2.3 Asymptotically flat space-times2 General Background2.1 Isolated systems

2.2 The emergence of the current picture 

The conformal treatment of infinity has grown out of many roots and it has a history which goes back to the middle of the past century. One of the main motivations was the desire to rigorously understand gravitational waves. The natural question arising was how to characterize wave-like solutions of Einstein's vacuum field equations (allowing for a possible cosmological constant tex2html_wrap_inline3731):

displaymath3733

The existence of gravitational waves was predicted by Einstein [35] shortly after he had found the general theory of relativity. However, due to the mathematical and physical complexity of the theory of gravitation there was confusion for a long time about whether the field equations really rigorously do admit solutions with a wave-like character. For instance, Rosen [123] came to the conclusion that there were no such solutions because in the class of plane symmetric waves every non-flat solution of the field equations became singular on a two-dimensional submanifold. This result was, however, due to the lack of understanding of the different kinds of singularities which can occur in a covariant theory and the singularity appearing in the plane wave space-times later turned out to be a coordinate singularity.

Thus, one of the early problems in the research area of gravitational waves was the invariant characterization of radiation. In 1957, Pirani [119] started the investigation with the suggestion that the algebraic properties of the Riemann (more specifically the Weyl) tensor should be considered as indicating the presence of radiation. In particular, under the assumption that gravitational radiation can in fact be characterized by the curvature tensor and that it propagates with the local speed of light, he proposed the definition that gravitational radiation is present if the curvature tensor has Petrov types tex2html_wrap_inline3735 or tex2html_wrap_inline3737 . He arrived at this characterization by the observation that a gravitational wave-front would manifest itself as a discontinuity of the Riemann tensor across null hypersurfaces, these being the characteristics for the Einstein equations. This result had previously been obtained by Lichnerowicz [90]. In his article, Pirani did not enforce the field equations, but towards the end of the paper he proposed to look at the equations tex2html_wrap_inline3739, which follow from the Bianchi identity for vacuum space-times. This is the first hint at the importance of the Bianchi identity for the study of gravitational waves. In [91] Lichnerowicz proposed a similar definition for a pure gravitational radiation field.

The next important step in the development of the subject was Trautman's study of the question of boundary conditions for the gravitational field equations [140]. He wanted to obtain conditions which were general enough to allow for gravitational radiation of an isolated system of matter, but still strong enough to guarantee uniqueness (appropriately defined) of the solution given ``reasonable'' initial data. He gave an asymptotic fall-off condition for the metric coefficients with respect to a certain class of coordinate systems. It was obtained by analogy to the situation with the scalar wave equation and Maxwell theory [139], where the fields can be required to satisfy Sommerfeld's ``Ausstrahlungsbedingung''. In those cases it is known that there exist unique solutions for given initial data, while in the gravitational case this was not known at the time and, in fact, remained unknown until only quite recently.

Trautman then went on to discuss some consequences of this boundary condition. He defined an energy-momentum four-vector at infinity which is well defined as a consequence of the boundary condition. It is obtained as the limit of an integral of the energy-momentum pseudo-tensor over a space-like manifold with boundary as it stretches out to infinity. By application of Stokes' theorem, the three-dimensional integral can be converted to a surface integral over the boundary, the sphere at infinity, of certain components of the so-called ``superpotentials'' for the energy-momentum pseudo-tensor. Nowadays these are recognized as certain special cases of the Nester-Witten two-form [37] and, hence, Trautmans energy-momentum integral coincides with the Bondi-Sachs expression or the ADM expression depending on how the limit to infinity is taken. This, however, is not explicitly specified in the paper. He considered the difference of energy-momentum between two space-like hypersurfaces and concluded that it must be due to radiation crossing the time-like cylinder which together with the two hypersurfaces bounds a four-dimensional volume. An estimate for the amount of radiation showed that it is non-negative. If the limit would have taken out to infinity along null directions, then this result would coincide with the Bondi-Sachs mass-loss formula.

Finally, Trautman observed that the definitions of pure radiation fields given by Pirani and Lichnerowicz are obeyed not exactly but only asymptotically by gravitational fields satisfying his boundary condition. Hence, he concluded that such solutions approach radiation fields in the limit of large distances to the source.

