binding energy is given by

=−





x =

arcsin(e)

. (10.11)

(a) Construct a relativistic spheroid by setting 2πψ

ρ ≡ 4πρ

as in exercise 10.1,

but now for the mass density ρ inside an oblate spheroid of collisionless particles

Abrahams et al. (1994).

Abrahams et al. (1995).

366 Chapter 10 Collapse of collisionless clusters in axisymmetry

momentarily at rest. As in exercise 10.1, the conformal factor satisﬁes ψ = 1 − 

where 

is the potential for the corresponding Newtonian spheroid. Show that

equations (10.6)and(10.7) apply once again, but for the parameter α given by

α =

5ae

arcsin(e). (10.12)

(b) Consider the inﬁnitesimally thin disk of radius a obtained by letting e → 1in

the oblate spheroid considered above. Show that for the homogeneous Newtonian

spheroid, the surface density satisﬁes

2πa



1 −



1/2

, (10.13)

while for the corresponding relativistic disk, the total mass density σ and rest-mass

surface density σ

are given by

σ = σ



−

ρr sinθ dθ =

. (10.14)

M =

1 +

(

1 + 6π M

/5a

)

1/2

. (10.15)

The collapse of a thin disk constructed in this fashion is homologous (i.e., self-similar),

and the solution is analytic, in Newtonian gravitation.

In general relativity, by contrast, one

has to integrate the full set of 2 + 1 equations to determine the spacetime. The simulation

summarized here was performed using a spatial grid of 300 radial and 16 angular zones

in the upper hemisphere, with the matter source sampled by 24 000 particles. Plots of

the particle positions at selected times during the collapse of a disk with initial radius

/M = 1.5areshowninFigure10.6 (recall that in isotropic coordinates, a Schwarzschild

black hole has a radius R

/M = 0.5). The location of the apparent horizon, which appears

at about t/M = 4.0, is also shown. The moving radial mesh algorithm, in which the

innermost zones follow the infall of the matter, enables the integrations to continue reliably

for a time t/M ≈ 15 after the appearance of the horizon. Beyond that, the effect of “grid

stretching” along the black hole throat induces numerical inaccuracies. Well before this

point, however, all of the matter is well inside the horizon.

Close limit approximation

Even though disk collapse is one of the simplest radiating systems with a matter source, it is

not possible to track the evolution long enough with this code to read off the full waveform

directly from the simulation. The problem is that a signiﬁcant amount of the radiation is still

in the near-zone, strong-ﬁeld region at the time the simulation breaks down. This difﬁculty

occurs even though all of the matter has long since disappeared into the black hole. This

See Abrahams et al. (1994), Section III.A, with h = 0 = ξ .

10.3 Disk collapse 367

1.5

0 .5 1 1.5

t = 9

t = 6

t = 0 t = 3

Figure 10.6 Snapshots of the particle positions for the collapse of a cold relativistic disk. Initially the radius is

/M = 1.5 and the particles are all at rest. The apparent horizon (dashed line) ﬁrst appears at t  4.0M.The

coordinate labels (t, x, y) are in units of M. [From Abrahams et al. (1994).]

problem can be overcome in one of several ways. Black hole excision or moving puncture

techniques are two approaches; we will discuss these methods in detail in Chapters 13.1

and 14.2.3. Here instead we discuss a different approach that is particularly useful when

the ﬁnal state of a dynamical system is a Schwarzschild black hole, as is the case here.

The method is based on black hole perturbation theory and approximates the spacetime at

late time as a single perturbed black hole. The perturbation of the black hole is computed

on a spatial slice from the numerical metric and extrinsic curvature using gauge-invariant

perturbation theory.

Waveforms are determined by evolving this perturbation to inﬁnity

using the Zerilli equation (9.90).

The method employed here for the collapsing disk is an extension of the “close limit”

approach introduced by Price and Pullin (1994) to treat radiation from merging black holes.

Price and Pullin considered Misner initial data representing two black holes at a moment of

time-symmetry.

