# Editorial note to: On the Newtonian limit of Einstein’s theory of gravitation (by Jürgen Ehlers)

## Abstract

We give an overview of literature related to Jürgen Ehlers’ pioneering 1981 paper on Frame theory—a theoretical framework for the unification of general relativity and the equations of classical Newtonian gravitation. This unification encompasses the convergence of one-parametric families of four-dimensional solutions of Einstein’s equations of General relativity to a solution of equations of a Newtonian theory if the inverse of a causality constant goes to zero. As such the corresponding light cones open up and become space-like hypersurfaces of constant absolute time on which Newtonian solutions are found as a limit of the Einsteinian ones. It is explained what it means to not consider the ‘standard-textbook’ Newtonian theory of gravitation as a complete theory unlike Einstein’s theory of gravitation. In fact, Ehlers’ Frame theory brings to light a modern viewpoint in which the ‘standard’ equations of a self-gravitating Newtonian fluid are Maxwell-type equations. The consequences of Frame theory are presented for Newtonian cosmological dust matter expressed via the spatially projected electric part of the Weyl tensor, and for the formulation of characteristic quasi-Newtonian initial data on the light cone of a Bondi–Sachs metric.

## Keywords

General relativity Newtonian limit Cosmological applications Bondi–Sachs metric Jürgen Ehlers Golden oldie## 1 The Newtonian Limit

To set the spirit of Jürgen’s paper [1]: whenever a notion such as “Newtonian limit of general relativity” appears to be used as if it were standard, trivial or just textbook knowledge, Jürgen’s fine sense of physics immediately raised the important question of whether this is indeed so, leaving wide open the possibility of understanding something else under this seemingly common sense expression. This happened often in personal interaction of one of us (TB^{1}) with Jürgen and it is this way of questioning that keeps the sense of physics awake. To give another example: in some work I (hereafter, I refers to TB) had hidden assumptions in what I called “relativistic ideal gas”. Jürgen did not just use the word “wrong”, but surprisingly raised the question of what this expression means to me, developing from scratch on the blackboard elements that should define an ideal gas, as if this were his first thoughts of curiosity about this notion.

The paper [1] is such an example of thoroughly asking this question. This pioneering article laid the ground for fruitful follow-up developments in rendering the notion of “Newtonian limit” precise. Indeed, Jürgen calls practical implementations of the Newtonian limit empirical (or heuristic), which is “useful for preliminary considerations [...] while making a suitable ansatz for the metric and matter variables in which small quantities are neglected [...] i.e. they are set to zero. [1, p. 2]”. He emphasizes in his introduction to the problem the absence of a corresponding mathematical justification in the literature. He spells out that empirical tests of general relativity already presuppose the existence of a limiting process.

Although later publications provide a shortcut or incorporate new developments of what is known as “Ehlers’ Frame Theory”, i.e., a theory that comprises both a Newtonian “theory” and general relativity within one framework,^{2} the present translation of his 1981 paper [1] reveals best Jürgen’s thoughts when generalizing the work by Élie Cartan, Kurt Friedrichs, and others.^{3} Therein, it allows the reader to enter his strategy of exploring and discovering afresh a notion that is thought—within a broad community that works empirically with the Newtonian limit—to not contain any ambiguous issue.

The reason why again the Newtonian “theory” is put into quotations above reflects Jürgen’s constant reminders that the equations of Newtonian gravitation in their continuous formulation do not represent a theory.^{4} There are elements that leave solutions to these equations non-unique, so that it makes no physical sense even to write down the Euler–Poisson system of equations without specifying and using three elements: (i) the introduction of a reference background space (see, however, [10] for the case of vacuum where *a priori* no background is present), (ii) boundary or fall-off conditions on the deviations from the reference background that need to be specified at all times, and (iii) translation invariance of Newton’s equations. A thorough investigation of these three elements that make Newton’s equations a theory can be found in [3]. This latter paper was largely motivated from Jürgen’s side by rendering the architecture of Newtonian cosmologies precise, guided by the uniqueness problem of Newton’s equations in continuum form, from my side motivated by the averaging problem in cosmology. This is also reflected by the *degeneracy* of the Newtonian limit, summarized in compact form in the language of fibre bundles and the contraction of the Lorentz group to the Galilei group, that Jürgen explains in the introduction. We will come back to this later.

Considering the Newtonian limit *in terms of equations and geometrical concepts* lies behind Frame theory, i.e., using the same geometrical language for both as the starting point, and assigning the parameter \(\lambda = c^{-2} > 0\) to general relativity and \(\lambda =0\) to the limiting theory. In other words, Frame theory renders the Newton–Cartan limit of general relativity mathematically precise. Jürgen’s approach to Frame theory is in fact a special (and less formal) application of a more general procedure [11] to consider the limit of space-times. As Jürgen also points out, Frame theory may help in improving perturbation schemes incorporating relativistic corrections—and indeed his work served later to improve on post-Newtonian expansions [8]. It already surfaces in [1] whose essential aim is to understand the limit of a family of solutions, and not only the limit of the physical laws. Frame theory allows us to explore both. As we know, some Newtonian solutions have no relativistic counterpart, and some general-relativistic space-times have no Newtonian solution as limit. Jürgen enriched this step by focussing on such solutions in his 1997 paper [7]. But, already at the end of [1, Sec. 4], we learn about solutions, e.g., that spherically symmetric isolated relativistic stars have a corresponding Newtonian star model as limit, and that Schwarzschild as well as Kerr black holes degenerate in the Newtonian limit to the field of a point mass, and Jürgen discusses the limit of a plane gravitational wave. Frame theory also led to an existence theorem for a class of stationary axisymmetric solutions of general relativity [12]. Jürgen emphasizes, however, that the interest in forming a notion of the limit through examples should rather focus on non-stationary situations with radiation. The first investigations in this regard were made by Winicour, soon after publication of [1], who discussed the construction of initial modes based on a Newtonian model for the radiative Bondi–Sachs metric of general relativity. The basic idea behind this construction is outlined in Sect. 6, and some consequences of such an initial model with regard to the (radiative) asymptotic structure of general relativity are briefly mentioned in Sect. 10.

## 2 Two theories

People have often pointed out that the concepts behind general relativity and Newtonian physics are entirely different, and one has to understand not only formally how the Newtonian equations arise through a limiting process from the geometrical theory of Einstein, but also how the content of concepts translates from one to the other. Here we do not necessarily enter such philosophical terrain with this question. Jürgen mentions related thoughts by Kuhn and Feyerabend, who suggest that the existence of a limiting process is not obvious: Einstein’s theory rather replaces Newton’s for the description of gravitation as the common physical object. It is true that Frame theory already uses concepts of general relativity and formulates Newton’s laws accordingly. Jürgen argues in [1] that with the transition to a formally simpler theory implications of the more complicated theory get lost; “the *formal* simplification is connected with a *conceptual* complication [1, p. 16]”. But, he also stresses that with the help of Frame theory the meanings of basic notions of the two theories can be compared, not only the formal structures, “despite Kuhn and Feyerabend [1, p. 16]”. Jürgen carefully collects common concepts and assigns them to the, within the Frame theory, common geometrical object of *connection*. He emphasizes that both theories are based on a symmetric linear connection for those aspects of the motion of matter that correspond to the notions of inertia or ponderousness (gravity), thereby revealing the common foundation of equivalence of inertial and ponderable (gravitational) mass.

