1.1 Notation and definitions

The following contains the notational conventions that will be in use throughout the course of this review.

Table 1: Glossary


+ ℑ, + ℑℂ

Future conformal null infinity, Complex future conformal null infinity

I+, I− ,I0

Future, Past conformal timelike infinity, Conformal spacelike infinity

𝕄, 𝕄 ℂ

Minkowski space, Complex Minkowski space

uB, uret

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

∂ f = f˙ uB

Derivation with respect to u B

∂ f = f′ uret

Derivation with respect to u ret


Affine parameter along null geodesics

(ζ, ¯ζ)

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

Ys (ζ, ¯ζ) li...j

Tensorial spin-s spherical harmonics

∂, ¯ ∂

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


Metric function on S2; often P = P0 ≡ 1 + ζ¯ζ


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

a a a a {l,n ,m , ¯m }

Null tetrad system; a a l na = − m ¯ma = 1


Null Geodesic Congruence


Newman–Penrose/Spin-Coefficient Formalism

A A {U,X ,ω,ξ }

Metric coefficients in the Newman–Penrose formalism

{ψ0,ψ1,ψ2, ψ3,ψ4 }

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


8πG- c4; Gravitational constant

¯ u = G (τ,ζ,ζ)

Cut function on + ℑ

¯ τ = s + iλ = T(u, ζ,ζ)

Complex auxiliary (CR) potential function

′ ∂τf = f

Derivation with respect to τ

2 ¯ 0 ¯ ∂(τ)G (τ,ζ,ζ) = σ (u,ζ, ζ)

Good-Cut Equation, describing asymptotically shear-free NGCs


Good-Cut Function

L(u,ζ,ζ¯) = ∂ G (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

three-dimensional CR Structure

A class of one-forms describing a real three-manifold of 2 ℂ


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

i i i 1 0 Dℂ = D E + iD M = 2ϕ 0i

Complex electromagnetic dipole

a η (uret)

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

Diℂ (grav) = Di(mass) + ic−1J i


    = − -√c2--ψ0i 6 2G 1

Complex gravitational dipole

a ξ (uret)

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


Universal Cut Function

u = X (τ,ζ, ¯ζ)

UCF; corresponding to the complex center-of-charge world line

Ψ ≡ ψ0 + ∂2σ-+ σσ˙= ¯Ψ 2

Bondi Mass Aspect

-c2-- 0 MB = − 2√2GΨ

Bondi mass

3 Pi = − c6GΨi

Bondi linear three-momentum

- Ji = √2c3 Im (ψ0i) 12G 1

Vacuum linear theory identification of angular momentum

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