Null Geodesic Congruences, AsymptoticallyFlat Spacetimes and Their Physical Interpretation
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Abstract
A priori, there is nothing very special about shearfree or asymptotically shearfree null geodesic congruences. Surprisingly, however, they turn out to possess a large number of fascinating geometric properties and to be closely related, in the context of general relativity, to a variety of physically significant effects. It is the purpose of this paper to try to fully develop these issues.
This work starts with a detailed exposition of the theory of shearfree and asymptotically shearfree null geodesic congruences, i.e., congruences with shear that vanishes at future conformal null infinity. A major portion of the exposition lies in the analysis of the space of regular shearfree and asymptotically shearfree null geodesic congruences. This analysis leads to the space of complex analytic curves in an auxiliary fourcomplex dimensional space, \({\mathcal H}\)space. They in turn play a dominant role in the applications.
The applications center around the problem of extracting interior physical properties of an asymptoticallyflat spacetime directly from the asymptotic gravitational (and Maxwell) field itself, in analogy with the determination of total charge by an integral over the Maxwell field at infinity or the identification of the interior mass (and its loss) by (Bondi’s) integrals of the Weyl tensor, also at infinity.
More specifically, we will see that the asymptotically shearfree congruences lead us to an asymptotic definition of the centerofmass and its equations of motion. This includes a kinematic meaning, in terms of the centerofmass motion, for the Bondi threemomentum. In addition, we obtain insights into intrinsic spin and, in general, angular momentum, including an angularmomentumconservation law with welldefined flux terms. When a Maxwell field is present, the asymptotically shearfree congruences allow us to determine/define at infinity a centerofcharge world line and intrinsic magnetic dipole moment.
Keywords
Complex Center Weyl Tensor Null Geodesic World Line Maxwell Field1 Introduction
Though from the very earliest days of Lorentzian geometries, families of null geodesics (null geodesic congruences (NGCs)) were obviously known to exist, it nevertheless took many years for their significance to be realized. It was from the seminal work of Bondi [16], with the introduction of null surfaces and their associated null geodesics used for the study of gravitational radiation, that the importance of NGCs became recognized. To analyze the differential structure of such congruences, Sachs [72] introduced the fundamental ‘tools’, known as the optical parameters, namely, the divergence, the twist (or curl) and the shear of the congruence. From the optical parameters one then could classify congruences by the vanishing (or the asymptotic vanishing) of one or more of these parameters. All the different classes exist in flat space but, in general, only special classes exist in arbitrary spacetimes. For example, in flat space, divergencefree congruences always exist, but for nonflat vacuum spacetimes they exist only in the case of certain high symmetries. On the other hand, twistfree congruences (null surfaceforming congruences) exist in all Lorentzian spacetimes. General vacuum spacetimes do not allow shearfree congruences, though all asymptoticallyflat spacetimes do allow asymptotically shearfree congruences, a natural generalization of shearfree congruences, to exist.
Our primary topic of study will be the cases of shearfree and asymptotically shearfree NGCs. In flat space the general shearfree congruences have been extensively studied. However, only recently has the special family of regular congruences been investigated. In general, as mentioned above, vacuum (or EinsteinMaxwell) metrics do not possess shearfree congruences; the exceptions being the algebraicallyspecial metrics, all of which contain one or two such congruences. On the other hand, all asymptoticallyflat spacetimes possess large numbers of regular asymptotically shearfree congruences. By a ‘regular congruence’ we mean a NGC that has all of its null geodesics coming from the interior of the spacetime and intersecting with future null infinity; none of its geodesics lie on future null infinity. This condition on the congruences play a fundamental role in the present work.
A priori there does not appear to be anything very special about shearfree or asymptotically shearfree NGCs. However, over the years, simply by observing a variety of topics, such as the classification of Maxwell and gravitational fields (algebraicallyspecial metrics), twistor theory, \({\mathcal H}\)space theory and asymptoticallyflat spacetimes, there have been more and more reasons to consider them to be of considerable importance. One of the earliest examples of this is Robinson’s [70] demonstration that a necessary condition for a curved spacetime to admit a null solution of Maxwell’s equation is that there be, in that space, a congruence of null, shearfree geodesics. Recent results have shown that the regular congruences — both the shearfree and the asymptotically shearfree congruences — have certain very attractive and surprising properties; each congruence is determined by a complex analytic curve in the auxiliary complex space that is referred to as \({\mathcal H}\)space. For asymptoticallyflat spacetimes, some of these curves contain a great deal of physical information about the spacetime itself [42, 40, 41].
It is the main purpose of this work to give a relatively complete discussion of these issues. However, to do so requires a digression.
A major research topic in general relativity (GR) for many years has been the study of asymptoticallyflat spacetimes. Originally, the term ‘asymptotically flat’ was associated with gravitational fields, arising from finite bounded sources, where infinity was approached along spacelike directions (e.g., [11, 74]). Then the very beautiful work of Bondi [16] showed that a richer and more meaningful idea to be associated with ‘asymptotically flat’ was to study gravitational fields in which infinity was approached along null directions. This led to an understanding of gravitational radiation via the Bondi energymomentum loss theorem, one of the profound results in GR. The Bondi energymomentum loss theorem, in turn, was the catalyst for the entire contemporary subject of gravitational radiation and gravitational wave detectors. The fuzzy idea of where and what is infinity was clarified and made more specific by the work of Penrose [62, 63] with the introduction of the conformal compactification (via the rescaling of the metric) of spacetime, whereby infinity was added as a boundary and brought into a finite spacetime region. Penrose’s infinity or spacetime boundary, referred to as Scri or ℑ, has many subregions: future null infinity, ℑ^{+}; past null infinity, ℑ^{−}; future and past timelike infinity, I^{+} and I^{−}; and spacelike infinity, I^{0} [23]. In the present work, ℑ^{+} and its neighborhood will be our primary arena for study.
A basic question for us is what information about the interior of the spacetime can be obtained from a study of the asymptotic gravitational field; that is, what can be learned from the remnant of the full field that now ‘lives’ or is determined on ℑ^{+}? This quest is analogous to obtaining the total interior electric charge or the electromagnetic multipole moments directly from the asymptotic Maxwell field, i.e., the Maxwell field at ℑ^{+}, or the Bondi energymomentum fourvector from the gravitational field (Weyl tensor) at ℑ^{+}. However, the ideas described and developed here are not really in the mainstream of GR; they may lie outside the usual interest and knowledge of many researchers. Nevertheless, they are strictly within GR: no new physics is introduced; only the vacuum Einstein or EinsteinMaxwell equations are used. The ideas come simply from observing (discovering) certain unusual and previously overlooked features of solutions to the Einstein equations and their asymptotic behavior.
These observations, as mentioned earlier, centered on the realization of the remarkable properties and importance of the special families of null geodesics: the regular shearfree and asymptotically shearfree NGCs.
The most crucial and striking of these overlooked features (mentioned now but fully developed later) are the following: in flat space every regular shearfree NGC is determined by the arbitrary choice of a complex analytic world line in complex Minkowski space, \(\mathbb{M}_{\mathbb{C}}\). Furthermore and more surprising, for every asymptoticallyflat spacetime, every regular asymptotically shearfree NGC is determined by the given Bondi shear (given for the spacetime itself) and by the choice of an arbitrary complex analytic world line in an auxiliary complex fourdimensional space, \({\mathcal H}\)space, endowed with a complex Ricciflat metric. In other words, the space of regular shearfree and asymptotically shearfree NGCs are both determined by arbitrary analytic curves in \(\mathbb{M}_{\mathbb{C}}\) and \({\mathcal H}\)space respectively [42, 40, 39].
Eventually, a unique complex world line in this space is singled out, with both the real and imaginary parts being given physical meaning. The detailed explanation for the determination of this world line is technical and reserved for a later discussion. However, a rough intuitive idea can be given in the following manner.
The idea is a generalization of the trivial procedure in electrostatics of first defining the electric dipole moment, relative to an origin, and then shifting the origin so that the dipole moment vanishes and thus obtaining the center of charge. Instead, we define, on ℑ^{+}, with specific Bondi coordinates and tetrad, the complex mass dipole moment (the real mass dipole plus ‘i’ times angular momentum) from certain components of the asymptotic Weyl tensor. (The choice of the specific Bondi system is the analogue of the choice of origin in the electrostatic case.) Then, knowing how the asymptotic Weyl tensor transforms under a change of tetrad and coordinates, one sees how the complex mass dipole moment changes when the tetrad is rotated to one defined from the asymptotically shearfree congruence. By setting the transformed complex mass dipole moment to zero, the unique complex world line, identified as the complex center of mass, is obtained. A similar process can be used in EinsteinMaxwell theory to obtain a complex center of charge.
This procedure, certainly unusual and perhaps appearing ambiguous, does logically hold together. The real justification for these identifications comes not from this logical structure though, but rather from the observed equivalence of the derived results from these identifications with wellknown classical mechanical and electrodynamical relations. These derived results involve both kinematical and dynamical relations. Though they will be discussed at length later, we mention that they range from a kinematic expression for the Bondi momentum of the form, P = Mv + …; a derivation of Newton’s second law, F = Ma; and a conservation law for angular momentum with a wellknown angular momentum flux, to the prediction of the Dirac value of the gyromagnetic ratio. We note that, for the charged spinning particle metric [53], the imaginary part of the world line is indeed the spin angular momentum, a special case of our results.
A major early clue that shearfree NGCs were important in GR was the discovery of the (vacuum or EinsteinMaxwell) algebraically special metrics. These metrics are defined by the algebraic degeneracy in their principle null vectors, which form (by the GoldbergSachs theorem [29]) a null congruence which is both geodesic and shearfree. For the asymptoticallyflat algebraicallyspecial metrics, this shearfree congruence (a very special congruence from the set of asymptotically shearfree congruences) determines a unique world line in the associated auxiliary complex \({\mathcal H}\)space. This shearfree congruence (with its associated complex world line) is a special case of the above argument of transforming to the complex center of mass. Our general asymptoticallyflat situation is, thus, a generalization of the algebraicallyspecial case. Much of the analysis leading to the transformation of the complex dipoles in the case of the general asymptoticallyflat spaces arose from generalizing the case of the algebraicallyspecial metrics.
To get a rough feeling (first in flat space) of how the curves in \(\mathbb{M}_{\mathbb{C}}\) are connected with the shearfree congruences, we first point out that the shearfree congruences are split into two classes: the twisting congruences and the twistfree ones. The regular twistfree ones are simply the null geodesics (the generators) of the light cones with apex on an arbitrary timelike Minkowski space world line. Observing backwards along these geodesics from afar, one ‘sees’ the world line. The regular twisting congruences are generated in the following manner: consider the complexification of Minkowski space, \(\mathbb{M}_{\mathbb{C}}\). Choose an arbitrary complex (analytic) world line in \(\mathbb{M}_{\mathbb{C}}\) and construct its family of complex light cones. The projection into the real Minkowski space, \(\mathbb{M}\), of the complex geodesics (the generators of these complex cones), yields the real shearfree twisting NGCs [7]. The twist contains or ‘remembers’ the apex on the complex world line, and looking backwards via these geodesics, one appears ‘to see’ the complex world line. In the case of asymptotically shearfree congruences in curved spacetime, one cannot trace the geodesics back to a complex world line. However, one can have the illusion (i.e., a virtual image) that the congruence is coming from a complex world line. It is from this property that we can refer to the asymptotically shearfree congruences as lying on generalized light cones. There is a duality between the real twisting congruences and the complex congruences coming from the complex world line: knowledge of one determines the other.
The analysis of the geometry of the asymptotically shearfree NGCs is greatly facilitated by the introduction of GoodCut Functions (GCFs). Each GCF is a complex slicing of ℑ^{+} from which the associated asymptotically shearfree NGC and world line can be easily obtained. For the special world line and congruence that leads to the complex center of mass, there is a unique GCF that is referred to as the UniversalCut Function (UCF).
Information about a variety of objects is contained in and can be easily calculated from the UCF: the unique complex world line; the direction of each geodesic of the congruence arriving at ℑ^{+}; and the Bondi asymptotic shear of the spacetime. The ideas behind the GCFs and UCF are due to some very pretty mathematics arising from the study of the ‘goodcut equation’ and its complex fourdimensional solution space, \({\mathcal H}\)space [49, 37]. In flat space almost every asymptotically vanishing Maxwell field determines its own Universal Cut Function, where the associated world line determines both the center of charge and the magnetic dipole moment. In general, for EinsteinMaxwell fields, there will be two different UCFs, (and hence two different world lines), one for the Maxwell field and one for the gravitational field. The physically interesting special case where the two world lines coincide will be discussed.
In this work, we seek to provide a comprehensive overview of the theory of asymptotically shearfree NGCs, as well as their physical applications to both flat and asymptoticallyflat spacetimes. The resulting theoretical framework unites ideas from many areas of relativistic physics and has a crossover with several areas of mathematics, which had previously appeared short of physical applications.
The main mathematical tool used in our description of ℑ^{+} is the NewmanPenrose (NP), or SpinCoefficient (SC), formalism [55]. Spherical functions are expanded in spins tensor harmonics [59]; in our approximations only the l = 0, 1, 2 harmonics are retained. Basically, the detailed calculations should be considered as expansions around the ReissnerNordström metric, which is treated as zeroth order; all other terms being small, i.e., at least first order. We retain terms only to second order.
In Section 2, we give a brief review of Penrose’s conformal null infinity ℑ along with an exposition of the NP formalism and its application to Maxwell theory and asymptoticallyflat spacetimes. There is then a description of ℑ^{+}, the stage on which most of our calculations take place. The Bondi mass aspect (a function on ℑ^{+}) is defined by the asymptotic Weyl tensor and asymptotic shear; from it we obtain the physical identifications of the Bondi mass and linear momentum. Also discussed is the asymptotic symmetry group of ℑ^{+}, the BondiMetznerSachs (BMS) group [16, 72, 56, 65]. The Bondi mass and linear momentum become basic for the physical identification of the complex centerofmass world line.
Section 3 contains the detailed analysis of shearfree NGCs in Minkowski spacetime. This includes the identification of the flat space GCFs from which all regular shearfree congruences can be found. We also show the intimate connection between the flat space GCFs, the (homogeneous) goodcut equation, and \(\mathbb{M}_{\mathbb{C}}\). As applications, we investigate the UCF associated with asymptoticallyvanishing Maxwell fields and in particular the shearfree congruences associated with the LiénardWiechert (and complex LiénardWiechert) fields. This allows us to identify a real (and complex) centerofcharge world line, as mentioned earlier.
In Section 4, we give an overview of the machinery necessary to deal with twisting asymptotically shearfree NGCs in asymptoticallyflat spacetimes. This involves a discussion of the theory of \({\mathcal H}\)space, the construction of the goodcut equation from the asymptotic Bondi shear and its complex fourparameter family of solutions. We point out how the simple Minkowski space of the preceding Section 3 can be seen as a special case of the more general theory outlined here. These results have ties to Penrose’s twistor theory and the theory of CauchyRiemann (CR) structures; an explanation of these crossovers is given in Appendices A and B.
Section 5 provides some examples of these ideas in action. We discuss linear perturbations off the Schwarzschild metric, RobinsonTrautman and twisting type II algebraically special metrics, as well as asymptotically stationary spacetimes, and illustrate how the goodcut equation can be solved and the UCF determined (explicitly or implicitly) in each case.
In Section 6, the methodology laid out in the previous Sections 3, 4 and 5 is applied to the general class of asymptoticallyflat spacetimes: vacuum and EinsteinMaxwell. Here, reviewing the material of the previous section, we use the solutions of the goodcut equation to determine all regular asymptotically shearfree NGCs by first choosing arbitrary world lines in the solution space and then singling out a unique one which determines the UCF (two world lines exist in the EinsteinMaxwell case, one for the gravitational field, the other for the Maxwell field). This identification of the unique lines comes from a study of the transformation properties, at ℑ^{+}, of the asymptoticallydefined mass and spin dipoles and the electric and magnetic dipoles. The work of Bondi, with the identification of energymomentum and its evolution, allows us to make a series of surprising further physical identifications and predictions. In addition, with a slightly different approximation scheme, we discuss our ideas applied to the asymptotic gravitational field with an electromagnetic dipole field as the source.
Section 7 contains an analysis of the gauge (or BMS) invariance of our results.
Section 8, the Discussion/Conclusion section, begins with a brief history of the origin of the ideas developed here, followed by comments on alternative approaches, possible physical predictions from our results, a summary and open questions.
Finally, we conclude with six appendices, which contain several mathematical crossovers that were frequently used or referred to in the text: twistor theory (A); CR structures (B); a brief exposition of the tensorial spherical harmonics [59] and their ClebschGordon product decompositions (C); an overview of the metric construction on \({\mathcal H}\)space (D); the description of certain real aspects of complex Minkowski space world lines (E); and a discussion of the ‘generalized goodcut equation’ with an arbitrary conformal factor (F).
1.1 Notation and definitions
 We use the symbols ‘l’, ‘m’, ‘n’ … 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, l^{ a }, m^{ a }, n^{ a } …, i.e., with an a, b, c … 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, l^{ a }, l^{#a}, l*^{ a } 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, \({\hat l^a}\), (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 \(\zeta, \,\bar \zeta\) sweep out the sphere, the \({\hat l^a}\) sweeps out the entire set of directions on the future null cone. The other members of the associated null tetrad are$${\hat l^a} = {{\sqrt 2} \over {2(1 + \zeta \bar \zeta)}}\left({1 + \zeta \bar \zeta ,\zeta + \bar \zeta ,i\bar \zeta  i\zeta ,  1 + \zeta \bar \zeta} \right) \equiv \left({{{\sqrt 2} \over 2}Y_0^0,{1 \over 2}Y_{1i}^0} \right).$$(1.1)$$\begin{array}{*{20}c} {{{\hat m}^a} = {{\sqrt 2} \over {2(1 + \zeta \bar \zeta)}}\left({0,1  {{\bar \zeta}^2},  i(1 + {{\bar \zeta}^2}),2\bar \zeta} \right),\quad \quad \quad} \\ {{{\hat n}^a} = {{\sqrt 2} \over {2(1 + \zeta \bar \zeta)}}\left({1 + \zeta \bar \zeta ,  (\zeta + \bar \zeta),i\zeta  i\bar \zeta ,1  \zeta \bar \zeta} \right).} \\ \end{array}$$(1.2)
 a)