I remember that Jürgen nuanced his view in discussions of the Newtonian limit later: “we should either work with general relativity *or* with Newtonian laws to describe a self-gravitating system”, which sounded as if there remain incommensurabilities of at least some conceptual aspects of a “limiting procedure”. It is also a matter of physical interpretation. For example, both on the level of equations and solutions, the same equations and solutions govern an expanding homogeneous-isotropic dust fluid in both theories, but in general relativity it is interpreted as an expanding space within a Lorentzian space-time, while in Newtonian cosmology it is a fluid expanding within an embedding Euclidean vector space. Although Frame theory delivers an unambiguous answer, mathematically, his statement may reflect aspects of the degeneracy of the limit, which we are going to discuss now.

## 3 Opening-up light cones—part I

Another element that is nowadays crucial for any theory is causality. That said, the Newtonian “limit” appears as a step back: why should we at all construct the limit where, in Jürgen’s words “the light cones open up [1, p. 3]”, thus sending the causal propagation speed to infinity and losing another essential element of a “theory”. Jürgen and I (TB) once discussed in private the ideas of gravitoelectromagnetism (see, e.g., [13]), and came to the observation that the first steps towards a finite propagation speed of gravitation may have been made by Heaviside [14] around the time of his derivations of key-stones of classical electrodynamics (such as his derivation of the Lorentz force). Especially, Heaviside’s idea of a gravitational analogy to electrodynamics pointed to a feature of Frame theory—Equations (15), (18) and (19) in [1]—by suggesting a Maxwellian form of the gravitational equations in the limit of setting the parameter \(\lambda = c^{-2}\) of the Frame theory to zero. Jürgen points out the existence of terms missing in standard writings of Newton’s equations, featuring a vector field \(\varvec{\omega }\), where the limiting theory assumes the classical Newtonian form iff \(\varvec{\omega }=0\).^{5} It has to be emphasized that Jürgen’s field \(\varvec{\omega }\) in Eqs. (15, 18, 19) results from non-inertial force terms which have to be added to the classical Newtonian system commonly written in non-rotating coordinates. The presence of these additional terms implies the non-existence of global inertial systems, which is also a property of Einstein’s theory. In Jürgen’s words the limiting theory for \(\lambda =0\) reduces to Newton’s equations, “...if the harmonic axial vector field \(\varvec{\omega }\), i.e. the vorticity field, is constant in every hypersurface \(t=const\), so that \(\varvec{\omega }\) depends only on *t*. That is to say, only in this case is there a subset of *non-rotating coordinate systems* in the class of Galilean coordinate systems, with respect to which \(\varvec{\omega }=0\), ... [1, p. 8]”. However, since the field \(\varvec{\omega }\) is a harmonic vector field, this is exactly the freedom that is fixed by the three elements mentioned above that render the Newtonian equations a theory [3]. Jürgen would have certainly added a remark after the results of the above paper that identifies the harmonic freedom in Newton’s equations. The harmonic part in the gravitational potential is for example fixed to be zero in the following case [3]: introducing a reference background, the solutions to Poisson’s equation for the deviations off this background are unique, \(\varPhi = \varPhi _b + \phi \), \(\phi ' = \phi + Z\), with the potential *Z* obeying the Laplace equation, iff the integral over the source \(\delta \varrho : = \varrho - \varrho _b\) vanishes. Imposing periodic boundary conditions on the deviation fields, i.e., assuming a \(3-\)torus topology for the deviation fields, leaves only spatially constant solutions of the Laplace equation. Translation invariance of Newton’s equations then allows one to set, without loss of generality, the harmonic potential to zero. A similar argument for a rigidly rotating \(3-\)torus holds for the harmonic vector field \(\varvec{\omega }\). In general relativity, exact statements on the harmonic parts are more involved and can be understood through Hodge-de Rham theory [15].

We finally quote Jürgen again: when formulating both theories geometrically, “the “only” difference in the formulation of the two theories is that the dimensional measures of space-time in Newton’s theory are given by a so-called Galilei Structure [...], while in Einstein’s Theory they are described by a Lorentz metric. [1, p. 2]”, i.e., a Lorentz-covariant Newtonian theory would alter the big picture essentially. But, is this really the whole story, and would this remove the degeneracy of the Newtonian limit? We will see below that the answer is not in the affirmative; this statement concerned the formal structure of the equations. There is a subtle difference that is related to the connection, which in Newtonian theory is *integrable*.

Before we discuss this important point, we take up the gravitoelectromagnetic analogy again as a first, and a transformation of the Newtonian equations to Lagrangian coordinates as a second, argument to illustrate that formal considerations of the Newtonian limit on the basis of equations depends on how we look at Newton’s equations, and this gives a higher voltage to Jürgen’s emphasis on considering the limit of *solutions*. Note in this context the first general result on the existence of a Newtonian limit for a large class of *solutions* of the Vlasov–Einstein system [16] and subsequent developments [8]. What we can learn from the Frame theory is the need for a covariant formulation as exemplified by the Newton–Cartan theory, simply because common considerations of Newton’s equations are based on writing the equations in coordinate components.

## 4 Newton’s theory written in coordinate components

In order to illustrate the various forms we could expect from writing Newton’s equations in coordinate components, we provide two forms presented in the following excursions. Through the trivial aspect of coordinate dependence we will discover important aspects related to the Newtonian limit, where the second form belongs to a development that Jürgen too followed in depth in the 1990s.

### 4.1 Eulerian form of Newton’s equations

*t*, evolving out of initial data at time \(t=t_{\mathrm {ini}}\), the evolution equations,

*ad hoc*to the field equation for \(\mathbf{g}\) the term corresponding to Heaviside’s “missing term” (the source term for the curl of the field strength \(\mathbf{g}\) that makes the analogy complete and the speed

*c*of propagation of the gravitational interaction finite, and which would render the gravitational force non-conservative). Note that this term is nothing but the gravitomagnetic analogy to Maxwell’s displacement current, on which Heaviside’s argument is based. We notice that, by dropping again this term (equivalent to sending

*c*to infinity), Newton’s equations still feature a gravitomagnetic field. Setting this field to zero or restricting it to a harmonic vector field would result in a highly restricted class of solutions to Newton’s equations (e.g., for irrotational flows the gradient of the density has to be aligned with the velocity). The above set of equations shows that Newton’s theory can come with a different face, the gravitomagnetic field arises only implicitly, rather than explicitly, in the limit of the Frame theory. As for the analogy to Maxwell’s displacement current we may speculate that Newton would have included it, had he lived at a later epoch, hence the question of the light cones opening-up would not arise in a formulation of the limiting process of general relativity to a Lorentz-covariant vector theory of gravitation.

