It is clear that Penrose’s method of conformally compactifying space-time does provide a convenient and elegant framework for discussing questions of asymp-totics. What is not obvious, however, is how to answer the question of existence of space-times which actually do possess the asymptotic structure suggested by the conformal picture. On the one hand, a very specific geometric property of the space-time is required, namely the possibility to attach a smooth conformal boundary, and on the other hand, the Einstein equation, a differential equation on the metric of the space-time, has to be satisfied. It is not clear whether these two different requirements are indeed compatible.
To be more precise, one would like to know how many asymptotically flat solutions of the field equations actually exist, and whether this class of solutions contains the physically interesting ones which correspond to radiative isolated gravitating systems. The only possible avenue to answering questions of existence consists of setting up appropriate initial value problems and proving existence theorems for solutions of the Einstein equations subject to the appropriate boundary conditions.
The conventional Cauchy problem which treats the Einstein equation as a second order partial differential equation for the metric field is already rather complicated by itself (see e.g. the review by Choquet-Bruhat and York [27]). But to obtain statements of the type mentioned above is further complicated by the fact that in order to discuss the asymptotic fall-off properties of solutions one would need to establish global (long time and large distance) existence together with detailed estimates about the fall-off behaviour of the solution.
The geometric characterization of the asymptotic conditions in terms of the conformal structure suggests to discuss also the existence problem in terms of the conformal structure. The general idea is as follows: Suppose we are given an asymptotically flat manifold which we consider as being embedded into an appropriate conformally related unphysical space-time. The Einstein equations for the physical metric imply conditions for the unphysical metric and the conformal factor relating these two metrics. It turns out that one can write down equations which are regular on the entire unphysical manifold, even at those points which are at infinity with respect to the physical metric. Existence of solutions of these “regular conformal field equations” on the conformal manifold then translate back to (semi-)global results for asymptotically flat solutions of the field equations in physical space. This approach towards the existence problem has been the programme followed by Friedrich since the late 1970’s.
In this section we will discuss the conformal field equations which have been derived by Friedrich, and the various subproblems which have been successfully treated by using the conformal field equations.
General properties of the conformal field equations
Before deriving the equations we need to define the arena where the discussion is taking place.Definition 2 A conformal space-time is a triple (\({\mathcal M}\), gab, Ω) such that
-
(i)
(\({\mathcal M}\), gab) is a (time- and space-orientable) Lorentz manifold;
-
(ii)
Ω is a smooth scalar field on \({\mathcal M}\) such that the set \(\tilde {\mathcal M} = \{ p \in {\mathcal M}:\Omega (p) > 0\} \) is non-empty and connected;
-
(iii)
the gravitational field Kabcd = Ω-1Cabcd extends smoothly to all of \({\mathcal M}\).
Two conformal space-times (\({\mathcal M}\), gab, Ω) and (\(\tilde {\mathcal M}\), ĝab, ̂Ω) are equivalent if \({\mathcal M}\) and \(\hat {\mathcal M}\) are diffeomorphic and if, after identification of \({\mathcal M}\) and \(\tilde {\mathcal M}\) with a suitable diffeomorphism, there exists a strictly positive scalar field θ on \({\mathcal M}\) such that ̂Ω = θΩ and ̂gab = θ2gab.
From this definition follows that \(\tilde {\mathcal M}\) is an open sub-manifold of \({\mathcal M}\) on which a metric ̃gab = Ω-2gab is defined, which is invariant in the sense that two equivalent conformal space-times define the same metric ̃gab.
The space-time (\(\tilde {\mathcal M}\), ̃gab) allows the attachment of a conformal boundary which is given by \(I = \{ p \in {\mathcal M}:\Omega (p) = 0,\;d\Omega \ne 0\} \). The above definition of conformal space-times admits much more general situations than those arising from asymptotically flat space-times; this generality is sometimes needed for numerical purposes.
