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, divergence-free congruences always exist, but for nonflat vacuum spacetimes they exist only in the case of certain high symmetries. On the other hand, twist-free congruences (null surface-forming congruences) exist in all Lorentzian spacetimes. General vacuum spacetimes do not allow shear-free congruences, though all asymptotically-flat spacetimes do allow asymptotically shear-free congruences, a natural generalization of shear-free congruences, to exist.

Our primary topic of study will be the cases of shear-free and asymptotically shear-free NGCs. In flat space the general shear-free congruences have been extensively studied. However, only recently has the special family of regular congruences been investigated. In general, as mentioned above, vacuum (or Einstein–Maxwell) metrics do not possess shear-free congruences; the exceptions being the algebraically-special metrics, all of which contain one or two such congruences. On the other hand, all asymptotically-flat spacetimes possess large numbers of regular asymptotically shear-free 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 shear-free or asymptotically shear-free NGCs. However, over the years, simply by observing a variety of topics, such as the classification of Maxwell and gravitational fields (algebraically-special metrics), twistor theory, -space theory and asymptotically-flat 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, shear-free geodesics. Recent results have shown that the regular congruences – both the shear-free and the asymptotically shear-free 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 -space. For asymptotically-flat 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 asymptotically-flat 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 energy-momentum loss theorem, one of the profound results in GR. The Bondi energy-momentum 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 , has many sub-regions: future null infinity, ; past null infinity, ; future and past timelike infinity, and ; and spacelike infinity, [23]. In the present work, 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 ? 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 , or the Bondi energy-momentum four-vector from the gravitational field (Weyl tensor) at . 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 Einstein–Maxwell 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 shear-free and asymptotically shear-free NGCs.

The most crucial and striking of these overlooked features (mentioned now but fully developed later) are the following: in flat space every regular shear-free NGC is determined by the arbitrary choice of a complex analytic world line in complex Minkowski space, . Furthermore and more surprising, for every asymptotically-flat spacetime, every regular asymptotically shear-free 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 four-dimensional space, -space, endowed with a complex Ricci-flat metric. In other words, the space of regular shear-free and asymptotically shear-free NGCs are both determined by arbitrary analytic curves in and -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 , with specific Bondi coordinates and tetrad, the complex mass dipole moment (the real mass dipole plus ‘’ 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 shear-free 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 Einstein–Maxwell 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 well-known 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, ; a derivation of Newton’s second law, ; and a conservation law for angular momentum with a well-known 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 shear-free NGCs were important in GR was the discovery of the (vacuum or Einstein–Maxwell) algebraically special metrics. These metrics are defined by the algebraic degeneracy in their principle null vectors, which form (by the Goldberg–Sachs theorem [29]) a null congruence which is both geodesic and shear-free. For the asymptotically-flat algebraically-special metrics, this shear-free congruence (a very special congruence from the set of asymptotically shear-free congruences) determines a unique world line in the associated auxiliary complex -space. This shear-free 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 asymptotically-flat situation is, thus, a generalization of the algebraically-special case. Much of the analysis leading to the transformation of the complex dipoles in the case of the general asymptotically-flat spaces arose from generalizing the case of the algebraically-special metrics.

To get a rough feeling (first in flat space) of how the curves in are connected with the shear-free congruences, we first point out that the shear-free congruences are split into two classes: the twisting congruences and the twist-free ones. The regular twist-free 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, . Choose an arbitrary complex (analytic) world line in and construct its family of complex light cones. The projection into the real Minkowski space, , of the complex geodesics (the generators of these complex cones), yields the real shear-free 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 shear-free 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 shear-free 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 shear-free NGCs is greatly facilitated by the introduction of Good-Cut Functions (GCFs). Each GCF is a complex slicing of from which the associated asymptotically shear-free 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 Universal-Cut 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 ; 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 ‘good-cut equation’ and its complex four-dimensional solution space, -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 Einstein–Maxwell 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 shear-free NGCs, as well as their physical applications to both flat and asymptotically-flat 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 is the Newman–Penrose (NP), or Spin-Coefficient (SC), formalism [55]. Spherical functions are expanded in spin- tensor harmonics [59]; in our approximations only the harmonics are retained. Basically, the detailed calculations should be considered as expansions around the Reissner–Nordström 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 along with an exposition of the NP formalism and its application to Maxwell theory and asymptotically-flat spacetimes. There is then a description of , the stage on which most of our calculations take place. The Bondi mass aspect (a function on ) 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 , the Bondi–Metzner–Sachs (BMS) group [16, 72, 56, 65]. The Bondi mass and linear momentum become basic for the physical identification of the complex center-of-mass world line.

Section 3 contains the detailed analysis of shear-free NGCs in Minkowski spacetime. This includes the identification of the flat space GCFs from which all regular shear-free congruences can be found. We also show the intimate connection between the flat space GCFs, the (homogeneous) good-cut equation, and . As applications, we investigate the UCF associated with asymptotically-vanishing Maxwell fields and in particular the shear-free congruences associated with the Liénard–Wiechert (and complex Liénard–Wiechert) fields. This allows us to identify a real (and complex) center-of-charge world line, as mentioned earlier.

In Section 4, we give an overview of the machinery necessary to deal with twisting asymptotically shear-free NGCs in asymptotically-flat spacetimes. This involves a discussion of the theory of -space, the construction of the good-cut equation from the asymptotic Bondi shear and its complex four-parameter 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 Cauchy–Riemann (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, Robinson–Trautman and twisting type II algebraically special metrics, as well as asymptotically stationary spacetimes, and illustrate how the good-cut 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 asymptotically-flat spacetimes: vacuum and Einstein–Maxwell. Here, reviewing the material of the previous section, we use the solutions of the good-cut equation to determine all regular asymptotically shear-free 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 Einstein–Maxwell 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 , of the asymptotically-defined mass and spin dipoles and the electric and magnetic dipoles. The work of Bondi, with the identification of energy-momentum 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 Clebsch–Gordon product decompositions (C); an overview of the metric construction on -space (D); the description of certain real aspects of complex Minkowski space world lines (E); and a discussion of the ‘generalized good-cut equation’ with an arbitrary conformal factor (F).

Living Rev. Relativity 15, (2012), 1
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