^{6}

### 4.2 Lagrangian form of Newton’s equations

*t*:

^{7}

*etc*. means any functional definition of other fields, written in terms of \({\mathbf {f}}\) and its derivatives, and where \(\varrho _{\mathrm {ini}} ({\mathbf {X}})\) and \(\varvec{\omega }_{\mathrm {ini}}({\mathbf {X}})\) stand for the initial values of the density and vorticity fields; an overdot denotes the partial time-derivative, and \(\varvec{\nabla }_\mathbf{0}\) the nabla operator with respect to the Lagrangian coordinates. Contrary to the Eulerian description, a single variable \({\mathbf {f}}\) describes the gravitational dynamics. In the Lagrangian approach, the Eulerian position \({\mathbf {x}}={\mathbf {f}}({\mathbf {X}},t)\) is no longer an independent variable; the independent variables are now \(({\mathbf {X}},t)\). According to the equivalence of inertial and gravitational masses, we can express the field strength in terms of the acceleration \({\mathbf {g}}={\mathbf {b}}=\ddot{{\mathbf {f}}}\). The Eulerian evolution equations (1) and (2) reduce to definitions, but we have to transform the Eulerian spatial derivatives in the field equation (4):

*Lagrange–Newton system*[4, 18, 19], appears to depend on the gradient of \({\mathbf {f}}\), but not on \({\mathbf {f}}\) itself. Since the constraints were transformed to evolution equations, the elliptic character of the Newtonian equations in Eulerian form seems to have got lost, but it re-surfaces in the construction of the initial data for the solutions of the Lagrange–Newton system.

*Lagrange–Newton system*in terms of properties of the

*Newtonian tidal tensor*\(E_{ij}\),

*irrotational dust*. Splitting the Einstein equations according to the \(3+1\) formalism, the (local) Lagrangian coordinates can be used as Gaussian normal coordinates. Expressed in the local exact basis \(\{{\mathbf {d}} X^i\}\), which spans the local cotangent spaces on the manifold, the dynamics of the fluid is no longer described by the gradient of the Newtonian trajectory function \({\mathbf {d}} f^{\mathbf {a}} = {f}^{\mathbf {a}}_{\ \vert i}{\mathbf {d}} X^i\), but by its relativistic counterpart: the non-integrable Cartan coframes \(\varvec{\eta }^{\mathbf {a}}= \eta ^{\mathbf {a}}_{\ i} {\mathbf {d}} X^i\). The 4-dimensional metric can be decomposed according to

*restriction*of a general one-form to an exact one, where exactness of the form implies the existence of an embedding vector space. Although \({\mathbf {d}}{\mathbf {d}} f^{\mathbf {a}} = {\mathbf {0}}\), compared to \({\mathbf {d}}\varvec{\eta }^{\mathbf {a}} \ne {\mathbf {0}}\) (leading to the Cartan connection one-form and, with another exterior derivative, to the curvature two-form in the Cartan structure equations), the Newtonian connection in the Lagrangian coordinate representation is nonzero [21, Sec. III.A.4],

*integrable*. Jürgen points out that [1, p. 2] “The decomposition of the gravitational connnection [...] into a flat gravitational connection and a vectorial gravitational force field is in fact locally always possible”, but “only unique under the certain condition—i.e. an additional global restriction, which is in general not valid for cosmological models—of an asymptotically flat gravitational connection.” He stresses here that this restricting condition is in general not possible in cosmological models (a remark which is echoed in the exact treatment of Newtonian cosmologies in [3]). Further below he says [1, p. 9]: “It seems that there is no local condition [...] such that Newton’s theory is a result of the restriction \(\lambda =0\) in Einstein’s theory with \(\lambda > 0\).”

Also note that Lagrangian observers are at rest, so that they do not “see” the light cone. The usual limiting process only applies to the \(4-\)dimensional coframes, where the restriction to exact forms represents the Einstein field equations in Minkowski space-time. This leaves the opening-up of the light cones as a separate limiting process in the above \(3+1\) formulation of the Newtonian limit. We conclude that, even if we consider a Lorentz-covariant form of Newton’s theory, the “limit” is degenerate in the sense that we have to impose *local* integrability of the spatial deformation with the consequence that the limiting theory is embedded into a vector space, together with corresponding *global* issues of non-uniqueness, unless boundary or fall-off conditions are specified.

## 5 What is Newtonian theory?

We propose a gedankenexperiment with the following troubling remark. Looking at the Lagrange–Newton system that depends on the gradient \({{\mathbf {d}}}{} \mathbf{f}\) in Eqs. (11), any realization (in terms of a numerical realization) would actually make the spatial deformation non-integrable.^{8} Hence, in practice, we would realize a non-exact one-form and consequently we would create a non-integrable spatial connection and spatial curvature. In fact, we would look at the electric part of Einstein’s equations (15).^{9} Both on the level of equations and the form of the solutions, the results would coincide and the difference would lie only in the Cauchy problem, i.e. the comparison of admissible initial data in general relativity and those initial data that are admissible in the elliptic Newtonian initial value problem [23]. Einstein’s theory delivers, together with Cartan’s structure equations, the complementary set of equations needed to fully determine the Cartan deformation, hence the metric.

We can reformulate the above in another context. We look at a further common sense expression. The Newtonian theory has a vanishing spatially projected magnetic part, \(H_{ij}\), of the Weyl tensor. From the geometrical perspective this is true and we can show this also in Frame theory [5]. However, Jürgen notes [1] that the local equation of motion follows from the field equations in Einstein’s theory, but for \(\lambda =0\), “the structures and laws are independent from each other in the limiting theory NG [1, p. 16]”. We will see that there is a non-vanishing term corresponding to the (vanishing) geometrical part \(H_{ij}\) in the equations of motion that, however, decouples from the geometry in the limit. To see this we recall that the Einstein equations for an irrotational dust model can be cast into a system of evolution equations and propagating constraints where the former reduce to a coupled system of ordinary differential equations for the restmass density \(\varrho \), the rate of expansion \(\varTheta \), the rate of shear \(\sigma ^2 : = (\sigma ^i_{\ j}\sigma ^j_{\ i})/2\), and the trace of the spatially projected electric part of the Weyl tensor \(E^2 : = (E^i_{\ j}E^j_{\ i})/2\) [24, 25] by setting the corresponding magnetic part to zero. These so-called *silent universe models* (see [26] and references to earlier work therein) are often used in cosmology. Now, the Newtonian theory also allows us to write the same set of evolution equations (where \(2 E^2\) is the tidal tensor multiplied by itself), but it is *not* “silent”, although the geometrical complement, the spatially projected magnetic part of the Weyl tensor, vanishes in the Newtonian limit. Instead, there is a term that links the system of evolution equations to the (non-local) constraints, the absence of which would again restrict the class of solutions drastically, see [5, 27].

These properties of Newtonian equations on the one hand illustrate what Jürgen means by the fact that the limiting procedure provides equations of motion that are decoupled from the geometry, or as already quoted above that “the *formal* simplification is connected to a *conceptual* complication [1, p. 16]”, but on the other hand they point to the reason for the decoupling of the evolution equations from the geometry: this is furnished by the restriction to integrability of Cartan deformations and connections, which is purported by the word “degenerate limit”, and which is strictly speaking independent of the limiting process of sending \(\lambda \) to zero.

While enjoying Jürgen’s paper, the reader may especially look at Jürgen’s list of questions in [1, Section 3] as a guide for reading. Jürgen answers aspects of these questions, but a number of issues remain open until today. We therefore contemplate some further aspects of Jürgen’s questions in what follows.

## 6 Jürgen’s fifth question

to which he added in his personal printout of the article, encoded in handwritten notes, what is the(5a) “Do approximation formulae, e.g., for asymptotic representations, for the fields \(g_{ab}(\lambda ), \ldots \) for \(\lambda \rightarrow 0\), follow from the existence of a Newtonian limit of a family of Einsteinian solutions?”

