a) Though in places, e.g., in Section 2.4, the symbols, , , …, i.e., with an can be thought of as the abstract representation of a null tetrad (i.e., Penrose’s abstract index notation [50]), in general, our intention is to describe vectors in a coordinate representation.
b) The symbols, , most often represent the coordinate versions of different null geodesic tangent fields, e.g., oneleg of a Bondi tetrad field or some rotated version.
c) The symbol, , (with hat) has a very different meaning from the others. It is used to represent the Minkowski components of a normalized null vector giving the null directions on an arbitrary light cone:
As the complex stereographic coordinates sweep out the sphere, the sweeps out the entire future null cone. The other members of the associated null tetrad areThe Bondi time, , is closely related to the retarded time, . The use of the retarded time, , is important in order to obtain the correct numerical factors in the expressions for the final physical results. Their derivatives are represented by
The , derivatives are denoted by the same prime () since it is always applied to functions with the same functional argument. Though we are interested in real physical spacetime, often the time variables () take complex values close to the real ( is always real). Rather than putting on ‘decorations’ to indicate when they are real or complex (which burdens the expressions with an overabundance of different symbols), we leave reality decisions to be understood from context. In a few places where the reality of the particular variable is manifestly first introduced (and is basic) we decorate the symbol by a superscript (), i.e., or . After their introduction we revert to the undecorated symbol.Note: At this point we are taking the velocity of light as and omitting it; later, when we want the correct units to appear explicitly, we restore the . This entails, via , , changing the prime derivatives to include the , i.e.,
Frequently, in this work, we use terms that are not in standard use. It seems useful for clarity to have some of these terms defined early.

where the indices, represent threedimensional Euclidean indices. To avoid extra notation and symbols we write scalar products and crossproducts without the use of an explicit Euclidean metric, leading to awkward expressions like

appears as the harmonics in the harmonic expansions. Thus, care must be used when lowering or raising the relativistic index, i.e., .
Symbol/Acronym 
Definition 
, 
Future conformal null infinity, Complex future conformal null infinity 
, 
Future, Past conformal timelike infinity, Conformal spacelike infinity 
, 
Minkowski space, Complex Minkowski space 
, 
Bondi time coordinate, Retarded Bondi time () 
Derivation with respect to 

Derivation with respect to 

Affine parameter along null geodesics 

; stereographic coordinates on 

Tensorial spin spherical harmonics 

, 
; spinweighted operator on the twosphere 
Metric function on ; often 

Application of operator to while the variable is held constant 

Null tetrad system; 

NGC 
Null Geodesic Congruence 
NP/SC 
Newman–Penrose/SpinCoefficient Formalism 
Metric coefficients in the Newman–Penrose formalism 

Weyl tensor components in the NewmanPenrose formalism 

Maxwell tensor components in the Newman–Penrose formalism 

Complex divergence of a null geodesic congruence 

Twist of a null geodesic congruence 

, 
Complex shear, Asymptotic complex shear of a NGC 
; Gravitational constant 

Cut function on 

Complex auxiliary (CR) potential function 

Derivation with respect to 

GoodCut Equation, describing asymptotically shearfree NGCs 

GCF 
GoodCut Function 
Stereographic angle field for an asymptotically shearfree NGC at 

CR equation, describing the embedding of into 

threedimensional CR Structure 
A class of oneforms describing a real threemanifold of 
space 
Complex fourdimensional solution space to the GoodCut Equation 
Complex electromagnetic dipole 

Complex centerofcharge world line, lives in space 



Complex gravitational dipole 

Complex centerofmass world line, lives in space 

UCF 
Universal Cut Function 
UCF; corresponding to the complex centerofcharge world line 

Bondi Mass Aspect 

Bondi mass 

Bondi linear threemomentum 

Vacuum linear theory identification of angular momentum 

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