1.1 Notation and definitions

• Throughout this work 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 [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., 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 future null cone. The other members of the associated null tetrad are
  
• Several different timelike variables (, ) and their derivatives 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. 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.,

  
• 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 early.

• 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) this is basically the “real center of mass plus ‘’ angular momentum” (or “real center of charge plus ‘’ magnetic moment”). There will be two different types of these “complex centers …”; 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 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) 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

  
  This, though easy to understand and keep track of, does run into the unpleasant fact that often the relativist four-vector,

  appears as the harmonics in the harmonic expansions. Thus, care must be used when lowering or raising the relativistic index, i.e., .

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Table 1: Glossary

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
,	; spin-weighted operator on the two-sphere
	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/Spin-Coefficient Formalism
	Metric coefficients in the Newman–Penrose formalism
	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
,	Complex shear, Asymptotic complex shear of a NGC
	; Gravitational constant
	Cut function on
	Complex auxiliary (CR) potential function
	Derivation with respect to
	Good-Cut Equation, describing asymptotically shear-free NGCs
GCF	Good-Cut Function
	Stereographic angle field for an asymptotically shear-free NGC at
	CR equation, describing the embedding of into
three-dimensional CR Structure	A class of one-forms describing a real three-manifold of
-space	Complex four-dimensional solution space to the Good-Cut Equation
	Complex electromagnetic dipole
	Complex center-of-charge world line, lives in -space
	Complex gravitational dipole
	Complex center-of-mass world line, lives in -space
UCF	Universal Cut Function
	UCF; corresponding to the complex center-of-charge world line
	Bondi Mass Aspect
	Bondi mass
	Bondi linear three-momentum
	Vacuum linear theory identification of angular momentum

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"Null Geodesic Congruences, Asymptotically-Flat Spacetimes and Their Physical Interpretation" Timothy M. Adamo and Carlos Kozameh and Ezra T. Newman http://www.livingreviews.org/lrr-2009-6

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