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Classical mechanics via general relativity and Maxwell’s theory: a bit of magic

  • Ezra NewmanEmail author
Editor’s Choice (Research Article)

Abstract

In the 1950’s Herman Bondi observed that a very effective way to study gravitational radiation was to use null surfaces as part of the coordinate system for analyzing the Einstein (Einstein–Maxwell) equations. A particular class of such surfaces, (referred to as Bondi null surfaces) with their associated null tetrad, has now been the main tool for this analysis for many years; their use—until recently—has been almost ubiquitous. Several years ago we realized that there was an attractive alternative to the use of Bondi coordinates, namely to use coordinates (in the asymptotic null future space–time region) that were as close to ordinary flat-space light-cones as possible—very different from Bondi surfaces. There were initially serious impediments to this program: these new null surfaces (referred to as asymptotically shear-free surfaces, ASF) were determined by solving a non-linear differential equation (the “good-cut” equation) whose solutions were most often complex. Eventually these problems were overcome and the program was implemented. In a series of papers, using the ASF null surfaces to study the asymptotically flat Einstein (or Einstein–Maxwell) equations, a variety of surprising (strange) results were obtained. Using only the Einstein and Maxwell equations, we found a large number of the basic relations of classical mechanics. They included very detailed conservation laws, well know kinematic relations and dynamic equations and even the Abraham–Lorentz–Dirac radiation reaction force and the rocket force. As interesting as these were, they came with a serious enigma. These relations from classical mechanics had no relationship with the physical space–time. The space for the action of these relations was the parameter space of solutions of the good-cut equation—a complex space, known as H-space. The enigma—what possible relationship did these standard appearing classical relations have with physical space–time? It is the purpose of this work to establish such a relationship—objects in H-space are related to structures in physical space–time. For example, a complex world-line in H-space becomes in physical space–time an asymptotically shear-free null geodesic congruence where its twist describes its intrinsic spin and if charged, its magnetic dipole.

Keywords

Asymptotically flat Einstein equations Maxwell equations Asymptotically shear-free Null geodesic congruences 

1 Introduction: prelude and variations

To begin, we emphasize that we are dealing here simply with classical general relativity coupled with Maxwell theory. There are no additional physical theories, constructs or assumptions that are included. This work extends an idea, developed in a series of recent papers [1, 2, 3, 4], concerning asymptotically flat (in the sense of Bondi–Sachs–Penrose) Einstein or Einstein–Maxwell space–times that led to a large variety of surprising relations that were identical to (or, if one preferred, mimicked) many of the dynamical laws and definitions of classical mechanics. In spite of the clarity of the final results and the clear visibility of their derivation (just sitting in the Bianchi identities) in the end we however were left with a troubling enigma concerning the physical meaning of these results. There was no obvious space–time interpretation of the results. For a long time this was unresolved. As almost all of the details and proofs (sometimes long and complicated) are already in this literature, we basically just give a summary of the relevant ideas, the issues and the problem—and finally at the end describe a (pretty but maybe qualified) resolution to the enigma in physical space–time terms.

The original idea was to replace, in the asymptotic null future of the space–time, the commonly used Bondi (null) coordinate systems with an alternative set of coordinates that were also based on null surfaces. The new null surfaces, near null infinity, were modeled on and closely resembled the standard null cones of Minkowski space. For light-cones in Minkowski space, the optical parameters of Sachs, i.e., their complex divergence and complex shear are \(\rho \; =r^{ -1}\;\) and \(\sigma =0\). Imitating them (in the neighborhood of future null infinity of the asymptotically flat space time) we used coordinates based on null surfaces with optical parameters, \(\rho \; = -r^{ -1} +O (r^{ -3})\) and \(\sigma =O (r^{ -4})\). These surfaces are referred to as asymptotically shear-free (ASF) surfaces. Note that for the Bondi null surfaces the optical parameters are, \(\rho \; = -\,r^{ -1} +O (r^{ -3})\) and \(\sigma =\sigma ^{0}/r^{ -2}\; +O (r^{4})\) (\(\sigma ^{0}\) is the time integral of the Bondi news function).