Several different time variables (u_{B}, u_{ret}, τ, s) and derivatives with respect to them are used.
The Bondi time, u_{B}, is closely related to the retarded time, \({u_{{\rm{ret}}}} = \sqrt 2 {u_{\rm{B}}}\) The use of the retarded time, u_{ret}, 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 byThe u_{ret}, τ, s, 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 (u_{ret}, u_{B}, τ) take complex values close to the real (s is always real). Rather than putting on ‘decorations’ to indicate when they are real or complex (which burdens the expressions with an overabundance of diferent 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 (R), i.e., \(u_{\rm{B}}^{(R)}\) or \(u_{{\rm{ret}}}^{(R)}\). After their introduction we revert to the undecorated symbol.$$\begin{array}{*{20}c} {{\partial _{{u_{\rm{B}}}}}K \equiv \dot K,\quad \quad \quad} \\ {{\partial _{{u_{{\rm{ret}}}}}}K \equiv K\prime = {{\sqrt 2} \over 2}\dot K.} \\ \end{array}$$(1.3)Remark 1. At this point we are taking the velocity of light as c = 1 and omitting it; later, when we want the correct units to appear explicitly, we restore the c. This entails, via τ → cτ, s → cs, changing the prime derivatives to include the c, i.e.,$${\rm{K\prime}} \rightarrow {c^{ 1}}{\rm{K\prime}}.$$(1.4) 
Often the angular (or sphere) derivatives, ð and \(\bar {\eth}\), are used. The notation ð_{(α)}K means, apply the ð operator to the function K while holding the variable (α) constant.

The complex conjugate is represented by the overbar, e.g., \(\bar \zeta\). When a complex variable, \(\tilde \zeta\), is close to the complex conjugate of ζ, but independent, we use \(\tilde \zeta \approx \bar \zeta\).

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 ‘massdipole plus “i” angular momentum’ (or ‘real electricdipole plus “i” 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, \({u_{\rm{B}}} = G(\tau, \zeta, \bar \zeta)\), referred to as GoodCut Functions, (GCFs) that are closely associated with shearfree NGCs. The details are given later. For any given asymptoticallyvanishing 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) UniversalCut Function (UCF). For vacuum asymptoticallyflat spacetimes, the Weyl tensor allows the choice of a unique member of the class referred to as the (gravitational) UCF. For EinsteinMaxwell 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 spins harmonics, as, e.g.,where the indices, i, j,k … 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$$\chi = {\chi ^0}{Y_0} + {\chi ^i}{Y_{1i}}(\zeta ,\bar \zeta) + {\chi ^{ij}}{Y_{2ij}}(\zeta ,\bar \zeta) + {\chi ^{ijk}}{Y_{3ijk}}(\zeta ,\bar \zeta) + \ldots ,$$This, though easy to understand and keep track of, does run into the unpleasant fact that often the fourvector,$$\begin{array}{*{20}c} {\vec \eta \cdot \vec \lambda \equiv {\eta ^i}{\lambda ^i} \equiv {\eta ^i}{\lambda _i},\quad \quad \quad \quad} \\ {{\mu ^k} = {{(\vec \eta \times \vec \lambda)}^k} \equiv {\eta ^i}{\lambda ^j}{\epsilon _{ijk}}.} \\ \end{array}$$appears as the l = 0, 1 harmonics in the harmonic expansions. Thus, care must be used when lowering or raising the relativistic index, i.e., η_{ ab }χ^{ a } = χ_{ b } = (x^{0}, − χ^{ i }).$${\chi ^a} = ({\chi ^0},{\chi ^i}),$$
 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 NewmanPenrose formalism. Although the identifications we make are the most natural in our framework, a numerical rescaling 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 byThe conventions used here were chosen so that the numerical coefficient of \(Q^{ij}_{\mathbb{C}}\) in \(\phi^{0}_{0}\) was equal to one; this rescaling 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.$$D_{\mathbb C}^i = D_{\mathbb C}^{i\,{\rm{physical}}},\qquad Q_{\mathbb C}^{ij} = {{\sqrt 2} \over 4}Q_{\mathbb C}^{ij\,{\rm{physical}}}.$$
1.2 Glossary of symbols and units
Glossary
Symbol/Acronym  Definition 

ℑ^{+}, \({\mathfrak{I}}_{\mathbb{C}}^{+}\)  Future null infinity, Complex future null infinity 
I^{+}, I^{−}, I^{0}  Future, Past timelike infinity, Spacelike infinity 
\(\mathbb{M},\;\mathbb{M}_{\mathbb{C}}\)  Minkowski space, Complex Minkowski space 
u_{B}, u_{ret}  Bondi time coordinate, Retarded Bondi time (\(\sqrt 2 {u_{\rm{B}}} = {u_{{\rm{ret}}}}\)) 
\({\partial _{{u_{\rm{B}}}}}f = \dot {f}\)  Derivation with respect to u_{B} 
\(\partial _{u_{\rm{ret}}}f = f{\prime}\)  Derivation with respect to u_{ret} 
r  Affine parameter along null geodesics 
(\(\zeta, \bar {\zeta}\))  (e^{ iϕ } cot(θ/2), e^{−iϕ} cot(θ/2)); stereographic coordinates on S^{2} 
\(Y_{li.j}^s(\zeta, \bar {\zeta})\)  Tensorial spins spherical harmonics 
ð, \(\bar{\eth}\)  \({P^{1  s}}{\partial \over {\partial \zeta}}{P^s},\;{P^{1 + s}}{\partial \over {\partial \bar \zeta}}{P^{ s}}\); spinweighted operator on the twosphere 
P  Metric function on S^{2}; often \(P = P_{0} \equiv 1 + \zeta \bar {\zeta}\) 
ð_{(α)}f  Application of ðoperator to f while the variable α is held constant 
\(\{{l^a},{n^a},{m^a},{\bar m^a}\}\)  Null tetrad system; \({l^a}{n_a} =  {m^a}{\bar m_a} = 1\) 
NGC  Null Geodesic Congruence 
NP/SC  NewmanPenrose/SpinCoefficient Formalism 
{U, X^{ A }, ω, ξ^{ A }}  Metric coefficients in the NewmanPenrose formalism 
{ψ_{0}, ψ_{1}, ψ_{2}, ψ_{3}, ψ_{4}}  Weyl tensor components in the NewmanPenrose formalism 
{ϕ_{0}, ϕ_{1}, ϕ_{2}}  Maxwell tensor components in the NewmanPenrose formalism 
ρ  Complex divergence of a null geodesic congruence 
∑  Twist of a null geodesic congruence 
σ, σ^{0}  Complex shear, Asymptotic complex shear of a NGC 
k  \({{2G} \over {{c^4}}}\); Gravitational constant 
\(\tau = s + i\lambda = T(u,\zeta, \bar {\zeta})\)  Complex auxiliary (CR) potential function 
δ_{ τ }f = f′  Derivation with respect to τ 
\(\eth_{(\tau)}^{2}G(\tau, \zeta, \bar{\zeta})=\sigma^{0}(G,\zeta,\bar{\zeta})\)  GoodCut Equation, describing asymptotically shearfree NGCs 
\({u_{\rm{B}}} = G(\tau, \zeta, \bar {\zeta})\)  GoodCut Function (GCF) on ℑ^{+} 
\(L(u_{\rm{B}},\zeta,\bar{\zeta})=\eth_{(\tau)}G\)  Stereographic angle field for an asymptotically shearfree NGC at ℑ^{+} 
\(\eth_{(u_{\rm{B}})}T+L\dot{T}=0\)  CR equation, describing the embedding of ℑ^{+} into ℂ^{2} 
\({\mathcal H}\)space  Complex fourdimensional solution space to the GoodCut Equation 
\(D_{\mathbb C}^i = D_E^i + iD_M^i = {1 \over 2}\phi _0^{0i}\)  Complex electromagnetic dipole 
η^{ a }(u_{ ret })  Complex centerofcharge world line, lives in \({\mathcal H}\)space 
\(Q_{\mathbb C}^{ij} = Q_E^{ij} + iQ_M^{ij} = {{\sqrt 2} \over 4}Q_{\mathbb C}^{ij\; {\rm{physical}}}\)  Complex electromagnetic quadrupole 
ξ^{ a }(u_{ ret })  Complex center of mass world line, lives in \({\mathcal H}\)space 
\(D_{({\rm{grav}})}^i = D_{({\rm{mass}})}^i + i{c^{ 1}}{J^i}{\rm{}} =  {{{c^2}} \over {6\sqrt 2 G}}\psi _1^{0i}\)  Complex gravitational dipole 
\(Q_{{\rm{Grav}}}^{ij} = Q_{{\rm{Mass}}}^{ij} + iQ_{{\rm{Spin}}}^{ij}\)  Complex gravitational quadrupole 
\({u_{\rm{B}}} = X(\tau, \zeta, \bar {\zeta})\)  Universal Cut Function (UCF) corresponding to the complex center of mass world line 
\({\xi ^{ij}} = {{\sqrt 2 G} \over {24{c^4}}}Q_{{\rm{Grav}}}^{ij}{\prime \prime}\)  Identification between l = 2 coefficient of the UCF and gravitational quadrupole 
\(\Psi \equiv \psi _2^0 + {\eth^2}\overline {{\sigma ^0}} + {\sigma ^0}\overset \cdot {\overline {{\sigma ^0}}} = \bar \Psi\)  Bondi Mass Aspect 
\({M_{\rm{B}}} =  {{{c^2}} \over {2\sqrt 2 G}}{\Psi ^0}\)  Bondi mass 
\({P^i} =  {{{c^3}} \over {6G}}{\Psi ^i}\)  Bondi linear threemomentum 
\({J^i} =  {{\sqrt 2 {c^3}} \over {12G}}{\rm{Im}}(\psi _1^{0i})\)  Vacuum linear theory identification of angular momentum 
Units
Quantity  Units 

