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

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7.3 Apparent horizons 241

In Chapter 1.4 we presented an instructive example in which an apparent horizon arises

during stellar collapse, namely the Oppenheimer–Snyder collapse of a homogeneous dust

sphere to a Schwarzschild black hole. The appearance of an apparent horizon is slicing-

dependent; in this example geodesic (Gaussian normal) slicing was employed. The adopted

slicing and coordinate conditions yield a spacetime that is completely analytic, including the

location of the trapped surfaces, the apparent horizon and the event horizon; see Figure 1.3.

The ﬁgure shows that trapped surfaces can appear suddenly and in a discontinuous manner,

but this feature depends on the slicing. In polar slicing, for example, no trapped surfaces

arise, as demonstrated in Exercise 7.13.

But it is already clear from this ﬁgure that the

absence of an apparent horizon does not imply the absence of an event horizon. Consider,

for example, a spatial slice at τ = 10M. This slice penetrates the event horizon, and hence

the interior of the nascent black hole, but nowhere does this slice exhibit any trapped

surfaces or an apparent horizon.

7.3.2 Axisymmetry

In axisymmetry, the function h is allowed to depend on θ, h = h(θ ). In equation (7.39)

only derivatives with respect to θ give a nonzero contribution, so that the apparent horizon

condition now becomes an ordinary differerential equation for h(θ ). This differential equa-

tion is still quite complicated (recall that both s

= λ(σ

− ∂

h)andλ contain derivatives

of h), but we will bring it into a tractable form as follows.

Adopting spherical polar coordinates, we may deﬁne

≡ ∂

τ = σ

− ∂

h = (1, −∂

h, 0), (7.43)

so that s

= λm

. At each point x

≡ (h,θ,φ) we introduce the tangent vector

≡ ∂

= (∂

h, 1, 0) (7.44)

(see Figure 7.4). Since for these vectors the φ components vanish, it is also useful to

introduce capital letter indices A, B,...that run only over r and θ.

The vector u

is the tangent vector to the r = h surface in the θ direction, and we can

compute the arc length s of its integral curves from



dθ



= γ

. (7.45)

We now ask the reader to derive two useful relationships.

Exercise 7.14 Show that

(2)

, (7.46)

See Petrich et al. (1986) for Oppenheimer–Snyder collapse in polar slicing and Chapter 8.2 for other examples of

spherical collapse in polar slicing.

242 Chapter 7 Locating black hole horizons

Figure 7.4 A schematic drawing of the surface r = h(θ), together with the vectors s

, σ

, m

and u

, deﬁning

an apparent horizon in axisymmetry.

where γ

(2)

= γ

θθ

− γ

rθ

is the determinant of the metric induced in the (r,θ)

hypersurface.

Exercise 7.15 Show that

dθ

(γ

(2)

)

1/2

. (7.47)

An immediate consequence of equation (7.46) is that the ﬁrst term in equation (7.39)

reduces to

λ∂

∂

h =

(2)

∂

h. (7.48)

We now make the assumption that γ

rφ

= γ

θφ

= 0, which can always be arranged in axisym-

metry. The contractions in equations (7.39) then decouple into (r,θ)andφ terms; for

example, m

= m

+ γ

φφ

. With equations (7.46) and (7.47) and the rela-

tion s

= λm

the apparent horizon condition (7.39) becomes

∂

h =−

−



dθ



φφ



φφ

−(γ

(2)

)

−1/2



dθ



− (γ

(2)

)

−1/2



dθ



φφ

. (7.49)

Derivatives ∂

h are still hidden in u

, m

and ds/dθ, but we have eliminated λ,which

depends on ∂

h in a rather complicated way.

See Eppley (1977), whose notation we adopt, for an alternative derivation originally due to D. Eardley.

7.3 Apparent horizons 243

As anticipated, this equation is an ordinary differential equation for the location r = h

of the apparent horizon as a function of θ. The coefﬁcients in this equation have to be

evaluated at the location h(θ). Exercise 7.16 provides an example for which this equation

assumes a much simpler form.

Exercise 7.16 Consider an axisymmetric spacetime at a moment of time symmetry

= 0) that is described by the conformally-ﬂat 3-metric

= ψ

(dr

(dθ

+ sin

θdφ

)), (7.50)

where ψ = ψ(r,θ). Show that equation (7.49) deﬁning the apparent horizon surface

r = h(θ) reduces to

∂

h =−



(∂

+ ∂



4∂

+ cot θ



+(∂



4∂



+ 2h + h

4∂

. (7.51)

Exercise 7.17 Verify that for a Schwarzschild black hole in isotropic coordinates

the apparent horizon condition (7.51) recovers equation (7.26), h = M/2.