They realized that when the two holes were sufﬁciently close, the system

could be treated as a single, perturbed black hole. By applying gauge invariant perturbation

theory, they calculated this perturbation and evolved it using the Zerilli equation. This

allowed them to compute asymptotic waveforms and emitted energies. Remarkably, for

Moncrief (1974); see discussion in Chapter 9.4.1.

See Chapter 3.1, equation (3.26) for Misner initial data.

368 Chapter 10 Collapse of collisionless clusters in axisymmetry

fairly small separations, the energies and waveforms agreed well with the results of full

2 + 1 numerical simulations, like the ones discussed in the previous section.

The remarkable success of the “close limit” approach can be explained, at least in part,

by the presence of the black hole horizon and its use as an inner boundary at which all

matter and radiation are purely ingoing. Once the horizon forms and the spacetime settles

down to a perturbed Schwarzschild black hole, data obtained by further evolution with a

2 + 1 numerical routine are no longer required to obtain the ﬁnal gravitational waveform.

The 2 + 1 simulation can be terminated at a relatively early epoch; instead of continuing

the simulation in order to propagate the exterior radiation pulse out to the weak-ﬁeld

extraction regime, a simpler set of black hole perturbation equations can be solved for the

waveform, using the ﬁelds on the last numerical time slice for initial data. However, this

“close limit” approach does require the formation of a black hole during the numerical

simulation: its horizon is essential for preventing all further evolution inside the black hole

from inﬂuencing the spacetime outside. The same approach does not work for, say, an

oscillating neutron star, where there is no horizon.

In the perturbation approach we treat the spacetime metric as a static Schwarzschild black

hole plus a perturbation (see Chapter 9.4.1). Since for our applications all perturbations

have even (i.e., polar) parity, we can restrict the analysis to the even-parity modes and

identify, from the perturbations, the gauge-invariant Moncrief functions R

,aswellas

their time derivatives, on a spatial slice (see equation 9.87). In axisymmetry, the only

nonvanishing modes are those with m = 0, R

. These functions now provide initial data

for the Zerilli equation (9.90)

∂

− ∂

∗

+ V

(e)

= 0, (10.16)

where r

∗

= r

+ 2M ln(r

/M − 1) is the tortoise coordinate, r

= r(1 + M/2r)

is the

Schwarzschild radius corresponding to (isotropic) radius r , and the Zerilli potential V

(e)

given by equation (9.91). Given that this equation depends on only one spatial coordinate,

it can be integrated on a ﬁne mesh from a small radius very close to the event horizon,

say r

∗

/M =−500, to a very large radius, say r

∗

/M = 1000, until the perturbation has

propagated out to a large distance, well beyond the peak of the potential just outside the

horizon. At large radius, the even-parity gravitational wave amplitude h

can be computed

from the R

as in equation (9.109). This perturbation waveform can be compared with

the waveform calculated by standard spacelike extraction of the 2 + 1 radiation data at

large distance.

In contrast to the perturbation result, the amplitude found from standard

extraction is cut short once the 2 + 1 integrations break down due to, e.g., grid stretching.

We can now mention another advantage of the perturbation method over wave extraction

at a ﬁnite radius. Integrating the Zerilli equation (10.16) to large radii automatically takes

into account the effects of gravitational wave backscatter off the black hole curvature.

See also Abrahams and Cook (1994), who extended this technique to initial data representing boosted black holes with

a common apparent horizon.

Chapter 9.2.2. The extraction algorithm of Abrahams and Evans (1990) was actually employed for the disk scenario

described here.

10.4 Collapse of rotating toroidal clusters 369

0.04

0.02

−0.02

r h+/sin

−0.04

02040

t − r

60 80 100

Figure 10.7 The gravitational waveform computed using the perturbation method for disk collapse is plotted as

a function of retarded time in units of M. For comparison, the gravitational waveform computed at a radius

r/M = 8 using a standard spacelike radiation extraction technique is also shown (crosses). [From Abrahams

et al. (1995).]