Based on the ideas of Pirani and Trautman and guided by his own investigations [130] of the structure of retarded linearized gravitational fields of particles, Sachs [125Jump To The Next Citation Point In The Article] proposed an invariant condition for outgoing gravitational waves. The intuitive idea was that at large distances from the source, the gravitational field, i.e. the Riemann tensor, of outgoing radiation should have approximately the same algebraic structure as does the Riemann tensor for a plane wave. As one approaches the source, deviations from the plane wave should appear. Sachs analyzed these deviations in detail and obtained rather pleasing qualitative insights into the behaviour of the curvature in the asymptotic regime.

In contrast to the earlier work, Sachs used more advanced geometrical methods. Based on his experience in the study of algebraically special metrics [34, 124, 86] he first analyzed the geometry of congruences of null curves. After the introduction of an appropriate null tetrad he used the Bianchi identity for the curvature tensor in a form which follows from the vacuum field equations to obtain the characteristic fall-off behaviour of the curvature components which has been termed the ``peeling property''. The Riemann tensor of a vacuum space-time has this property, if for any given null geodesic with an affine parameter r which extends to infinity, the curvature falls off along the curve in such a way that to order 1/ r it is null (Petrov type N or tex2html_wrap_inline3745) with a quadruple PND (principal null direction) along the curve. To order tex2html_wrap_inline3747 it has type 3 (tex2html_wrap_inline3737) with the triple PND pointing along the curve, and to order tex2html_wrap_inline3751 it has type 2 (tex2html_wrap_inline3735) with the double PND oriented along the curve. To order tex2html_wrap_inline3755 it is algebraically general (type 1 or tex2html_wrap_inline3757) but one of the PND's lies in the direction of the geodesic. To order tex2html_wrap_inline3759 the curvature is not related to the geodesic. Symbolically one can express this behaviour in the form

  equation100

where C on the left hand side denotes the Weyl tensor, and each tex2html_wrap_inline3763 on the right hand side stands for a tensor which has an r -fold PND along the null geodesic and which is independent of r . The important point is that the part of the curvature which has no relation to the null direction of the outgoing geodesic goes as tex2html_wrap_inline3759 .

Sachs postulated the outgoing radiation condition to mean that a bounded source field is free of mixed (i.e. a non-linear superposition of in- and outgoing) radiation at large distances if and only if the field has the peeling property. Later it was realized [26, 102Jump To The Next Citation Point In The Article] that this condition does not exclude ingoing radiation. Instead it is possible to have an ingoing wave profile provided that it falls off sufficiently fast as a function of an advanced time parameter.

The study of gravitational waves and the related questions was the main area of research of the group around H. Bondi and F. Pirani at King's College, London, in the years between 1955 and 1967. In a series of papers [92, 93, 24, 120, 94, 125, 25Jump To The Next Citation Point In The Article, 127Jump To The Next Citation Point In The Article] they analyzed several problems related to gravitational waves of increasing complexity. The most important of their results certainly is the work on axisymmetric radiating systems by Bondi, van der Burg and Metzner [25]. They used a different approach to the problem of outgoing gravitational waves. Instead of looking at null geodesics they focused on null hypersurfaces, and instead of analyzing the algebraic structure of the curvature using the Bianchi identity they considered the full vacuum field equations. Their work was concerned with axi-symmetric systems but shortly afterwards Sachs [127] removed this additional assumption.

The essential new ingredient was the use of a retarded time function. This is a scalar function u whose level surfaces are null hypersurfaces opening up towards the future. Based on the assumption that such a function exists, one can introduce an adapted coordinate system, so-called Bondi coordinates, by labeling the generators of the null hypersurfaces with coordinates on the two-sphere and introducing the luminosity distance r (essentially the square root of the area of outgoing wave fronts) along the null generators. The metric, when written in this kind of coordinate system, contains only six free functions.

Asymptotic conditions were imposed to the effect that one should be able to follow the null geodesics outwards into the future for arbitrarily large values of r . Then the metric was required to approach the flat metric in the limit of infinite distance. Additionally, it was assumed that the metric functions and other quantities of interest (in particular the curvature) were analytic functions of 1/ r .