Under the conditions of Definition 2, it follows that the Weyl tensor vanishes on ℐ because the gravitational field (i.e. the rescaled Weyl tensor) is smooth on \({\mathcal M}\). Note that we make no assumptions about the topology of ℐ. If each null geodesic which starts from the inside of \(\tilde {\mathcal M}\) has a future and a past endpoint on ℐ, then \(\tilde {\mathcal M}\) is asymptotically simple in the sense of Definition 1. If, in addition, the metric ̃gab is a vacuum metric then ℐ has the implied topology S2 × ℝ. Note also that it is quite possible to have situations, where ̃gab is a vacuum metric and where the topology of ℐ is not S2 × ℝ, but e.g. T2 × ℝ. Then, necessarily, there must exist null geodesics which do not reach ℐ.
In the special case when ℐ is empty, the conformal factor Ω is strictly positive, i.e. \(\tilde {\mathcal M} = {\mathcal M}\), and the conformal space-time is isometric to the physical space-time (choosing θ = Ω-1).
Our goal is to express the vacuum equations in \(\tilde {\mathcal M}\) in terms of geometric quantities on the unphysical space-time. Consider first the Einstein vacuum equation for the metric ̃gab = Ω2gab. When expressed in terms of unphysical quantities it reads (see the formulae of Appendix 7)
$$0=\displaystyle \tilde{G}_{ab}=G_{ab}-\frac{2}{\Omega}(\nabla_{a}\nabla_{b}\Omega-g_{ab}\square \Omega)-\frac{3}{\Omega^{2}}g_{ab}\nabla_{c}\Omega\nabla^{c}\Omega.$$
((13))
This equation can be interpreted as the Einstein equation for the metric gab with a source term which is determined by the conformal factor. If we assume Ω to be known, then it is a second order equation for gab, which is formally singular on ℐ, where Ω vanishes. Therefore, it is very hard to make any progress towards the existence problem using this equation. To remedy this situation, Friedrich [43, 44, 45] suggested to consider a different system of equations on \({\mathcal M}\) which can be derived from the geometric structure on \({\mathcal M}\), the conformal transformation properties of the curvature and the vacuum Einstein equation on \(\tilde {\mathcal M}\). It consists of equations for a connection ∇a, its curvature and certain other fields obtained from the curvature and the conformal factor.
Let us assume that ∇a is a connection on \({\mathcal M}\) which is compatible with the metric gab so that
$${\nabla _c}{g_{ab}} = 0$$
((14))
holds. This condition does not fix the connection. Let Tabc and Rabcd denote the torsion and curvature tensors of ∇a. We will write down equations for the following unknowns:
-
— the connection ∇a,
-
— the conformal factor Ω, a one-form Σa and a scalar function S,
-
— a symmetric trace-free tensorfield Φab, and
-
— a completely trace-free tensorfield Kabcd which has the symmetries of the Weyl tensor.
We introduce the zero-quantity
$$Z = (T_{bc}^a,\:{{\mathcal Q}_{abcd}},\:{{\mathcal P}_{abc}},\:{{\mathcal B}_{bcd}},\:{{\mathcal S}_a},\:{{\mathcal S}_{ab}},\:{{\mathcal D}_a}),$$
((15))
where T
abc
is the torsion tensor of ∇a and the other components of Z are defined in terms of the unknowns by
$${{\mathcal Q}_{abcd}} \equiv {R_{abcd}} - \Omega {K_{abcd}} + 2{g_{c[a}}{\Phi _{b]d}} - 2{g_{d[a}}{\Phi _{b]c}} - 4{g_{c[a}}{g_{b]d}}\Lambda ,$$
((16))
$${{\mathcal P}_{abc}} \equiv 2{\nabla _{[c}}{\Phi _{a]b}} + 2{g_{b[c}}{\nabla _{a]}}\Lambda - {K_{cab}}^d{\Sigma _d},$$
((17))
$${\mathcal B_{bcd}} \equiv {\nabla _a}{K^a}_{bcd},$$
((18))
$${{\mathcal S}_a} \equiv {\nabla _a}\Omega - {\Sigma _a},$$
((19))
$${{\mathcal S}_{ab}} \equiv {\nabla _a}{\Sigma _b} - {g_{ab}}S + \Omega {\Phi _{ab}},$$
((20))
$${{\mathcal D}_a} \equiv {\nabla _a}S + {\Phi _{ab}}{\Sigma ^b} - \Omega {\nabla _a}\Lambda - 2\Lambda {\Sigma _a}.$$
((21))
In addition, we consider the scalar field
$${\mathcal T} \equiv 2\Omega S - 2{\Omega ^2}\Lambda - {\Sigma _a}{\Sigma ^a}$$
((22))
on \({\mathcal M}\). The equations Z = 0 are the regular conformal vacuum field equations. They are first order equations. In contrast to Equation (13) this system is regularFootnote 1 on \({\mathcal M}\), even on ℐ because there are no terms containing Ω-1.
