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

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Appendix C The surface element on the apparent horizon 611

If the original coordinates x

are Cartesian, we ﬁnd the more complicated expressions





1 − (l

)

(1 + l

∂

h + l

∂

h) −l



1 − (l

)

∂





1 − (l

)

(1 + l

∂

h + l

∂

h) −l



1 − (l

)

∂



)



1 − (l

)

∂

h + l

∂

h) −



1 − (l

)

(1 + l

∂





1 − (l

)



∂

h − l

∂

h) −l







1 − (l

)



∂

h − l

∂

h) +l







1 − (l

)

∂

h − l

∂

(Cartesian coordinates),

(C.11)

where l

= ∂x

/∂r

= x

= (sin θ cos φ,sin θ sin φ,cos θ).

Scalar, vector and tensor

spherical harmonics

In this appendix we provide a partial list of useful properties of scalar, vector and tensor spherical

harmonics. We focus on those properties that we use in our treatment of gravitational wave

extraction in Chapter 9.4.1. We adopt the notation of Nagar and Rezzolla (2005) and refer to

Thorne and Campolattaro (1967), Sandberg (1978) and Thorne (1980) for additional details and

references.

Here it is useful to depart from our general notation convention. Instead of separating spatial

indices from spacetime indices, it is more convenient, in the context of the spherical polar

coordinates used here, to separate the angular coordinates θ and φ from the remaining spacetime

coordinates r and t. Following this convention, we label the former (θ and φ) with lower-case

letters a, b,..., and the latter (t and r) with upper-case letters A, B,... We also write the

two-dimensional metric on the unit sphere S

= dθ

+ sin

θdφ

. (D.1)

Scalar spherical harmonics

The scalar spherical harmonics Y

(θ,φ) satisfy the equations

∇

=−l(l + 1)Y

(D.2)

and

∂

= imY

(D.3)

and can be expressed in terms of associated Legendre polynomials P

(cos θ )as

(θ,φ) =

2l + 1

4π

)

(l − m)!

(l + m)!

(cos θ )e

imφ

. (D.4)

Under a space inversion, spherical harmonics have even (or polar) parity (see our discussion in

Chapter 9.1.2)

(π − θ,π + φ) = (−1)

(θ,φ). (D.5)

We also note the property

l −m

(θ,φ) = (−1)

∗

(θ,φ). (D.6)

The ﬁrst few scalar spherical harmonics, up to l = 2, are listed in Table D.1.

The derivative ∂

can be computed in terms of the associated Legendre polynomials as

∂

(θ,φ) =−

2l+1

4π

)

(l−m)!

(l+m)!



(l + m)(l − m + 1)P

m−1

(cos θ ) + m cot θ P

(cos θ )



imφ

(D.7)

612

Appendix D Scalar, vector and tensor spherical harmonics 613

Table D.1 The scalar spherical harmonics Y

up to l = 2.

l = 0 l = 1 l = 2

m =−2



2π

sin

θe

−2iφ

m =−1



2π

sin θe

−iφ



2π

sin θ cos θ e

−iφ

m = 0



cos θ



(3 cos

θ − 1)

m = 1 −



2π

sin θe

iφ

−



2π

sin θ cos θ e

iφ

m = 2



2π

sin

θe

2iφ

The scalar spherical harmonics satisfy the orthogonality relation



∗



d = δ



. (D.8)

A generalization of the scalar spherical harmonics are the spin-weighted spherical harmonics,

which can be computed from the regular spherical harmonics by applying certain spin-raising

or lowering operators. Other than the s = 0 harmonics, which are the usual scalar spherical

harmonics, the s =−2 spin-weighted harmonics,

−2

(θ,φ) =

)

(l − 2)!

(l + 2)!



