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

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A.1 The Lie derivative 601

In this sense the Lie derivative is a tensorial generalization of the partial derivative. In fact, this

relationship is sometimes used as a starting point for deﬁning the Lie derivative.

It is an important property of the Lie derivative that, for any symmetric afﬁne connection

(i.e., 

= 

(bc)

) we can replace all the partial derivatives with covariant derivatives.

Exercise A.1 Show that the expression

= X

∇

− T

∇

+ T

∇

, (A.11)

where ∇

denotes a covariant derivative

with a symmetric connection, is equivalent

to equation (A.8).

As we have emphasized, our derivation holds only for coordinate or holonomic bases, for

which the afﬁne connection associated with the metric is symmetric. But for these bases we can

replace all the partial derivatives with covariant derivatives, obtaining covariant (coordinate-free)

expressions for the Lie derivative. Accordingly, these covariant expressions for the Lie derivative,

like equation (A.11), hold even in noncoordinate bases, and are hence more general. In the

following we will therefore use covariant derivatives when writing out a Lie derivative.

For a scalar f , the Lie derivative naturally reduces to the partial derivative

f = X

∇

f = X

∂

f, (A.12)

since there are no free indices on f to take care of. For a vector ﬁeld v

, we ﬁnd from (A.8)

= X

∇

− v

∇

= [X,v]

, (A.13)

which is the commutator of the two vector ﬁelds, while for a 1-form ω

we have

= X

∇

+ ω

∇

. (A.14)

Note that the Lie derivative satisﬁes the Leibnitz rule for outer products.

Exercise A.2 Show that

) = v

+ ω

. (A.15)

Also, the Lie derivative commutes with the exterior derivative of a p-form



Ω.

Exercise A.3 Show that for a p-form



Ω



Ω =



Ω. (A.16)

Exercise A.4 Let x

(λ) be the integral curves of a vector ﬁeld X

, and let Y

be a

second vector ﬁeld. Show that if Y

is Lie dragged along X

= 0, (A.17)

then it will connect points of equal λ along the congruence x

(λ).

The last exercise provides a geometrical interpretation of a property of the Lie derivative that

is very important in constructing 3 + 1 evolution equations in Chapter 2. If a vector Y

resides in

We depart from our convention in the rest of the book and denote by ∇

a covariant derivative in any number of

dimensions.

602 Appendix A Lie derivatives, Killing vectors, and tensor densities

a spacelike hypersurface, then it connects, in the language of Chapter 2, points in spacetime that

have the same coordinate time t.Exercise(A.4) shows that if we Lie drag Y

along the vector

αn

, where α is the lapse function and n

the normal vector, then it will still connect points of

equal t, and hence will continue to reside in our spacelike hypersurfaces.

A.2 Killing vectors

As an important application, consider the Lie derivative along X

of a metric g

= X

∇

+ g

∇

+ g

∇

. (A.18)

If ∇

is compatible with the metric, the ﬁrst term vanishes, and we ﬁnd

=∇

+∇

. (A.19)

A Killing vector ﬁeld ξ

can now be deﬁned by

= 0. (A.20)

In other words, a Killing ﬁeld ξ

generates an isometry of the spacetime, and a displacement

along ξ

leaves the metric invariant. From (A.19), we immediately recover Killing’s equation

∇

+∇

= 0. (A.21)

In some cases it is very easy to identify a Killing vector. If the metric components are indepen-

dent of a coordinate x

, then it follows from the property (A.10) that the coordinate basis vector

(i)

is a Killing vector. An example is the ﬂat metric of a unit sphere in spherical polar coordinates

θ and φ,

= diag



1, sin



. (A.22)

The above metric is independent of φ, making e

(φ)

a Killing vector.

In general it is much less trivial to identify Killing vectors. If we happen to know a Killing

vector ξ

and its derivatives at one point of a manifold, the following relation can help to ﬁnd ξ

everywhere.

Exercise A.5 Show that a Killing vector ﬁeld ξ

satisﬁes

∇

=−R

bca

, (A.23)

where R

bca

is the Riemann tensor.

We can now deﬁne the quantity L

≡∇

. If both ξ

and L

are given at a point P of a

manifold, then we can ﬁnd ξ

and L

at another point Q by integrating

∇

= v

(A.24)

∇

=−R

bca

, (A.25)

along a curve connecting P and Q, where v

denotes the tangent to this curve.

Among other useful properties, Killing vectors can be used to identify conserved quantities.