[G]  L^{3}M^{−1}T^{−2} 
[q]  \({{\rm{M}}^{{1 \over 2}}}{{\rm{L}}^{{3 \over 2}}}{{\rm{T}}^{ 1}}\) 
[k] = [Gc^{−4}]  M^{−1}L^{−1}T^{2} 
\([c\tau ] = [c{u_{\rm{B}}}] = [G(\tau, \zeta, \bar \zeta)]\)  L 
[ξ^{ i }(τ)] = [η^{ i }(τ)] = [ξ^{ ij }(τ)]  L 
\([D^{i}_{\mathbb{C}}]\)  \({{\rm{M}}^{{1 \over 2}}}{{\rm{L}}^{{5 \over 2}}}{{\rm{T}}^{ 1}}\) 
\([D^{i}_{(\rm{grav})}]\)  ML 
\([Q^{ij}_{\mathbb{C}}]\)  \(\rm{M}^{\frac{1}{2}}\rm{L}^{\frac{7}{2}}\rm{T}^{1}\) 
\([Q^{ij}_{\rm{Grav}}]\)  ML^{2} 
[J^{ i }]  ML^{2}T^{−1} 
[ϕ_{0}] = [ϕ_{1}] = [ϕ_{2}]  \(\rm{M}^{\frac{1}{2}}\rm{L}^{\frac{1}{2}}\rm{T}^{1}\) 
\([\phi^{0}_{0}]\)  \(\rm{M}^{\frac{1}{2}}\rm{L}^{\frac{5}{2}}\rm{T}^{1}\) 
\([\phi^{0}_{1}]\)  \(\rm{M}^{\frac{1}{2}}\rm{L}^{\frac{3}{2}}\rm{T}^{1}\) 
\([\phi^{0}_{2}]\)  M^{½}L^{½}T^{−1} 
[ψ_{0}] = [ψ_{1}] = [ψ_{2}] = [ψ_{3}] = [ψ_{4}]  L^{−2} 
\([\psi^{0}_{0}]\)  L^{3} 
\([\psi^{0}_{1}]\)  L^{2} 
\([\psi^{0}_{2}]\)  L 
\([\psi^{0}_{3}]\)  1 
\([\psi^{0}_{4}]\)  L^{−1} 
2 Foundations
In this section, we review several of the key ideas and tools that are indispensable in our later discussions. We keep our explanations as concise as possible, and refrain from extensive proofs of any propositions. The reader will be directed to the appropriate references for the details. In large part, much of what is covered in this section should be familiar to many workers in GR.
2.1 Asymptotic flatness and ℑ^{+}
Ever since the work of Bondi [16] illustrated the importance of null hypersurfaces in the study of outgoing gravitational radiation, the study of asymptoticallyflat spacetimes has been one of the more important research topics in GR. Qualitatively speaking, a spacetime can be thought of as (future) asymptotically flat if the curvature tensor vanishes at an appropriate rate as infinity is approached along the futuredirected null geodesics of the null hypersurfaces. The type of physical situation we have in mind is an arbitrary compact gravitating source (perhaps with an electric charge and current distribution), with the associated gravitational (and electromagnetic) field. The task is to gain information about the interior of the spacetime from the study of farfield features, multipole moments, gravitational and electromagnetic radiation, etc. [60]. The arena for this study is on what is referred to as future null infinity, ℑ^{+}, the future boundary of the spacetime. The intuitive picture of this boundary is the set of all endpoints of futuredirected null geodesics.
A precise definition of null asymptotic flatness and the boundary was given by Penrose [62, 63], whose basic idea was to rescale the spacetime metric by a conformal factor, which approaches zero asymptotically: the zero value defining future null infinity. This process leads to the boundary being a null hypersurface for the conformallyrescaled metric. When this boundary can be attached to the interior of the rescaled manifold in a regular way, then the spacetime is said to be asymptotically flat.
 (A):
For both the asymptoticallyflat vacuum Einstein equations and the EinsteinMaxwell equations, ℑ^{+} is a null hypersurface of the conformally rescaled metric.
 (B):
ℑ^{+} is topologically S^{2} × ℝ
 (C):
The Weyl tensor \(C_{bcd}^a\) vanishes at ℑ^{+}, with the peeling theorem describing the speed of its falloff (see below).
Property (B) allows an easy visualization of the boundary, ℑ^{+}, as the past light cone of the point I^{+}, future timelike infinity. As mentioned earlier, ℑ^{+} will be the stage for our study of asymptotically shearfree NGCs.
2.2 Bondi coordinates and null tetrad
Proceeding with our examination of the properties of ℑ^{+}, we introduce, in the neighborhood of ℑ^{+}, what is known as a Bondi coordinate system: \({u_{\rm{B}}},r,\zeta, \bar \zeta\). In this system, u_{B}, the Bondi time, labels the null surfaces, r is the affine parameter along the null geodesics of the constant u_{B} surfaces and ζ = e^{ iϕ } cot(θ/2) the complex stereographic coordinate labeling the null geodesics of ℑ^{+}. To reach ℑ^{+}, we simply let r → ∞, so that ℑ^{+} has coordinates \({u_{\rm{B}}}, \zeta, \bar \zeta\) The time coordinate u_{B}, the topologically ℝ portion of ℑ^{+}, labels ‘cuts’ of ℑ^{+}. The stereographic coordinate ζ accounts for the topological generators of the S^{2} portion of ℑ^{+}, i.e., the null generators of ℑ^{+}. The choice of a Bondi coordinate system is not unique, there being a variety of Bondi coordinate systems to choose from. The coordinate transformations between any two, known as BondiMetznerSachs (BMS) transformations or as the BMS group, are discussed later in this section.
There remains the issue of both coordinate and tetrad freedom, i.e., local Lorentz transformations. Most of the time we work in one arbitrary but fixed Bondi coordinate system, though for special situations more general coordinate systems are used. The more general transformations are given, essentially, by choosing an arbitrary slicing of ℑ^{+}, written as \({u_{\rm{B}}} = G(s,\zeta, \bar \zeta)\) with s labeling the slices. To keep conventional coordinate conditions unchanged requires a rescaling of r : r → r^{′} = (∂_{ s }G)^{−1}r. It is also useful to be able to shift the origin of r by \({r\prime} = r  {r_0}({u_{\rm{B}}},\zeta, \bar \zeta)\) with arbitrary \({r_0}({u_{\rm{B}}},\zeta, \bar \zeta)\).
Eventually, by the appropriate choice of the function \(L({u_{\rm{B}}},\zeta, \bar \zeta)\) the new null vector, l*^{ a }, can be made into the tangent vector of an asymptotically shearfree NGC.
An example would be to take a vector on ℑ^{+}, say η^{ a }, and form the spinweightone quantity, η_{(1)} = η^{ a }m_{ a }.
Comment: For later use we note that \(L({u_{\rm{B}}},\zeta, \bar \zeta)\) has spin weight, s = 1.
For each s, spins functions can be expanded in a complete basis set, the spins harmonics, \(_s{Y_{lm}}(\zeta, \bar \zeta)\) or spins tensor harmonics, \(Y_{l\,\,i \ldots j}^{(s)}(\zeta, \bar \zeta) \Leftrightarrow_s{Y_{lm}}(\zeta, \bar \zeta)\) (cf. Appendix C).
2.3 The optical equations
Since this work concerns NGCs and, in particular, shearfree and asymptotically shearfree NGCs, it is necessary to first define them and then study their properties.
2.4 The NewmanPenrose formalism
Though the NP formalism is the basic working tool for our analysis, this is not the appropriate venue for its detailed exposition. Instead we will simply give an outline of the basic ideas followed by the results found, from the application of the NP equations, to the problem of asymptoticallyflat spacetimes.
Remark 2. We mention that much of the physical content and interpretations in the present work comes from the study of the lowest spherical harmonic coefficients in the leading terms of the farfield expansions of the Weyl and Maxwell tensors.
The NP version of the vacuum (or EinsteinMaxwell) equations consists of three sets (or four sets) of nonlinear firstorder coupled partial differential equations for the variables: the tetrad components, the spin coefficients, the Weyl tensor (and Maxwell field when present). Though there is no hope that they can be solved in any general sense, many exact solutions have been found from them. Of far more importance, large classes of asymptotic solutions and perturbation solutions can be found. Our interest lies in the asymptotic behavior of the asymptoticallyflat solutions. Though there are some subtle issues, integration in this class is not difficult [55, 61]. With no explanation of the integration process, except to mention that we use the Bondi coordinate and tetrad system of Eqs. (2.3), (2.5), and (2.7) and asymptotic flatness (ψ_{0} ∼ O(r^{−5}) and certain uniform smoothness conditions on sideways derivatives), we simply give the final results.
 The Weyl tensor:$$\begin{array}{*{20}c} {{\psi _0} = \psi _0^0{r^{ 5}} + O({r^{ 6}}),} \\ {{\psi _1} = \psi _1^0{r^{ 4}} + O({r^{ 5}}),} \\ {{\psi _2} = \psi _2^0{r^{ 3}} + O({r^{ 4}}),} \\ {{\psi _3} = \psi _3^0{r^{ 2}} + O({r^{ 3}}),} \\ {{\psi _4} = \psi _4^0{r^{ 1}} + O({r^{ 2}}).} \\ \end{array}$$(2.36)
 The Maxwell tensor:$$\begin{array}{*{20}c} {{\phi _0} = \phi _0^0{r^{ 3}} + O({r^{ 4}}),} \\ {{\phi _1} = \phi _1^0{r^{ 2}} + O({r^{ 3}}),} \\ {{\phi _2} = \phi _2^0{r^{ 1}} + O({r^{ 2}}).} \\ \end{array}$$(2.37)
 The spin coefficients and metric variables:$$\begin{array}{*{20}c} {\kappa = \pi = \epsilon = 0,\qquad \tau = \bar \alpha + \beta ,\quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\rho = \bar \rho =  {r^{ 1}}  {\sigma ^0}{{\bar \sigma}^0}{r^{ 3}} + O({r^{ 5}}),\quad \quad \quad \quad \quad \quad \quad \quad} \\ {\sigma = {\sigma ^0}{r^{ 2}} + \left({{{({\sigma ^0})}^2}{{\bar \sigma}^0}  \psi _0^0/2} \right){r^{ 4}} + O({r^{ 5}}),\quad \quad \quad \quad} \\ {\alpha = {\alpha ^0}{r^{ 1}} + O({r^{ 2}}),\qquad \beta = {\beta ^0}{r^{ 1}} + O({r^{ 2}}),\quad \quad \quad \quad} \\ {\gamma = {\gamma ^0}  \psi _2^0{{(2{r^2})}^{ 1}} + O({r^{ 3}}),\qquad \lambda = {\lambda ^0}{r^{ 1}} + O({r^{ 2}}),\quad} \\ {\mu = {\mu ^0}{r^{ 1}} + O({r^{ 2}}),\qquad \nu = {\nu ^0} + O({r^{ 1}}),\quad \quad \quad \quad \quad \quad} \\ \end{array}$$(2.38)$$\begin{array}{*{20}c} {A = \zeta \,{\rm{or}}\,\bar \zeta ,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{\xi ^A} = {\xi ^{0A}}{r^{ 1}}  {\sigma ^0}{{\bar \xi}^{0A}}{r^{ 2}} + {\sigma ^0}{{\bar \sigma}^0}{\xi ^{0A}}{r^{ 3}} + O({r^{ 4}}),} \\ {\omega = {\omega ^0}{r^{ 1}}  ({\sigma ^0}{{\bar \omega}^0} + \psi _1^0/2){r^{ 2}} + O({r^{ 3}}),\quad \quad} \\ {{X^A} = (\psi _1^0{{\bar \xi}^{0A}} + \bar \psi _1^0{\xi ^{0A}}){{(6{r^3})}^{ 1}} + O({r^{ 4}}),\quad \quad \quad \quad \quad} \\ {U = {U^0}  ({\gamma ^0} + {{\bar \gamma}^0})r  (\psi _2^0 + \bar \psi _2^0){{(2r)}^{ 1}} + O({r^{ 2}}).} \\ \end{array}$$
 The functions of integration are determined, using coordinate conditions, as:$${\xi ^{0\zeta}} =  P,\quad {\bar \xi ^{0\zeta}} = 0,$$(2.39)$${\xi ^{0\bar \zeta}} = 0,\quad {\bar \xi ^{0\bar \zeta}} =  P,$$(2.40)$$P = 1 + \zeta \bar \zeta ,$$(2.41)$${\alpha ^0} =  {\bar \beta ^0} =  {\zeta \over 2},$$(2.42)$${\gamma ^0} = {\nu ^0} = 0,$$(2.43)$${\omega ^0} =  \bar \eth {\sigma ^0},$$(2.44)$${\lambda ^0} = {\dot \bar \sigma ^0},$$(2.45)$${\mu ^0} = {U^0} =  1,$$(2.46)$$\psi _4^0 =  {\ddot \bar \sigma ^0},$$(2.47)$$\psi _3^0 = \eth \dot {\bar {\sigma}}^{0},$$(2.48)$$\psi _2^0  \bar \psi _2^0 = {\bar \eth^2}{\sigma ^0}  {\eth^2}{\bar \sigma ^0} + {\bar \sigma ^0}{\lambda ^0}  {\sigma ^0}{\bar \lambda ^0}.$$(2.49)
 The mass aspect,satisfies the physically very important reality condition:$$\Psi \equiv \psi _2^0 + {\eth^2}{\bar \sigma ^0} + {\sigma ^0} \dot {\bar {\sigma}}^{0},$$(2.50)$$\Psi = \bar \Psi .$$(2.51)
 Finally, from the asymptotic Bianchi identities, we obtain the dynamical (or evolution) relations:$$\dot \psi _2^0 =  \eth \psi _3^0 + {\sigma ^0}\psi _4^0 + k\phi _2^0\bar \phi _2^0,$$(2.52)$$\dot \psi _1^0 =  \eth \psi _2^0 + 2{\sigma ^0}\psi _3^0 + 2k\phi _1^0\bar \phi _2^0,$$(2.53)$$\dot \psi _0^0 =  \eth \psi _1^0 + 3{\sigma ^0}\psi _2^0 + 3k\phi _0^0\bar \phi _2^0,$$(2.54)$$\dot \phi _1^0 =  \eth \phi _2^0,$$(2.55)$$\dot \phi _0^0 =  \eth \phi _1^0 + {\sigma ^0}\phi _2^0;$$(2.56)$$k = 2G{c^{ 4}}.$$(2.57)
Remark 3. These last five equations, the first of which contains the beautiful Bondi energymomentum loss theorem, play the fundamental role in the dynamics of our physical quantities.
From these results, the characteristic initial problem can roughly be stated in the following manner. At u_{B} = u_{B0} we choose the initial values for \(\psi _0^0,\, \psi _1^0,\, \psi _2^0\), i.e., functions only of \(\zeta, \bar \zeta\). The characteristic data, the complex Bondi shear, \({\sigma ^0}({u_{\rm{B}}},\,\zeta, \bar \zeta)\), is then freely chosen. Since \(\psi _3^0\) and \(\psi _4^0\) are functions of σ^{0}, Eqs. (2.45), (2.47) and (2.48) and its derivatives, all the asymptotic variables can now be determined from Eqs. (2.52)–(2.56).
An important consequence of the NP formalism is that it allows simple proofs for many geometric theorems. Two important examples are the GoldbergSachs theorem [29] and the peeling theorem [73]. The peeling theorem is essentially given by the asymptotic behavior of the Weyl tensor in Eq. (2.36) (and Eq. (2.37)). The GoldbergSachs theorem is discussed in some detail in Section 2.6. Both theorems are implicitly used later.
In our case, where we consider only quadrupole gravitational radiation, the quadratic correction terms do in fact vanish and hence Eq. (2.62), modified by the Maxwell terms, is correct as it is stated.
2.5 The BondiMetznerSachs group
The group of coordinate transformations at ℑ^{+} that preserves the Bondi coordinate conditions, the BMS group, is the same as the asymptotic symmetry group that arises from approximate solutions to Killing’s equation as ℑ^{+} is approached. The BMS group has two parts: the homogeneous Lorentz group and the supertranslation group, which contains the Poincaré translation subgroup. Their importance to us lies in the fact that all the physical quantities arising from our identifications must transform appropriately under these transformations [65, 42].
2.6 Algebraicallyspecial metrics and the GoldbergSachs theorem
Theorem (GoldbergSachs). For a nonflat vacuum spacetime, if there is an NGC that is shearfree, i.e., there is a null vector field with (κ = 0, σ = 0), then the spacetime is algebraically special and, conversely, if a vacuum spacetime is algebraically special, there is an NGC with (κ = 0, σ = 0).
In particular, this means that for all algebraically special metrics there is an everywhere shearfree NGC, and a null tetrad exists such that ψ_{0} = ψ_{1} = 0. The main idea of this review is an asymptotic generalization of this statement: for all asymptotically flat metrics, there exists a null tetrad such that the l = 0 and l = 1 harmonic coefficients of the asymptotic Weyl tensor components ψ_{0} and ψ_{1} (namely, \(\psi^{0i}_{0}\) and \(\psi^{0i}_{1}\)) vanish. Note that this is in reality a nontrivial condition only on \(\psi^{0i}_{1}\), since the other three components vanish automatically when we recall that ψ_{0} and ψ_{1} are spinweight two and one respectively.
3 ShearFree NGCs in Minkowski Space
The structure and properties of asymptotically shearfree NGCs (our main topic) are best understood by first looking at the special case of congruences that are shearfree everywhere (except at their caustics). Though shearfree congruences are also found in algebraicallyspecial spacetimes, in this section only the shearfree NGCs in Minkowski spacetime, \(\mathbb{M}\), are discussed [7]
3.1 The flatspace goodcut equation and goodcut functions