The field equations in Bondi coordinates have a rather nice hierarchical structure which is symptomatic for the use of null coordinates and which allows for a simpler formal analysis compared to the related Cauchy problem. Bondi et al. and Sachs were able to solve the field equations asymptotically for large distances. In essence their procedure amounts to the formulation of a certain characteristic initial value problem (see [128Jump To The Next Citation Point In The Article, 129]) and the identification of the free data. It turns out that the freely specifiable data are two functions, essentially components of the metric, on an initial null hypersurface tex2html_wrap_inline3779 and two similar functions at `` tex2html_wrap_inline3781 ''. These latter functions are Bondi's news functions whose non-vanishing is taken to indicate the presence of gravitational radiation.

The results of this analysis were very satisfactory and physically reasonable. The most important consequence is the demonstration that outgoing gravitational waves carry away energy from the source and hence diminish its mass. This is the consequence of the Bondi-Sachs mass loss formula which relates the rate of the mass decrease to the integral over the absolute value of the news. Another consequence of the analysis was the peeling property: For space-times which satisfy the vacuum field equations and the Bondi-Sachs boundary conditions, the curvature necessarily has the asymptotic behaviour (1Popup Equation) as predicted by Sachs' direct analysis of the Riemann tensor using the vacuum Bianchi identity. Thus, the Bondi-Sachs conditions imply the covariant outgoing radiation condition of Sachs and also the boundary condition proposed by Trautman.

The group of coordinate transformations which preserve the form of the metric and the boundary conditions was determined. This infinite dimensional group which became known as the BMS group is isomorphic to the semi-direct product of the homogeneous Lorentz group with the Abelian group of so-called super-translations. The emergence of this group came as a surprise because one would have expected the Poincaré group as the asymptotic symmetry group but one obtained a strictly larger group. However, the structure of the BMS group is sufficiently similar to the Poincaré group. In particular, it contains a unique Abelian normal subgroup of four dimensions, which can be identified with the translation group. This result forms the basis of further investigations into the nature of energy-momentum in general relativity. The BMS group makes no reference to the metric which was used to derive it. Therefore, it can be interpreted as the invariance group of some universal structure which comes with every space-time satisfying the Bondi-Sachs boundary conditions. The BMS group has been the subject of numerous further investigations since then. For some of them we refer to [95, 96, 97, 101, 111Jump To The Next Citation Point In The Article, 118Jump To The Next Citation Point In The Article, 126].

At about the same time Newman and Penrose [100Jump To The Next Citation Point In The Article] had formulated what has become known as the NP formalism. It combined the spinor methods which had been developed earlier by Penrose [107] with the (null-)tetrad calculus used hitherto. Newman and Penrose applied their formalism to the problem of gravitational radiation. In particular, they constructed a coordinate system which was very similar to the ones used by Bondi et al. and Sachs. The only difference was their use of an affine parameter instead of luminosity distance along the generators of the null hypersurfaces of constant retarded time. Based on these coordinates and an adapted null frame they showed that the single assumption tex2html_wrap_inline3783 (and the technical assumption of the uniformity of the angular derivatives) already implied the peeling property as stated by Sachs. The use of tex2html_wrap_inline3785 as the quantity whose properties are specified on a null hypersurface was in accordance with a general study of characteristic initial value problems for spinor equations and in particular for general relativity undertaken by Penrose [113].

An important point in this work was the realization that the Bianchi identity could be regarded as a field equation for the Weyl tensor. It might be useful here to point out that it is a misconception to consider the Bianchi identity as simply a tautology and to ignore it as contributing no further information, as it is done even today. It is an important piece of the structure on a Riemannian or Lorentzian manifold which relates the (derivatives of the) Ricci and Weyl tensors. If the Ricci tensor is restricted by the Einstein equations to equal the energy-momentum tensor, then the Bianchi identity provides a differential equation for the Weyl tensor. Its structure is very similar to the familiar zero rest-mass equation for a particle with spin 2. In fact, in a sense one can consider this equation as the essence of the gravitational theory.

