1.2 Glossary of symbols and units

In this work we make use of substantial notational machinery. The most frequently used symbols and acronyms are gathered here for easy reference:

Table 1: Glossary


+ ℑ, + ℑℂ

Future null infinity, Complex future null infinity

I+, I−, I0

Future, Past timelike infinity, Spacelike infinity

𝕄, 𝕄 ℂ

Minkowski space, Complex Minkowski space

uB, uret

Bondi time coordinate, Retarded Bondi time (√ -- 2uB = uret)

∂uBf = f˙

Derivation with respect to uB

∂uretf = f′

Derivation with respect to uret


Affine parameter along null geodesics

(ζ, ¯ζ)

(eiϕcot(𝜃∕2), e−iϕcot(𝜃∕2)); stereographic coordinates on S2

Ysli...j(ζ, ¯ζ)

Tensorial spin-s spherical harmonics

∂, ∂¯

P 1−s ∂-P s, P 1+s ∂-P −s ∂ζ ∂¯ζ; spin-weighted operator on the two-sphere


Metric function on 2 S; often ¯ P = P0 ≡ 1 + ζζ

∂ f (α)

Application of ∂-operator to f while the variable α is held constant

{la,na, ma, ¯ma }

Null tetrad system; lan = − mam¯ = 1 a a


Null Geodesic Congruence


Newman–Penrose/Spin-Coefficient Formalism

{U,XA, ω,ξA}

Metric coefficients in the Newman–Penrose formalism

{ψ ,ψ ,ψ ,ψ ,ψ } 0 1 2 3 4

Weyl tensor components in the Newman–Penrose formalism


Maxwell tensor components in the Newman–Penrose formalism


Complex divergence of a null geodesic congruence


Twist of a null geodesic congruence

σ, σ0

Complex shear, Asymptotic complex shear of a NGC


2G4- c; Gravitational constant

τ = s + iλ = T(u, ζ, ¯ζ)

Complex auxiliary (CR) potential function

∂τf = f′

Derivation with respect to τ

∂2 G (τ,ζ, ¯ζ) = σ0(G, ζ, ¯ζ) (τ)

Good-Cut Equation, describing asymptotically shear-free NGCs

u = G (τ,ζ, ¯ζ) B

Good-Cut Function (GCF) on ℑ+

L(u ,ζ, ¯ζ) = ∂ G B (τ)

Stereographic angle field for an asymptotically shear-free NGC at + ℑ

˙ ∂(uB )T + LT = 0

CR equation, describing the embedding of + ℑ into 2 ℂ


Complex four-dimensional solution space to the Good-Cut Equation

i i i 1 0i Dℂ = D E + iD M = 2ϕ 0

Complex electromagnetic dipole

a η (uret)

Complex center-of-charge world line, lives in ℋ-space

√ - Qiℂj= QijE + iQijM = --2Qiℂjphysical 4

Complex electromagnetic quadrupole


Complex center of mass world line, lives in ℋ-space

Di = Di + ic−1J i (grav) (mass)


    c2 0i = − 6√2G-ψ1

Complex gravitational dipole

QiGjrav = QijMass + iQijSpin

Complex gravitational quadrupole

u = X (τ,ζ, ¯ζ) B

Universal Cut Function (UCF) corresponding to the complex center of mass world line

ij √2G ij′′ ξ = 24c4Q Grav

Identification between l = 2 coefficient of the UCF and gravitational quadrupole

Ψ ≡ ψ0 + ∂2σ0-+ σ0σ˙0-= Ψ¯ 2

Bondi Mass Aspect

-c√2-- 0 MB = − 2 2GΨ

Bondi mass

3 Pi = − c6GΨi

Bondi linear three-momentum

√- Ji = − -2c3Im (ψ0i) 12G 1

Vacuum linear theory identification of angular momentum

In much of what follows, we use simplified units where c = 1. However, in Section 6 we will revert to a notation which makes dependence upon numerical constants explicit for the sake of comparing our results with well-known quantities in classical mechanics and electromagnetism. We therefore include the following reference table for the units of several prominent objects in our calculations to ease in verifying that correct powers of dimensional constants (e.g., c, G) appear in our final results. Here [⋅] stands for the units of a given quantity, and

L = [length], M = [mass ], T = [time ].

Table 2: Units
Quantity Units
[G ] 3 −1 − 2 L M T
[q] 1 3 M 2L2T −1
[k] = [Gc− 4] M − 1L −1T2
[cτ] = [cuB] = [G (τ,ζ, ¯ζ)] L
[ξi(τ)] = [ηi(τ)] = [ξij(τ)] L
[Di ] ℂ M 12L52T −1
[Di ] (grav) ML
ij [Q ℂ] 1 7 −1 M 2L2T
[QijGrav] ML2
[J i] ML2T −1
[ϕ ] = [ϕ ] = [ϕ ] 0 1 2 M 12L− 12T −1
0 [ϕ 0] 12 52 −1 M L T
0 [ϕ 1] 1 3 −1 M 2L2T
[ϕ02] 1 1 M 2L2T −1
[ψ0 ] = [ψ1] = [ψ2 ] = [ψ3] = [ψ4] L −2
[ψ00] L3
[ψ01] L2
[ψ02] L
[ψ03] 1
[ψ0 ] 4 L −1

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