In Section 2, we saw that in the NP formalism, two of the complex spin coefficients, the optical parameters ρ and σ of Eqs. (2.23) and (2.24), play a particularly important role in their description of an NGC; namely, they carry the information of the divergence, twist and shear of the congruence.
There are several important comments to be made about Eq. (3.2). The first is that there is a simple geometric meaning to the parameters (\({u_{\rm{B}}},\zeta, \bar {\zeta}\)): they are the values of the Bondi coordinates of ℑ^{+}, where each geodesic of the congruence intersects ℑ^{+}. The second concerns the geometric meaning of L. At each point of ℑ^{+}, consider the past light cone and its sphere of null directions. Coordinatize that sphere (of null directions) with stereographic coordinates. The function \(L({u_{\rm{B}}},\zeta, \bar {\zeta})\) is the stereographic angle field on ℑ^{+} that describes the null direction of each geodesic intersecting ℑ^{+} at the point (\({u_{\rm{B}}},\zeta, \bar {\zeta}\)). The values L = 0 and L = ∞ represent, respectively, the direction along the Bondi l^{ a } and n^{ a } vectors. This stereographic angle field completely determines the NGC.
Thus, we see that all information about the NGC can be obtained from the cut function \(G(\tau, \zeta, \bar {\zeta})\).
Thus, we have our first major result: every regular shearfree NGC in Minkowski space is generated by the arbitrary choice of a complex world line in what turns out to be complex Minkowski space. See Eq. (2.66) for the connection between the l = (0, 1) harmonics in Eq. (3.15) and the Poincare translations. We see in the next Section 4 how this result generalizes to regular asymptotically shearfree NGCs.
3.2 Real cuts from the complex good cuts, I
Though our discussion of shearfree NGCs has relied, in an essential manner, on the use of the complexification of ℑ^{+} and the complex world lines in complex Minkowski space, it is the real structures that are of main interest to us. We want to find the intersection of the complex GCF with real ℑ^{+}, i.e., what are the real points and real cuts of \({u_{\rm{B}}} = G(\tau, \zeta, \bar {\zeta}),\,\,(\tilde \zeta = \bar \zeta)\), and what are the values of τ that yield real u_{B}. These reality structures were first observed in [7] and recently there have been attempts to study them in the framework of holographic dualities (cf. [8] and Section 8).
Continuing, with small values for the imaginary part of \(\xi^{a}(\tau)=\xi_{R}^{a}(\tau)+i\xi_{I}^{a}(\tau)\), (\(\xi_{R}^{a}(\tau),\ \xi_{I}^{a}(\tau)\) both real analytic functions) and hence small \(\Lambda (s,\zeta, \bar {\zeta})\), it is easy to see that \(\Lambda (s,\zeta, \bar {\zeta})\) (for fixed value of s) is a bounded smooth function on the (\(\zeta, \bar {\zeta}\)) sphere, with maximum and minimum values, \({\lambda _{\max}} = \Lambda (s,{\zeta _{\max}},{\bar \zeta _{\max}})\) and \({\lambda _{\min}} = \Lambda (s,{\zeta _{\min}},{\bar \zeta _{\min}})\). Furthermore on the (\(\zeta, \bar {\zeta}\)) sphere, there are a finite linesegments worth of curves (circles) that lie between (\(\zeta_{\min},\overline{\zeta}_{\min}\)) and (\({\zeta _{\max}},{\bar \zeta _{\max}}\)) such that \(\Lambda (s,\zeta, \bar {\zeta})\) is a monotonically increasing function on the family of curves. Hence there will be a family of circles on the (\(\zeta, \bar {\zeta}\))sphere where the value of λ is a constant, ranging between λ_{max} and λ_{min}.
Summarizing, we have the result that in the complex τplane there is a ribbon or strip given by all values of s and line segments parametrized by λ between λ_{min} and λ_{max} such that the complex lightcones from each of the associated points, ξ^{ a }(s + iλ), all have some null geodesics that intersect real ℑ^{+}. More specifically, for each of the allowed values of τ = s + iλ there will be a circle’s worth of complex null geodesics leaving the point ξ^{ a }(s + iλ), reaching real ℑ^{+}. It is the union of these null geodesics, corresponding to the circles on the (\(\zeta, \bar {\zeta}\))sphere from the line segment, that produces the real family of cuts, Eq. (3.26).
The real structure associated with a complex world line is then this oneparameter family of slices (cuts) Eq. (3.26).
It is the complex point of view of the complex lightcones coming from the complex world line that dominates our discussion.
3.3 Approximations
Due to the difficulties involved in the intrinsic nonlinearities and the virtual impossibility of exactly inverting arbitrary analytic functions, it often becomes necessary to resort to approximations. The basic approximation will be to consider the complex world line ξ^{ a }(τ) as being close to the straight line, \(\xi _0^a(\tau) = \tau \delta _0^a\); deviations from this will be considered as first order. We retain terms up to second order, i.e., quadratic terms. Another frequently used approximation is to terminate spherical harmonic expansions after the l = 2 terms.
3.4 Asymptoticallyvanishing Maxwell fields
3.4.1 A prelude
The basic starting idea in this work is simple. It is in the generalizations and implementations where difficulties arise.
We emphasize that this is done in a fixed Lorentz frame and only the origin is moved. In different Lorentz frames there will be different complex centers of charge.
Later, directly from the general asymptotic Maxwell field itself (satisfying the Maxwell equations), we define the asymptotic complex dipole moment and give its transformation law, including transformations between Lorentz frames. This yields a unique complex center of charge independent of the Lorentz frame.
3.4.2 Asymptoticallyvanishing Maxwell fields: General properties
In this section, we describe how a complex center of charge for asymptotically vanishing Maxwell fields in flat spacetime can be found by using the shearfree NGCs, constructed from solutions of the homogeneous goodcut equation, to transform certain Maxwell field components to zero. Although this serves as a good example for our later methods in asymptotically flat spacetimes, the reader may wish to skip ahead to Section 4, where we go directly to gravitational fields in a setting of greater generality.
Later in this section it will be shown that we can find a unique complex world line, ξ^{ a }(τ) = (ξ^{0}, ξ^{ i }), (the world line associated with a shearfree NGC), that is closely related to the η^{ i }(u_{ret}) From this complex world line we can define the intrinsic complex dipole moment, \(D_{\mathcal{I}\mathbb{C}}^i = q{\xi ^i}(s)\).
However, we first discuss a particular Maxwell field, F^{ ab }, where one of its eigenvectors is a tangent field to a shearfree NGC. This solution, referred to as the complex LiénardWiechert field is the direct generalization of the ordinary LiénardWiechert field. Though it is a real solution in Minkowski space, it can be thought of as arising from a complex world line in complex Minkowski space.
3.4.3 A coordinate and tetrad system attached to a shearfree NGC
The use and insight given by this coordinate/tetrad system is illustrated by its application to a special class of Maxwell fields. We consider, as mentioned earlier, the Maxwell field where one of its principle null vectors, l*^{ a }, (an eigenvector of the Maxwell tensor, \({F_{ab}}{l^{{\ast}a}} = \lambda l_b^{\ast}\)), is a tangent vector of a shearfree NGC. Thus, it depends on the choice of the complex world line and is therefore referred to as the complex LiénardWiechert field. (If the world line was real it would lead to the ordinary LiénardWiechert field.) We emphasize that though the source can formally be thought of as a charge moving on the complex world line, the Maxwell field is a real field on real Minkowski space. It will have a real (distributional) source at the caustics of the congruence. Physically, its behavior is very similar to real LiénardWiechert fields, the essential difference is that the electric dipole is now replaced by the combined electric and magnetic dipoles. The imaginary part of the world line determines the magnetic dipole moment.
3.4.4 Complex LiénardWiechert Maxwell field
The present section, included as an illustration of the general ideas and constructions in this work, is rather technical and complicated and can be omitted without loss of continuity.
Remark 8. In this case of the complex LiénardWiechert Maxwell field, the ξ^{ a } determines the intrinsic centerofcharge world line, rather than the relative centerofcharge line.
These remaining equations depend only on \(L({u_{\rm{B}}},\zeta, \bar {\zeta})\), which, in turn, is determined by ξ^{ a }(τ). In other words, the solution is driven by the complex line, ξ^{ a }(τ). As they now stand, Eqs. (3.61) appear to be difficult to solve, partially due to the implicit description of the \(L({u_{\rm{B}}},\zeta, \bar {\zeta})\).
Though we now have the exact solution, unfortunately it is in complex coordinates where virtually every term depends on the complex variable τ, via ξ^{ a }(τ). This is a severe impediment to a full description and understanding of the solution in the real Minkowski space.
This example was intended to show how physical meaning could be attached to the complex world line associated with a shearfree NGC. In this case and later in the case of asymptoticallyflat spacetimes, when the GCF is singled out by either the Maxwell field or the gravitational field, it will be referred to it as a UCF. For either of the two cases, a flatspace asymptoticallyvanishing Maxwell field (with nonvanishing total charge) and for a vacuum asymptoticallyflat spacetime, there will be a unique UCF. In the case of the EinsteinMaxwell fields there will, in general, be two UCFs: one for each field.
3.4.5 Asymptotically vanishing Maxwell fields & shearfree NGCs
3.4.5.1 The (non)uniqueness of spherical harmonic expansions
An important observation, obvious but easily overlooked, concerning the spherical harmonic expansions is that, in a certain sense, they lack uniqueness. As this issue is significant, its clarification is important.
Though it is clear that extracting \(\phi _{0i}^{0 {\ast}}(\tau)\) with this relationship is available in principle, in practice it is impossible to do it exactly and all examples are done with approximations: essentially using slow motion for the complex world line.
Remark 9. If by some accident the Maxwell field was a complex LiénardWiechert field, a world line ξ^{ a }(τ) could be chosen so that from the associated complex null cones we would have \(\phi_{0}^{\ast 0}=0\). However, though this cannot be done in general, the l = 1 harmonics of \(\phi _0^{\ast 0}\) can be made to vanish by the appropriate choice of the ξ^{ a }(τ). This is the means by which a unique world line is chosen.
3.4.6 The complex center of charge
In Section 5, these ideas are applied to GR, with the complex electric and magnetic dipoles being replaced by the complex combination of the mass dipole and the angular momentum.
4 The GoodCut Equation and \({\mathcal H}\)Space
In Section 3, we discussed NGCs in Minkowski spacetime that were shearfree. In this section we consider asymptotically shearfree NGCs in asymptoticallyflat spacetimes. That is to say, we consider NGCs that have nonvanishing shear in the interior of the spacetime but where, as null infinity is approached, the shear vanishes. Whereas fully shearfree NGCs almost never occur in general asymptotically flat spacetimes, asymptotically shearfree congruences always exist. The case of algebraicallyspecial spacetimes is the exception; they do allow one or two shearfree congruences.
We begin by reviewing the shearfree condition and follow with its generalization to the asymptotically shearfree case. From this we derive the generalization of the homogeneous goodcut equation to the inhomogeneous goodcut equation. Almost all the properties of the shearfree and asymptotically shearfree NGCs come from the study of these equations and virtually all the attributes of shearfree congruences are shared by the asymptotically shearfree congruences. It is from the use of these shared attributes that we will be able to extract physical identifications and information (e.g., complex center of mass/charge, Bondi mass, linear and angular momentum, equations of motion, etc.) from the asymptotic gravitational fields.
Though again the use of the formal complexification of ℑ^{+}, i.e., \(\mathfrak{I}_{\mathbb{C}}^{+}\), is essential for our analysis, it is the extraction of the real structures that is important.
4.1 Asymptotically shearfree NGCs and the goodcut equation
It is this pair of equations, (4.1) and (4.2), that will now be generalized to asymptoticallyflat spacetimes.
4.2 \({\mathcal H}\)space and the goodcut equation
Later in this section, by choosing an arbitrary complex analytic world line in \({\mathcal H}\)space, z^{ a } = ξ^{ a }(τ), we describe how to construct the shearfree angle field, \(L({u_{\rm{B}}},\zeta, \tilde \zeta)\). First, however, we discuss properties and the origin of Eq. (4.18).
We note that using this form of the solution implies that we have set stringent coordinate conditions on the \({\mathcal H}\)space by requiring that the first four spherical harmonic coefficients be the four \({\mathcal H}\)space coordinates. Arbitrary coordinates would just mean that these four coefficients were arbitrary functions of other coordinates. How these special coordinates change under the BMS group is discussed later.
4.2.1 Solutions to the shearfree equation
A Brief Summary: The description and analysis of the asymptotically shearfree NGCs in asymptoticallyflat spacetimes is remarkably similar to that of the flatspace regular shearfree NGCs. We have seen that all regular shearfree NGCs in Minkowski space and asymptoticallyflat spaces are generated by solutions to the goodcut equation, with each solution determined by the choice of an arbitrary complex analytic world line in complex Minkowski space or \({\mathcal H}\)space. The basic governing variables are the complex GCF, \({u_{\rm{B}}} = G(\tau, \zeta, \tilde \zeta)\), and the stereographic angle field on \(\mathfrak{I}_{\mathbb{C}}^ +, \, L({u_{\rm{B}}},\, \zeta,\, \tilde \zeta)\), restricted to real ℑ^{+}. In every sense, the flatspace case can be considered as a special case of the asymptoticallyflat case.
4.3 Real cuts from the complex good cuts, II
As varies we obtain a oneparameter family of cuts. If these cuts do not intersect with each other we say that the complex world line ξ^{ a }(τ) is by definition ‘timelike.’ This occurs when the time component of the real part of the complex velocity vector, υ^{ a }(τ) = dξ^{ a }(τ)/dτ, is sufficiently large.
4.4 Summary of Real Structures