Boundary conditions for equation (7.49) arise from the requirement that the surface

S be smooth (e.g., no cusps at poles or equator). This requirement implies that the ﬁrst

derivative ∂

h has to vanish at the poles θ = 0andθ = π, and, in the case of equatorial

symmetry, at θ = π/2.

Exercise 7.18 The term cot θ in equation (7.51) is irregular at the poles. Use a Taylor

expansion of h(θ ) about θ = 0 to show that, to lowest order in θ, the differential

equation (7.51) near the poles can be written as

∂

h = h + 2h

∂

, (7.52)

demonstrating that the coordinate singularity there can be removed.

Using the above formalism we can now ﬁnd the apparent horizon for the boosted black

hole solution that we discussed in Chapter 3.2.

Exercise 7.19 (a) Consider the boosted black hole solution of Chapter 3.2 and

ﬁnd the location of the apparent horizon to leading order in the correction due

to P. Speciﬁcally, write h as h = h

+ h

+ O(P

) where, from exercise 7.17,

= M/2, and show that to linear order in P equation (7.49) reduces to

∂

− h

+ (cot θ )∂

−

cos θ = 0, (7.53)

where P = (P

)

1/2

is the magnitude of the 3-momentum P

,andwherewe

have taken P

to be aligned with the polar axis.

See Appendix A.2 of Dennison et al. (2006); see also Cook and York, Jr. (1990).

244 Chapter 7 Locating black hole horizons

(b) Show that this equation, together with the boundary conditions at the poles, is

solved by

=−

cos θ. (7.54)

Hint: Compute K

= ψ

−2

from equation (3.80); also note that, according to

exercise 3.13, corrections to ψ enter only at second order in P, so that they can be

neglected here.

To lowest order in P, the boost of the black hole therefore results in a translation of

the apparent horizon in the direction opposite to the boost. We can now use this result to

compute the irreducible mass (7.2) of this boosted black hole – at least approximately. The

irreducible mass (7.2) is deﬁned in terms of the proper area of the event horizon, and we, of

course, have only located the apparent horizon. For stationary spacetimes the two horizons

coincide, and for small deviations from stationarity, as is the case here, the assumption is

that the two are very close to each other. Making this assumption, we will approximate the

irreducible mass by its value in terms of the apparent horizon.

Exercise 7.20 (a) Show that, for a conformally ﬂat 3-metric in axisymmetry (equa-

tion 7.50), the proper area of the black hole apparent horizon is given by

A =



2π





1 +



∂





1/2

sin θdθdφ. (7.55)

Hint: Treat the horizon as a “surface of revolution” r = h(θ ) about the z-axis and

use elementary calculus, or use the results of Appendix C.

(b) Expand both h and ψ to second order in P, i.e., write

h = h

+ h

(7.56)

and

ψ = ψ

(2)

= 1 +

(2)

(7.57)

to show that

= 16



(2)



. (7.58)

Notice that the second-order perturbation of the horizon, h

, has cancelled out of

this expression, so that the results of exercises 3.13 and 7.19 are sufﬁcient to compute

the irreducible mass to second order.

A = 16πM + 16π P



(2)

sin θdθ + 16π



sin θdθ

+π



16(∂

)

sin θdθ. (7.59)

7.3 Apparent horizons 245

Finally insert the results of exercises 3.13 and 7.19, carry out the integrations, and

use (7.2) to show that

irr

= M



1 +



. (7.60)

Theresultofexercise7.20 now helps to resolve an issue that we encountered in exercise

3.24, where we found the expression

ADM

= M +

+ O(P

) (7.61)

for the ADM mass of this boosted black hole solution (see equation 3.155). At ﬁrst sight,

this result seems surprising because we might have expected a factor of 1/2 instead of

5/8 in front of the second term in order to obtain the correct Newtonian expression

for the kinetic energy (see also the paragraph following exercise 3.24). The culprit is

the mass parameter

M, which, in general, is not a physical measure of the black hole

mass. Now we can use equation (7.60) to replace

M with M

irr

in equation (7.61). This

yields

ADM

= M

irr

+ O(P

), (7.62)

which is very reassuring. This example illustrates that the ADM mass of a boosted,

nonspinning black hole differs from its irreducible mass by its kinetic energy, which is all

the mass-energy that can be extracted from such a black hole.