(θ,φ) − i

(θ,φ)

sin θ



, (D.9)

are of special interest for our purposes in Chapter 9.4.1. Here the functions W

and X

are

deﬁned in terms of derivatives of the scalar spherical harmonics by



∂

− cot θ∂

−

sin

∂



(D.10)

and

= 2∂

(∂

− cot θ)Y

. (D.11)

The function W

can also be expressed as

= l(l + 1)Y

+ 2∂

=−l(l + 1)Y

− 2 cot θ∂

sin

. (D.12)

We will see below that the functions X

and W

are also very useful in the context of the tensor

spherical harmonics (see equation D.25 below). The spin-weighted spherical harmonics satisfy

the same orthogonality relation as the spherical harmonics,



∗



d = δ



. (D.13)

Exercise D.1 Compute the s =−2 spin-weighted spherical harmonic

−2

and

show that it has the same angular behavior as the h

polarization of a l = 2, m = 0

linearized wave at large radii r (see equation 9.57;theh

polarization vanishes

identically for this mode).

614 Appendix D Scalar, vector and tensor spherical harmonics

Vector spherical harmonics

In addition to the familiar scalar spherical harmonics, we can also deﬁne vector and tensor

spherical harmonics. Vector spherical harmonics form a basis for vectors on S

. Evidently, we

need two linearly independent vectors to form a basis on S

, and we can create such a basis

by taking two different derivatives of the Y

. The electric vector spherical harmonics E

are

deﬁned as gradients of the scalar spherical harmonics

≡∇

, (D.14)

and have even (or polar) parity under space inversion, while the magnetic vector spherical

harmonics S

are deﬁned as the duals

of the gradients,

≡ 

∇

, (D.15)

and have odd (or axial) parity. Here 

is the 2-dimensional Levi-Civita tensor on S

, which has



θφ

=−

φθ

= sin θ (D.16)

as the only nonvanishing components.

Exercise D.2 Show that that the electric and magnetic vector spherical harmonics

are orthogonal,

= σ

= 0. (D.17)

Since the scalar spherical harmonics Y

are scalars, the covariant derivatives in the above

expressions can be expressed in terms of partial derivatives, and can be evaluated with the help

of equations (D.3) and (D.7).

The vector spherical harmonics satisfy the orthogonality relations



)

∗



d = l(l + 1)δ



(D.18)

and



)

∗



d = l(l + 1)δ



. (D.19)

Exercise D.3 Show that the electric and magnetic l = 2, m = 0 vector spherical

harmonics are





−

cos θ sin θ





and S





−

cos θ sin





. (D.20)

Tensor spherical harmonics

It will also be useful to have a basis for symmetric, traceless, rank-2 tensors on S

.Intwo

dimensions, these objects again have only two independent components, and we can therefore

In two dimensions the dual of a vector is again a vector.

Appendix D Scalar, vector and tensor spherical harmonics 615

create a basis, in complete analogy to the vector spherical harmonics, by taking two different

second derivatives of the scalar spherical harmonics. In particular, we deﬁne

≡∇

∇

l(l + 1)

, (D.21)

which have even (or polar) parity under space inversion, and

≡∇

, (D.22)

which have odd (or axial) parity. By construction, both Z

and S

are symmetric.

Exercise D.4 (a) Show that the two tensor spherical harmonics S

and Z

are

orthonormal,

= 0. (D.23)

(b) Show that the tensor spherical harmonics S

and Z

are traceless,

= 0andσ

= 0. (D.24)

Both Z

and S

are symmetric, traceless, two-dimensional rank-2 tensors, meaning that

each one has only two independent components. The orthogonality relation (D.23) reduces the

combined number of independent components to three. One of these three can be accounted for

by an arbitrary normalization, so that we are left with only two independent components for the

and S

. That is not surprising, of course, since we have constructed the Z

and S

as a

basis for symmetric, traceless, 2-dimensional rank-2 tensors, which only have two independent

components. As it turns out, we can conveniently express these two independent components in

terms of the functions X

and W

deﬁned by equations (D.10) and (D.11).