In Minkowski space, the future directed lightcones emanating from a real timelike world line, x^{ a } = ξ^{ a }(s), intersect future null infinity, ℑ^{+}, on a oneparameter family of spherical nonintersecting cuts.

The complex lightcones emanating from a timelike complex analytic curve in complex Minkowski space, z^{ a } = ξ^{ a }(τ) parametrized by the complex parameter τ = s + iλ, has for each fixed value of s and λ a limited set of null geodesics that reach real ℑ^{+}. However, for a ribbon in the complex τplane (i.e., a region topologically ℝ × I, with s ∈ ℝ and λ ∈ I = [λ_{min}, λ_{max}]), there will be many null geodesics intersecting ℑ^{+}. Such null geodesics were referred to as ‘real’ geodesics. More specifically, for a fixed s, there is a specific range of λ values such that all the real null geodesics intersect ℑ^{+} in a full cut, leading to a oneparameter family of real (distorted sphere) slicings of ℑ^{+}. The ribbon is the generalization of the real world line and the slicings are the analogues of the spherical slicings. When the ribbon shrinks to a line it degenerates to the real case. We can consider the ribbon as a generalized world line and the ‘real’ null geodesics from a constant s portion of the ribbon as a generalized lightcone.

For the case of asymptotically flat spacetimes, the real lightcones from interior points are replaced by the virtual lightcones generated by the asymptotically shearfree NGCs. These cones emanate from a complex virtual world line z^{ a } = ξ^{ a }(τ) in the associated \({\mathcal H}\)space. As in the case of complex Minkowski space, there is a ribbon in the τplane where the ‘real’ null geodesics originate from. The ‘real’ null geodesics coming from a crosssection of the strip at fixed s (as in the complex Minkowski case), intersect ℑ^{+} in a cut; the collection of cuts yielding a oneparameter family. The situation is exactly the same as in the complex Minkowski space case except that the spherical harmonic decomposition of these cuts is in general more complicated.
4.4.1 Example: the (charged) Kerr metric
Though we are certainly not making the claim that one can in reality ‘observe’ these complex world lines that arise from (asymptotically) shearfree congruences, we nevertheless claim that they can be observed in a different sense. In the next two sections our goal will be to show that, just as a complex center of charge world line in \({\mathbb {M}}_{\mathbb {C}}\) can be selected, so too can a complex center of mass world line be singled out in \({\mathcal H}\)space. As we will see, some surprising physical identifications arise from this program, and it is in this sense which the footprints of these complex world lines can be observed.
5 Simple Applications
In this section we give four simple examples of the use of shearfree and asymptotically shearfree NGCs in GR. The first is for asymptoticallylinearized perturbations off the Schwarzschild metric, while the next two are from the class of algebraicallyspecial metrics, namely the RobinsonTrautman metric and the type II twisting metrics; the fourth is for asymptotically static/stationary metrics.
5.1 Linearized off Schwarzschild
As a first example, we describe how the shearfree NGCs are applied in linear perturbations off the Schwarzschild metric. The ideas used here are intended to clarify the more complicated issues in the full nonlinear asymptotic theory. We will see that these linear perturbations greatly resemble our results from Section 3.4 on the determination of the intrinsic center of charge in Maxwell theory, when there were small deviations from the Coulomb field.
We begin with the Schwarzschild spacetime, treating the Schwarzschild mass, M_{sch} ≡ M_{B}, as a zerothorder quantity, and integrate the linearized Bianchi identities for the linear Weyl tensor corrections. Though we could go on and find the linearized connection and metric, we stop just with the Weyl tensor. The radial behavior is given by the peeling theorem, so that we can start with the linearized asymptotic Bianchi identities, Eqs. (2.52)–(2.54).
The linearization off Schwarzschild, with our identifications, lead to a stationary spinning spacetime object with the standard classical mechanics kinematic and dynamic description. It was the linearization that let to such simplifications, and in Section 6, when nonlinear terms are included (in similar calculations), much more interesting and surprising physical results are found.
5.2 The RobinsonTrautman metrics
5.3 Type II twisting metrics
It was pointed out in the previous section that the RT metrics are the general relativistic analogues of the (real) LiénardWiechert Maxwell fields. The type II algebraicallyspecial twisting metrics are the gravitational analogues of the complex LiénardWiechert Maxwell fields described earlier. Unfortunately they are far more complicated than the RT metrics. In spite of the large literature and much effort there are very few known solutions and much still to be learned [41, 58, 46]. We give a very brief description of them, emphasizing only the items of relevance to us.
Recently, the type II EinsteinMaxwell equations were studied using a slowmotion perturbation expansion around the ReissnerNördstrom metric, keeping spherical harmonic contributions up to l = 2. It was found that the abovementioned world line coincides in this case with that given by the AbrahamLorentzDirac equation, prompting us to consider such spacetimes as ‘type II particles’ in the same way that one can refer to ReissnerNördstromSchwarzschild or KerrNewman ‘particles’ [52].
5.4 Asymptotically static and stationary spacetimes
By defining asymptotically static or stationary spacetimes as those asymptoticallyflat spacetimes where the asymptotic variables are ‘time’ independent, i.e., u_{B} independent, we can look at our procedure for transforming to the complex center of mass (or complex center of charge). This example, though very special, has the huge advantage in that it can be done exactly, without the use of perturbations [3].
These results for the lower multipole moments, i.e., l = 0, 1, are identical to those of the Kerr metric presented earlier! The higher moments are still present (appearing in higher r^{−1} terms in the Weyl tensor) and are not affected by these results.
6 Main Results
We saw in Sections 3 and 4 how shearfree and asymptotically shearfree NGCs determine arbitrary complex analytic world lines in the auxiliary complex \({\mathcal H}\)space (or complex Minkowski space). In the examples from Sections 3 and 5, we saw how, in each of the cases, one could pick out a special GCF, referred to as the UCF, and the associated complex world line by a transformation to the complex center of mass or charge by requiring that the complex dipoles vanish. In the present section we consider the same problem, but now perturbatively for the general situation of asymptoticallyflat spacetimes satisfying either the vacuum Einstein or the EinsteinMaxwell equations in the neighborhood of future null infinity. Since the calculations are relatively long and complicated, we give the basic ideas in outline form and then present the final results for EinsteinMaxwell spacetimes without detailed steps.
We begin with the ReissnerNordström metric, considering both the mass and the charge as zerothorder quantities, and perturb from it. The perturbation data is considered to be first order and the perturbations themselves are general in the class of analytic asymptoticallyflat spacetimes. Though our considerations are for arbitrary mass and charge distributions in the interior, we look at the fields in the neighborhood of ℑ^{+}. The calculations are carried to second order in the perturbation data. Throughout we use expansions in spherical harmonics and their tensor harmonic versions, but terminate the expansions after l = 2. ClebschGordon expansions are frequently used; see Appendix C.
6.1 A brief summary — Before continuing
For the case of the EinsteinMaxwell fields, in general there will be two complex world lines and two associated UCFs, one for the center of charge, the other for the center of mass. For later use we note that the gravitational world line will be denoted by ξ^{ a }, while the electromagnetic world line by η^{ a }. Later we consider the special case when the two world lines and the two UCFs coincide, i.e., τ^{ a }=η^{ a }.
This equation, though complicated and unattractive, is our main source of information concerning the complex centerofmass world line. The information is extracted in the following way: Considering only the l = 1 harmonics at constant τ in Eq. (6.28), we set the l = 1 harmonics of \(\psi_{1}^{{\ast} 0}\) (with constant τ) to zero (i.e., \(\psi _1^{{\ast} 0i} = 0\)). The three resulting relations are used to determine the three spatial components, ξ^{ k }(τ), of ξ^{ a }(τ) (with ξ^{0} = τ). This fixes the complex center of mass in terms of \(\psi _1^{0i},\;{\Psi ^0},{\Psi ^i}\), and other data which is readily interpreted physically. Alternatively it allow us to express \(\psi_{1}^{0i}\) in terms of the ξ^{ a }(τ).
Extracting this information takes a bit of effort.
6.2 The complex centerofmass world line
 1.
As previously noted, Eq. (6.28) is a function of both τ (via the ξ^{ i }, ξ^{ ij }) and u_{ret} (via the \(\psi _{1}^{0i}\) and Ψ). The extraction of the l = 1 part of \(\psi_{1}^{{\ast} 0}\) must be taken on the constant τ cuts. In other words u_{ret} must be eliminated by using Eq. (6.2).
 2.This elimination of u_{B} (or u_{ret}) is done in the linear terms via the expansion:In the nonlinear terms we can simply use$$\begin{array}{*{20}c} {\eta ({u_{{\rm{ret}}}}) = \eta \left({\tau  {{\sqrt 2} \over 2}\xi _R^i(\tau)Y_{1i}^0 + \sqrt 2 \xi _R^{ij}(\tau)Y_{2ij}^0} \right)\quad \quad \quad \quad}\\ {\approx \eta (\tau)  {{\sqrt 2} \over 2}\eta (s)\prime \left[ {\xi _R^i(s)Y_{1i}^0  2\xi _R^{ij}(s)Y_{2ij}^0} \right]} \\ {{u_{{\rm{ret}}}} = \tau  {{\sqrt 2} \over 2}{\xi ^i}(\tau)Y_{1i}^0 + \sqrt 2 {\xi ^{ij}}(\tau)Y_{2ij}^0 + \ldots \quad \quad \quad \;} \\ \end{array}$$(6.30)$${u_{{\rm{ret}}}} = \tau .$$
 3.
In the ClebschGordon expansions of the harmonic products, though we need both the l = 1 and l = 2 terms in the calculation, we keep at the end only the l = 1 terms for the \(\psi _1^{0i}\). (Note that there are no l = 0 terms since \(\psi_{1}^{{\ast} 0}\) is spin weight s = 1.)
 4.
For completeness, we have included into the calculations Maxwell fields with both a complex dipole (electric and magnetic), \(D_{\mathbb{C}}^{i}=q\eta^{i}=q(\eta_{R}^{i}+i\eta_{I}^{i})\) and complex quadrupole (electric and magnetic) fields \(Q_{\mathbb{C}}^{kj}=Q_{E}^{kj}+iQ_{M}^{kj}\).
We will see shortly that there is also a great deal of physical content to be found in the nonlinear terms of Eq. (6.34).
6.3 The evolution of the complex center of mass
Remark 11. In the calculation leading to Eq. (6.49) , nonlinear terms with P^{ i } (or its derivatives) were replaced by the linear expression \({P^i} \approx {M_{\rm{B}}}\xi _R^{i}{\prime}  {2 \over 3}{c^{ 3}}{q^2}\eta _R^{i}{\prime \prime}\). Additionally, we have neglected the time derivatives of the Bondi mass, \(M{\prime}_{\rm{B}}\); this is because, as we shall see momentarily, these derivatives are themselves second order quantities and hence give a vanishing contribution to Eq. (6.49) at our level of approximation.
6.3.1 Physical Content

The first term of P^{ i } is the standard Newtonian kinematic expression for the linear momentum, \(M\,\,\vec \upsilon\).

The second term, \( {2 \over 3}{c^{ 3}}{q^2}\eta _R^{i}{\prime \prime}\), which is a contribution from the second derivative of the electric dipole moment, \(q\eta_{R}^{i}\), plays a special role for the case when the complex center of mass coincides with the complex center of charge, η^{ a } = ξ^{ a }. In this case, the second term is exactly the contribution to the momentum that yields the classical radiation reaction force of classical electrodynamics [43].

Many of the remaining terms in P^{ i }, though apparently second order, are really of higher order when the dynamics are considered. Others involve quadrupole interactions, which contain high powers of c^{−1}.

In the expression for J^{ i } we have already identified, in earlier discussions, the first two terms \({M_{\rm{B}}}c\xi _{I}^{j}\) and \(M_{\rm{B}}\xi_{R}^{k}{\prime}\xi_{R}^{i}\epsilon_{ikj}\) as the intrinsic spin angular momentum and the orbital angular momentum. The further terms, a spinspin, spinquadrupole and quadrupolequadrupole interaction terms, are considerably smaller.

As mentioned earlier, in Eq. (6.46) we see that there are five flux terms, the second is from the gravitational quadrupole flux, the third and fifth are from the classical electromagnetic dipole and electromagnetic quadrupole flux, while the fourth come from dipolequadrupole coupling. The Maxwell dipole part is identical to that derived from pure Maxwell theory [43]. We emphasize that this angular momentum flux law has little to do directly with the chosen definition of angular momentum. The imaginary part of the Bianchi identity (6.42), with the reality condition \(\Psi = \overline {\Psi}\), is the angular momentum conservation law. How to identify the different terms, i.e., identifying the time derivative of the angular momentum and the flux terms, comes from different arguments. The identification of the Maxwell contribution to total angular momentum and the flux contain certain arbitrary assignments: some terms on the lefthand side of the equation, i.e., terms with a time derivative, could have been moved onto the righthand side and been called ‘flux’ terms. However, our assignments were governed by the question of what terms appeared most naturally to be on different sides. The first term appears to be a new prediction.

The angular momentum conservation law can be considered as the evolution equation for the imaginary part of the complex world line, i.e., \(\xi _I^i({u_{\rm{ret}}})\). The evolution for the real part is found from the Bondi energymomentum loss equation.
 In the special case where the complex centers of mass and charge coincide, τ = ξ^{ a }, we have a rather attractive identification: since now the magnetic dipole moment is given by \(D_M^i = q\xi _I^i\) and the spin by \({S^i} = {M_{\rm{B}}}c\xi _I^i\), we have that the gyromagnetic ratio isleading to the Dirac value of g, i.e., g = 2.$${{\vert {S^i}\vert} \over {\vert D_M^i\vert}} = {{{M_{\rm{B}}}c} \over q}$$(6.50)
6.4 The evolution of the Bondi energymomentum
Remark 12. The Bondi mass, \({M_{\rm{B}}} =  {{{c^2}} \over {2\sqrt 2 G}}{\Psi ^0}\), and the original mass of the ReissnerNordström (Schwarzschild) unperturbed metric, \({M_{{\rm{RN}}}} =  {{{c^2}} \over {2\sqrt 2 G}}\psi _2^{0\,0}\), i.e., the l = 0 harmonic of \(\psi _2^0\), differ by a quadratic term in the shear, the l = 0 part of \(\sigma \dot \bar {\sigma}\). This suggests that the observed mass of an object is partially determined by its timedependent quadrupole moment.
6.4.1 Physical Content
 If the complex world line associated with the Maxwell center of charge coincides with the complex center of mass, i.e., if η^{ i } = ξ^{ i }, the termbecomes the classical electrodynamic radiation reaction force.$${2 \over 3}{c^{ 3}}{q^2}\xi _R^i\prime \prime \prime$$(6.57)

This result follows directly from the EinsteinMaxwell equations. There was no model building other than requiring that the two complex world lines coincide. Furthermore, there was no mass renormalization; the mass was simply the conventional Bondi mass as seen at infinity. The problem of the runaway solutions, though not solved here, is converted to the stability of the EinsteinMaxwell equations with the ‘coinciding’ condition on the two world lines. If the two world lines do not coincide, i.e., the Maxwell world line forms independent data, then there is no problem of unstable behavior. This suggests a resolution to the problem of the unstable solutions: one should treat the source as a structured object, not a point, and centers of mass and charge as independent quantities.