Two things to note: these ASF surfaces are determined from solutions of a differential equation (the “good-cut” equation [1]) where the solution space is a complex four-dimensional manifold and (unhappily) the solutions are, in general, complex. In other words the set of ASF surfaces are parametrized by four complex parameters, \(z^{a}\), and the surfaces themselves are complex. By taking a complex world-line in the \(z^{a}\) space, \(z^{a} =\xi ^{a} (\tau )\), we had (from the null generators of the cones), an ASF null geodesic congruence—that was complex. For many years these results were a severe impediment to the use of the ASF null surfaces—what meaning was there to such “complex congruences” and surfaces? This complexity presented serious interpretation problems but simultaneously offered a rich opportunity for exploring new structures.

We realized [1, 2] that this complexity issue or situation also arose naturally in the case of real Minkowski space. Consider—purely formally—complex Minkowski space with complex coordinates, za, complex metric, \(ds^2=\eta _{ab} d z^{a} d z^{b}\), and a complex “world-line”, \(z^{a} =\xi ^{a} (\tau ) ,\) with its associated set of complex light-cones with apex on the “line”. The null generators of the cones formed a complex ASF null geodesic congruence. The important realization was that this complex congruence could be mapped into the real space yielding a real null congruence. It was still shear-free, \(\sigma =0\;\) but now it had a new structure, twist \(\Sigma \), with divergence, \(\varrho =(i \Sigma -r)/(r^{2} +\Sigma ^{2})\). The twist \(\Sigma \) is a (real) measure of how far (or where) in the complex plane, the world line was: \(\Sigma =\xi _{I}^{a} (\tau ) (n_{a} -l_{a})\) with \(\xi _{I}^{a} (\tau )\) the imaginary part of the world-line. For each value of the “world-line” parameter, \(\tau \Rightarrow t\) (now real after the mapping), there is an \(S^{2}\) set of geodesics which we refer to as a sub-congruence—a distortion of the light-cone rays due to the twist. The (now) real rays do not focus back to a point but rather to caustics.

This projection or mapping of a Minkowski space complex ASF congruence into the real was generalized to the asymptotically flat space–time case. Similarly to the flat situation, an arbitrary complex world-line in the \(z^{a}\)-space, \(z^{a} =\xi ^{a} (\tau )\), and its associated complex congruence was mapped into the real asymptotically flat space–time yielding (in the real space–time), a real three-parameter family of ASF null geodesics—i.e., a real null geodesic congruence. They again were not surface forming, i.e., they were (in general) twisting. As in the flat case, the twist was again a measure of where in the complex the world-line was. For each value of the (real) parameter \(\tau \Rightarrow t\;\) there was an \(S^{2}\) set of null geodesics, again a sub-congruence. As t varied, we would have in the real space–time itself, a one parameter set of two-dimensional sub-congruences, i.e., as was just noted, a real null congruence.

Remark A (real) ASF coordinate system (replacing the Bondi system) is determined by such a projected real world-line—essentially the real part of \(\xi ^{a} (\tau )\). The coordinates would be the curve parameter t, the complex stereographic coordinates on the \(S^{2}\) of the sub-congruence and the affine parameter, r, along the null geodesics.

Digression We remark that (in either case, flat or asymptotically flat), the complex parameter space of the \(z^{a}\) (referred to as H-space [5]) has a very interesting structure in its own right. In a natural sense, H-space has its own (holomorphic) metric structure—it is Ricci flat and self-dual. It is the natural space of Penrose’s non-linear graviton [6]. We do not use these properties here.

Digression We point out that the mapping of the complex structures into the real is non-trivial. One can not simply take the real parts of relevant expressions and consider that to be the mapping—often one must first perform derivative operations before finding the real parts [1].