Exercise D.5 (a) Show that



−sin

θ W



and S

2sinθ



−X

sin

θ W

sin

θ W

sin

θ X



(D.25)

(b) Show that for l = 2, m = 0, the tensor spherical harmonics are







sin

θ 0

0 −

sin







and S







sin

θ 0







(D.26)

Finally, the tensor spherical harmonics satisfy the orthogonality relations



)

∗



d =

(l − 1)l(l + 1)(l + 2)



(D.27)

and



)

∗



d =

(l − 1)l(l + 1)(l + 2)



. (D.28)

Post-Newtonian

results

The central premise of this book is that, for many situations of greatest physical interest, Einstein’s

ﬁeld equations (1.32),

= 8π T

, (E.1)

do not admit exact, analytic solutions. In order to treat these situations we therefore need to

construct approximate solutions. This book is an introduction to numerical relativity, which

aims at constructing such solutions by numerical means. The intrinsic accuracy of this approach

depends on the reliability of the numerical algorithms and the adopted computational resources,

but it can handle gravitational ﬁelds that are arbitrarily strong and speeds v of the sources that

are arbitrarily close to the speed of light.

Post-Newtonian methods provide an alternative approach to constructing approximate solu-

tions to Einstein’s equations (E.1). In this approach a solution is constructed iteratively, starting

with the corresponding Newtonian solution. In each step of the iteration a new correction term

of order v

times the previous one is added so that the resulting solution is effectively a Taylor

expansion in the parameter v

about the Newtonian solution. Given that this expansion is car-

ried out analytically, this approach does not introduce any numerical error per se. Instead, its

accuracy is limited by the number of terms retained and the rate of convergence of the expan-

sion. Practically speaking, then, the method is limited both by the strength of the gravitational

ﬁeld and the speed of the sources, which are required to be small for any ﬁnite expansion to be

accurate.

For compact binaries, numerical relativity and post-Newtonian approaches complement each

other very well. For moderately large binary separations the binary inspirals very slowly, and the

orbital speeds and tidal deformations are small (see the discussion of binary inspiral in Chapter

12.1). Post-Newtonian approximations based on point-mass companions are very accurate in this

regime, while numerical models are severely challenged by the vast dynamic range of the prob-

lem that stretches computational resources. For close binary separations, on the other hand, the

different time and length scales characterizing binary inspiral all become comparable. The orbital

speeds become large and approach the speed of light, nonlinear gravitation, including the tidal

interaction between the conﬁgurations, becomes signiﬁcant, and the evolution drives the sources

far from equilibrium. In this regime, post-Newtonian models become inaccurate, while numerical

methods become both crucial and well-suited to tracking the dynamical behavior. In the intermedi-

ate regimes, both approaches are reliable and comparisons between post-Newtonian and numerical

calculations provide very valuable checks (see Chapter 13.2 for some examples). Furthermore,

616

Appendix E Post-Newtonian results 617

realistic numerical binary initial data, which by practical necessity consist of binaries at relatively

small separation, can incorporate post-Newtonian results that encode effects of the prior inspiral.

In this appendix we sketch how post-Newtonian expansions are constructed, and list some of

the most important results for compact binaries.

Post-Newtonian expansions

Einstein’s equations (E.1) form a set of differential equations for the spacetime metric g

.We

may express this metric in terms of new variables

≡

√

−gg

− η

, (E.2)

which are sometimes referred to as the gravitational ﬁeld perturbation amplitudes.Hereg is

the determinant of the 4-metric, g = det(g

), and η

is the Minkowski metric. Evidently, for a

ﬂat spacetime we have h

= 0, so that h

measures deviations from Minkowski spacetime. In

general, however, h

is neither zero, nor small; in strong-ﬁeld regions we might have |h

∼

Einstein’s equations cannot determine a solution for the metric uniquely, since they allow for

a choice of four coordinate conditions. If we choose harmonic coordinates, also known as de

Donder coordinates,

∂

= 0(E.3)

(see Chapter 4.3), we can bring Einstein’s equations (E.1) into the form

h

= 16πτ

. (E.4)

Here  ≡ η

∂

is the ﬂat d’Alembertian operator, and the source term

≡−gT

16π



(E.5)

contains contributions from both the stress-energy tensor T

as well as terms that are nonlinear

in h

, which we have absorbed in a new quantity 

. For our purposes it is sufﬁcient to state

that 

,thestress-energy pseudo-tensor, contains terms that are at least quadratic in h

as well

as its ﬁrst two derivatives.