The \(F_{{\rm{recoil}}}^{i}\) is the recoil force from momentum radiation.

The \(\Xi ^{i} \prime =  F_{{\rm{RR}}}^i\) can be interpreted as the gravitational radiation reaction.

The first term in F^{ i }, i.e., \( M{\prime} _{\rm{B}}\xi _R^{i}\prime\), is identical to a term in the classical LorentzDirac equations of motion. Again it is nice to see it appearing, but with the use of the mass loss equation it is in reality third order.
6.5 Other related results
The ideas involved in the identification, at future null infinity, of interior physical quantities that were developed in the proceeding sections can also be applied to a variety of different perturbation schemes. Bramson, Adamo and Newman [19, 2, 4] have investigated how gravitational perturbations originating solely from a Maxwell radiation field can be carried through again using the asymptotic Bianchi identities to obtain, in a different context, the same identifications: a complex centerofmass/charge world line, energy and momentum loss, as well as an angular momentum flux law that agrees exactly with the predictions of classical electromagnetic field theory. This scheme yields (up to the order of the perturbation) an approximation for the metric in the interior of the perturbed spacetime.
We briefly describe this procedure. One initially chooses as a background an exact solution of the Einstein equations; three cases were studied: flat Minkowski spacetime, the Schwarzschild spacetime with a ‘small’ mass and the Schwarzschild spacetime with a finite, ‘zero order’, mass. For such backgrounds, the set of spin coefficients is known and fixed. On this background the Maxwell equations were integrated to obtain the desired electromagnetic field that acts as the gravitational perturbation. Bramson has done this for a pure electric dipole solution [19, 2] on the Minkowski background. Recent work has used an electric and magnetic dipole field with a Coulomb charge [4]. The resulting Maxwell field, in each case, is then inserted into the asymptotic Bianchi identities, which, in turn, determine the behavior of the perturbed asymptotic Weyl tensor, i.e., the Maxwell field induces nontrivial changes to the gravitational field. Treating the Maxwell field as first order, the calculations were done to second order, as was done earlier in this review.
Using the just obtained Weyl tensor terms, one can proceed to the integration of the spincoefficient equations and the secondorder metric tensor. For example, one finds that the dipole Maxwell field induces a secondorder Bondi shear, σ^{0}. (This in principle would lead to a fourthorder gravitational energy loss, which in our approximation is ignored.)
Returning to the point of view of this section, the perturbed Weyl tensor can now be used to obtain the same physical identifications described earlier, i.e., by employing a null rotation to set \(\psi _1^{0 \ast i} = 0\), equations of motion and asymptotic physical quantities, (e.g., center of mass and charge, kinematic expressions for momentum and angular momentum, etc.) for the interior of the system could be found. Although we will not repeat these calculations here, we present a few of the results. Though the calculations are similar to the earlier ones, they differ in two ways: there is no firstorder freely given Bondi shear and the perturbation term orders are different.
The familiarity of such results is an exhibit in favor of the physical identification methods described in this review, i.e., they are a confirmation of the consistency of the identification scheme.
7 Gauge (BMS) Invariance
The issue of gauge invariance, the understanding of which is not obvious or easy, must now be addressed. The claim is that the work described here is in fact gauge (or BMS) invariant.
First of all we have, \({\mathfrak{I}}_{\mathbb{C}}^{+}\), or its real part, ℑ^{+}. On \({\mathfrak{I}}_{\mathbb{C}}^{+}\), for each choice of spacetime interior and solution of the EinsteinMaxwell equations, we have its UCF, either in its complex version, \({u_{\rm{B}}} = X(\tau, \zeta, \tilde {\zeta})\), or its real version, Eq. (6.21). The geometric picture of the UCF is a oneparameter family of slicings (complex or real) of \({\mathfrak{I}}_{\mathbb{C}}^{+}\) or ℑ^{+}. This is a geometric construct that has a different appearance or description in different Bondi coordinate systems. It is this difference that we must investigate. We concentrate on the complex version.
Before discussing the relevant effects of the Lorentz transformations on our considerations we first digress and describe an important technical issue concerning representation of the homogeneous Lorentz group.
Of major interest for us is not so much the invariant subspaces but instead their interactions with their compliments (the full vector space modulo the invariant subspace). Under the action of the Lorentz transformations applied to a general vector in the representation space, the components of the invariant subspaces remain in the invariant subspace but in addition components of the complement move into the invariant subspace. On the other hand, the components of the invariant subspaces do not move into the complement subspace: the transformed components of the compliment involve only the original compliment components. The transformation thus has a nontrivial Jordan form.
Rather than give the full description of these invariant subspaces we confine ourselves to the few cases of relevance to us.
7.1 The GoodCut function
Though our interest is primarily in the negative integer representations, we first address the positive integer case of the s = 0 and w = 1, [(n_{1}, n_{2}) = (2, 2)], representation. The harmonics, l = (0, 1) form the invariant subspace. The cut function, \(X(\tau, \zeta, \bar \zeta)\), for each fix values of τ, lies in this space.
7.2 The Mass Aspect
This is the justification for calling the l = (0, 1) harmonics of the mass aspect a Lorentzian fourvector. (Technically, the Bondi fourmomentum is a cofactor but we have allowed ourselves a slight notational irregularity.)
7.3 The Complex Dipole Moment
This allows us to assign Lorentzian invariant physical meaning to our identifications of the complex mass dipole moment and angular momentum vector, \(D_{({\rm{mass}})}^i + i{J^i}\).
7.4 General Invariants
Our last example is a general discussion of how to construct Lorentzian invariants from the representation spaces. Though we will confine our remarks to just the cases of s = 0, it is easy to extend them to nonvanishing by having the two functions have respectively spinweight and −s.
8 Discussion/Conclusion
8.1 History/background
The work reported in this document has had a very long gestation period. It began in 1965 [53] with the publication of a paper where a complex coordinate transformation was performed on the Schwarzschild/ReissnerNordström solutions. This, in a precise sense, moved the ‘point source’ onto a complex world line in a complexified spacetime. It thereby led to a derivation of the spinning and the chargedspinning particle metrics. How and why this procedure worked was considered to be rather mysterious and a great deal of effort by a variety of people went into trying to unravel it. In the end, the use of the complex coordinate transformation for the derivation of these metrics appeared as if it was simply an accident; that is, a trick with no immediate significance. Nevertheless, the idea of a complex world line, appearing in a natural manner, was an intriguing thought, which frequently returned. Some years later, working on an apparently unrelated subject, we studied and found the condition for a regular NGC, in asymptoticallyflat spacetime, to have a vanishing asymptotic shear [12]. This led to the realization that a regular NGC was generated by a complex world line, though originally there was no relationship between the two complex world lines. This condition (our previously discussed shearfree condition, Eq. (4.12)) was eventually shown to be closely related to Penrose’s asymptotic twistor theory, and in the flatspace case it led to the Kerr theorem and totally shearfree NGCs. From a very different point of view, searching for asymptotically shearfree complex null surfaces, the goodcut equation was found with its fourcomplex parameter solution space, leading to the theory of \({\mathcal H}\)space.
Years later, the different strands came together. The shearfree condition was found to be closely related to the goodcut equation; namely, that one equation could be transformed into the other. The major surprise came when we discovered that the regular solutions of either equation were generated by complex world lines in an auxiliary space [39]. These complex world lines were interpreted as being complex analytic curves in the associated \({\mathcal H}\)space. The deeper meaning of this remains a major question still to be fully resolved; it is this issue which is partially addressed in the present work.
At first, these complex world lines were associated with the spinning, charged and uncharged particle metrics — type D algebraically special metrics, but now can be seen as just special cases of the asymptotically flat solutions. Since these metrics were algebraically special, among the many possible asymptotically shearfree NGCs there was (at least) one totally shearfree (rather than asymptotically shearfree) congruence coming from the GoldbergSachs theorem. Their associated world lines were the ones first discovered in 1965 (coming, by accident, from the complex coordinate transformation), and became the complex centerofmass world line (which coincided with the complex center of charge in the charged case.). This observation was the clue for how to search for the generalization of the special world line associated with algebraicallyspecial metrics and thus, in general, how to look for the special world line (and congruence) to be identified with the complex center of mass for arbitrary asymptotically flat spacetimes.
For the algebraicallyspecial metrics, the null tetrad system at ℑ^{+} with one leg being the tangent null vector to the shearfree congruence leads to the vanishing of the asymptotic Weyl tensor component, i.e., \(\psi _0^{\ast}= \psi _1^{\ast}= 0\). For the general case, no tetrad exists with that property but one can always find a null tetrad with one leg being tangent to an asymptotically shearfree congruence so that the l = 1 harmonics of \(\psi _1^{0{\ast}}\) vanish. It is precisely that choice of tetrad that led to our definition of the complex center of mass.
8.2 Other choices for physical identification
The question of whether our definition of the complex center of mass is the best possible definition, or even a reasonable one, is not easy to answer. We did try to establish a criteria for choosing such a definition: (i) it should predict already known physical laws or reasonable new laws, (ii) it should have a clear geometric foundation and a logical consistency and (iii) it should agree with special cases, mainly the algebraicallyspecial metrics or analogies with flatspace Maxwell theory. We did try out several other possible choices [42] and found them all failing. This clearly does not rule out others that we did not think of, but at the present our choice appears to be both natural and effective in making contact with physical phenomena. There still remains the mystery of the physical meaning of \({\mathcal H}\)space or why it leads to such reasonable physical results. Possible resolutions appear to lie in the duality between the complex world lines and real but twisting NGCs [8].
8.3 Predictions
Our equations of motion are simultaneously satisfactory and unsatisfactory: they yield the equations of motion for an isolated object with a great deal of internal structure (timedependent multipoles with the emission of gravitational and electromagnetic radiation) in the form of Newton’s second law. In addition, they contain a definition of angular momentum with an angular momentum flux. The dipole part of the angular momentum flux agrees with classical electromagnetic theory. Unfortunately, there appears to be no immediate way to study or describe interacting particles in this manner.
There are other unfamiliar terms that can be thought of as predictions of this theoretical construct, though how to possibly measure them is not at all clear.
8.4 Summary of results
 1.
From the asymptotic Weyl and Maxwell tensors, with their transformation properties, we were able (via the asymptotically shearfree NGC) to obtain two complex world lines — a complex ‘center of mass’ and ‘complex center of charge’ in the auxiliary \({\mathcal H}\)space. When ‘viewed’ from a Bondi coordinate and tetrad system, this led to an expression for the real center of mass of the gravitating system and a kinematic expression for the total angular momentum (including intrinsic spin and orbital angular momentum), as seen from null infinity. It is interesting to observe that the kinematical expressions for the classical linear momentum and angular momentum came directly from the gravitational dynamical laws (Bianchi identities) for the evolution of the Weyl tensor.
 2.From the real parts of one of the asymptotic Bianchi identities, Eq. (2.52), we found the standard kinematic expression for the Bondi linear momentum, \(P = M{\xi\prime _R} + \ldots\), with the radiation reaction term \({{2{q^2}} \over {3{c^3}}}\upsilon _R^{k}\,{\prime}\) among others. The imaginary part was the angular momentum conservation law with a very natural looking flux expression of the form:The first flux term is identical to that calculated from classical electromagnetic theory$$J\prime = {\rm{Flu}}{{\rm{x}}_{{\rm{E\& M}}\,{\rm{dipole}}}} + {\rm{Flu}}{{\rm{x}}_{{\rm{Grav}}}} + {\rm{Flu}}{{\rm{x}}_{{\rm{E\& M quad}}}}$$
 3.
Using the kinematic expression for the Bondi momentum in a second Bianchi identity (2.53), we obtained a secondorder ODE for the center of mass world line that could be identified with Newton’s second law with radiation reaction forces and recoil forces, M_{B}ξ″_{ R } = F.
 4.
From Bondi’s mass/energy loss theorem we obtained the correct energy flux from the electromagnetic dipole and quadrupole radiation as well as the gravitational quadrupole radiation.
 5.From the specialized case where the two world lines coincide and the definitions of spin and magnetic moment, we obtained the Dirac gyromagnetic ratio, g = 2. In addition, we find the classical electrodynamic radiation reaction term with the correct numerical factors. In this case we have the identifications of the meaning of the complex position vector: \({\xi ^i} = \xi _R^i + i\xi _I^i\).$$\begin{array}{*{20}c} {\xi _R^i = {\rm{center  of  mass}}\,{\rm{position}}\quad \quad \quad} \\ {{S^i} = Mc\xi _I^i = {\rm{spin}}\,{\rm{angular}}\,{\rm{momentum}}} \\ {D_E^i = q\xi _R^i = {\rm{electric}}\,{\rm{dipole}}\,{\rm{moment}}\quad} \\ {D_M^i = q\xi _I^i = {\rm{magnetic}}\,{\rm{dipole}}\,{\rm{moment}}} \\ \end{array}$$