Soon after publication of Jürgen’s article, Jeffrey Winicour presented a series of articles [28, 29, 30] answering a modified version of Jürgen’s fifth question, and he also makes statements on the differentiability with respect to \(\lambda \) at \(\lambda =0\). Winicour’s motivation is the numerical application of such approximation formulae for the characteristic formulation of general relativity and especially its relation to the Bondi–Sachs formulation [31, 32] (see e.g. [33, 34] for review) of the latter. His work was commented by his contemporary colleague Bernard Schutz as [35] – “...The null cone approach to the Newtonian limit described by Winicour is one of the most original ideas to have emerged in this subject recently...”.(5b) “differentiability for \(\lambda \) at \(\lambda =0\)?”

Provided the \(\varrho _N\), \(v^i_N\) and \(\sigma _N\) are the data for the Einstein-fluid equation of a Bondi–Sachs metric \(g_{ab}(\lambda )\) on an interval \([u_0,u_1]\) of retarded times \(u\in [u_0,u_1]\), and \(\varrho _N\), \(v^i_N\) and \(\sigma _N\) are the density, the three velocity \(v^i_N\) and the gravitational shear of the corresponding Newtonian limit of \(g_{ab}(\lambda )\) at \(\lambda =0\), then [28]:(1) What is the appropriate gravitational data such that the limit \(\lambda \rightarrow 0\) gives the Newtonian gravitational structure for this fluid?

(2) Do \(\varrho _N\), \(v^i_N\) and \(\sigma _N\) remain valid throughout this time interval, or does the \(\lambda =0\) limit of the Einsteinian fluid evolve away from its Newtonian counterpart?

An exemplary system with a Newtonian limit is a core-collapse supernova scenario, where the initial system, say (I), is an degenerated iron core of an isolated star of mass^{10} \(M<8M_\odot \) (see e.g., [38, 39, for an astrophysical review] [40, in the context of the Bondi–Sachs formalism] and [41, where a star collapses to a black hole via formation of an intermediate (proto-)neutron star (see Fig. 1)].^{11} The state (I) is ‘clearly’ Newtonian as can be seen by calculating the relevant compactness \(C = GM / (c^2R)\) of the object, i.e., \(C\lesssim 10^{-6}\). The iron-core collapses due to the lack of radiation pressure, caused by the stopping of the nuclear fusion in its centre, to balance the gravity of the fluid. During the collapse phase, (II), of the fluid, gravitational waves are in general emitted from the region where the fluid resides. This is because the collapsing fluid behaves to lowest order as a time-varying quadrupole since the initial model (I) is in general rotating.^{12} The final remnant of stage (III) after the collapse is a proto-neutron star of mass \(M\lesssim 1.4 M_\odot \), which is a weak relativistic system where \(C\sim 10^{-1}\). In this example, the physically investigated system gives rise to a space-time that does not have “holes” in the sense required by Jürgen [1, see Sec. 3]. That is, space-time has a compact space containing the stellar fluid and an empty region commencing at the boundary of the fluid.

Since all of the mass is confined to a compact space, it is possible to single out a centre-of-mass world line of the fluid. It is thus natural to choose freely falling (Fermi) observers [42] along this world line, and choose the outgoing null cones emanating from this world line to define a one-parametric family of Bondi–Sachs metrics \(g_{ab}(\lambda )\) (such a construction is demonstrated in the axisymmetric vacuum case in [43]). Studying the time evolution of \(g_{ab}(\lambda )\) from stage (I) via the radiative period (II) to the final stage (III) has mainly been done numerically, because on the one hand the equation of state for the fluid matter is in general provided in tabular form, while on the other hand the underlying Einstein-fluid equations are highly nonlinear. However, the numerical solution provides an answer to Jürgen’s question stated above if \(\lambda =0\) is chosen for the numerical solution of the one-parametric family of metrics \(g_{ab}(\lambda )\).

Despite the problems of nonlinearity and the issue with the equation of state, Winicour is able to make qualitative statements on the evolution of the initial data. But, before a numerical study can be made, the initial data at stage (I) for the Bondi–Sachs metric at some retarded time \(u=0\) must be provided. This requires the ‘correct’ determination of the gravitational shear \(\sigma \) for the Newtonian system in the (stationary) stage (I). Winicour’s two main articles [28, 29] deal with this question. For its answer, the following assumptions on the Newtonian initial model at stage (I) are made: (i) the fluid is a perfect fluid with matter density \(\varrho \) and an equation of state \(p(\varrho ,\lambda )\), (ii) \(w^i\), the three-dimensional velocity field of the matter, (iii) \(\varPhi \), there is a Newtonian gravitational (connection) potential, (iv) the matter is coupled with the connection potential via the Poisson equation, and (v) the (non-relativistic) Euler equations.

## 7 Opening up light cones—part II

*t*. The limit of (17) for \(\lambda =0\) provides an empty Newtonian space-time structure with

^{13}

*u*, if \(\lambda =1\), and to the Newtonian ‘absolute time’, if \(\lambda =0\). The gradient to the surfaces \(u=const\) is

*c*(

*t*) passing through all of the points where \(x^i=0\) in the hyperplanes \(t=const\). The function

*r*measures the distance between the vertex of a given cone \(u=0\) and points on this respective cone. The three-dimensional vector \(r_i\) measures the direction cosines with respect to a parallel propagated frame along

*c*(

*t*). As for the relation \(x^i = rr^i\), it is useful to parameterize \(r^i = r^i(y^A)\) by two angular coordinates \(y^A\) that are constant along the generators of the cone. Then \(y^A\) measure the direction angles of light rays emanating from the vertex along the light cone \(u=0\). The ‘propagation velocity’ of light along these rays is \(\lambda ^{-1/2}\). Now, consider one arbitrary light cone \(u=0\) at proper time \(t=0\) on

*c*(

*t*). Letting \(\lambda \rightarrow 0\), we observe in the behaviour of the tangent vector (18) the ‘opening up’ of the light cones mentioned in Jürgen’s introduction. In this limit, \(k_a\rightarrow t_a\), and

*r*measures the space-like distance in the hyperplane \(t=0\) with respect to the point \(x^i=0\), while \(y^A\) measure spherical angles with respect to the origin \(x^i=0\).

*c*(

*t*) is found by a simple coordinate transformation of (17) to null polar coordinates \(y^a = (u, y^1=r, y^A)\), so that

## 8 One-parametric families of Bondi–Sachs metrics \(g_{ab}(\lambda , y^c)\)

For the investigation of the Newtonian limit of an Einstein-fluid system, the above construction needs to be done at the centre-of-mass world line of the initial model at stage (I). The regularity requirements of the metric along the centre-of-mass world line imposed by a freely falling observer introduces constraints on the metric at the vertices of the null cones that have to be maintained throughout the evolution of the system from stage (I) to stage (III). As such, the boundary conditions are fixed at the origin of the system, which is contrary to fixing the boundary conditions at infinity as proposed by Jürgen [1, see Theorem 1 and the conjecture in Sec. 3]. The reason for the former is that the Einstein equations in their Bondi–Sachs formulation are integrated along the null rays emanating from the vertex, and the vertices of the null cones must follow a regular world line. Moreover, setting up a Bondi evolution at null infinity is ill-defined as it requires data, the news function \(N:=\frac{1}{2}\sigma _{,u}\), in the future because *N* is a retarded time derivative.