Consider the equation \({{\mathcal B}_{bcd}} = {\nabla _a}{K^a}_{bcd} = 0\). This subsystem lies at the heart of the full system of conformal field equations because it feeds back into all the other parts. It was pointed out in Section 2.2 that the importance of the Bianchi identity had been realized by Sachs. However, it was first used in connection with uniqueness and existence proofs only by Friedrich [45, 44]. Its importance lies in the fact that it splits naturally into a symmetric hyperbolic system of evolution equationsFootnote 2 and constraint equations. Energy estimates for the symmetric hyperbolic system naturally involve integrals over a certain component of the Bel-Robinson tensor [52], a well known tensor in general relativity which has certain positivity properties.
The usefulness of the conformal field equations is documented inTheorem 1 Suppose that ∇a is compatible with gab and that Z = 0 on \({\mathcal M}\). If \({\mathcal T} = 0\) at one point of \({\mathcal M}\), then \({\mathcal T} = 0\) everywhere and, furthermore, the metric Ω-2gab is a vacuum metric on \(\tilde {\mathcal M}\).Proof: From the vanishing of the torsion tensor it follows that ∇a is the Levi-Civita connection for the metric gab. Then, \({{\mathcal Q}_{abcd}} = 0\) is the decomposition of the Riemann tensor into its irreducible parts which implies that the Weyl tensor Cabcd = ΩKabcd, that Φab is the trace-free part of the Ricci tensor, and that Λ = 24R. The equation \({{\mathcal S}_a} = 0\) defines Σa in terms of Ω, and the trace of the equation \({{\mathcal S}_ab} = 0\) defines S = 1/4□Ω. The trace-free part of that equation is the statement that ̃Φab = 0, which follows from the conformal transformation property (110) of the trace-free Ricci tensor. With these identifications the equations \({{\mathcal B}_{abc}} = 0\) resp. \({{\mathcal P}_{abc}} = 0\) do not yield any further information because they are identically satisfied as a consequence of the Bianchi identity on (\({\mathcal M}\), g), resp. (\(\tilde {\mathcal M}\), ̃g).
Finally, we consider the field \({\mathcal T}\). Taking its derivative and using \({{\mathcal S}_{ab}} = 0\) and
\({{\mathcal D}_a} = 0\), we obtain \({\nabla _a}{\mathcal T} = 0\). Hence, \({\mathcal T}\) vanishes everywhere if it vanishes at one point. It follows from the transformation (111) of the scalar curvature under conformal rescalings that \({\mathcal T} = 0\) implies ̃Λ = 0. Thus, ̃gab is a vacuum metric.
It is easy to see that the conformal field equations are invariant under the conformal rescalings of the metric specified in Definition 2 and the implied transformation of the unknowns. The conformal invariance of the system implies that the information it contains depends only on the equivalence class of the conformal space-time.
The reason for the vanishing of the gradient of \({\mathcal T}\) is essentially this: If we impose the equation ̃Φab = 0 for the trace-free part of the Ricci tensor of a manifold, then by use of the contracted Bianchi identity we obtain ̃∇ãΛ = 0. Expressing this in terms of unphysical quantities leads to the reasoning in Theorem 1. The special case Ω = 1 reduces to the standard vacuum Einstein equations, because then we have Kabcd = Cabcd and Σa =0. Then \({{\mathcal S}_{ab}} = 0\) implies Φab = 0 and S = 0, while \({\mathcal T} = 0\) forces Λ = 0. The other equations are identically satisfied.