Remark 13. In the past, most discussions of angular momentum make use of group theoretical ideas with Noether theorem type arguments, via the BMS group and the Lorentz subgroup, to define angular momentum. Unfortunately this has been beset with certain difficulties; different authors get slightly different numerical factors in their definitions, with further ambiguities arising from the supertranslation freedom of the BMS group. (See the discussion after Eq. (2.62) ) Our approach is very different from the group theoretical approach in that we come to angular momentum directly from the dynamics of the Einstein equations (the asymptotic Bianchi Identities). We use the angular momentum definition from linear theory, Eq. (2.62), (agreed to by virtually all) and then supplement it via conservation equations (the flux law) obtained directly from the Bianchi Identities. We have a unique oneparameter family of cuts coming from the complex world line defining the complex center of mass. This is a geometric structure with no ambiguities. However, another ambiguity does arise by asking which Bondi frame should be used in the description of angular momentum; this is the ambiguity of what coordinate system to use.
8.5 Issues and open questions
 1.
A particularly interesting issue raised by our equations is that of the runaway (unstable) behavior of the equations of motion for a charged particle (with or without an external field). We saw in Eq. (6.54) that there was a driving term in the equation of motion depending on the electric dipole moment (or the real center of charge). This driving term was totally independent of the real center of mass and thus does not lead to the classical instability. However, if we restrict the complex center of charge to be the same as the complex center of mass (a severe, but very attractive restriction leading to g = 2), then the innocuous driving dipole term becomes the classical radiation reaction term — suggesting instability. (Note that in this coinciding case there was no model building — aside from the coinciding lines — and no mass renormalization.)
A natural question then is: does this unstable behavior really remain? In other words, is it possible that the large number of extra terms in the gravitational radiation reaction or the momentum recoil force might stabilize the situation? Answering this question is extremely difficult. If the gravitational effects do not stabilize, then — at least in this special case — the EinsteinMaxwell equations are unstable, since the runaway behavior would lead to an infinite amount of electromagnetic dipole energy loss.
An alternative possible resolution to the classical runaway problem is simply to say that the classical electrodynamic model is wrong; and that one must treat the center of charge as different from the center of mass with its own dynamics.
 2.
In our approximations, it was assumed that the complex world line yielded cuts of ℑ^{+} that were close to Bondi cuts. At the present we do not have any straightforward means of finding the world lines and their associated cuts of ℑ^{+} that are far from the Bondi cuts.
 3.
As mentioned earlier, when the gravitational and electromagnetic world lines coincide we find the rather surprising result of the Dirac value for the gyromagnetic ratio. Though this appears to be a significant result, we unfortunately do not have any deeper understanding of it.
 4.
Is it possible that the complex structures that we have been seeing and using are more than just a technical device to organize ideas, and that they have a deeper significance? One direction to explore this is via Penrose’s twistor and asymptotic twistor theory. It is known that much of the material described here is closely related to twistor theory; an example is the fact that asymptotic shearfree NGCs are really a special case of the Kerr theorem, an important application of twistor theory (see Appendix A). This connection is being further explored.
 5.
With much of the kinematics and dynamics of ordinary classical mechanics sitting in our results, i.e., in classical GR, is it possible that ordinary particle quantization could play a role in understanding quantum gravity? Attempts along this line have been made [25, 15] but, so far, without much success.
 6.
An interesting issue, not yet fully explored but potentially important, is what more can be said about the \({\mathcal H}\)space structures associated with the special regions (the \({\mathcal H}\)space ribbon of (4.40) that are related to the real cuts of null infinity. We touch on this briefly below.
 7.
Another issue to be explored comes from the duality between the complex \({\mathcal H}\)space lightcones and the real shearfree but twisting NGCs in the real physical spacetime. From either one the other can be determined. It appears as if one might be able to reinterpret (almost) all the \({\mathcal H}\)space structures in terms of real structures associated with the optical parameters of the twisting NGCs and the real slicings associated with the ribbons. This reinterpretation would likely result in lost geometric simplicity and elegance — but perhaps would avoid the mysterious use of the complex \({\mathcal H}\)space for physical identifications.
 8.
As a final remark, we want to point out that there is an issue that we have ignored: do the asymptotic solutions of the Einstein equations that we have discussed and used throughout this work really exist? By ‘really exist’ we mean the following: are there, in sufficiently general circumstances, Cauchy surfaces with physicallygiven data such that their evolution yields these asymptotic solutions? We have tacitly assumed throughout, with physical justification but no rigorous mathematical justification, that the full interior vacuum Einstein equations do lead to these asymptotic situations. However, there has been a great deal of deep and difficult analysis [24, 20, 21] showing, in fact, that large classes of solutions to the Cauchy problem with physicallyrelevant data do lead to the asymptotic behavior we have discussed. Recently there has been progress made on the same problem for the EinsteinMaxwell equations.
8.6 New interpretations and future directions
Throughout this review, we have focused on classical general relativity as expressed in the NewmanPenrose (or spincoefficient) formalism; yet despite this highly classical setting, the geometric structures discussed here bear a striking resemblance to ideas from more ambitious areas of theoretical physics. In particular, the relationship between real, twisting asymptotically shearfree NGCs in asymptotically flat spacetime and complex, twistfree NGCs in (complex) \({\mathcal H}\)space appears to form a dual pair. The complexified asymptotic boundary \({\mathfrak{I}}_{\mathbb{C}}^{+}\) acts as the translator between these two descriptions: from \({\mathfrak{I}}_{\mathbb{C}}^{+}\) we determine the complexified congruence in \({\mathcal H}\)space via the angle fields L and \(\tilde{L}\) or the real congruence (in the real spacetime) via L and \(\bar{L}\). Imposing reality conditions in the former case gives an open world sheet (the ‘\({\mathcal H}\)space ribbon’ discussed in the text) for the NGC’s complex source. In the real case, the twisting congruence’s caustic set in Minkowski space (which is interpreted as its real source) forms a closed loop propagating in real time [7], or a closed world sheet. In the asymptotically flat case, we cannot follow twisting congruences back to their real caustic set, and they must be represented by the dual ‘ribbon’.
In both cases, we see that the congruence’s source is a structure which has a natural interpretation as a classical string, with the boundary \({\mathfrak{I}}_{\mathbb{C}}^{+}\) interpolating between the two descriptions. This is suggestive of the socalled ‘holographic principle’, which aims to equate a theory in a ddimensional compact space with another theory defined on its d−1dimensional boundary [77, 17]. In practice, this can allow one to interpolate between a ‘physical’ theory in one space and a ‘dual’ theory living on its boundary (or vice versa). In our case, the ‘physical’ information is the real, twisting NGC in the asymptotically flat spacetime; \({\mathfrak{I}}_{\mathbb{C}}^{+}\) acts as a lens into \({\mathcal H}\)space, which serves as the virtual image space where physical data (such as the mass, linear momentum, angular momentum, etc.) is computed by the methods reviewed here. Hence, it is tempting to refer to \({\mathfrak{I}}_{\mathbb{C}}^{+}\) as the ‘holographic screen’ for some application of the holographic principle to general relativity. The presence of classical stringlike structures on both sides of the duality makes such a possibility all the more intriguing.
This should be contrasted with the most wellknown instance of the holographic principle: the AdS/CFT correspondence [47, 78, 9]. Here, the AdSboundary acts as the holographic screen between a type IIB string theory in AdS_{5} × S^{5} (the virtual image space) and maximally supersymmetric YangMills theory in real fourdimensional Minkowski spacetime (other versions exist, but all involve some supersymmetry). It is interesting that we appear to be describing an instance of the holographic principle that requires no supersymmetry, although ’t Hooft’s original work relating the planar limit of gauge theories to stringtype theories did not use supersymmetry either [76]. In ’t Hooft’s work, an extra dimension for string propagation enters the picture due to anomaly cancellation in the same way that the extra dimensions of AdS_{5} × S^{5} allow for anomaly cancellation in a full supersymmetric string theory. In our investigations, one can think of the analytic continuation of ℑ^{+} to \({\mathfrak{I}}_{\mathbb{C}}^{+}\) in the same manner: instead of canceling a (quantum) anomaly, the ‘extra dimensions’ arising from analytic continuation allow us to reinterpret the real twisting NGC in terms of a simpler geometric structure, namely the complex lightcones in \({\mathcal H}\)space.
It is worth noting that this is not the first time that there has been a suggested connection between structures in asymptotically flat spacetimes and the holographic principle. Most prior studies have attempted to understand such a duality in terms of the BMS group, which serves as the symmetry group of the asymptotic boundary [10]. Loosely speaking, these studies take their cue more directly from the AdS/CFT correspondence: by studying fields living on ℑ^{+} which carry representations of the BMS group, one hopes to reconstruct the full interior of the spacetime ‘holographically’. It would be interesting to see how, if at all, our methodology relates to this program of research.
Additionally, as we have mentioned throughout this review (and further elaborated in Appendix A), the nature of many of the objects studied here is highly twistorial. This is essentially because \({\mathcal H}\)space is a complex vacuum spacetime equipped with an antiselfdual metric, and hence possesses a curved twistor space by Penrose’s nonlinear graviton construction [32]. It would be interesting to know if our procedure for identifying the complex center of mass (and/or charge) in an asymptotically flat spacetime could be phrased purely in terms of twistor theory. Furthermore, the past several years have seen dramatic progress in using twistor theory to study gauge theories and their scattering amplitudes [1]. These techniques may provide the most promising route for connecting our work with any quantized version of gravity, as illustrated by the recent twistorial derivation of the treelevel MHV graviton scattering amplitude [48].
While the interpretations we have suggested here are far from precise, they suggest a myriad of further directions which research in this area could take. It would be truly fascinating for a topic as old as asymptotically flat spacetimes to make meaningful contact with ambitious new areas of mathematics and physics such as holography or twistorial scattering theory.
Notes
Acknowledgments
We are happy to note and acknowledge the great deal of detailed help and understanding that we have received over the years from Gilberto SilvaOrtigoza, our coauthor on many of the earlier papers on the present subject. Many others have contributed to our understanding of asymptoticallyflat spacetimes. Prime among them are Roger Penrose, Andrzej Trautman, Jerzy Lewandowski, Pawel Nurowski, Paul Tod, Lionel Mason, Jörg Frauendiener and Helmut Friedrich. We thank them all.
We also want to point out the very early work of Brian Bramson [18], William Hallidy and Malcolm Ludvigsen [30]. Their ideas, though not fully developed, are close to the ideas developed here, and should be viewed as precursors or preliminary formulations of our theory. We apologize for not having noticed them earlier and giving them more credit.
References
 [1]Adamo, T.M., Bullimore, M., Mason, L. and Skinner, D., “Scattering amplitues and Wilson loops in twistor space”, J. Phys. A: Math. Theor., 44, 454008, (2011). [DOI], [arXiv:1104.2890]. (Cited on page 74.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [2]Adamo, T.M. and Newman, E.T., “The gravitational field of a radiating electromagnetic dipole”, Class. Quantum Grav., 25, 245005, (2008). [DOI], [arXiv:0807.3537]. (Cited on page 63.)ADSCrossRefzbMATHGoogle Scholar
 [3]Adamo, T.M. and Newman, E.T., “Asymptotically stationary and static spacetimes and shear free null geodesic congruences”, Class. Quantum Grav., 26, 155003, (2009). [DOI], [arXiv:0906.2409]. (Cited on pages 46 and 52.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [4]Adamo, T.M. and Newman, E.T., “Electromagnetically induced gravitational perturbations”, Class. Quantum Grav., 26, 015004, (2009). [DOI], [arXiv:0807.3671]. (Cited on pages 25 and 63.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [5]Adamo, T.M. and Newman, E.T., “Vacuum nonexpanding horizons and shearfree null geodesic congruences”, Class. Quantum Grav., 26, 235012, (2009). [DOI], [arXiv:0908.0751]. (Cited on page 85.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [6]Adamo, T.M. and Newman, E.T., “The Generalized Good Cut Equation”, Class. Quantum Grav., 27, 245004, (2010). [DOI], [arXiv:1007.4215]. (Cited on pages 46, 75, and 85.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [7]Adamo, T.M. and Newman, E.T., “The real meaning of complex Minkowskispace worldlines”, Class. Quantum Grav., 27, 075009, (2010). [DOI], [arXiv:0911.4205]. (Cited on pages 9, 27, 30, 31, 73, 78, and 84.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [8]Adamo, T.M. and Newman, E.T., “Light cones in relativity: Real, complex and virtual, with applications”, Phys. Rev. D, 83, 044023, (2011). [DOI], [arXiv:1101.1052]. (Cited on pages 30, 31, 39, 60, and 70.)ADSCrossRefGoogle Scholar
 [9]Aharony, O., Gubser, S.S., Maldacena, J.M., Ooguri, H. and Oz, Y., “Large N field theories, string theory and gravity”, Phys. Rep., 323, 183–386, (2000). [DOI], [hepth/9905111]. (Cited on page 73.)ADSMathSciNetCrossRefGoogle Scholar
 [10]Arcioni, G. and Dappiaggi, C., “Exploring the holographic principle in asymptotically flat spacetimes via the BMS group”, Nucl. Phys. B, 674, 553–592, (2003). [DOI], [hepth/0306142]. (Cited on page 73.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [11]Arnowitt, R., Deser, S. and Misner, C.W., “Energy and the Criteria for Radiation in General Relativity”, Phys. Rev., 118, 1100–1104, (1960). [DOI], [ADS]. (Cited on page 7.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [12]Aronson, B. and Newman, E.T., “Coordinate systems associated with asymptotically shearfree null congruences”, J. Math. Phys., 13, 1847–1851, (1972). [DOI]. (Cited on pages 28, 42, 49, and 69.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [13]Ashtekar, A., Beetle, C. and Lewandowski, J., “Geometry of generic isolated horizons”, Class. Quantum Grav., 19, 1195–1225, (2002). [DOI], [grqc/0111067]. (Cited on page 85.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [14]Ashtekar, A. and Krishnan, B., “Isolated and Dynamical Horizons and Their Applications”, Living Rev. Relativity, 7, lrr200410, (2004). URL (accessed 28 April 2011): http://www.livingreviews.org/lrr200410. (Cited on page 85.)