For example, given a stress energy tensor T

and a Killing vector ﬁeld ξ

, we can show that the

ﬂux J

= T

is conserved,

∇

=∇

) = ξ

∇

+ T

∇

= 0. (A.26)

A.3 Tensor densities 603

Here the ﬁrst term vanishes because the stress energy tensor is divergence free (conservation of

stress-energy), and the second term vanishes as a consequence of Killing’s equation (A.21): the

term ∇

is antisymmetric, so that the contraction with T

, which is symmetric, vanishes.

Exercise A.6 Let ξ

be a Killing vector ﬁeld and let p

be the 4-momentum vector

of a particle (or photon) moving on a geodesic curve. Show that ξ

is conserved

along the geodesic.

Since freely falling particles (or photons) follow geodesics, we see that each symmetry of the

spacetime, i.e., each linearly independent Killing vector ﬁeld, gives rise to a conserved quantity

for such particles. Speciﬁcally, if the metric is independent of some coordinate x

, then that

coordinate basis vector e

(b)

is a Killing vector and the particle momentum conjugate to that

coordinate, p

, is conserved.

Before leaving this section we brieﬂy discuss conformal Killing vectors.

Exercise A.7 Use the conformal transformation laws for the connection coefﬁcients

(3.7) to show that a Killing vector ξ

of the metric g

, i.e., a vector that satisﬁes

Killing’s equation (A.21), also satisﬁes the conformal Killing’s equation

∇

−

¯γ

∇

= 0, (A.27)

where

∇

is the covariant derivative associated with the conformally related metric

= ψ

−4

,andwheren = δ

is the number of dimensions.

A vector ﬁeld ξ

that satisﬁes the conformal Killing’s equation (A.27)iscalledaconformal

Killing vector.Exercise(A.7) demonstrates that a Killing vector of the metric g

is automatically

a conformal Killing vector of the conformally related metric

We point out that in three dimensions the differential operator in (A.27) is identical to the

longitudinal operator, or vector gradient,

(

LW)

∇

−

¯γ

∇

(A.28)

deﬁned in (3.50), which is therefore also known as the conformal Killing operator.

A.3 Tensor densities

So far we have restricted our attention to tensors, which are deﬁned by their transformation

properties from one coordinate system x

to another x



. A rank (

) tensor, for example, is

deﬁned by transforming as



∂x



∂x



(A.29)

under a coordinate transformation. There is a related class of objects that behaves similar to

tensors, except that they pick up a certain power of the Jacobian

J ≡ det



∂x





(A.30)

604 Appendix A Lie derivatives, Killing vectors, and tensor densities

during the transformation. An example of such an object is the determinant of the metric. The

spacetime metric transforms as



∂x



∂x



, (A.31)

so taking the determinant of this equation yields



= J

g. (A.32)

We now deﬁne a tensor density of weight W and rank (

) as an object that transforms as



= J

∂x



∂x



. (A.33)

Tensors of different rank transform in an analogous way, namely as the normal tensor transforma-

tion times J to the power W . Normal “plain-old” tensors are therefore tensor densities of weight

zero. The determinant of the spacetime metric, on the other hand, is a scalar density of weight

W =+2. Another example of a tensor density is the “natural” conformal 3-metric (3.6)

¯γ

= γ

−1/3

(A.34)

that we encounter in several places throughout this book.

Given that γ = det(γ

) is a scalar

density of weight +2, ¯γ

is then a tensor density of weight −2/3.

We now construct the Lie derivative of a tensor density. Following the procedure of Section

A.1 we drag T

from a point P to a point Q, assuming that the tensor transforms as if under

an inﬁnitesimal coordinate transformation generated by X

. That means that we simply have to

replace the transformation (A.5) for a tensor with (A.33) for a tensor density. To ﬁrst order in δλ

the Jacobian of the coordinate transformation (A.4)is

J = 1 − δλ ∂

+ O(δλ

), (A.35)

so that (A.5) now becomes



) = J

∂x



∂x



(x)

= (1 − W δλ ∂

)(δ

+ δλ ∂

)(δ

− δλ ∂

(x) + O(δλ

) (A.36)

= T

(x) + δλ



∂

(x) − ∂

(x) − W ∂

(x)



+ O(δλ

The Taylor expansion (A.6) is unaffected by T

now being a tensor density, so inserting (A.36)

together with (A.6) into the deﬁnition of the Lie derivative (A.7) we ﬁnd

= X

∂

− T

∂

+ T

∂

+ W T

∂

. (A.37)

This expression can be generalized for a tensor of any other rank. The rule is that the Lie derivative

of a tensor density of weight W is the same as that of a tensor, except that we have to add a term

e.g., in the context of the minimal distortion gauge condition in Chapter 4.5 and the BSSN formulation in Chapter 11.5.