In Sect. 2 we will review earlier results [1, 2, 3, 4] where we chose a special world-line in H-space (for our coordinate system near \(I^{ +})\), that we referred to as the complex center of mass and complex center of charge line. This leads to a wide variety of standard relations: definitions, conservation laws and dynamics of classical mechanics. These results (a large generalization of the Bondi–Sachs mass/momentum relations) include, among many others, the standard energy/momentum/angular momentum conservations laws, Newton’s second law, the Abraham–Lorentz–Dirac radiation reaction force, etc. That these results were simply sitting there, to be observed in the Bianchi identities was a huge surprise. No less of a surprise was the resulting enigma; what possible meaning could be given to the physical dynamics and relationships taking place in the complex H-space with no mention or use of the physical space time? We had wrestled with this enigma for many years.

In Sect. 3, we address the enigma and give as we said earlier a simple resolution—embarassingly simple—but pretty.

2 Review of recent results and main development

In order to try to make this work largely self-contained, some of the standard results of asymptotically flat space–times, described in the NP formalism [7], are given. For more details see the earlier references [1, 4].

Our major interest centers on the Weyl (and Maxwell) tensors; their asymptotic behavior, their physical meaning and evolution. The transformation properties of these tensors between the Bondi systems and those based on the ASF congruences, played a huge role. Unfortunately the calculations were long and quite complicated. They involved Taylor series expansions and multiple usage of Clebsch–Gordon expansions. Little will be said about their details. Physical identifications are made with the harmonic coefficients.

We begin by integrating (in a Bondi system) the NP version of the Bianchi identities (BI’s) and Maxwell equations. Using the five complex self-dual NP components of the Weyl tensor and three complex Maxwell components,
$$\begin{aligned} \Psi _{0}= & {} -\,C_{abcd} l^{a} m^{b} l^{c} m^{d}, \end{aligned}$$
(1)
$$\begin{aligned} \Psi _{1}= & {} -\,C_{abcd} l^{a} n^{b} l^{c} m^{d} , \end{aligned}$$
(2)
$$\begin{aligned} \Psi _{2}= & {} -\,C_{abcd} l^{a} m^{b} \bar{m}^{c} n^{d}, \end{aligned}$$
(3)
$$\begin{aligned} \Psi _{3}= & {} -\,C_{abcd} l^{a} n^{b} \bar{m}^{c} n^{d}, \end{aligned}$$
(4)
$$\begin{aligned} \Psi _{4}= & {} -\,C_{abcd} n^{a} \bar{m}^{b} \bar{m}^{c} n^{d}, \end{aligned}$$
(5)
$$\begin{aligned} \phi _{0}= & {} F_{ab} l^{a} m^{b}, \\ \phi _{1}= & {} \frac{1}{2} F_{ab}\,\left( l^{a} n^{b} +m^{a} \bar{m}^{b}\right) , \\ \phi _{2}= & {} F_{ab} n^{a}. \end{aligned}$$
the integration of the radial BI’s and Maxwell equations, led to their asymptotic behavior (the so-called peeling theorem):
$$\begin{aligned} \Psi _{0}= & {} \Psi _{0}^{0} r^{ -5} +O \left( r^{ -6}\right) , \quad \Psi _{1} = \Psi _{1}^{0} r^{ -4} +O \left( r^{ -5}\right) , \quad \Psi _{2} = \Psi _{2}^{0} r^{ -3} +O \left( r^{ -4}\right) , \\ \Psi _{3}= & {} \Psi _{3}^{0} r^{ -2} +O \left( r^{ -3}\right) , \quad \Psi _{4} =\Psi _{4}^{0} r^{ -1} +O \left( r^{ -2}\right) ,\\ \phi _{0}= & {} \phi _{0}^{0} r^{ -3} +O \left( r^{ -4}\right) , \quad \phi _{1} = \phi _{1}^{0} r^{ -2} +O \left( r^{ -3}\right) , \quad \phi _{2} = \phi _{2}^{0} r^{ -1} +O \left( r^{ -2}\right) , \end{aligned}$$
with
$$\begin{aligned} \Psi _{n}^{0} = \Psi _{n}^{0} \left( u ,\zeta ,\bar{\zeta }\right) , \quad \phi _{n}^{0} = \phi _{n}^{0} \left( u ,\zeta ,\bar{\zeta }\right) . \end{aligned}$$
The remaining (non-radial) BI’s and Maxwell equations yield the evolution equations:where \(k =2Gc^{ -4}\) and an overdot denotes the u-derivative.

