- We use the symbols ‘’, ‘’, ‘’ …with several different ‘decorations’ but always meaning
a null tetrad or a null tetrad field.
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 [66]), 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., one-leg 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 set of directions on the future null cone. The other members of the associated null tetrad are - Several different time variables (, ) and derivatives with respect to them are
used.
The 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. Derivatives with respect to these variables 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. - Often the angular (or sphere) derivatives, and , are used. The notation means, apply the operator to the function while holding the variable constant.
- The complex conjugate is represented by the overbar, e.g., . When a complex variable, , is close to the complex conjugate of , but independent, we use .

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 from the outset:

- As mentioned earlier, we use the term ‘generalized light cones’ to mean (real) NGCs that appear to have their apexes on a world line in the complexification of the spacetime. A detailed discussion of this will be given in Sections 3 and 4.
- The term ‘complex center of mass’ (or ‘complex center of charge’) is frequently used. Up to the choice of constants (to give correct units) they basically lead to the ‘mass-dipole plus “” angular momentum’ (or ‘real electric-dipole plus “” magnetic dipole moment’). There will be two different types of these ‘complex centers of …’; one will be geometrically defined or intrinsic, i.e., independent of the choice of coordinate system, the other will be relative, i.e., it will depend on the choice of (Bondi) coordinates. The relations between them are nonlinear and nonlocal.
- A very important technical tool used throughout this work is a class of complex analytic functions, , referred to as Good-Cut Functions, (GCFs) that are closely associated with shear-free NGCs. The details are given later. For any given asymptotically-vanishing Maxwell field with nonvanishing total charge, the Maxwell field itself allows one, on physical grounds, to choose a unique member of the class referred to as the (Maxwell) Universal-Cut Function (UCF). For vacuum asymptotically-flat spacetimes, the Weyl tensor allows the choice of a unique member of the class referred to as the (gravitational) UCF. For Einstein–Maxwell there will be two such functions, though in important cases they will coincide and be referred to as UCFs. When there is no ambiguity, in either case, they will simple be UCFs.
- A notational irritant arises from the following situation. Very often we expand functions on the sphere
in spin- harmonics, as, e.g.,
where the indices, represent three-dimensional Euclidean indices. To avoid extra notation and symbols we write scalar products and cross-products 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., .

- Throughout this review (and especially in Section 6), we will invoke comparisons between our results
and those of classical electromagnetism and relativity (cf. [43]). This process rests upon our
identifications of the electric and magnetic dipole and quadrupole moments in the spherical harmonic
expansions of the Maxwell tensor in the Newman–Penrose formalism. Although the identifications we
make are the most natural in our framework, a numerical re-scaling is required to obtain the physical
formulae in some cases: in terms of the complex dipole and quadrupole moments used for the
electromagnetic field, this is given by
The conventions used here were chosen so that the numerical coefficient of in was equal to one; this re-scaling can simply be viewed as choosing a different (perhaps less natural) identification for the electromagnetic quadrupole moment, or as a sort of gauge choice for our results.

Living Rev. Relativity 15, (2012), 1
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