Newman and Unti [104] carried the calculations started in [100] further and managed to solve the full vacuum field equations asymptotically for large distances. The condition of asymptotic flatness was imposed not on the metric but directly on the Weyl tensor in the form suggested by Newman and Penrose, namely that the component tex2html_wrap_inline3785 of the Weyl tensor should have the asymptotic behaviour tex2html_wrap_inline3789 . From this assumption alone (and some technical requirements similar to the ones mentioned above) they obtained the correct peeling behaviour of the curvature, the form of the metric up to the order of tex2html_wrap_inline3759, in particular its flatness at large distances, and also the Bondi-Sachs mass loss formula. Later, the procedure developed by Newman and Unti to integrate the vacuum field equations asymptotically has been analyzed by Dixon [33], who showed that it can be carried out consistently to all orders of 1/ r .

It is remarkable how much progress could be made within such a short time (only about four years). The trigger seems to have been the use of the structure of the light cones in one form or another in order to directly describe the properties of the radiation field: the introduction of the retarded time function, the use of an adapted null-tetrad, and the idea to ``follow the field along null directions''. This put the emphasis onto the conformal structure of space-times.

The importance of the conformal structure became more and more obvious. Schücking had emphasized the conformal invariance of the massless free fields, a fact which had been established much earlier by Bateman [15] and Cunningham [31] for the wave equation and the free Maxwell field, and by McLennan [98] for general spin. This had led to the idea that conformal invariance might play a role also in general relativity and, in particular, in the asymptotic behaviour of the gravitational radiation field (see [116] for a personal account of the development of these ideas). Finally, Penrose [108] outlined a completely different point of view on the subject, arrived at by taking the conformal structure of space-time as fundamental. He showed that if one regarded the metric of Minkowski space-time to be specified only up to conformal rescalings tex2html_wrap_inline3795 for some arbitrary function tex2html_wrap_inline3721, then one could treat points at infinity on the same basis as finite points. Minkowski space-time could be completed to a highly symmetrical conformal manifold by adding a ``null-cone at infinity''. The well known zero rest-mass fields which transform covariantly under conformal rescalings of the metric are well defined on this space and the condition that they be finite on the null-cone at infinity translates into reasonable fall-off conditions for the fields on Minkowski space. On the infinite null-cone one could prescribe characteristic data for the fields which correspond to the strength of their radiation field. He suggested that asymptotically flat space-times should share at least some of these properties.

This point of view proved successful. In a further paper by Penrose [109Jump To The Next Citation Point In The Article], the basic qualitative picture we have today is developed. Roughly speaking (see the next section for a detailed account), the general idea is to attach boundary points to the ``physical'' space-time manifold which idealize the end-points at infinity reached by infinitely extended null geodesics. This produces a manifold with boundary, the ``unphysical'' manifold, whose interior is diffeomorphic to the physical manifold. Its boundary is a regular hypersurface whose causal character depends on the cosmological constant. The unphysical manifold is equipped with a metric which is conformal to the physical metric with a conformal factor tex2html_wrap_inline3721 which vanishes on the boundary. In addition, the structure of the conformal boundary is uniquely determined by the physical space-time.

Let us illustrate this with a simple example. The metric of Minkowski space-time in polar coordinates is

  equation134

where tex2html_wrap_inline3801 is the metric of the unit sphere. To perform the conformal rescaling we introduce null coordinates u = t - r and v = t + r . This puts the Minkowski line-element into the form

equation137

where tex2html_wrap_inline3801 is the metric of the unit sphere. The coordinates u and v each range over the complete real line, subject only to the condition tex2html_wrap_inline3813 . This infinite range is compactified by transforming with an appropriate function, e.g.

equation140

thus introducing new null coordinates U and V, in terms of which the metric takes the form

equation142

The coordinates U, V both range over the open interval tex2html_wrap_inline3823 with the restriction tex2html_wrap_inline3825 . Obviously, the Minkowski metric is not defined at points with tex2html_wrap_inline3827 or tex2html_wrap_inline3829 . Any extension of the metric in this form will be singular.

Now we define a different metric tex2html_wrap_inline3719, conformally related to tex2html_wrap_inline3717 by the conformal factor tex2html_wrap_inline3835 . Thus,

  equation145

This metric is perfectly regular at the points mentioned above and, in fact, g is the metric of the Einstein cylinder tex2html_wrap_inline3839 . This can be verified by defining an appropriate time and radius coordinate. With T = U + V, R = V - U we have

  equation148

  

Click on thumbnail to view image

Figure 1: The embedding of Minkowski space into the Einstein cylinder tex2html_wrap_inline3839 .

