Jump to ContentJump to Main Navigation
Introduction to 3+1 Numerical Relativity$

Miguel Alcubierre

Print publication date: 2008

Print ISBN-13: 9780199205677

Published to Oxford Scholarship Online: September 2008

DOI: 10.1093/acprof:oso/9780199205677.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 14 December 2017

(p.413) APPENDIX D SPIN-WEIGHTED SPHERICAL HARMONICS

(p.413) APPENDIX D SPIN-WEIGHTED SPHERICAL HARMONICS

Source:
Introduction to 3+1 Numerical Relativity
Publisher:
Oxford University Press

Consider the Laplace operator written in spherical coordinates as

(D.1)
2 f = 1 r 2 r 2 ( r 2 f ) + 1 r 2 L 2 f ,

where L 2 is the angular operator

(D.2)
L 2 f : = 1 sin θ θ ( sin θ θ f ) + 1 sin 2 θ φ 2 f .

By separation of variables, we can show that f will be a solution of the Laplace equation ∇2 f=0 if it can be written as

(D.3)
f ( r , θ , φ ) = r l g 1 ( θ , φ ) + 1 r l + 1 g 2 ( θ , φ ) ,

where g 1,2(θ,ϕ) are eigenfunctions of the L 2 operator such that

(D.4)
L 2 g = l ( l + 1 ) g .

The solutions of the last equation are known as spherical harmonics. They are usually denoted by Y l,m(θ,ϕ) and have the form

(D.5)
Y l , m ( θ , φ ) = [ ( 2 l + 1 ) 4 π ( l m ) ! ( l + m ) ! ] 1 / 2 P l , m ( cos θ ) e i m φ ,

where l and m are integers such that ∣m∣≤l, and P l,m(z) are the associated Legendre polynomials. The Y l,m(θ,ϕ) are orthogonal to each other when integrated over a sphere, and the normalization chosen above is such that

(D.6)
Y l , m ( θ , φ ) Y ¯ l , m ( θ , φ ) d Ω = δ l l δ m m ,

where dΩ=sin θdθdϕ is the standard area element of the sphere. Using the fact that the associated Legendre functions are such that

(D.7)
P l , m ( z ) = ( 1 ) m ( l m ) ! ( l + m ) ! P l , m ( z ) ,

we can show that the complex conjugate of the Y l,m is given by

(D.8)
Y ¯ l , m ( θ , φ ) = ( 1 ) m Y l , m ( θ , φ ) .

When we work with non-scalar functions defined on the sphere we introduce the so-called spin-weighted spherical harmonics as generalizations of the ordinary (p.414) spherical harmonics. Spin-weighted spherical harmonics were first introduced by Newman and Penrose [219] for the study of gravitational radiation, but they can also be used to study solutions of the Maxwell equations, the Dirac equation, or in fact dynamical equations for fields of arbitrary spin.

Consider a complex function f on the sphere that might correspond to some combination of components of a tensorial (or spinorial) object in the orthonormal basis ê r,ê θ,ê ϕ) induced by the spherical coordinates (r,θ,ϕ). We will say that f has spin weight s if, under a rotation of the angular basis ê θ,ê ϕ) by an angle ψ, it transforms as fe −isψ f. A trivial example is a scalar function whose spin weight is clearly zero. A more interesting example corresponds to a three-dimensional vector v with components ( v r ^ , v θ ^ , v ϕ ^ ) . Notice that these components are different from those in the coordinate basis (which is not orthonormal), and are related to them through ( v r ^ , v θ ^ , v ϕ ^ ) = ( v r , r v θ , r sin θ   v ϕ ) . Define now two unit complex vectors as

(D.9)
e ^ ± : = ( e ^ θ i e ^ φ ) / 2 .

The vector v can then be written as

(D.10)
v = v 0 e ^ r + v + e ^ + + v e ^ ,

where

(D.11)
v 0 : = v r ^ , v + : = ( v θ ^ + i v ϕ ^ ) / 2 , v : = ( v θ ^ i v ϕ ^ ) / 2

By considering a rotation of the vectors ê θ,ê ϕ by an angle ψ it is now easy to see that v 0 has spin weight zero, while the spin weight of v ± is ±1.

The spin-weighted spherical harmonics, denoted by s Y ¯ l , m ( θ , φ ) , form a basis for the space of functions with definite spin weight s. They can be introduced in a number of different ways. We can start by defining the operators

(D.12)
ð f : = sin s θ ( θ + i sin θ φ ) ( f sin s θ ) = ( θ + i sin θ φ s cot θ ) f ,
(D.13)
ð ¯ f : = sin s θ ( θ i sin θ φ ) ( f sin s θ ) = ( θ i sin θ φ + s cot θ ) f ,

where s is the spin weight of f. The spin-weighted spherical harmonics are then defined for ∣m∣≤l and l≥∣s in terms of the standard spherical harmonics as

(D.14)
s Y l , m : = [ ( l s ) ! ( l + s ) ! ] 1 / 2 ð s ( Y l , m ) , + l s 0 ,
(D.15)
s Y l , m : = ( 1 ) s [ ( l + s ) ! ( l s ) ! ] 1 / 2 ð ¯ s ( Y l , m ) , l s 0.
(p.415) In particular we have 0 Y l,m=Y l,m. The above definition implies that
(D.16)
ð ( s Y l , m ) = + [ ( l s ) ( l + s + 1 ) ] 1 / 2 s + 1 Y l , m ,
(D.17)
ð ¯ ( s Y l , m ) = [ ( l + s ) ( l s + 1 ) ] 1 / 2 s 1 Y l , m .