The \(\sigma ^{0}\), referred to as the asymptotic shear or free radiation data, is the main part of the shear of the Bondi null surfaces, \(\sigma =\sigma ^{0}/r^{ -2} +O (r^{ -4})\) (it usually has the form of a mass and spin quadrupole, \(\sigma ^{0} (u ,\zeta ,\zeta ) =(Q_{M}^{i j} +i Q_{S}^{i j})\; Y_{2ij}^{2} (\zeta ,\bar{\zeta }))\).

Equations (6)–(10) contain our mechanical equations of motion. Before starting into the details, we remark that these functions of (u,\(\zeta ,\bar{\zeta }\)) can be expanded in spin-s spherical harmonics with coefficients being functions of u. Our physical identifications lie with many of the lower harmonic coefficients, \(\ell =0,1,2\).

The first items (of physical identification) are the well known coefficients of the \(\ell =0\) and 1 harmonics of \(\Psi _{2}^{0}\). They are, by definition, proportional to the Bondi–Sachs energy–momentum four-vector [8, 9], \((M_{B} ,P_{B}^{i})\):
$$\begin{aligned} \Psi _{2}^{0}\; =\; -\frac{2 \sqrt{2} G}{c^{2}} M_{B}\; -\frac{6 G}{c^{3}} P_{B}^{i}\; Y_{1 i\;}^{0} +\cdots \end{aligned}$$
(11)
We define (and this is the only other physical identification that is given by definition) the \(\ell =1\) harmonic coefficient of \(\Psi _{1}^{0}\) as proportional to the complex mass dipole moment, \(D_{\mathrm{cmd}}^{i} \equiv D_{{\mathrm{mass}}}^{i} +i c^{ -1} J^{i}\),
$$\begin{aligned} \Psi _{1}^{0} = -6 \sqrt{2} G c^{ -2}D_{\mathrm{cmd}}^{i}\, Y_{1i}^{1} +\cdots , \end{aligned}$$
(12)
where \(D_{\mathrm{mass}}^{i}\) is the mass dipole moment and \(J^{i}\) the total angular momentum.
The \(\ell =1\) coefficient of \(\phi _{0}^{0}\) is proportional to the (conventional) complex electromagnetic dipole moment,
$$\begin{aligned} D_{\mathrm{cemd}}^{i} =D_{E}^{i} +i D_{M}^{i}, \end{aligned}$$
(13)
with \((D_{E}^{i} ,D_{M}^{i})\) the electric and magnetic dipole moments.

The relationships given by the Bianchi identities, Weyl and Maxwell components and definitions are transferred from Bondi coordinates to an ASF coordinate system via an unknown but to be determined world-line, \(\xi ^{a} (\tau )\). Extreme care must be taken going back and forth between independent variables u and \(\tau \).