*V*and \(h_{AB}\) are represented by

^{14}The \(\lambda \)-factors in (21) are chosen such that the Einstein equations ensure a smooth limit for the fields \(\beta , W, U_A\) and \(\gamma _{AB}\) at \(\lambda =0\). This partially answers Jürgen’s (handwritten) question on the differentiability for \(\lambda =0\). The conformal metric \(h_{AB}\) has the determinant of the unit sphere \(\det (h_{AB}) = \det (q_{AB}) = \sin ^2\theta \). Setting \(\lambda =1\) in (21) yields the standard form of the Bondi–Sachs metric [31, 32, 33]. The symmetric \(2-\)tensor \(\gamma _{AB}\) is transverse-traceless and, due to the choice of coordinates, it gives rise to the shear tensor \(c_{AB}\) of the outgoing null cones with tangent vector \(k^a\),

## 9 The Newtonian limit of a family of null cones

Due to the implications of the twice contracted Bianchi identities and due to the requirement of the regularity conditions imposed at the vertices of the null cones, only six of the ten Einstein equations need to be considered for the Bondi–Sachs metric (21) (see e.g. [33] for demonstration). Four of these six equations are hypersurface equations not containing any time-derivatives, and the remaining two equations describe the retarded time-evolution of \(\sigma \). Only the general structure of these equations is required here, not their particular form.

### 9.1 Consequences of the hypersurface equations

*p*of the form:

*W*are known at the vertices \(r=0\),

^{15}then this provides knowledge of \(\sigma \) and \(T_{ra} (\varrho , v_a)\) everywhere on a null cone \(u=u_0\); the equations (27) can then be solved in a hierarchical manner. For example, given \(\sigma \) on a cone \(u_0\) and initial values for \(h_{AB}\) at the vertex, (27a) can be solved for \(h_{AB}\) on the entire cone \(u_0\); then, with the so-obtained \(h_{AB}\), \(\beta \) can be found on the cone \(u_0\) using (27b), provided we have knowledge of \(T_{rr}\) on all of \(u_0\) and an initial value for \(\beta \). This hierarchical scheme proceeds until

*W*is found after integration of (27e). In particular, we note that

*W*is completely determined by the shear \(\sigma \), the fluid data \(\varrho \) and \(v_a\), that is

*W*must necessarily restrict the gravitational shear \(\sigma \), meaning

*O*(

*r*) at the vertices \(r=0\), and which allows for such an inversion. On top of that, this scalar field has the desired decomposition of the connection [1, Eq. (39)],

*r*, \(\theta \) and \(\phi \)) and a space-like vectorial gravitational field \(g^{ae}\varPhi ^*_{,e}\) in the Newtonian limit \(\lambda =0\), which also obeys the Poisson equation in the limit \(\lambda =0\). Denoting with \(Y_{lm}(y^A)\) the standard spherical harmonics, \(\varPhi ^*\) differs from the usual Newtonian potential \(\varPhi \) by monopole-dipole terms:

Relation (32) is in principle the answer to Winicour’s question (1) and it gives rise to a ‘simple’ algorithm to determine the initial data, i.e., \(\sigma \), for a quasi-Newtonian system at a null cone with a Newtonian limit for \(\lambda =0\): first, the Poisson equation for the potential \(\varPhi \) is solved for the Newtonian system; second, the gravitational shear \(\sigma \) is determined from (31) and (32), where the boundary conditions are fixed at the origin. Note that, as \(\sigma \) is a complex field of spin-weight-two, it can always be written as \(\sigma = q^Aq^B\eth _A\eth _B \varSigma \), where \(\varSigma \) is a complex scalar field of spin-weight-zero. The real and imaginary parts of \(\varSigma \) are called the electric and magnetic parts of \(\varSigma \), respectively. This nomenclature is adopted from the electric/magnetic decomposition of the Faraday tensor of the electromagnetic field, and corresponds to a curl/gradient decomposition of tensor fields on a sphere (see e.g. [47] for a related discussion with regard to gravitational waves).

### 9.2 Consequences of the evolution equations

^{16}To some extent this observation also gives a further answer to Jürgen’s question (5b), indicating that the differentiability in \(\lambda \) is necessary to obtain a bona fide Newtonian limit of his “Frame theory”.

The answer to Winicour’s second question (2) on the validity of the initial data with regard to its Newtonian limit during a retarded time-evolution requires consideration of the evolution Eq. (33) and the local conservation law of energy-momentum, \(\nabla _aT^a_{b}=0\), at leading order in \(\lambda \). Energy-momentum conservation in the limit \(\lambda =0\) ensures that \(\varrho _N\) and \(v^i_N\) remain valid throughout the time interval \([u_0, u_1]\). To investigate the time-evolution of \(\sigma _N\) for the evolution on the retarded time interval \([u_0, u_1]\), all fields \({{\mathcal {F}}\in \{\beta , h_{AB}, U^A, V, \varrho , p, v_a\}}\) are assumed to have a smooth \(\lambda -\)expansion of the type \({\mathcal {F}}(\lambda ) = \sum _n {\mathcal {F}}_{(n)} \lambda ^n\). Inserting these expansions into the evolution equation (33) yields an iterative scheme, where the retarded time-derivative of the shear \(\sigma _{(n)}\) is determined by the \(\sigma _{(n-1)}\) coefficient. The \(\sigma _{(1)}\) coefficient is unconstrained and may be used to fix the initial data on the cone \(u=u_0\). The successive determination of the \(\sigma _{(n)}\) from a Newtonian model \((\varPhi , \varrho , p(\varrho ), V_r, V_A)\) provides the gravitational initial data for the Bondi–Sachs system, preserving the Newtonian limit for \(\lambda =0\). Proper determination of the \(\sigma _{(n)}\)’s is crucial for ruling out, in Jürgen’s words, “...incoming fields from the exterior. [1, p. 9]”. The algorithm to calculate \(\sigma _{(n)}\) up to \(n=2\) is given in [29], and an example for Newtonian dust at \(\lambda =0\) can be found in [49].

We finally note an important observation regarding the \(\lambda \)-expansion of \({\mathcal {F}}\): since all fields become \(\lambda \)-dependent for \(u>u_0\) during the evolution of the centre-of-mass world line of the Newtonian system, this system will in general move away from the geodesic along which the null cones of the Bondi–Sachs metric are constructed. This is because of the \(\lambda \)-dependence of the pressure, whose gradient introduces an acceleration towards fluid elements at the origin of the geodesic of the Newtonian theory at \(\lambda =0\).