Because of this, ð and ð ¯ are known as the spin raising and spin lowering operators. We also find that

(D.18)
ð ¯ ð ( s Y l , m ) = l [ ( l + 1 ) s ( s + 1 ) ] s Y l , m ,
(D.19)
ð ð ¯ ( s Y l , m ) = [ l ( l + 1 ) s ( s 1 ) ] s Y l , m ,

so the s Y l,m are eigenfunctions of the operators ð ¯ ð and ð ð ¯ , which are generalizations of L 2. For a function with zero spin weight we in fact find that L 2 f = ð ¯ ð f = ð ð ¯ f .

From the above definitions it is possible to show that the complex conjugate of the s Y l,m is given by

(D.20)
s Y ¯ l , m ( θ , φ ) = ( 1 ) s + m s Y l , m ( θ , φ ) ,

which is just the generalization of (D.8) to the case of non-zero spin.

We can also find generalizations of the standard angular momentum operators for the case of non-zero spin weight by looking for operators J ^ z and J ^ ± such that (here we are ignoring the factor i that normally appears in quantum mechanics) [112]

(D.21)
J ^ z   s Y l , m = i m s Y l , m ,
(D.22)
J ^ ± s Y l , m = i [ ( l m ) ( l + 1 ± m ) ] 1 / 2 s Y l , m ± 1 .

We then find that such operators must have the form

(D.23)
J ^ z = φ ,
(D.24)
J ^ ± = e ± i φ [ ± i θ cot θ φ i s csc θ ] .

The operators for the x and y components of the angular momentum are then simply obtained from J ^ ± = J ^ x ± i J ^ y , so that we find:

(D.25)
J ^ x = ( J ^ + + J ^ ) / 2 , J ^ y = i ( J ^ + J ^ ) / 2 ,

The s Y l,m can also be constructed in terms of the so-called Wigner rotation matrices d m s l , which are defined in quantum mechanics as the following matrix elements of the operator for rotations around the y axis:

(D.26)
d m s l ( θ ) : = < l , m | e i J ^ y θ | l , s > .
(p.416) We then find that the s Y l,m have the form
(D.27)
s Y l , m ( θ , ϕ ) = ( 1 ) m ( 2 l + 1 4 π ) 1 / 2 e i m ϕ d m s l ( θ ) .

Closed expressions for the rotation matrices d m s l , as well as their principal properties, are well known but we will not go into the details here (the interested reader can look at standard textbooks on quantum mechanics, e.g. [202, 294]).

There are several very important properties of the spin-weighted spherical harmonics that can be obtained directly from the properties of the rotation matrices d m s l . In the first place, just as the ordinary spherical harmonics, the different s Y l,m are orthonormal,

(D.28)
s Y l , m ( θ , φ ) s Y ¯ l , m ( θ , φ ) d Ω = δ s s δ u δ m m .

Also, for a given value of s, the s Y l,m form a complete set. This property can be expressed in the form

(D.29)
l , m s Y l , m ( θ , φ ) s Y ¯ l , m ( θ , φ ) = δ ( φ φ ) δ ( cos θ cos θ ) .

The integral of three spin-weighted spherical harmonics is also frequently needed (for example in the calculation of the momentum flux of gravitational waves) and can be expressed in general as

(D.30)
s 1 Y l 1 , m 1 ( θ , φ ) s 2 Y l 2 , m 2 ( θ , φ ) s 3 Y l 3 , m 3 ( θ , φ ) d Ω = [ ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ( 2 l 3 + 1 ) 4 π ] 1 / 2 ( l 1 l 2 l 3 s 1 s 2 s 3 ) ( l 1 l 2 l 3 m 1 m 2 m 3 ) .

In the above expression we have used the Wigner 3-lm symbols, which are related to the standard Clebsch–Gordan coefficients 〈l 1,m 1,l 2,m 2j 3,m 3〉 through

(D.31)
( l 1 l 2 l 3 m 1 m 2 m 3 ) = ( 1 ) l 1 l 2 m 3 2 l 3 + 1 < l 1 , m 1 , l 2 , m 2 | l 3 , m 3 > ,

or equivalently

(D.32)
< l 1 , m 1 , l 2 , m 2 | l 3 , m 3 > = ( 1 ) l 1 l 2 + m 3 2 l 3 + 1 ( l 1 l 2 l 3 m 1 m 2 m 3 ) .