By now requiring that \(\xi ^{a} (\tau )\) be the complex center of mass world-line, \(\xi ^{a} (\tau ) =\xi _{\mathrm{CofM}}^{a}(\tau )\), the complex mass dipole moment, given in Eq. (12), must vanish from the meaning of ‘center of mass’. We thus set, \(D_{\mathrm{cmd}}^{i} =0\) in the ASF system. Working backwards to the Bondi system, neglecting high powers of \(c^{ -1}\) and quadrupole interactions, the complex mass dipole moment becomes
$$\begin{aligned} D_{\mathrm{cmd}}^{i} \equiv D_{\mathrm{mass}}^{i} +i c^{ -1} J^{i} = -c M_{B} \xi _{\mathrm{CofM}}^{i} +i P_{B}^{k} \xi _{\mathrm{CofM}}^{j} \epsilon _{k j i}+\cdots \end{aligned}$$
(14)
By writing \(\xi _{\mathrm{CofM}}^{i} =\xi _{R}^{i} +i \xi _{I}^{i}\), we have from the real and imaginary parts our first pair of results:
$$\begin{aligned} D_{{\mathrm{mass}}}^{i}= & {} M_{B} \xi _{R}^{i} -c^{ -1} P_{B}^{k} \xi _{I}^{j}\epsilon _{jki} \end{aligned}$$
(15)
$$\begin{aligned} J^{i}= & {} c M_{B} \xi _{I}^{i} +P_{B}^{k} \xi _{R}^{j} \epsilon _{jki} +\cdots \end{aligned}$$
(16)
The second term in Eq. (15) is rather anomalous: in one form or another it appears in several published definitions of center of mass. The \(J^{i}\) in Eq. (16) is composed of the spin term (same as in the Kerr metric) and the standard orbital angular momentum term, \(\mathbf{r} \times \mathbf{P}\). Note that these are derived results and not definitions. They are observed directly as terms in the BI’s.

This was surprising—standard classical mechanics relationships appearing in H-space...

Next, substituting Eqs. (15) and (16) into the Bianchi identity (7), then decomposing it into real and imaginary parts, leads directly (by observation) to the second pair of results:
$$\begin{aligned} P_{B}^{i}= & {} M_{B}\, \xi _{R}^{i\, \prime } -\frac{2 q^{2}}{3 c^{3}}\, \xi _{R}^{i\, \prime \prime } +\cdots ,\end{aligned}$$
(17)
$$\begin{aligned} J^{i \,\prime }= & {} -\frac{2 q^{2}}{3 c^{3}}\, \xi _{I}^{i \,\prime \prime } +\frac{2 q^{2}}{3 c^{3}}\, \big (\xi _{R}^{j \,\prime }\, \xi _{R}^{k\, \prime \prime } +\xi _{I}^{k\, \prime } \, \xi _{I}^{k\, \prime \prime }\big )\, \epsilon _{k j i} +\cdots \end{aligned}$$
(18)
The first of these is the kinematic expression for linear momentum including the term associated with radiation reaction. The radiation reaction term shows up later in the equations of motion. Note that this was simply sitting in the Bianchi Identity and required no further assumptions. The second equation, modulo the derivative term \({2 q^{2}} \xi _{I}^{i \,\prime \prime }/({3 c^{3}})\), which could appear on either side of the equation, is exactly the Landau–Lifschitz [10] angular momentum conservation law with an additional term arising from spin loss.
The third set of results arise directly from the Bianchi Identity, Eq. (6). From the \(\ell =0\) harmonic coefficient, using the Bondi–Sachs \(M_{B}\) and \(P_{B}\) with Eqs. (16) and (18), we obtain the Bondi mass loss equation augmented by the classical electromagnetic dipole and quadrupole energy loss, plus new terms from the spin-quadrupole energy loss:
$$\begin{aligned} M_{B}^{ \prime }= & {} -\frac{G}{5 c^{7}}\, Q_{\mathrm{mass}}^{j k \,\prime \prime \prime }\, Q_{\mathrm{mass}}^{j k \,\prime \prime \prime } -\frac{4 q^{2}}{3 c^{5}} \big (\xi _{R}^{i\, \prime \prime } \,\xi _{R}^{i\, \prime \prime } +\xi _{I}^{i \,\prime \prime }\, \xi _{I}^{i\, \prime \prime }\big ) \nonumber \\&-\frac{4}{45 c^{7}} \big (Q_{E}^{j k\, \prime \prime \prime } \,Q_{E}^{j k\, \prime \prime \prime } +Q_{M}^{j k\, \prime \prime \prime }\, Q_{M}^{j k \,\prime \prime \prime } \big ). \end{aligned}$$
(19)
The \(\ell =1\) harmonic coefficients yield the momentum loss,
$$\begin{aligned} P_{B}^{i\, \prime } =F_{\mathrm{recoil}}^{i}, \end{aligned}$$
(20)
where \(F_{\mathrm{recoil}}^{i}\) is composed of many non-linear radiation terms involving the time derivatives of the gravitational quadrupole and the electromagnetic dipole and quadrupole moments whose details are not relevant for us.
By substituting Eq. (17) into (20), we obtain Newton’s second law,
$$\begin{aligned} M_{B} \xi _{R}^{i\, \prime \prime } =F^{i} =M_{B}^{ \prime }\, \xi _{R}^{i\, \prime } +\frac{2 q^{2}}{3 c^{3}}\, \xi _{R}^{i\, \prime \prime \prime } +F_{recoil}^{i}. \end{aligned}$$
(21)
A further item worth noting: the imaginary part of the complex center of mass position, \(\xi _{I}^{i}\), is used to determine both the spin-angular momentum \(S^{i}=mc\, \xi _{I}^{i} \) and the magnetic dipole moment \(D^{i}_{M} =q\,\xi _{M}^{i}\). They lead to the Dirac value of the gyromagnetic ration, \(g =2\).