Given a smooth solution of the conformal field equations on a conformal manifold, Theorem 1 implies that on \(\tilde {\mathcal M}\) we obtain a solution of the vacuum Einstein equation. In particular, since the Weyl tensor of gab vanishes on ℐ due to the smoothness of the gravitational field, this implies that the Weyl tensor has the peeling property in the physical space-time. Therefore, if existence of suitable solutions of the conformal field equations on a conformal manifold can be established, one has automatically shown existence of asymptotically flat solutions of the Einstein equations. The main advantage of this approach is the fact that the conformal compactification supports the translation of global problems into local ones.
Note that the use of the conformal field equations is not limited to vacuum space-times. It is possible to include matter fields into the conformal field equations provided the equations for the matter have well-defined and compatible conformal transformation properties. This will be the case for most of the interesting fundamental field equations (Maxwell, Yang-Mills [52], scalar wave [77, 78] etc.)
The reduction process for the conformal field equations
We have set up the conformal field equations as a system of equations for the conformal geometry of a conformal Lorentz manifold. As such they are invariant under general diffeomorphisms and, as we have seen, under conformal rescalings of the metric. In this form they are not yet very useful for treating questions of existence of solutions or even for numerical purposes. For existence results and numerical evolution the geometric equations have to be transformed into partial differential equations for tensor components which can then be used to set up well-posed initial value problems for hyperbolic systems of evolution equations.
This process, sometimes referred to as “hyperbolic reduction” consists of several steps. First, one needs to break the invariance of the equations. By imposing suitable gauge conditions one can specify a coordinate system, a linear reference frame and a conformal factor. Then the equations can be written as equations for the components of the geometric quantities with respect to the chosen frame in the chosen conformal gauge and as functions of the chosen coordinates. In the next step, one needs to extract from the equations a subsystem of propagation equations which is hyperbolic so that it has a well-posed initial value problem. It is often referred to as the “reduced equations”. Finally, one has to make sure that solutions of the reduced system give rise to solutions of the full system. This step may involve the verification that the gauge conditions imposed are compatible with the propagation equations, or that other equations (constraints) not included in the reduced system are preserved under the propagation. The first two steps, choice of gauge and extraction of the reduced system, are very much related. Gauge conditions should be imposed such that they lead to a hyperbolic reduced system. Furthermore, the gauge conditions should be such that they can be imposed locally without loss of generality.
The gauge freedom present in the conformal field equations can easily be determined. The freedom to choose the coordinates amounts to four scalar functions while the linear reference frame, which we take to be orthogonal, can be specified by a Lorentz rotation, which amounts to six free functions. Finally, the choice of a conformal factor contributes another free function. Altogether, there are eleven functions which can be chosen at will.
Once the geometric equations have been transformed into equations for components, the next step is to extract the reduced system. These are equations for the components of the geometric quantities defined above as well as for gauge-dependent quantities: the components of the frame with respect to the coordinate basis, the components of the connection with respect to the given frame and the conformal factor.
There are several well-known choices for coordinates (harmonic, Gauß, Bondi, etc.), as well as for frames (Fermi-Walker transport, Newman-Penrose, etc.). These are usually “hard-wired” into the equations and one has no further control on the properties of the gauge. Gauß coordinates for instance have the tendency to become singular when the geodesic congruence which is used for their definition starts to self-intersect. Similarly, Bondi coordinates are attached to null-hypersurfaces which have the tendency to self-intersect thus destroying the coordinate system. In the context of existence proofs and the numerical evolution of the equations it is of considerable interest to have additional flexibility in order to prevent the coordinates or the frame from becoming singular. The goal is to “fix the gauge” in as flexible a manner as possible and to obtain reduced equations which still have useful properties.