We can now write the formal solution to Einstein’s equations (E.4) in terms of the Green

function for the ﬂat d’Alembertian operator,

(t, x) =−4





, t −|x −x



|)d



|x − x



, (E.6)

We brieﬂy discuss some of these approaches in Chapter 12.

Our brief summary of post-Newtonian methods and results is based on the review by Blanchet (2006), to which we

refer the reader for a more detailed treatment, as well as a list of references. We also refer the reader to Blanchet et al.

(2008) for further results on 3PN gravitational wave polarizations.

Note that some authors introduce these quantities with the opposite sign, which leads to sign differences in many of the

following equations.

Compare with the Lorentz gauge condition (9.3), which we introduced in the context of perturbation theory and

linearized waves in Chapter 9.1.1.

For a precise deﬁnition, see Blanchet (2006).

618 Appendix E Post-Newtonian results

where, for convenience, we have denoted the three spatial coordinates with a bold-face x. Unfortu-

nately, this solution is not of very much immediate help, since the source term τ

in the integrand

depends on the integral h

. It forms, however, the basis for the iteration that is involved in con-

structing post-Newtonian approximations.

We start the iteration with some known Newtonian solution. For example, for a binary this

solution could describe two point-masses in a circular orbit. Using this Newtonian solution we

can compute a Newtonian stress-energy tensor T

= T

Newt

and set the corresponding Newtonian

values of the gravitational ﬁeld perturbation amplitudes to zero, h

= 0. This determines the

Newtonian source term τ

, which we may insert into equation (E.6). Solving the integral then

yields the ﬁrst correction to the gravitational ﬁeld perturbations, h

. Given these, we can re-

evaluate the equations of motion, compute τ

, insert these into the integrand of equation (E.6),

and compute the next correction, h

. In general, we can obtain the (n + 1)th correction by

inserting the previous one into the integrand of equation (E.6),

n+1

(t, x) =−4





, t −|x −x



|)d



|x − x



. (E.7)

If all goes well, this iteration converges to give the correct solution h

(t, x).

While our crude recipe grossly over-simpliﬁes the problem, it does lay out a starting point

for constructing a post-Newtonian expansion. Each iteration in the above procedure adds a new

correction that improves the previous one by an order 1/c

, where, in a rare reappearance in this

volume, c is the speed of light. An nth-order post-Newtonian expansion therefore includes terms

up to order 1/c

; a “3PN” expansion, for example, goes to order 1/c

Starting out with a purely Newtonian expression we obtain correction terms that are even

powers in 1/c. These terms capture only the so-called conservative effects, but not the radiation-

reaction effects. The latter ﬁrst appear at the odd order 1/c

, or in a 2.5PN expansion. We have

seen this, for example, in exercise 9.3, where we computed the leading-order gravitational wave

luminosity dE/dt for a binary in circular orbit using the weak-ﬁeld, slow-velocity quadrupole

formula. There we expressed the result (9.41) in geometrized units with c = G = 1, where G is

the gravitational constant; if we now restore these constants we ﬁnd

=−



, (E.8)

demonstrating that the leading-order radiation-reaction effects indeed appear in a 2.5PN expan-

sion. When referring to post-Newtonian expansions of the gravitational radiation, an nPN expan-

sion usually means a correction of order 1/c

beyond the leading order quadrupole formula

above.

Results for compact binaries

In lieu of pursuing the construction of post-Newtonian expansions in greater detail here, we

shall simply summarize some of the most important post-Newtonian results for compact binary

inspiral. Binary inspiral has been the focus of considerable attention, given its importance as a

promising source of gravitational radidation.

We refer the reader to the review article by Blanchet (2006) for a much more detailed discussion and references.