See exercise 11.3 for an example of how the Lie derivative of an even more general object can be constructed.

A.3 Tensor densities 605

W T ∂

, where T is the tensor density of whatever rank we are considering. From the deﬁnition

(A.7) we note that the Lie derivative of a tensor density of weight W is again a tensor density of

the same weight.

As in exercise A.1 it might be desirable to express the Lie derivative (A.37) in terms of covariant

derivatives instead of partial derivatives. The attentive reader may object that we do not yet know

what the covariant derivative of a tensor density is. We may counter that, by construction, the Lie

derivative is independent of the connection, so that we can express it in terms of any connection

(as we have seen in Section A.1). In fact, requiring that the Lie derivative of a tensor density

take the same form as (A.37) when expressed in terms of covariant derivatives will guide us in

deﬁning the covariant derivative of a tensor density.

Since X

is a vector, we have

∇

= ∂

+ X



. (A.38)

Using equation (A.38) to substitute for all the partial derivatives of X

in equation (A.37)

gives

= X



∂

+ T



− T



− W T





−T

∇

+ T

∇

+ W T

∇

. (A.39)

We now identify the ﬁrst line in (A.39) with the covariant derivative of a tensor density of weight

W and rank (

)as

∇

= ∂

+ T



− T



− W T



. (A.40)

The generalization to other tensors of arbitrary rank is straightforward following this rule: the

covariant derivative of a tensor density T

a...

b...

of weight W is the same as that of a tensor T

a...

b...

of the same rank, except that we have to subtract a term W T

a...

b...



in the case of the tensor

density.

Exercise A.8 Use the identity

∂

g = gg

∂

(A.41)

to show that

∇

g = 0. (A.42)

Exercise A.9 Show that for a vector X

we have

∇

(|g|

1/2

) = ∂

(|g|

1/2

), (A.43)

and hence

∇

|g|

1/2

∂

(|g|

1/2

). (A.44)

With the deﬁnition (A.40) we can now, by construction, write the Lie derivative of a tensor

density in terms of any covariant derivative

= X

∇

− T

∇

+ T

∇

+ W T

∇

. (A.45)

606 Appendix A Lie derivatives, Killing vectors, and tensor densities

Exercise A.10 Show that the Lie derivative of the conformal metric (A.34)is

¯γ

∇

−

¯γ

∇

= (

LX)

, (A.46)

where we have lowered the indices of X

with ¯γ

Exercise A.10 provides an alternative deﬁnition of a conformal Killing vector. The deﬁnition

(A.20) states that the Lie derivative of the metric along a Killing vector vanishes, while (A.46)

together with (A.27) states that the Lie derivative of the conformal metric along a conformal

Killing vector vanishes.

Solving the

vector Laplacian

In this appendix we discuss two approaches to solving the vector Poisson equation in ﬂat space,

(

ﬂat

W )

= S

, (B.1)

which in Cartesian coordinates becomes

∂

= S

. (B.2)

We have encountered this operator in several places in this book, starting with the momentum

constraint (3.60) in the conformal transverse-traceless decomposition in Chapter 3.2.

The ﬁrst approach

to solving this equation entails writing the vector ﬁeld W

as a sum of a

vector ﬁeld V

and the gradient of a scalar ﬁeld U,

= V

+ ∂

U. (B.3)

Inserting this decomposition into equation (B.2) yields

∂

+ ∂

∂

U +

∂

U = S

. (B.4)

We can now choose U in such a way that the two terms involving U in equation (B.4) cancel the

second term,

∂

U =−

∂

, (B.5)

so that equation (B.4) reduces to

∂

= S

. (B.6)

We have thereby rewritten the vector Poisson equations (B.2) as a set of four coupled scalar

Poisson equations for U and the three components of V

. We use this decomposition to construct

the solution for a spinning black hole in Chapter 3.2.

A second approach

adopts the ansatz

−



∂

U + x

∂



. (B.7)

Inserting this decomposition into equation (B.2) yields

∂

−

∂

U −

∂

= S

. (B.8)

If we now choose U so that it satisﬁes

∂

U =−S

, (B.9)

Bowen and York, Jr. (1980).

Shibata (1999b); see also Oohara et al. (1997).

607

608 Appendix B Solving the vector Laplacian

then equation (B.8) reduces to

∂

−

∂

, (B.10)

which is solved by

∂

= S

. (B.11)

The vector equation (B.2) now has been reduced to a set of four decoupled scalar Poisson equations

(B.9) and (B.11). In Chapter 3.2 we use this decomposition to construct boosted black holes.