A scheme to obtain the reduced equations in symmetric hyperbolic form while still allowing for arbitrary gauges has been devised by Friedrich [48] (see also [55] for various examples). The idea is based on the following observation. Cartan’s structure equations which express the torsion and curvature tensors in terms of tetrad and connection coefficients are two-form equations: They are skew on two indices and the information contained in the equations is not enough to fix the tetrad and the connection by specifying the torsion and the curvature. The additional information is provided by fixing a gauge. Normally, this is achieved by reducing the number of variables, in this case the number of tetrad components and connection coefficients. However, one can just as well add appropriate further equations to have enough equations for all unknowns. The additional equations should be chosen so that the ensuing system has “nice” properties.
We illustrate this procedure by a somewhat trivial example. Consider, in flat space with coordinates (xμ) = (t, x1, x2, x3), a one- form ω which we require to be closed:
$${\partial _\mu }{\omega _\nu } - {\partial _\nu }{\omega _\mu } = 0.$$
From this equation we can extract three evolution equations, namely
$$\begin{array}{l}
{\partial _t}{\omega _1} - {\partial _1}{\omega _0} = 0,\\
{\partial _t}{\omega _2} - {\partial _2}{\omega _0} = 0,\\
{\partial _t}{\omega _3} - {\partial _3}{\omega _0} = 0.
\end{array}$$
Obviously, these three equations are not sufficient to reconstruct ω from appropriate initial data. One possibility to proceed from here is to specify one component of ω freely and then obtain equations for the other three. However, it is easily seen that only by specifying ω0 we can achieve a pure evolution system. Otherwise, we get mixtures of evolution and constraint equations. So we may note that proceeding in this way leads to a restriction of possibilities as to which components should be specified freely and, in general, it also entails that derivatives of the specified component appear.
Another possible procedure is to enlarge the system by adding an equation for the time derivative of ω0. Doing this covariantly implies that we should add an equation in the form of a divergence
$${\partial ^\mu }{\omega _\mu } = F,$$
where F is an arbitrary function. This results in the system
$$\begin{array}{*{20}{l}}
{{\partial _t}{\omega _0} - {\partial _1}{\omega _1} - {\partial _2}{\omega _2} - {\partial _3}{\omega _3} = F,} \\
{\;\;\;\;\;\;\;\;\;{\partial _t}{\omega _1} - {\partial _1}{\omega _0} = 0,} \\
{\;\;\;\;\;\;\;\;\;{\partial _t}{\omega _2} - {\partial _2}{\omega _0} = 0,} \\
{\;\;\;\;\;\;\;\;\,{\partial _t}{\omega _3} - {\partial _3}{\omega _0} = 0,}
\end{array}$$
which is symmetric hyperbolic for any choice of F. Note also that F appears as a source term and only in undifferentiated form. Clearly, our influence on the component ω0 is now very indirect via the solution of the system, while before we could specify it directly.
In a similar way, one proceeds in the present case of the conformal field equations. Note, however, that this way of fixing a gauge is not at all specific to these equations. Since it depends essentially only on the form of Cartan’s structure equations it is applicable in all cases where these are part of the first order system. The Cartan equations can be regarded as exterior equations for the one-forms ω
μa
dual to a tetrad e
aμ
and the connection one-forms \(\omega _{{a^\nu }}^\mu = \omega _c^\mu {\nabla _a}e_\nu ^c\). Similar to the system above, the equations involve only the exterior derivative of the one-forms and so we expect that we should add equations in divergence form, namely
$$\begin{array}{l}
{\nabla ^a}\omega _a^\mu = {F^\mu },\\
{\nabla ^a}\omega _{{a^\nu }}^\mu = {F^\mu }_\nu ,
\end{array}$$
((23))
with arbitrary gauge source functions Fμ for fixing coordinates and Fμv for choosing a tetrad. Note that ω
μa
= ∇axμ implies Fμ = □xμ;.