The Clebsch–Gordan coefficients arise from the addition of angular momentum in quantum mechanics, and correspond to the coefficients of the expansion of an eigenstate ∣L,M〉 with total angular momentum L and projection M, in terms (p.417) of a basis formed by the product of the individual eigenstates ∣l 1,m 1〉∣l 2,m 2〉. These coefficients are always real, and are only different from zero if

(D.33)
l i < m i < l i , | l 1 l 2 | < l 3 < l 1 + l 2 .

The Clebsch–Gordan coefficients have some important symmetries, though these symmetries are easier to express in terms of the 3-lm symbols. In particular, the 3-lm symbols are invariant under an even permutation of columns, and pick up a factor ( 1 ) l 1 + l 2 + l 3 under an odd permutation. Also,

(D.34)
( l 1 l 2 l 3 m 1 m 2 m 3 ) = ( 1 ) l 1 + l 2 + l 3 ( l 1 l 2 l 3 m 1 m 2 m 3 ) .

A closed expression for the Clebsch–Gordan coefficients was first found by Wigner (see e.g. [299]). This expression is somewhat simpler when written in terms of the 3-lm coefficients, and has the form

(D.35)
( l 1 l 2 l 3 m 1 m 2 m 3 ) = ( 1 ) l 1 m 1 δ m 1 + m 2 , m 3 × [ ( l 1 + l 2 l 3 ) ! ( l 1 + l 3 l 2 ) ! ( l 2 + l 3 l 1 ) ! ( l 3 + m 3 ) ! ( l 3 m 3 ) ! ( l 1 + l 2 + l 3 + 1 ) ! ( l 1 + m 1 ) ! ( l 1 m 1 ) ! ( l 2 + m 2 ) ! ( l 2 m 2 ) ! ] 1 / 2 × k 0 ( 1 ) k k ! [ ( l 2 + l 3 + m 1 k ) ! ( l 1 m 1 + k ) ! ( l 3 l 1 + l 2 k ) ! ( l 3 m 3 k ) ! ( l 1 l 2 + m 3 + k ) ! ] .

In the above expression the sum runs over all values of k for which the arguments inside the factorials are non-negative. Also, if the particular combination of {l i,m i} is such that the arguments of the factorials outside of the sum are negative, then the corresponding coefficient vanishes. A more symmetric (though longer) expression that is equivalent to (D.35) was later derived by Racah [234], but we will not write it here.

In the general case, (D.35) is rather complicated, but this is not a serious problem as we can find tables of the most common coefficients in the literature, and even web-based “Clebsch–Gordan calculators. Moreover, in some special cases the coefficients simplify considerably. For example, in the case where m 1=l 1, m 2=l 2, and l 3=m 3=l 1+l 2 we find

(D.36)
( l 1 l 2 l 1 + l 2 l 1 l 2 l 1 + l 2 ) = 1 2 ( l 1 + l 2 ) + 1 < l 1 , l 1 , l 2 , l 2 | l 1 + l 2 , l 1 + l 2 > = 1.

Another particularly interesting case corresponds to taking l 3=m 3=0 (i.e. zero total angular momentum in quantum mechanics). In that case we find

(D.37)
( l 1 l 2 0 m 1 m 2 0 ) = < l 1 , m 1 , l 2 , m 2 | 0 , 0 > = ( 1 ) l 1 m 1 2 l 1 + 1 δ l 1 , l 2 δ m 1 , m 2 .

Taking this result, together with (D.20) and the fact that 0 Y 00 = 1 / 4 π , we can easily recover the orthonormality condition (D.28) from the integral of three s Y l , m , (D.30).

(p.418) The cases with l 3=1 are also interesting as they appear in the expression for the momentum radiated by gravitational waves. We find

(D.38)
( l 1 l 2 1 m 1 m 2 0 ) = ( 1 ) l 1 m 1 δ m 1 + m 2 , 0 × [ δ l 1 , l 2 ( 2 m 1 ( 2 l 1 + 2 ) ( 2 l 1 + 1 ) ( 2 l 1 ) ) + δ l 1 , l 2 + 1 ( ( l 1 + m 1 ) ( l 1 m 1 ) l 1 ( 2 l 1 + 1 ) ( 2 l 1 1 ) ) 1 / 2 δ l 1 + 1 , l 2 ( ( l 2 m 2 ) ( l 2 + m 2 ) l 2 ( 2 l 2 + 1 ) ( 2 l 2 1 ) ) 1 / 2 ] ,
(D.39)
( l 1 l 2 1 m 1 m 2 ± 1 ) = ( 1 ) l 1 m 1 δ m 1 + m 2 , 1 × [ ± δ l 1 , l 2 ( ( l 1 m 1 ) ( l 1 m 2 ) l 1 ( 2 l 1 + 2 ) ( 2 l 1 + 1 ) ) 1 / 2 + δ l 1 , l 2 + 1 ( ( l 1 m 1 ) ( l 1 ± m 2 ) 2 l 1 ( 2 l 1 + 1 ) ( 2 l 1 1 ) ) 1 / 2 + δ l 1 + 1 , l 2 ( ( l 2 m 2 ) ( l 2 ± m 1 ) 2 l 2 ( 2 l 2 + 1 ) ( 2 l 2 1 ) ) 1 / 2 ] .