## 10 Closing remarks

Jürgen conjectured [1, Sec. 3] that space-times that are asymptotically flat at space-like and null infinity have a Newtonian limit. This fixes the boundary conditions of the fields at infinity. Asymptotic flatness at null infinity implies the peeling property [32] of the Weyl tensor \(C_{abcd}\) in the limit \(r\rightarrow \infty \). The ten independent components of \(C_{abcd}\) are most conveniently expressed in terms of the five complex Newman-Penrose Weyl scalars \(\varPsi _n, n\in \{0,...4\}\) [50, 51], and the peeling property of asymptotically flat space-times is given by \(\varPsi _n(y^a) = \varPsi ^{(0)}_n(u, y^A)/r^{5-n} + O(r^{-6})\). Winicour, however, fixed the boundary conditions of the fields at a geodesic inside a Newtonian matter distribution. This is necessary to obtain a well-defined characteristic initial value formulation. In turn, no conditions at null infinity can be fixed. Surprisingly, further investigations of Winicour and collaborators show, for an exemplary system with a Newtonian limit at \(\lambda =0\), that the Weyl tensor does not need to have the peeling property. Especially due to the presence of a logarithmic term in \(\sigma \) implied by such a Newtonian limit, the asymptotic behaviour of the Weyl tensor is spoiled [52], as it does not behave as \(C_{abcd}=O(r^{-1})\), as implied by the peeling theorem, but it behaves as \(O(r^{-1} \ln r)\). Thus, peeling cannot be recovered, but it is possible to establish a weaker version of the peeling theorem, where both the electric and the magnetic parts of the Weyl tensor are \(O(r^{-1})\) in the limit \(r\rightarrow \infty \) [53]. Nevertheless, the famous Einstein quadrupole formula is unaffected by this logarithmic behaviour and has a Newtonian limit for \(\lambda =0\) [30, 52]. Although, the finding of logarithmic terms at null infinity came as a surprise when mapping a Newtonian system onto a null cone of a quasi-Newtonian relativistic system, logarithmic terms can also be found by considering the global asymptotics of the characteristic initial value formulation of general relativity [54, 55]. Indeed, further investigations [54] showed the existence of an entire ‘zoo’ of polyhomogeneous space-times (i.e., space-times where the Weyl tensors are \(O(r^{-k}\ln ^{m}r)\) with \(k,m \ge 1\)), and whose mathematical meaning is not yet fully understood. One might speculate that some of them relate to higher-order corrections to the null cone initial data when Newtonian models of a Newton-Cartan theory at \(\lambda =0\) are mapped onto a null cone.

Jürgen raises some doubts (“For Einstein’s Theory many *asymptotic flatness* conditions were proposed [...] but their suitability for non stationary space-times is still in doubt [1, p. 9]”) about whether asymptotic flatness for isolated systems at null infinity is in fact a completed chapter. Winicour’s approximation model for a one-parametric family of Bondi–Sachs metrics \(g_{ab}(\lambda )\) with a Newtonian system at \(\lambda =0\) together with the zoo of polyhomogeneous space-times is proof that Jürgen’s doubt on asymptotically flat space-times is still standing. In particular, Jürgen’s advice to not take for granted generally imposed assumptions (as noted in [1] “...textbook and monographs suggest the existence of a well-understood [Newtonian] limit \((\rightarrow )\) ... [1, p. 2]”) should be taken as a motto for any scientist, not only working in general relativity.

## 11 Biography: Jürgen Ehlers

A biography of Jürgen Ehlers was published in Gen. Relativ. Gravit. 41, 1899 (2009), in the form of the obituary reprinted from the Biennial Report 2006/2007 of the Albert Einstein Institute.

## Footnotes

- 1.
- 2.
- 3.
For references see section 1 in Jürgen’s paper [1].

- 4.
Rather, we may call it an “incomplete theory” for the reasons listed below. Jürgen thought of the hyperbolic character of a theory that completely determines the system from initial data only. We may add the property of Einstein’s theory being

*background-free*, in contrast to Newtonian gravitation. (As the constraint equations of Einstein’s theory require the specification of boundary conditions (in addition to the initial conditions) the global topology is to be specified too, see the discussion in [9]. Einstein’s theory in its classical formulation determines the global topology for all times by the topology of the initial Cauchy hypersurface; generally, however, one does not consider the possibility of a dynamical topology change.) - 5.
As an aside we may acknowledge that the set of equations (18) and (19) in [1] suggest the presence of a non-inertial repulsive term \(\omega ^2\) in the field equation, where a cosmologist of the “dark energy era” would be tempted to analyze a possible replacement of a cosmological term, and it suggests that the force field is non-conservative featuring a source for the curl of the gravitational field strength.

- 6.
The reader may consult the rich literature on gravitoelectromagnetism, e.g. [13], emphasizing that the notion of Newtonian limit also crucially depends on the considered families of space-time splits.

- 7.
Latin indices in the middle of the alphabet \(i,j,k,\ldots \) denote coordinate indices \(i,j,k,\ldots \in \{1,2,3\}\) for three-dimensional coordinates \(X^i\). A vertical slash \({}_{\vert i}\) stands for the partial spatial derivative with respect to Lagrangian coordinates \(X^i\), while a comma denotes partial spatial derivative with respect to Eulerian coordinates \(x^i\). Antisymmetrization of an index pair is given by \([ij] = (ij-ji)/2\), and \(\epsilon _{ijk} \) is the Levi-Civita symbol. Latin bold face indices in the beginning of the alphabet \(\mathbf {a,b,c, }\ldots \) label the components of a triad \(\mathbf {a,b,c,}\ldots \in \{1,2,3\}\). Later, latin indices

*a*,*b*,*c*are used as four-dimensional space-time indices \(a,b,c,\ldots \in \{0,1,2,3\}\), while latin capital indices*A*,*B*,*C*... are two-dimensional, \(A,B,C,\ldots \in \{2,3\}\), for angular coordinates of a \(2-\)sphere. The four-dimensional covariant derivative is denoted with the ‘nabla’ symbol \(\nabla \). We use the Einstein summation convention throughout. To be in tune with [1], we choose the \(4-\)metric signature as \((+,-,-,-)\). - 8.
That is, the impossibility of numerically prescribing the components of the matrix (\(f^{\mathbf {a}}_{\ \vert i}\)) such that they exactly derive from vector components. The Lagrange–Newton metric components in the Lagrangian basis, \({}^N g_{ij} = \delta _{\mathbf {ab}}f^{\mathbf {a}}_{\ \vert i}f^{\mathbf {b}}_{\ \vert j}\), can only then be transformed via the inverse mapping \(\mathbf{f}^{-1}\) to the Euclidean metric components \({}^N g_{ij} \equiv \delta _{ij}\) in the Eulerian basis.

- 9.
This insight can be exploited to construct relativistic solutions from Newtonian ones [22].

- 10.
The solar mass \(M_\odot \) is \(M_\odot \approx 2\times 10^{30}\)kg.

- 11.
In this case the quasi-Newtonian initial data of [28, 29] can also be employed. However, the final space-time will contain a Kerr black hole whose Newtonian limit in Frame theory is a point mass [1, p. 9] and [7]. As such Fig. 1 is only valid up until the formation of the (quasi-Newtonian) proto-neutron star.

- 12.
Only in (the astrophysically unlikely case of) a spherically symmetric scenario, are no gravitational waves emitted.

- 13.
In units where the speed of light equals unity.

- 14.
- 15.
These values follow from the requirement of having a freely falling origin at \(r=0\).

- 16.
It is intriguing to remark at this stage that most physically relevant quantities are related to the electric (gradient) type of a tensor field rather than the magnetic (curl) type. For example, in cosmological applications (e.g. [5] and Eq. (12)) it is the electric part of the Weyl tensor driving most of the dynamics, the relevant initial data \(\sigma \) on a null cone for an isolated system are of electric type and in the global asymptotic properties of gravitational radiation [47, 48], too, only the electric type seems to play a fundamental role.

