Null Geodesic Congruences, Asymptotically-Flat Spacetimes and Their Physical Interpretation

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 asymptotically-flat 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 future-directed 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 far-field 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 future-directed 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 conformally-rescaled 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.

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 shear-free 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 S² 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 Bondi-Metzner-Sachs (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)⁻¹r. 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 shear-free NGC.

An example would be to take a vector on ℑ⁺, say η ^a , and form the spin-weight-one 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, spin-s functions can be expanded in a complete basis set, the spin-s harmonics, \(_s{Y_{lm}}(\zeta, \bar \zeta)\) or spin-s 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, shear-free and asymptotically shear-free NGCs, it is necessary to first define them and then study their properties.

2.4 The Newman-Penrose 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 asymptotically-flat 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 far-field expansions of the Weyl and Maxwell tensors.

The NP version of the vacuum (or Einstein-Maxwell) equations consists of three sets (or four sets) of nonlinear first-order 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 asymptotically-flat 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 (ψ₀ ∼ O(r⁻⁵) and certain uniform smoothness conditions on sideways derivatives), we simply give the final results.

Remark 3. These last five equations, the first of which contains the beautiful Bondi energy-momentum 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 σ⁰, 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 Goldberg-Sachs 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 Goldberg-Sachs 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 Bondi-Metzner-Sachs 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 sub-group. 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 Algebraically-special metrics and the Goldberg-Sachs theorem

Theorem (Goldberg-Sachs). For a nonflat vacuum spacetime, if there is an NGC that is shear-free, 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 shear-free NGC, and a null tetrad exists such that ψ₀ = ψ₁ = 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 ψ₀ and ψ₁ (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 ψ₀ and ψ₁ are spin-weight two and one respectively.

3.1 The flat-space good-cut equation and good-cut 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 shear-free 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 shear-free NGCs.

3.2 Real cuts from the complex good cuts, I

Though our discussion of shear-free 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 line-segments 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 τ-plane 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 light-cones 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 one-parameter family of slices (cuts) Eq. (3.26).

It is the complex point of view of the complex light-cones 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 Asymptotically-vanishing Maxwell fields

4.1 Asymptotically shear-free NGCs and the good-cut equation

It is this pair of equations, (4.1) and (4.2), that will now be generalized to asymptotically-flat spacetimes.

4.2 \({\mathcal H}\)-space and the good-cut 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 shear-free 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.

4.3 Real cuts from the complex good cuts, II

As varies we obtain a one-parameter 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

5.1 Linearized off Schwarzschild

As a first example, we describe how the shear-free 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 zeroth-order 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 Robinson-Trautman 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) Liénard-Wiechert Maxwell fields. The type II algebraically-special twisting metrics are the gravitational analogues of the complex Liénard-Wiechert 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 Einstein-Maxwell equations were studied using a slow-motion perturbation expansion around the Reissner-Nördstrom metric, keeping spherical harmonic contributions up to l = 2. It was found that the above-mentioned world line coincides in this case with that given by the Abraham-Lorentz-Dirac equation, prompting us to consider such spacetimes as ‘type II particles’ in the same way that one can refer to Reissner-Nördstrom-Schwarzschild or Kerr-Newman ‘particles’ [52].

5.4 Asymptotically static and stationary spacetimes

By defining asymptotically static or stationary spacetimes as those asymptotically-flat 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.

6.1 A brief summary — Before continuing

For the case of the Einstein-Maxwell 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 center-of-mass 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 ξ⁰ = 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 center-of-mass world line

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.

6.4 The evolution of the Bondi energy-momentum

Remark 12. The Bondi mass, \({M_{\rm{B}}} = - {{{c^2}} \over {2\sqrt 2 G}}{\Psi ^0}\), and the original mass of the Reissner-Nordström (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 time-dependent quadrupole moment.

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 center-of-mass/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 spin-coefficient equations and the second-order metric tensor. For example, one finds that the dipole Maxwell field induces a second-order Bondi shear, σ⁰. (This in principle would lead to a fourth-order 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 first-order 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.

7.1 The Good-Cut 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, [(n₁, n₂) = (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 four-vector. (Technically, the Bondi four-momentum is a co-factor 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 spin-weight and −s.

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/Reissner-Nordström 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 charged-spinning 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 asymptotically-flat 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 shear-free condition, Eq. (4.12)) was eventually shown to be closely related to Penrose’s asymptotic twistor theory, and in the flat-space case it led to the Kerr theorem and totally shear-free NGCs. From a very different point of view, searching for asymptotically shear-free complex null surfaces, the good-cut equation was found with its four-complex parameter solution space, leading to the theory of \({\mathcal H}\)-space.

Years later, the different strands came together. The shear-free condition was found to be closely related to the good-cut 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 shear-free NGCs there was (at least) one totally shear-free (rather than asymptotically shear-free) congruence coming from the Goldberg-Sachs theorem. Their associated world lines were the ones first discovered in 1965 (coming, by accident, from the complex coordinate transformation), and became the complex center-of-mass 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 algebraically-special 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 algebraically-special metrics, the null tetrad system at ℑ⁺ with one leg being the tangent null vector to the shear-free 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 shear-free 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 algebraically-special metrics or analogies with flat-space 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 (time-dependent 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

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 one-parameter 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

8.6 New interpretations and future directions

Throughout this review, we have focused on classical general relativity as expressed in the Newman-Penrose (or spin-coefficient) 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 shear-free NGCs in asymptotically flat spacetime and complex, twist-free 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 so-called ‘holographic principle’, which aims to equate a theory in a d-dimensional compact space with another theory defined on its d−1-dimensional 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 string-like structures on both sides of the duality makes such a possibility all the more intriguing.

This should be contrasted with the most well-known instance of the holographic principle: the AdS/CFT correspondence [47, 78, 9]. Here, the AdS-boundary acts as the holographic screen between a type IIB string theory in AdS₅ × S⁵ (the virtual image space) and maximally super-symmetric Yang-Mills theory in real four-dimensional 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 string-type 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 AdS₅ × S⁵ 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 light-cones 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 anti-self-dual 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 tree-level 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.