Following these examples we now return to a more general treatment of the apparent

horizon in axisymmetry. We have reduced the problem of ﬁnding apparent horizons in

axisymmetry to a standard two point boundary value problem that can be solved with

standard numerical techniques, e.g., spectral methods, the shooting method, or relaxation

methods.

In a variant of the spectral method we can write the location h(θ) in a power series in

cos θ,

h(θ ) =

max



n=0

cos

θ, (7.63)

which automatically satisﬁes the boundary conditions at the poles. Inserting this expansion

into equation (7.49) yields an equation for the coefﬁcients c

that has to hold for all θ ,say

P(c

, c

,...,c

,θ) = 0. (7.64)

We could solve this equation by evaluating it at a certain set of points, which yields a set

of equations for the c

that can then be solved by matrix inversion (see Chapter 6.3). In an

See Appendix A.3 of Dennison et al. (2006).

See, e.g., Press et al. (2007).

246 Chapter 7 Locating black hole horizons

alternative iterative method we can vary each c

, keeping all other coefﬁcients constant,

until the integral



dθ assumes a minimum.

An apparent horizon will have been

located provided this integral becomes zero to within a desired tolerance.

Alternatively we can adopt a simple “shooting” technique. Starting at the pole θ = 0, we

can set ∂

h = 0 and take a trial value h(0) = h

as initial data and integrate equation (7.49)

either to the equator (assuming equatorial symmetry) or to the other pole (in the absence

of equatorial symmetry). We can then vary h

from values much larger than the suspected

location of the apparent horizon to values much smaller than this value and search for a

sign change of ∂

h at the equator or the other pole. If there is no sign change, there is no

apparent horizon. If there is a sign change, we can iterate over h

until ∂

h = 0, which

indicates that we have located the apparent horizon.

Finally, the apparent horizon condition (7.49) can be ﬁnite differenced and then solved

by relaxation.

An interesting application of apparent horizon ﬁnders in axisymmetry is to examine the

horizons in Brill–Lindquist initial data for two black holes (see Chapter 3.1). For a moment

of time symmetry (K

= 0), a vacuum solution to the constraint equations describing two

equal-mass black holes momentarily at rest with respect to each other is given by

= ψ

(7.65)

where

ψ = 1 +

(7.66)

(see equation (3.24) and the related discussion). We can choose the two black holes to

be aligned along the z-axis and the origin to lie midway between them. In this case we

have r

1,2

= (x

+ y

+ (z ± z

)

1/2

,wherez

is the coordinate separation between each

singularity and the origin.

For large separations z

 M each black hole is surrounded by its own “disjoint”

apparent horizon. For small separations there is also a “common” apparent horizon that

surrounds both black holes. It is of interest to locate the critical separation z

crit

at which

this common horizon ﬁrst appears. Numerical integrations give

crit

= 0.767M,atwhich

point the common horizon has an area of

A = 62.5πM

7.3.3 General case: no symmetry assumptions

In the absence of any symmetry assumptions the operator m

∂

that acts on h in

equation (7.39)or(7.40) is a 2-dimensional Laplace operator with respect to the metric

Eppley (1977).

Nakamura et al. (1988); Shapiro and Teukolsky (1992b). Shapiro and Teukolsky (1992b) reintroduce the parameter s

(arc length) and integrate ordinary differential equations for h(s)andθ(s) simultaneously, in lieu of equation (7.49).

Cook and York, Jr. (1990).

Brill and Lindquist (1963);

Cade

z (1974); Bishop (1982).

7.3 Apparent horizons 247

induced on the 2-surface S. We can use several different numerical techniques to

integrate either one of these partial differential equations, including spectral methods,

ﬁnite difference methods, and a ﬂow method in which the elliptic operator is converted

into a parabolic operator.

We implement a spectral method by expanding the location of the horizon r = h(θ,φ)

in spherical harmonics,

h(θ,φ) =

max



l=0



m=−l

(θ,φ), (7.67)

and then ﬁnding an iterative algorithm to determine the expansion coefﬁcients a

. Spher-

ical harmonics are eigenfunctions of the ﬂat-space Laplace operator L

on a 2-sphere,

i.e., L

=−l(l + 1)Y

. However, they are not eigenfunctions of the operator m

∂

appearing in the apparent horizon equation (7.39)or(7.40). Nevertheless, we can take

advantage of the eigenfunction property of the spherical harmonics by recasting the appar-

ent horizon condition into an equation for L

h,where

h ≡ ∂

∂

h + cot θ∂

h + sin

−2

θ∂

∂

h. (7.68)

Suppose then that we rewrite the apparent horizon condition  = 0as

h = ρ+ L

h, (7.69)

where we substitute equation (7.39)for. Let us choose the scalar function ρ so that

the partial derivative ∂

∂

h, which appears in both  and L

h, exactly cancels on the

right-hand side of equation (7.69).