This particular result required us to assume that the complex center of charge coincided with the complex center of mass. This is a special assumption and need not be the case.

It must be emphasized that much detail has been left out in this condensed and truncated presentation and discussion of the origin and the appearance of our results in the Einstein–Maxwell equations. Considerable use of Maxwell’s equations was omitted here as well as the use of several different types of ASF coordinate systems. The technical details of the transformations—back and forth—between the Bondi and the ASF coordinates, though very important, would, if presented, lead us far astray. The hope was to give an overview of the strange results—the well-known physical identifications and equations of motion—apparently taking place in H-space—without any physical space–time. They were an enigma—attested to by comments from many referees. One referee did not want us to publish until we understood what connection there was with real space–time.

The enigma is addressed and (at least partially) resolved in the next section.

3 Resolution

Trying to understand our version of Classical Mechanics and what possible meaning it could have, with no obvious space–time in its description, has troubled us for many years. Hours of thought, discussions with friends and colleagues went unrewarded and produced no insights.

Suddenly with not much thought, we understood much of it—a relatively simple (partial?) resolution.

We first recall that points in H-space, i.e., each \(z^{a}\), label an ASF null surface in the complexified asymptotically flat space–time. A curve, \(z^{a} =\xi ^{a} (\tau )\), in the H-space thus represents (in the complexified space–time) a (complex) family of moving ASF surfaces. Unfortunately they are complex and do not represent real space–time objects. However, quoting from Sect. 1: “...an arbitrary complex world-line in the \(z^{a}\)-space, \(z^{a} =\xi ^{a} (\tau )\), and its associated complex congruence, was mapped into the real asymptotically flat space–time yielding (in the real space–time), a real three-parameter family of ASF null geodesics—i.e., a real null geodesic congruence. It again was not surface forming, i.e., it was (in general) twisting. As in the flat case, the twist was again a measure of where in the complex plane, the world-line was. For each value of the (now real) parameter \(\tau \Rightarrow t\) there was an \(S^{2}\) set of null geodesics, again a sub-congruence. As t varied, we would have, in the real space–time itself, a one parameter set of two dimensional sub-congruences, a real twisting null geodesic congruence.”