Appendix E Post-Newtonian results 619

Label the masses of the two stars m

and m

, and deﬁne the total mass as M = m

+ m

and

the reduced mass as µ = m

/M. It will also be useful to introduce the symmetric mass ratio

ν ≡

+ m

)

, (E.9)

which takes its maximum value of ν

max

= 1/4 for equal-mass binaries and approaches zero for

extreme mass ratios. We will focus on nonspinning binaries

in quasicircular orbit, a problem

which, at the time of this writing, has been solved up to order 3.5PN for the orbital phase evolution

and to order 3PN for the gravitational waveform.

We ﬁrst write the binary’s orbital angular velocity ω as a function of the binary separation R,

where R is the coordinate separation in the harmonic gauge (equation E.3). Following Blanchet

(2006), we introduce a dimensionless measure of the binary separation

γ ≡

= O





. (E.10)

In terms of this ratio we then obtain



1 +



− 3 + ν



γ +



6 +

ν + ν





− 10 +



−

75707

840

+ 22 ln

(

)



ν +

+ ν





+ O





(E.11)

This is the 3.5PN version of Kepler’s third law in harmonic coordinates; the leading-order term is

recognizable as the Newtonian expression.

Upon further thought, we realize that the harmonic

coordinate conditions (E.3) do not fully specify the coordinates uniquely.

This gauge ambiguity

is encoded in the term ln(R/R

) in the above expression, which includes as yet an undetermined

length scale R

The energy E of a circular orbit to the same order is given by

E =−

µc



1 +



−



γ +



−

ν +





−

235



46031

2240

−

123

(

)



ν +





+ O





(E.12)

We again recognize the Newtonian result as the leading-order term.

Both the angular velocity ω and the energy E are gauge-invariant quantities. In equations (E.11)

and (E.12), however, both of these quantities still appear to depend on the gauge-dependent length

Effects of spin have been included up to order 2.5PN, see Faye et al. (2006); Blanchet et al. (2006) (and errata).

See Blanchet et al. (2008); recall that a 3PN expansion of the gravitational waveforms includes corrections up to order

1/c

beyond the weak-ﬁeld, slow-velocity quadrupole formula (E.8).

Recall that in terms of the areal radius R

, the angular velocity of a test particle in circular orbit about a Schwarzschild

black hole is given by ω

= GM/R

exactly, to all orders of γ .

Compare the discussion following equation (9.4) in the context of the Lorentz gauge for linearized waves.

620 Appendix E Post-Newtonian results

scale R

. The dependence is an artifact that results from having expressed ω and E in terms of

the harmonic coordinate binary separation R, which itself is a gauge-dependent quantity. To

eliminate this gauge dependence we express E in terms of ω. For this purpose it is convenient to

introduce a dimensionless angular velocity according to

x ≡



GMω



2/3

= O





. (E.13)

We can now invert equation (E.11) to ﬁnd γ in terms of x. Substituting this result into equa-

tion (E.12) then yields

E =−

µc



1 +



−



x +



−

ν −





−

675



34445

576

−

205



ν −

155

−

5184





+ O





, (E.14)

which, as expected, is gauge-invariant, and independent of the length scale R

The above expressions hold for binaries in quasicircular orbit. To model the slow, adiabatic

inspiral, caused by the emission of gravitational radiation, we also need to compute the gravita-

tional radiation luminosity

=−

. (E.15)

In order to keep this expression consistent with the 3.5PN expressions above, we need to compute

this luminosity beyond the weak-ﬁeld, slow-velocity expression (E.8) to order 3.5PN. The result,

when expressed in terms of x,is



1 +



−

1247

336

−



x + 4π x

3/2



−

44711

9072

9271

504

ν +





−

8191

672

−

583



π x

5/2



6643739519

69854400

−

1712

105

C −

856

105

ln(16x)



−

134543

7776



ν −

94403

3024

−

775

324





−

16285

504

214745

1728

ν +

193385

3024



π x

7/2

+ O







. (E.16)

Here C = 0.57722 ... is the Euler constant. It is easy to verify that the leading-order term is the

familiar quadrupole formula (E.8).

We can now use the above results to obtain the binary’s orbital phase  as a function of time.

The idea is to integrate

dE/dt

dE/dx

=−

dE/dx

(E.17)