Substituting equation (7.67) and using the eigenvalue

equation for Y

on the left-hand side of equation 7.69, we can then multiply both sides of

this equation by Y

∗

and integrate over S to obtain

−l(l + 1) a



∗

(ρ+ L

h) d, (7.70)

where d = sin θ dθdφ, and where we have used the orthogonality property of the spherical

harmonics. A priori, this new equation is not very helpful, since the right hand side must

be evaluated on S, whose location depends on the very coefﬁcients a

that we are trying

to determine. However, equation (7.70) can be used to establish an iteration procedure by

which the integral on the right hand side is evaluated using a previous guess for the set

in order to determine a new, improved set, a

n+1

. Clearly this algorithm works only for

l ≥ 1, while for l = m = 0 the left-hand side of equation (7.70) vanishes. In this case we

may consider the integral on the right-hand side a function of the coefﬁcient a

through

, and vary this coefﬁcient until a root of the right-hand side has been located. This

determines the improved value a

n+1

See, e.g., Nakamura et al. (1984, 1985); Kemball and Bishop (1991); Gundlach (1998)

See Gundlach (1998) for a generalization.

248 Chapter 7 Locating black hole horizons

In an alternative approach we can determine the expansion coefﬁcients a

with the help

of a multidimensional minimization method.

Consider the integral of 

over S,

S =





dσ. (7.71)

Since  depends on h,andh is expressed in terms of the expansion coefﬁcients a

, S

is also a function of these coefﬁcients. We can then use a standard minimization method,

for example Powell’s method or a Davidson–Fletcher–Powell algorithm,

to vary the a

until a mimimum of S has been found. An apparent horizon has been located if S can be

brought sufﬁciently close to zero within a speciﬁed tolerance.

We can also solve the apparent horizon condition (7.39)or(7.40) with ﬁnite difference

methods.

We can again write the condition as in equation (7.69), but now cover S with a

ﬁnite difference grid (θ

,φ

), on which h(θ,φ) is represented as h

i, j

. The operator (7.68)

can then be represented as

i, j

i+1, j

− 2h

i, j

+ h

i−1, j

(θ)

+ cot θ

i+1, j

− h

i−1, j

2θ

sin

−2

i, j+1

− 2h

i, j

+ h

i, j−1

(φ)

. (7.72)

As before, we can solve equation (7.69) using an iterative algorithm. On the right-hand

side, the operator L

acting on h (equation 7.72) can be evaluated for a previous set of

values h

i, j

. On the left-hand side, the same operator acts on the new values h

n+1

i, j

. Evaluating

equation (7.69) at all gridpoints (θ

,φ

) then yields a coupled set of linear equations for

the h

n+1

i, j

, which can be solved with standard techniques of matrix inversion. Using the

representation (7.69) which employs L

simpliﬁes the matrix that needs to be inverted,

but alternatively we can ﬁnite difference and invert the operator m

∂

directly, without

employing equation (7.69).

Yet another method that has been used for the locating apparent horizons in three spatial

dimensions is the curvature ﬂow method.

This method is related to solving an elliptic

equation by converting it into a parabolic equation in an artiﬁcial “time” coordinate – we

have described a similar approach for solving the maximal slicing condition in Chapter 4.2.

During evolution in “time”, the solution of such a parabolic problem settles down to

equilibrium, which furnishes the solution to the original elliptic equation. In this way, we

can deform a trial surface S according to

∂x

∂λ

=−s

, (7.73)

See Libson et al. (1996); Baumgarte et al. (1996); Anninos et al. (1998) who expand h in terms of symmetric traceless

tensors instead of spherical harmonics, which is completely equivalent, but more convenient in Cartesian coordinates.

See, e.g., Press et al. (2007).

Shibata (1997); also Shibata and Ury

u (2000).

See, e.g., Thornburg (1996); Huq et al. (2002); Schnetter (2003).

Tod (1991).

7.4 Isolated and dynamical horizons 249

where λ is the “time” parameter. For time-symmetric data with K

= 0wehave

√

2 = D

, so that the expansion becomes proportional to the trace of the extrinsic

curvature of S in , D

. The apparent horizon then satisﬁes D

= 0 and is therefore a

minimal surface,

for which this method is known to converge. For general data, the ﬂow

equation (7.73) is no longer guaranteed to converge, but numerical experience shows that

it typically does.