We have described, in the above paragraph, an interpretation of H-space world-lines as non-local objects in the physical asymptotically flat space–time. In other words, a complex H-space world-line is represented (in the asymptotic region) as the entire twisting null geodesic congruence that is based on an associated real world-line. At this stage we know nothing in general about their focusing (but it is certainly often with caustics) or their intersections with physical sources. If the H-space world-line was the complex center of mass line then the associated geodesic congruence would be a non-local description of the evolution of the center of mass. It would satisfy the classic equations, the kinematics, the conservation laws and the dynamics, described in the previous section. The spin angular momentum would be essentially, \(M_{B} \Sigma \), mass times the twist—and in the Maxwell case the magnetic dipole would be the charge times twist, \(q\Sigma \). There is no implication in this description of center of mass and/or center of charge that a real source is associated with the congruence. The Kerr–Newman metric presents a detailed model of the above scenario—but in that case the caustics do represent the source. It would be attractive and make sense if this picture could be generalized. But perhaps it does not even need a generalization, e.g., every asymptotically flat space–time has this structure.

Although we have given a physical space model of H-space and thus, to a certain extent, resolved or given meaning to many of the earlier results, we still remain with serious issues and questions. The \(\xi _{\mathrm{CofM}}(\tau )\) is interpreted as the twisting space–time congruence, yet its appearance in the orbital angular moment or in the force laws is still rather strange and inexplicable. It seems to be screaming out for a local position meaning—which we do not have.

This has all been done with straight GR/Maxwell theory and temporary use of complexification (modulo the gyromagnetic ratio assumption which did not affect the other relations)—the results follow directly from the Einstein–Maxwell equations. Is there any physical manifestation or prediction from this physical model of the center of mass? Is there something that can be observed from it? What happens with the radiation reaction force and its known (associated) instability? Where would any of this fit with quantum gravity?

At the moment we do not know.

Notes

Acknowledgements

We thank Timothy Adamo for hours of wonderful discussions and collaboration on earlier manuscripts where many of the ideas were developed. Roger Penrose is due—for his insights, encouragement and suggestions—more thanks than we can possibly express.

References

  1. 1.
    Adamo, T.M., Newman, E.T., Kozameh, C.: Living Rev. Relativ. 15, 1 (2012)ADSCrossRefGoogle Scholar
  2. 2.
    Newman, E.T.: Class. Quant. Grav. 33, 145006 (2016)ADSCrossRefGoogle Scholar
  3. 3.
    Newman, E.T.: Class. Quant. Grav. 34, 135004 (2017)ADSCrossRefGoogle Scholar
  4. 4.
    Newman, E.T.: Gen. Relativ. Gravit. 49, 102 (2017)ADSCrossRefGoogle Scholar
  5. 5.
    Hansen, R.O., Newman, E.T., Penrose, R., Tod, K.P.: Proc. R. Soc. Lond. A 363, 445 (1978)ADSCrossRefGoogle Scholar
  6. 6.
    Penrose, R.: Gen. Relativ. Gravit. 7, 171 (1976)ADSCrossRefGoogle Scholar
  7. 7.
    Newman, E.T., Penrose, R.: Scholarpedia 4, 7445 (2009)ADSCrossRefGoogle Scholar
  8. 8.
    Bondi, H., van der Burg, M.G.J., Metzner, A.W.K.: Proc. R. Soc. Lond. A 269, 21 (1962)ADSCrossRefGoogle Scholar
  9. 9.
    Sachs, R.K.: Proc. R. Soc. Lond. A 270, 103 (1963)ADSCrossRefGoogle Scholar
  10. 10.
    Landau, L., Lifschitz, E.M.: Classical Theory of Fields. Addison-Wesley, Reading (1962)Google Scholar

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Authors and Affiliations

  1. 1.Department of Physics and AstronomyUniversity of PittsburghPittsburghUSA

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