In a given gauge (i.e., coordinates and frame field are specified) the gauge sources can be determined from
$$F^{\mu}=\square x^{\mu},$$
((24))
$${F^\mu }_\nu = {\nabla ^a}(\omega _c^\mu {\nabla _a}e_\nu ^c).$$
((25))
In fact, these equations are exactly the same equations as (23) except that they are written in a more invariant form. Now it is obvious that the gauge sources contain information about the coordinates and the frame used. What needs to be shown is that any specification of the gauge sources fixes a gauge. In fact, suppose we are given functions ̂Fμ and ̂Fμv on ℝ4 then there exist (locally) coordinates ̂xμ and a frame ̂e
aμ
so that in that coordinate system the gauge sources are just the prescribed functions ̂Fμ and ̂Fμv. This follows from the equations
$$\square \hat{x}^{\mu}=\hat{F}^{\mu}(\hat{x}^{\nu}) ,$$
((26))
$${\nabla ^a}(\hat \omega _c^\mu {\nabla _a}\hat e_\nu ^c) = {\hat F^\mu }_{\;\;\nu }({\hat x^\rho }).$$
((27))
These are semi-linear wave equations which determine a unique solution from suitably given initial data close to the initial surface. Note that on the right hand side of (26) there is a function of the ̂xμ, and not a source term. The equations can be solved in steps. Once the coordinates ̂xμ have been determined from (26), the right hand side of (27) can be considered as a source term.
Finally, we need to discuss the gauge freedom in the choice of the conformal factor Ω. In many discussions of asymptotic structure the conformal factor is chosen in such a way that null-infinity is divergence-free, in addition to the vanishing of its shear, which is a consequence of the asymptotic vacuum equations. That means that infinitesimal area elements remain unchanged in size as they are parallelly transported along the generators of ℐ. Since they also remain unchanged in form due to the vanishing shear of ℐ, they remain invariant and hence they can be used to define a unique metric on the space of generators of ℐ. This choice simplifies many calculations on ℐ, still leaving the conformal factor quite arbitrary away from ℐ. Yet, in numerical applications this choice of the conformal factor may be too rigid and so one needs a flexible method for fixing the conformal factor.
It turns out that one can introduce a gauge source function for the conformal gauge as well. Consider the change of the scalar curvature under the conformal rescaling gab ↦ ̂gab = θ2gab, Ω ↦ ̂Ω = θΩ,: It transforms according to
$$\Lambda \mapsto \hat \Lambda = {\theta ^{ - 2}}(\Lambda + \frac{{\square \theta }}{{4\theta }})\:.$$
Reading this transformation law as an equation for θ we obtain
$$\square \theta = 4\theta ({\theta ^2}\hat \Lambda - \Lambda ).$$
((28))
It follows from this equation that we may regard the scalar curvature as a gauge source function for the conformal factor: For, suppose we specify the function ̂Λ arbitrarily on \({\mathcal M}\), then Equation (28) is a non-linear wave equation for θ which can be solved given suitable initial data. This determines a unique θ, hence a unique ̂Ω and ̂gab such that the scalar curvature of the rescaled metric ̂gab has scalar curvature ̂Λ. Note that these considerations are local. They show that locally the gauges can be fixed arbitrarily. However, the problem of identifying and fixing a gauge globally is very difficult but also very important because only when the gauges are globally known one can really compare two different space-times.
Having established that the gauge sources do in fact, locally, fix a unique gauge we can now split the system of conformal field equations into evolution equations and constraints. The resulting system of equations is exhibited below. The reduction process is rather straightforward but tedious. It is sketched in Appendix 6. Here, we only describe it very briefly. We introduce an arbitrary time-like unit vectorfield ta which has a priori no relation to the tetrad field used for framing. We split all the tensorial quantities into the parts which are parallel and orthogonal to that vector field using the projector hab = δ
ba
- tatb. The connection coefficients for the four-dimensional connection ∇a are treated differently. We introduce the covariant derivatives of the vectorfield ta by
$${\chi ^a}_b = {h^c}_a{\nabla _c}{t^a},\;\;\;\;\;{\chi ^a} = {t^c}{\nabla _c}{t^a}.$$
They account for half (9+3) of the four-dimensional connection coefficients. The other half is captured by defining a covariant derivative ∂a which has the property that it annihilates both ta and ta and agrees with ∇a when acting on tensors orthogonal to ta (see Equations (40)). Note that we have not required that ta be the time-like member of the frame, nor have we assumed that it be hypersurface orthogonal. In the latter case, χab is the extrinsic curvature of the family of hypersurfaces orthogonal to ta and hence it is symmetric. Furthermore, the derivative ∂a agrees with the Levi-Civita connection of the metric hab induced on the leaves by the metric gab.