Thus, we may consider Minkowski space to be conformally embedded into the Einstein cylinder. This is shown in Figure  1 . The Minkowski metric determines the structure of the boundary, namely the two 3-dimensional null-hypersurfaces tex2html_wrap_inline3847 and tex2html_wrap_inline3849 which represent (future and past) ``null-infinity''. This is where null-geodesics ``arrive''. They are given by the conditions tex2html_wrap_inline3851 (tex2html_wrap_inline3847) and tex2html_wrap_inline3855 (tex2html_wrap_inline3849). The points tex2html_wrap_inline3859 are given by tex2html_wrap_inline3861 . They represent ``future and past time-like infinity'', the start respectively the end point of time-like geodesics, while tex2html_wrap_inline3683 is a point with tex2html_wrap_inline3865 . It is the start and end point of all space-like geodesics, hence it represents ``space-like infinity''.

The conformal boundary of Minkowski space-time consists of the pieces tex2html_wrap_inline3867, tex2html_wrap_inline3859, and tex2html_wrap_inline3683 . These are fixed by the Minkowski metric. In contrast to this, the conformal manifold into which Minkowski space-time is embedded (here the Einstein cylinder) is not fixed by the metric. Obviously, had we chosen a different conformal factor tex2html_wrap_inline3873 with some arbitrary positive function tex2html_wrap_inline3875 we would not have obtained the metric of the Einstein cylinder but a different one. We see from this that although the conformal boundary is unique, the conformal extension beyond the boundary is not.

The conformal compactification process is useful for several reasons. First of all it simplifies the discussion of problems at infinity which would involve complicated limit procedures when viewed with respect to the physical metric. Transforming to the unphysical metric attaches a boundary to the manifold so that issues which arise at infinity with respect to the physical metric can be analyzed by local differential geometric arguments in the neighbourhood of the boundary.

This is particularly useful for discussing solutions of conformally invariant field equations on space-time. The basic idea is the following: Consider a space-time which allows us to attach a conformal boundary, thus defining an unphysical manifold conformally related to the given space-time. Suppose we are also given a solution of a conformally invariant equation on this unphysical manifold. Because of the conformal invariance of the equation, there exists a rescaling of that unphysical field with a power of the conformal factor, which produces a solution of the equation on the physical manifold. Now suppose that the unphysical field is smooth on the boundary. Then the physical solution will have a characteristic asymptotic behaviour which is entirely governed by the conformal weight of the field, i.e. by the power of the conformal factor used for the rescaling. Thus, the regularity requirement of the unphysical field translates into a characteristic asymptotic fall-off or growth behaviour of the physical field, depending on its conformal weight.

Penrose used this idea to show that solutions of the zero rest-mass equations for arbitrary spin on a space-time, which can be compactified by a conformal rescaling, exhibit the peeling property in close analogy to the gravitational case as discovered by Sachs. Take as an example the spin-2 zero rest-mass equation for a tensor tex2html_wrap_inline3877 with the algebraic properties of the Weyl tensor

  equation160

This is the equation for linear perturbations of the gravitational fields propagating on a fixed background. It is conformally covariant, in the sense that it remains unchanged provided the field is rescaled as tex2html_wrap_inline3879, i.e., it is a conformal density with weight -1.

Using the geometric technique of conformal compactification, Penrose was able to establish the peeling property also for general (non-linear) gravitational fields. We will discuss this result explicitly in the following section. Furthermore, he showed that the group of transformations of the conformal boundary leaving the essential structure invariant was exactly the BMS group. This geometric point of view suggested that the asymptotic behaviour of the gravitational field of an isolated radiating gravitational system can be described entirely in terms of its conformal structure. The support for this suggestion was overwhelming from an aesthetical point of view, but a rigorous support for this claim was provided essentially only from the examination of the formal expansion type solutions of Bondi-Sachs and Newman-Unti and the analysis of explicit stationary solutions of the field equations.

The geometric point of view outlined above is the foundation on which many modern developments within general relativity are based. Let us now discuss the notion of asymptotically flat space-times and some of their properties in more detail.



2.3 Asymptotically flat space-times2 General Background2.1 Isolated systems

image Conformal Infinity
Jörg Frauendiener
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