Various variations and combinations of the above methods have been implemented,

but it is not clear whether any one of these methods is preferable to the others for all

situations.

We will encounter several examples of apparent horizon identiﬁcation in

nonaxisymmetric, 3 + 1 dimensional spacetimes when we discuss simulations of binary

black holes in Chapters 12 and 13 and black hole–neutron star binaries in Chapter 17.

7.4 Isolated and dynamical horizons

The formalism of isolated and dynamical horizons combines in many respects the dif-

ferent advantages of event and apparent horizons as black hole diagnostics. Like appar-

ent horizons, isolated and dynamical horizons are quasilocal and do not require global

knowledge of the spacetime. Like event horizons, but unlike apparent horizons, they

furnish insight into the evolution of a black hole. For example, isolated and dynam-

ical horizons provide a framework for a quasilocal formulation of black hole ther-

modynamics. This framework furnishes a useful diagnostic of the physical properties

(e.g., the mass and spin) of any black hole that may be present in a numerical simula-

tion. It also provides natural boundary conditions for initial data describing black holes in

quasiequilibrium.

Here we will restrict our discussion of isolated and dynamical horizons to a brief and

qualitative sketch of their deﬁnitions and properties. We shall refer the reader to the

literature for a more comprehensive discussion and proofs.

Consider a sequence of apparent horizons S on neighboring spatial slices . Since appar-

ent horizons can “jump” discontinuously on neighboring slices,

the resulting worldtube

H may not be continuous. Let us disregard these jumps, and instead focus on smooth

sections of H. The worldtube H can then be either spacelike or null. If matter or radiation

is falling into the horizon, the black hole is growing in mass, its horizon is expanding, H

is spacelike, and we call it a dynamical horizon. If no matter or radiation is falling into the

black hole, H becomes null and we call it a nonexpanding horizon. The deﬁnition of an

isolated horizon requires some additional mathematical structure, but for our purposes it

See discussion in Chapter 4.2.

See also Gundlach (1998); Shoemaker et al. (2000) for implementations of related methods.

See Baumgarte and Shapiro (2003c) for more details.

See the review article by Ashtekar and Krishnan (2004)andDreyer et al. (2003) for discussion, proofs and references,

and Schnetter et al. (2006) for an introduction in the context of numerical relativity.

See Chapter 1.4 and Figure 1.3 for an example.

250 Chapter 7 Locating black hole horizons

Figure 7.5 Spacetime diagram distinguishing an isolated vs. a dynamical horizon (see text for details).

is sufﬁcient to identify a nonexpanding horizon with an isolated horizon.

The concepts

of dynamical and isolated horizons are illustrated in Figure 7.5.

First consider dynamical horizons. Since H is spacelike, we can apply, at least locally, the

same formalism that we developed to describe the spatial slices  in Chapter 2. In analogy

with the normal vector n

on , we can deﬁne τ

as a timelike unit vector that is normal

to H and satisﬁes g

=−1. We can then deﬁne the spatial metric q

= g

+ τ

induced on H (cf. equation 2.27), as well as the extrinsic curvature K

=−q

∇

(cf.

equation 2.49). Evidently, q

and K

have to satisfy the Hamiltonian constraint (2.132)

and the momentum constraint (2.133). We also deﬁne a spatial normal on cross sections

S of H in analogy to s

; while s

is the normal to S in  (see Figure 7.3), we now deﬁne

as the normal to S in H (see Figure 7.5). We also construct outgoing and ingoing null

vectors k

and l

in terms of τ

and r

, in analogy with equation (7.13).

We can now use this machinery to compute the angular momentum contained within S;

i.e., the spin of the black hole. To do so we need a Killing vector ﬁeld φ

of q

,which

is deﬁned on H and is tangent to all cross sections S.

Similar to the angular momentum

integral (3.187) we can deﬁne the angular momentum as

8π

(

, (7.74)

where dS

= d

√

and where m is the determinant of the induced metric on S.Asit

turns out, we may replace K

in the integrand with K

(the extrinsic curvature of )and

See Ashtekar et al. (2000); Ashtekar and Krishnan (2003) for precise deﬁnitions.

In the absence of an exact Killing vector ﬁeld, an approximate angular momentum can also be computed from an

approximate Killing vector ﬁeld; see, e.g., Cook and Whiting (2007).