While the second approach (B.7) seems a little more complicated, it has the advantage that

the equations are decoupled and the source terms in equations (B.9) and (B.11) are nonzero only

where S

is nonzero. In some cases this later feature results in compact sources,

which can have

advantages for numerical implementations. In the approach (B.3), on the other hand, the source

term of equation (B.5) is never compact. Other approaches have been suggested,

and we refer

the reader to the literature

for a comparison of the numerical performance of these different

methods.

By “compact” we mean that the sources are nonzero only on a bounded subset of the coordinate space.

See, e.g., Oohara and Nakamura (1999).

See Grandcl

ement et al. (2001).

The surface element on the

apparent horizon

In this appendix we outline a method for computing the surface element on a closed 2-surface

S embedded in some 3-dimensional spatial hypersurface . The approach is useful for diverse

numerical applications that require areas of closed 2-surfaces on spatial slices, but our prime

motivation here is to compute the proper area of an apparent horizon of a black hole or a system

of black holes. Recall that for stationary spacetimes, the apparent horizon coincides with the

event horizon and its area then determines the irreducible mass (7.2) of a black hole.

One approach to computing this surface element is to construct the induced metric m

on the

surface S (see Chapter 7). Typically. however, the resulting line element will not be expressed

in terms of coordinates that are convenient for setting up the surface element. One way of

constructing such a coordinate system is the following:

We start with the line element for the

spatial hypersurface ,

= γ

, (C.1)

where the x

denote the spatial coordinates that have been used to label points on  in the

numerical calculation. We can now transform to spherical polar coordinates, centered on some

ﬁducial point C

. The coordinate separation between nearby points as expressed in the two

coordinate systems will then be related by the usual transformation,

∂x

∂r

∂x

∂θ

dθ +

∂x

∂φ

dφ. (C.2)

Using the notation of Chapter 7, we deﬁne the closed 2-surface S around C

as the level surface

τ (x

) = r

) − h(θ,φ) = 0, (C.3)

where r

is the coordinate separation between the point x

and the point C

. For example, in

Cartesian coordinates, we have r

= x

+ y

+ z

, provided the centers of the polar and Cartesian

coordinate systems coincide. The function h(θ,φ) then measures the coordinate distance from

to the 2-surface S in the (θ,φ) direction.

In the following we assume that we have already

determined h(θ,φ) by means, for example, of the techniques described in Chapter 7 in the case

where S is an apparent horizon.

On S, where r

= h(θ,φ), we must have

∂h

∂θ

dθ +

∂h

∂φ

dφ. (C.4)

See Appendix D in Baumgarte et al. (1996).

By our notation h(θ,φ) we mean that h can be expressed in terms of the coordinates θ and φ via, for example, a

superposition of spherical harmonics (cf. Chapter 7.3.3). In Cartesian coordinates, it may be more convenient to express

h in terms of combinations of x , y and z via, for example, a superpositions of symmetric, tracefree “STF” tensors.

609

610 Appendix C The surface element on the apparent horizon

Inserting this into equation (C.2) we obtain



∂x

∂r

∂h

∂θ

∂x

∂θ



dθ +



∂x

∂r

∂h

∂φ

∂x

∂φ



dφ. (C.5)

We now deﬁne the vectors





∂x

∂r

∂h

∂θ

∂x

∂θ



(C.6)

and



sin θ



∂x

∂r

∂h

∂φ

∂x

∂φ



, (C.7)

both of which are tangent to S.

Exercise C.1 Show that both 

and 

are orthogonal to m

= D

τ (i.e.,

= ∂

τ ), which demonstrates that both are tangent to S.

With the help of 

and 

we can now write the line element on S as

= γ



dθ + r

sin θ

dφ)(r



dθ + r

sin θ

dφ)

= r

(



dθ

+ 2sinθ



dθ dφ + 



sin

θφ

), (C.8)

where we have lowered the indices on 

and 

with γ

. In the above equation we may, of

course, replace r

with h. We have now expressed the line element on S in terms of the polar

coordinates θ and φ. The outward-oriented surface element on the apparent horizon is then given

by dS

= s



(2)

γ dθdφ (cf. Chapter 3.5), which yields

= s



(



)(



) − (



)



1/2

sin θ dθdφ. (C.9)

Here s

= λD

τ is the unit normal on S (see equation 7.33), λ is a normalization factor, and

(2)

is the determinant of the 2-metric appearing in equation (C.8).

What remains to be done is computing the vectors 

and 

, whose form depends on the

original coordinate system x

. If these coordinates were already spherical polar coordinates

centered on C

, then they take the particularly simple form



(∂

h, 1, 0)



sin θ

(∂

h, 0, 1)

(spherical polar coordinates). (C.10)