## Notes

### Acknowledgements

We wish to thank Malcolm MacCallum for inviting this editorial note and for constant assistance. In particular, TM thanks Malcolm MacCallum for help on some subtle grammatical issues as well as ‘false friends’ appearing in a first draft of the translation of [15]. We thank Frank Steiner for valuable remarks.

## References

- 1.Ehlers, J.: Über den Newtonschen Grenzwert der Einsteinschen Gravitationstheorie. In: Nitsch, J., Pfarr, J., Stachow, E.-W. (eds.) Grundlagenprobleme der modernen Physik: Festschrift für Peter Mittelstaedt zum 50. Geburtstag, pp. 65-84. Bibliographisches Institut, Mannheim, Wien, Zürich (1981). Translation: Gen. Relativ. Gravit. 51 (2019)Google Scholar
- 2.Buchert, T., Ehlers, J.: Lagrangian theory of gravitational instability of Friedmann–Lemaître cosmologies—second-order approach: an improved model for non-linear clustering. Mon. Not. R. Astron. Soc.
**264**, 375 (1993)ADSGoogle Scholar - 3.Buchert, T., Ehlers, J.: Averaging inhomogeneous Newtonian cosmologies. Astron. Astrophys.
**320**, 1 (1997). arXiv:astro-ph/9510056 ADSGoogle Scholar - 4.Ehlers, J., Buchert, T.: Newtonian cosmology in Lagrangian formulation: foundations and perturbation theory. Gen. Relativ. Gravit.
**29**, 733 (1997). arXiv:astro-ph/9609036 ADSMathSciNetzbMATHGoogle Scholar - 5.Ehlers, J., Buchert, T.: On the Newtonian limit of the Weyl tensor. Gen. Relativ. Gravit.
**41**, 2153 (2009). arXiv:0907.2645 ADSMathSciNetzbMATHGoogle Scholar - 6.Ehlers, J.: The Newtonian limit of general relativity. In: Ferrarese, G. (ed.) Classical Mechanics and Relativity: Relationship and Consistency, pp. 95–106. Bibliopolis, Napoli (1991)Google Scholar
- 7.Ehlers, J.: Examples of Newtonian limits of relativistic space-times. Class. Quantum Gravity
**14**, A119–A126 (1997)ADSzbMATHGoogle Scholar - 8.Oliynyk, T.A., Schmidt, B.: Existence of families of space-times with a Newtonian limit. Gen. Relativ. Gravit.
**41**, 2093 (2009)ADSzbMATHGoogle Scholar - 9.Levin, J., Scannapieco, E., Silk, J.: The topology of the universe: the biggest manifold of them all. Class. Quantum Gravity
**15**, 2689 (1998). arXiv:gr-qc/9803026 ADSMathSciNetzbMATHGoogle Scholar - 10.Steiner, F.: Do Black Holes exist in a finite universe having the topology of a flat 3-torus? In: Arendt, W. et al. (eds.) Ulmer Seminare 2016/2017, vol. 20, pp. 331–351. Universität Ulm (2018). arXiv:1608.03133
- 11.Geroch, R.: Limits of spacetimes. Commun. Math. Phys.
**13**(3), 180 (1969)ADSMathSciNetGoogle Scholar - 12.Heilig, U.: On the existence of rotating stars in General Relativity. Commun. Math. Phys.
**166**, 457 (1995)ADSMathSciNetzbMATHGoogle Scholar - 13.Jantzen, R.T., Carini, P., Bini, D.: The many faces of gravitoelectromagnetism. Ann. Phys.
**215**, 1 (1992). arXiv:gr-qc/0106043 ADSMathSciNetGoogle Scholar - 14.Heaviside, O.: A gravitational and electromagnetic analogy. Electr. vol. 31, 281 (part I), 359 (part II) (1893)Google Scholar
- 15.Al Roumi, F., Buchert, T., Wiegand, A.: Lagrangian theory of structure formation in relativistic cosmology. IV. Lagrangian approach to gravitational waves. Phys. Rev. D
**96**, 123538 (2017). arXiv:1711.01597 ADSMathSciNetGoogle Scholar - 16.Rendall, A.D.: The Newtonian limit for asymptotically flat solutions of the Vlasov–Einstein system. Commun. Math. Phys.
**163**, 89 (1994). arXiv:gr-qc/9303027 ADSMathSciNetzbMATHGoogle Scholar - 17.Buchert, T.: An exact Lagrangian integral for the Newtonian gravitational field strength. Phys. Lett. A
**354**, 8 (2006). arXiv:gr-qc/0508059 ADSMathSciNetzbMATHGoogle Scholar - 18.Buchert, T., Götz, G.: A class of solutions for self-gravitating dust in Newtonian gravity. J. Math. Phys.
**28**, 2714 (1987)ADSMathSciNetzbMATHGoogle Scholar - 19.Buchert, T.: Lagrangian perturbation approach to the formation of large-scale structure, In: Bonometto, S., Primack, J., Provenzale, A. (eds.) International School of Physics Enrico Fermi, Course CXXXII: Dark Matter in the Universe, Varenna 1995, pp. 543–564. IOP Press, Amsterdam (1996). arXiv:astro-ph/9509005
- 20.Buchert, T.: Toward physical cosmology: focus on inhomogeneous geometry and its non-perturbative effects. Class. Quantum Gravity
**28**, 164007 (2011). arXiv:1103.2016 ADSMathSciNetzbMATHGoogle Scholar - 21.Buchert, T., Ostermann, M.: Lagrangian theory of structure formation in relativistic cosmology I: Lagrangian framework and definition of a nonperturbative approximation. Phys. Rev. D
**86**, 023520 (2012). arXiv:1203.6263 ADSGoogle Scholar - 22.Alles, A., Buchert, T., Al Roumi, F., Wiegand, A.: Lagrangian theory of structure formation in relativistic cosmology. III. Gravitoelectric perturbation and solution schemes at any order. Phys. Rev. D
**92**, 023512 (2015). arXiv:1503.02566 ADSMathSciNetGoogle Scholar - 23.Lottermoser, M.: A convergent post-Newtonian approximation for the constraints in General Relativity. Ann. Inst. Henri Poincaré
**57**, 279 (1992)MathSciNetzbMATHGoogle Scholar - 24.Ehlers, J.: Akad. Wiss. Lit. (Mainz); Abh. Math.-Nat. Kl.
**No. 11**, 793 (1961); translation: Contributions to the relativistic mechanics of continuous media, Gen. Relativ. Gravit.**25**, 1225 (1993)Google Scholar - 25.Ellis, G.F.R., van Elst, H.: Cosmological models (Cargèse lectures 1998), Theoretical and observational cosmology. In: Lachièze–Rey, M. (ed .) Proceedings of the NATO Advanced Study Institute on Theoretical and Observational Cosmology, Cargèse, France, 1998, vol. 541, pp. 1–116. Kluwer Academic, NATO science series C, Mathematical and physical sciences, Boston (1999). arXiv:gr-qc/9812046 Google Scholar
- 26.Van Elst, H., Uggla, C., Lesame, W.M., Ellis, G.F.R., Maartens, R.: Integrability of irrotational silent cosmological models. Class. Quantum Gravity