These effects can only be incorporated approximately with standard extraction methods,

if an integration over a timelike cylinder is used to separate off near-zone effects.

The perturbation and standard extraction waveforms are compared for disk collapse

in Figure 10.7.

The perturbation results used simulation data at t/M ≈ 4.7. Since the

simulation becomes inaccurate within t/M ≈ 20M there is little time for the wave to reach

the numerical extraction radius at r/M = 8. The standard method can only measure the

wave from the initial infall phase and the ﬁrst black hole oscillation before the simulation

breaks down. The two waveforms are in good agreement during this epoch. But the

perturbation waveform continues to track the black hole ringdown radiation, exhibiting the

dominant quasinormal mode oscillations (of wavelength λ/M ≈ 16.8 for the lowest l = 2

mode) that characterize a perturbed Schwarzschild black hole. Simulations that employ

either black hole excision or the moving puncture method can track the fully nonlinear

evolution of black holes, from formation to ringdown, to arbitrarily late times.

10.4 Collapse of rotating toroidal clusters

Most objects found in nature have spin, including relativistic conﬁgurations like neutron

stars and black holes. So it is imperative that we learn how to evolve numerical spacetimes

with net rotation. For an axisymmetric spacetime with rotation, the form of the metric

Abrahams et al. (1994).

370 Chapter 10 Collapse of collisionless clusters in axisymmetry

given by equation (10.1) for the nonrotating case must be generalized. The spatial line

element in the quasi-isotropic gauge in spherical polar coordinates now becomes

= A

(dr

dθ

) + B

(ξdθ + sinθdφ)

, (10.17)

and all three components of the shift vector, (β

,β

), may be present. Each of the metric

components is a function of t, r and θ , but not φ. Once again the variable η represents

the even-parity radiative polarization state, as η → h

at large distances from the source.

In the case of rotation, the odd-parity polarization state can also be present. At large distance

the variable ξ is a measure of the odd-parity wave amplitude, as ∂ξ/∂r →−∂h

/∂t when

there is purely outgoing radiation.

The adopted form of the 2 + 1 ﬁeld equations, and the mean-ﬁeld, particle simulation

scheme used to integrate them, are generalizations of the ones used for the nonrotating

systems described in the previous sections of this chapter and summarized in Appendix F.

The scheme is fully constrained: the Hamiltonian constraint is used to obtain a 3-metric

component, and the three momentum constraints are used to compute components of the

extrinsic curvature. The two remaining 3-metric components and two extrinsic curvature

components are evolved. The shift vector components β

are used to maintain the quasi-

isotropic spatial gauge condition. The lapse function α is determined by the maximal

time slicing condition K = ∂

K = 0. The Hamiltonian constraint and lapse equations are

elliptic equations, while the shift equations comprise a mixed parabolic-elliptic system of

equations.

The above scheme has been used in a number of applications, including a study of the

stability of rotating polytropic and toroidal clusters. The formation of Kerr black holes

following the collapse of unstable clusters has been demonstrated. The collapse of highly

rotating conﬁgurations is particularly interesting, since it is impossible to form a Kerr

black hole with angular momentum J/M

≥ 1. In axisymmetry, where angular momen-

tum cannot be radiated, such collapses must result either in new, stationary, nonsingular

equilibrium conﬁgurations, or Kerr black holes surrounded by rapidly rotating disks, or

naked singularities. In a parameter study of collapsing toroidal clusters of varying J/M

clusters with J/M

≤ 1 all collapse to black holes, while those with J /M

≥ 1 all collapse

to new equilibrium conﬁgurations. While not unexpected theoretically, it is reassuring to

obtain this outcome numerically. We will return to this same issue in Chapter 14.2.3 when

we treat collapse of ﬂuid stars.

Toroidal black holes and topological censorship

One of the most interesting outcomes of the simulations of the collapse of rotating toroidal

clusters of collisionless particles to Kerr black holes is the emergence of the black hole

See Abrahams et al. (1994) for the full set of ﬁeld and collisionless matter equations for rotating spacetimes in

axisymmetry and a summary of the numerical method.