 [15]Bergmann, P.G., “NonLinear Field Theories”, Phys. Rev., 75, 680–685, (1949). [DOI], [ADS]. (Cited on page 72.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [16]Bondi, H., van der Burg, M.G.J. and Metzner, A.W.K., “Gravitational Waves in General Relativity. VII. Waves from AxiSymmetric Isolated Systems”, Proc. R. Soc. London, Ser. A, 269, 21–52, (1962). [DOI], [ADS]. (Cited on pages 7, 10, 17, and 24.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [17]Bousso, R., “The holographic principle”, Rev. Mod. Phys., 74, 825–874, (2002). [DOI], [hepth/0203101]. (Cited on page 73.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [18]Bramson, B.D., “Relativistic Angular Momentum for Asymptotically Flat EinsteinMaxwell Manifolds”, Proc. R. Soc. London, Ser. A, 341, 463–490, (1975). [DOI]. (Cited on page 74.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [19]Bramson, B., “Do electromagnetic waves harbour gravitational waves?”, Proc. R. Soc. London, Ser. A, 462, 1987–2000, (2006). [DOI]. (Cited on page 63.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [20]Chruściel, P.T. and Friedrich, H., eds., The Einstein Equations and the Large Scale Behavior of Gravitational Fields: 50 Years of the Cauchy Problem in General Relativity, (Birkhäuser, Basel; Boston, 2004). [Google Books]. (Cited on page 72.)zbMATHGoogle Scholar
 [21]Corvino, J. and Schoen, R.M., “On the asymptotics for the vacuum Einstein constraint equations”, J. Differ. Geom., 73, 185–217, (2006). [grqc/0301071]. (Cited on page 72.)MathSciNetzbMATHGoogle Scholar
 [22]Dragomir, S. and Tomassini, G., Differential Geometry and Analysis on CR Manifolds, (Birkhäuser, Boston; Basel; Berlin, 2006). [Google Books]. (Cited on pages 77 and 78.)zbMATHGoogle Scholar
 [23]Frauendiener, J., “Conformal Infinity”, Living Rev. Relativity, 7, lrr20041, (2004). URL (accessed 31 July 2009): http://www.livingreviews.org/lrr20041. (Cited on pages 8 and 17.)
 [24]Friedrich, H., “On the Existence of nGeodesically Complete or Future Complete Solutions of Einstein’s Field Equations with Smooth Asymptotic Structure”, Commun. Math. Phys., 107, 587–609, (1986). [DOI]. (Cited on page 72.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [25]Frittelli, S., Kozameh, C.N., Newman, E.T., Rovelli, C. and Tate, R.S., “Fuzzy spacetime from a nullsurface version of general relativity”, Class. Quantum Grav., 14, A143–A154, (1997). [DOI], [grqc/9603061]. (Cited on page 72.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [26]Frittelli, S. and Newman, E.T., “PseudoMinkowskian coordinates in asymptotically flat spacetimes”, Phys. Rev. D, 55, 1971–1976, (1997). [DOI], [ADS]. (Cited on page 64.)ADSMathSciNetCrossRefGoogle Scholar
 [27]Gel’fand, I.M., Graev, M.I. and Vilenkin, N.Y., Generalized Functions, Vol. 5: Integral geometry and representation theory, (Academic Press, New York; London, 1966). (Cited on pages 64 and 65.)zbMATHGoogle Scholar
 [28]Goldberg, J.N., Macfarlane, A.J., Newman, E.T., Rohrlich, F. and Sudarshan, E.C.G., “Spins Spherical Harmonics and ð”, J. Math. Phys., 8, 2155–2161, (1967). [DOI]. (Cited on pages 19 and 79.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [29]Goldberg, J.N. and Sachs, R.K., “A Theorem on Petrov Types”, Acta Phys. Pol., 22, 13–23, (1962). Republished as DOI:10.1007/s1071400807225. (Cited on pages 9, 24, and 26.)MathSciNetzbMATHGoogle Scholar
 [30]Hallidy, W. and Ludvigsen, M., “Momentum and Angular Momentum in the HSpace of Asymptotically Flat, EinsteinMaxwell SpaceTimes”, Gen. Relativ. Gravit., 10, 7–30, (1979). [DOI]. (Cited on page 74.)ADSCrossRefGoogle Scholar
 [31]Hansen, R.O. and Newman, E.T., “A complex Minkowski space approach to twistors”, Gen. Relativ. Gravit., 6, 361–385, (1975). [DOI]. (Cited on pages 41 and 43.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [32]Hansen, R.O., Newman, E.T., Penrose, R. and Tod, K.P., “The Metric and Curvature Properties of \({\mathcal H}\)Space”, Proc. R. Soc. London, Ser. A, 363, 445–468, (1978). [DOI], [ADS]. (Cited on pages 43, 44, and 74.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [33]Held, A., Newman, E.T. and Posadas, R., “The Lorentz Group and the Sphere”, J. Math. Phys., 11, 3145–3154, (1970). [DOI]. (Cited on pages 64, 65, 67, and 79.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [34]Hill, C.D., Lewandowski, J. and Nurowski, P, “Einstein’s equations and the embedding of 3dimensional CR manifolds”, Indiana Univ. Math. J., 57, 3131–3176, (2008). [DOI], [arXiv:0709.3660]. (Cited on page 78.)MathSciNetCrossRefzbMATHGoogle Scholar
 [35]Hugget, S.A. and Tod, K.P., An Introduction to Twistor Theory, London Mathematical Society Student Texts, 4, (Cambridge University Press, Cambridge; New York, 1994), 2nd edition. [Google Books]. (Cited on page 75.)CrossRefGoogle Scholar
 [36]Ivancovich, J., Kozameh, C.N. and Newman, E.T., “Green’s functions of the edh operators”, J. Math. Phys., 30, 45–52, (1989). [DOI]. (Cited on page 43.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [37]Ko, M., Newman, E.T. and Tod, K.P., “\({\mathcal H}\)Space and Null Infinity”, in Esposito, F.P. and Witten, L., eds., Asymptotic Structure of SpaceTime, Proceedings of a Symposium on Asymptotic Structure of SpaceTime (SOASST), held at the University of Cincinnati, Ohio, June 14–18, 1976, pp. 227–271, (Plenum Press, New York, 1977). (Cited on page 9.)CrossRefGoogle Scholar
 [38]Kozameh, C.N. and Newman, E.T., “Electromagnetic dipole radiation fields, shearfree congruences and complex centre of charge world lines”, Class. Quantum Grav., 22, 4667–4678, (2005). [DOI], [grqc/0504093]. (Cited on page 34.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [39]Kozameh, C.N. and Newman, E.T., “The large footprints of Hspace on asymptotically flat spacetimes”, Class. Quantum Grav., 22, 4659–4665, (2005). [DOI], [grqc/0504022]. (Cited on pages 8, 28, 29, 42, and 69.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [40]Kozameh, C.N., Newman, E.T., SantiagoSantiago, J.G. and SilvaOrtigoza, G., “The universal cut function and type II metrics”, Class. Quantum Grav., 24, 1955–1979, (2007). [DOI], [grqc/0612004]. (Cited on pages 7, 8, 27, 28, 29, 35, and 54.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [41]Kozameh, C.N., Newman, E.T. and SilvaOrtigoza, G., “On the physical meaning of the RobinsonTrautmanMaxwell fields”, Class. Quantum Grav., 23, 6599–6620, (2006). [DOI], [grqc/0607074]. (Cited on pages 7, 44, and 51.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [42]Kozameh, C.N., Newman, E.T. and SilvaOrtigoza, G., “On extracting physical content from asymptotically flat spacetime metrics”, Class. Quantum Grav., 25, 145001, (2008). [DOI], [arXiv:0802.3314]. (Cited on pages 7, 8, 25, 67, 70, and 80.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [43]Landau, L.D. and Lifshitz, E.M., The classical theory of fields, (Pergamon Press; AddisonWesley, Oxford; Reading, MA, 1962), 2nd edition. (Cited on pages 13, 35, 60, 61, 64, and 67.)zbMATHGoogle Scholar
 [44]Lewandowski, J. and Nurowski, P., “Algebraically special twisting gravitational fields and CR structures”, Class. Quantum Grav., 7, 309–328, (1990). [DOI]. (Cited on page 77.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [45]Lewandowski, J., Nurowski, P. and Tafel, J., “Einstein’s equations and realizability of CR manifolds”, Class. Quantum Grav., 7, L241–L246, (1990). [DOI]. (Cited on page 77.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [46]Lind, R.W., “Shearfree, twisting EinsteinMaxwell metrics in the NewmanPenrose formalism”, Gen. Relativ. Gravit., 5, 25–47, (1974). [DOI]. (Cited on page 51.)ADSMathSciNetCrossRefGoogle Scholar
 [47]Maldacena, J.M., “The LargeN Limit of Superconformal Field Theories and Supergravity”, Adv. Theor. Math. Phys., 2, 231–252, (1998). [DOI], [hepth/9711200]. (Cited on page 73.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [48]Mason, L.J. and Skinner, D., “Gravity, Twistors and the MHV Formalism”, Commun. Math. Phys., 294, 827–862, (2010). [DOI], [arXiv:0808.3907]. (Cited on page 74.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [49]Newman, E.T., “Heaven and Its Properties”, Gen. Relativ. Gravit., 7, 107–111, (1976). [DOI]. (Cited on pages 9 and 44.)ADSMathSciNetCrossRefGoogle Scholar
 [50]Newman, E.T., “Maxwell fields and shearfree null geodesic congruences”, Class. Quantum Grav., 21, 3197–3221, (2004). [DOI]. (Cited on pages 34, 36, 37, and 40.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [51]Newman, E.T., “Asymptotic twistor theory and the Kerr theorem”, Class. Quantum Grav., 23, 3385–3392, (2006). [DOI], [grqc/0512079]. (Cited on pages 75 and 76.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [52]Newman, E.T., “Newton’s second law, radiation reaction and type II EinsteinMaxwell fields”, Class. Quantum Grav., 28, 245003, (2011). [DOI], [arXiv:1109.4106]. (Cited on page 52.)ADSCrossRefzbMATHGoogle Scholar
 [53]Newman, E.T., Couch, E., Chinnapared, K., Exton, A., Prakash, A. and Torrence, R., “Metric of a Rotating, Charged Mass”, J. Math. Phys., 6, 918–919, (1965). [DOI]. (Cited on pages 9, 49, and 69.)ADSMathSciNetCrossRefGoogle Scholar
 [54]Newman, E.T. and Nurowski, P., “CR structures and asymptotically flat spacetimes”, Class. Quantum Grav., 23, 3123–3127, (2006). [DOI], [grqc/0511119]. (Cited on page 78.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [55]Newman, E.T. and Penrose, R., “An Approach to Gravitational Radiation by a Method of Spin Coefficients”, J. Math. Phys., 3, 566–578, (1962). [DOI], [ADS]. (Cited on pages 10, 21, and 22.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [56]Newman, E.T. and Penrose, R., “Note on the BondiMetznerSachs Group”, J. Math. Phys., 7, 863–870, (1966). [DOI], [ADS]. (Cited on pages 10 and 79.)ADSMathSciNetCrossRefGoogle Scholar
 [57]Newman, E.T. and Penrose, R., “Spincoefficient formalism”, Scholarpedia, 4(6), 7445, (2009). URL (accessed 30 July 2009): http://www.scholarpedia.org/article/Spincoefficient_formalism. (Cited on page 21.)ADSCrossRefGoogle Scholar
 [58]Newman, E.T. and Posadas, R., “Motion and Structure of Singularities in General Relativity”, Phys. Rev., 187, 1784–1791, (1969). [DOI], [ADS]. (Cited on page 51.)ADSCrossRefzbMATHGoogle Scholar
 [59]Newman, E.T. and SilvaOrtigoza, G., “Tensorial spins harmonics”, Class. Quantum Grav., 23, 497–509, (2006). [DOI], [grqc/0508028]. (Cited on pages 10, 11, 79, 80, and 81.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [60]Newman, E.T. and Tod, K.P., “Asymptotically flat spacetimes”, in Held, A., ed., General Relativity and Gravitation: One Hundred Years After the Birth of Albert Einstein, 2, pp. 1–36, (Plenum Press, New York, 1980). (Cited on pages 17, 18, 19, 21, 52, and 65.)Google Scholar
 [61]Newman, E.T. and Unti, T.W.J., “Behavior of Asymptotically Flat Empty Spaces”, J. Math. Phys., 3, 891–901, (1962). [DOI], [ADS]. (Cited on page 22.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [62]Penrose, R., “Asymptotic Properties of Fields and SpaceTimes”, Phys. Rev. Lett., 10, 66–68, (1963). [DOI], [ADS]. (Cited on pages 8 and 17.)ADSMathSciNetCrossRefGoogle Scholar
 [63]Penrose, R., “Zero RestMass Fields Including Gravitation: Asymptotic Behaviour”, Proc. R. Soc. London, Ser. A, 284, 159–203, (1965). [DOI], [ADS]. (Cited on pages 8 and 17.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [64]Penrose, R., “Twistor Algebra”, J. Math. Phys., 8, 345–366, (1967). [DOI]. (Cited on page 75.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [65]Penrose, R., “Relativistic symmetry groups”, in Barut, A.O., ed., Group Theory in NonLinear Problems, Proceedings of the NATO Advanced Study Institute, held in Istanbul, Turkey, August 7–18, 1972, NATO ASI Series C, 7, pp. 1–58, (Reidel, Dordrecht; Boston, 1974). (Cited on pages 10 and 25.)CrossRefGoogle Scholar
 [66]Penrose, R. and Rindler, W., Spinors and spacetime, Vol. 1: Twospinor calculus and relativistic fields, Cambridge Monographs on Mathematical Physics, (Cambridge University Press, Cambridge; New York, 1984). [Google Books]. (Cited on pages 11, 21, 22, 25, and 26.)CrossRefzbMATHGoogle Scholar
 [67]Penrose, R. and Rindler, W., Spinors and spacetime, Vol. 2: Spinor and twistor methods in spacetime geometry, Cambridge Monographs on Mathematical Physics, (Cambridge University Press, Cambridge; New York, 1986). [Google Books]. (Cited on pages 29, 75, and 76.)CrossRefzbMATHGoogle Scholar
 [68]Petrov, A.Z., “The Classification of Spaces Defining Gravitational Fields”, Gen. Relativ. Gravit., 32, 1665–1685, (2000). [DOI]. (Cited on page 26.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [69]Pirani, F.A.E., “Invariant Formulation of Gravitational Radiation Theory”, Phys. Rev., 105(3), 10891099, (1957). [DOI]. (Cited on page 26.)MathSciNetCrossRefzbMATHGoogle Scholar
 [70]Robinson, I., “Null Electromagnetic Fields”, J. Math. Phys., 2, 290–291, (1961). [DOI]. (Cited on page 7.)ADSMathSciNetCrossRefGoogle Scholar
 [71]Robinson, I. and Trautman, A., “Some spherical gravitational waves in general relativity”, Proc. R. Soc. London, Ser. A, 265, 463–473, (1962). [DOI]. (Cited on pages 44 and 50.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [72]Sachs, R.K., “Gravitational Waves in General Relativity. VIII. Waves in Asymptotically Flat SpaceTime”, Proc. R. Soc. London, Ser. A, 270, 103–126, (1962). [DOI], [ADS]. (Cited on pages 7, 10, and 20.)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 [73]Sachs, R.K., “Gravitational radiation”, in DeWitt, C.M. and DeWitt, B., eds., Relativity, Groups and Topology, Lectures delivered at Les Houches during the 1963 session of the Summer School of Theoretical Physics, University of Grenoble, pp. 523–562, (Gordon and Breach, New York, 1964). (Cited on page 24.)Google Scholar
 [74]Sommers, P., “The geometry of the gravitational field at spacelike infinity”, J. Math. Phys., 19, 549–554, (1978). [DOI], [ADS]. (Cited on page 7.)ADSCrossRefGoogle Scholar
 [75]Szabados, L.B., “QuasiLocal EnergyMomentum and Angular Momentum in General Relativity”, Living Rev. Relativity, 12, lrr20094, (2009). URL (accessed 31 July 2009): http://www.livingreviews.org/lrr20094. (Cited on pages 25, 48, and 49.)
 [76]’t Hooft, G., “A Planar Diagram Theory for Strong Interactions”, Nucl. Phys. B, 72, 461, (1974). [DOI]. (Cited on page 73.)ADSCrossRefGoogle Scholar
 [77]’t Hooft, G., “Dimensional reduction in quantum gravity”, in Ali, A., Ellis, J. and RandjbarDaemi, S., eds., Salamfestschrift, A Collection of Talks from the Conference on Highlights of Particle and Condensed Matter Physics, ICTP, Trieste, Italy, 8–12 March 1993, World Scientific Series in 20th Century Physics, 4, (World Scientific, Singapore; River Edge, NJ, 1994). [arXiv:grqc/9310026]. (Cited on page 73.)Google Scholar
 [78]Witten, E., “Antide Sitter space and holography”, Adv. Theor. Math. Phys., 2, 253–291, (1998). [hepth/9802150]. (Cited on page 73.)ADSMathSciNetCrossRefzbMATHGoogle Scholar