**14**, 1151 (1997). arXiv:gr-qc/9611002 ADSMathSciNetzbMATHGoogle Scholar - 27.Kofman, L., Pogosyan, D.: Dynamics of gravitational instability is nonlocal. Astrophys. J.
**442**, 30 (1995). arXiv:astro-ph/9403029 ADSGoogle Scholar - 28.Winicour, J.H.: Newtonian gravity on the null cone. J. Math. Phys.
**24**, 1193 (1983)ADSMathSciNetzbMATHGoogle Scholar - 29.Winicour, J.H.: Null infinity from a quasi-Newtonian view. J. Math. Phys.
**25**, 2506 (1984)ADSMathSciNetGoogle Scholar - 30.Winicour, J.H.: The quadrupole radiation formula. Gen. Relativ. Gravit.
**19**, 281 (1987)ADSMathSciNetGoogle Scholar - 31.Bondi, H., van der Burg, M.G.J., Metzner, A.W.K.: Gravitational waves in general relativity. VII. Waves from axi-symmetric isolated systems. Proc. R. Soc. Lond. Ser. A
**269**, 21 (1962)ADSMathSciNetzbMATHGoogle Scholar - 32.Sachs, R.K.: Gravitational waves in general relativity. VIII. Waves in asymptotically flat space-time. Proc. R. Soc. Lond. Ser. A
**270**, 103 (1962)ADSMathSciNetzbMATHGoogle Scholar - 33.Mädler, T., Winicour, J.H.: Bondi–Sachs Formalism. Scholarpedia
**11**(12), 33528 (2016). arXiv:1609.01731 ADSGoogle Scholar - 34.Winicour, J.H.: Characteristic evolution and matching. Living Rev. Relativ.
**15**, 2 (2012). arXiv:0810.1903 ADSzbMATHGoogle Scholar - 35.Schutz, B.: Symposium 14. Radiative Spacetimes and Approximation Methods, In: MacCallum, M.A.H. (ed.) General Relativity and Gravitation: Proceedings of the 11th International Conference on General Relativity and Gravitation, pp. 369–376. Cambridge University Press, Cambridge (1987)Google Scholar
- 36.Penrose, R.: Asymptotic properties of fields and space-times. Phys. Rev. Lett.
**10**, 66 (1963)ADSMathSciNetGoogle Scholar - 37.Tamburino, L.A., Winicour, J.H.: Gravitational fields in finite and conformal Bondi frames. Phys. Rev.
**150**, 1039 (1966)ADSGoogle Scholar - 38.Müller, B.: The status of multi-dimensional core-collapse supernova models. Publ. Astron. Soc. Aust.
**33**, e048 (2016). arXiv:1608.03274 ADSGoogle Scholar - 39.Cerdá-Durán, P., Elias-Rosa, N.: Neutron stars formation and core collapse supernovae. In: Rezzolla L., Pizzochero P., Jones D., Rea N., Vidaña I. (eds.) The Physics and Astrophysics of Neutron Stars. Astrophysics and Space Science Library, vol 457. Springer, Cham. arXiv:1806.07267
- 40.Siebel, F., Font, J.A., Müller, E., Papadopoulos, P.: Axisymmetric core collapse simulations using characteristic numerical relativity. Phys. Rev. D
**67**, 124018 (2003). arXiv:gr-qc/0301127 ADSMathSciNetGoogle Scholar - 41.Cerdá-Durán, P., DeBrye, N., Aloy, M.A., Font, J.A., Obergaulinger, M.: Gravitational wave signatures in black hole forming core collapse. Astrophys. J. Lett.
**779**, L18 (2013)ADSGoogle Scholar - 42.Manasse, F.K., Misner, C.W.: Fermi normal coordinates and some basic concepts in differential geometry. J. Math. Phys.
**4**, 735 (1963)ADSMathSciNetzbMATHGoogle Scholar - 43.Mädler, T., Müller, E.: The Bondi–Sachs metric at the vertex of a null cone: axially symmetric vacuum solutions. Class. Quantum Gravity
**30**, 055019 (2013). arXiv:1211.4980 ADSMathSciNetzbMATHGoogle Scholar - 44.Thorne, K.S.: The theory of gravitational radiation–an introductory review, in Gravitational radiation. In: Deruelle, N., Piran, T. (eds.) Proceedings of the Advanced Study Institute, Les Houches, Haute-Savoie, France, June 2–21, 1982 (A84–35026 16–90), pp. 1–57. North-Holland Publishing, Amsterdam (1985)Google Scholar
- 45.Mädler, T., Winicour, J.H.: Boosted Schwarzschild metrics from a Kerr–Schild perspective. Class. Quantum Gravity
**35**, 035009 (2018). arXiv:1708.08774 ADSMathSciNetzbMATHGoogle Scholar - 46.Mädler, T., Winicour, J.H.: Kerr black holes and nonlinear radiation memory. Class. Quant. Grav
**36**, 095009 (2019). https://doi.org/10.1088/1361-6382/ab1187, arXiv:1811.04711 ADSMathSciNetGoogle Scholar - 47.Mädler, T., Winicour, J.H.: The sky pattern of the linearized gravitational memory effect. Class. Quantum Gravity
**33**, 175006 (2016). arXiv:1605.01273 ADSMathSciNetzbMATHGoogle Scholar - 48.Mädler, T., Winicour, J.H.: Radiation memory, boosted Schwarzschild spacetimes and supertranslations. Class. Quantum Gravity
**34**, 115009 (2017). arXiv:1701.02556 ADSMathSciNetzbMATHGoogle Scholar - 49.Isaacson, R.A., Welling, J.S., Winicour, J.H.: Gravitational radiation from dust. J. Math. Phys.
**26**, 2859 (1985)ADSMathSciNetzbMATHGoogle Scholar - 50.Newman, E.T., Penrose, R.: An approach to gravitational radiation by a method of spin coefficients. J. Math. Phys.
**3**, 566 (2004)ADSMathSciNetzbMATHGoogle Scholar - 51.Newman, E.T., Penrose, R.: Spin-coefficient formalism. Scholarpedia
**4**(6), 7445 (2009)ADSGoogle Scholar - 52.Isaacson, R.A., Welling, J.S., Winicour, J.H.: Extension of the Einstein quadrupole formula. Phys. Rev. Lett.
**53**, 1870 (1984)ADSMathSciNetGoogle Scholar - 53.Winicour, J.H.: Logarithmic asymptotic flatness. Found. Phys.
**15**, 605 (1985)ADSMathSciNetGoogle Scholar - 54.Chruściel, P.T., MacCallum, M.A.H., Singleton, D.B.: Gravitational waves in general relativity. XIV: Bondi expansions and the “Polyhomogeneity” of Scri. Philos. Trans. R. Soc. Lond. A
**350**, 19950004 (1995). https://doi.org/10.1098/rsta.1995.0004, arXiv:gr-qc/9305021 - 55.Christodoulou, D.: The global initial value problem in general relativity. In: Gurzadyan, V., Jantzen, R., Ruffini, R. (eds.) Proceedings of the Ninth Marcel Grossman Meeting on General Relativity, pp. 44–54. World Scientific, Singapore (2002)Google Scholar