We write the equations in terms of the derivative ∂a and the “time derivative” ∂ which is defined in a way similar to ∂a (see Equation (40)), because in this form it is quite easy to see the symmetric hyperbolicity of the equations.
As they stand, the Equations (90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103) form a symmetric hyperbolic system of evolution equations for the collection of 65 unknowns
$$\{c^{\mu},c_{a}^{\mu},\chi^{a},\Gamma_{{a}^{b}_{c}},\Lambda^{a}{\;_{b}},\phi_{ab},\phi_{a},E_{ab},B_{ab},\Omega,\sigma_{a},\sigma,S\}.$$
This property is present irrespective of the particular gauge. For any choice of the gauge source functions Fμ, Fa, Fab and Λ, the system is symmetric hyperbolic. The fact that the gauge sources appear only in undifferentiated form implies that one can specify them not only as functions of the space-time coordinates but also as functions of the unknown fields. In this way, one can feed information about the current status of the evolution back into the system in order to influence the future development.
Other ways of specifying the coordinate gauge, including the familiar choice of a lapse function and a shift vector, are not as flexible because then not only these functions themselves appear in the equations, but also their derivatives. Specifying them as functions of the unknown fields alters the principal part of the system and, hence, the propagation properties of the solution. This may not only corrupt the character of the system but it may also be disastrous for the numerical applications because an uncontrolled change of the local propagation speeds implies that the stability of a numerical scheme can break down due to violation of the CFL condition (see [39] for a more detailed discussion of these issues). However, due to the intuitive meaning of lapse and shift they are used (almost exclusively) in numerical codes.
There are several other ways of writing the equations. Apart from various possibilities to specify the gauges which result in different systems with different numbers of unknowns, one can also set up the equations using spinorial methods. This was the method of choice in almost all of Friedrich’s work (see e.g. [52] and also [38]). The ensuing system of equations is analogous to those obtained here using the tetrad formalism. The main advantages of using spinors is the fact that the reduction process automatically leads to a symmetric hyperbolic system, that the variables are components of irreducible spinors which allows for the elimination of redundancies, and that variables and equations become complex and hence easier to handle.
Another possibility is to ignore the tetrad formalism altogether (or, more correctly, to choose as a basis for the tangent spaces the natural coordinate frame). This also results in a symmetric hyperbolic system of equations (see [55, 81]), in which the gauge dependent variables are not the frame components with their corresponding connection coefficients but the components of the spatial metric together with the usual Christoffel symbols and the extrinsic curvature.
The fact that the reduced equations form a symmetric hyperbolic system leads, via standard theorems, to the existence of smooth solutions which evolve uniquely from suitable smooth data given on an initial surface. We have theTheorem 2 [Friedrich [48]] For functions Fμ, Fa, Fab, Λ on ℝ4 and data given on some initial surface let u be the solution of the reduced equations. If u satisfies the conformal field equations (16, 17, 18, 19, 20, 21) on the initial surface then, in fact, it satisfies them on the entire domain of dependence of the initial surface in the space-time defined by u.
The proof of this theorem relies on the existence of a “subsidiary system” of equations for the zero-quantity Z (see Equation (15)), whose vanishing indicates the validity of the conformal field equations. This system turns out to be linear, symmetric hyperbolic and homogeneous. Thus, one has uniqueness of the solutions so that Z vanishes in the domain of dependence of the initial surface if it vanishes on the surface. Hence, the conformal field equations hold. It can be shown that solutions obtained from different gauge source functions are in the same conformal class, so they lead to the same physical space-time.
Initial value problems
From the physical point of view the most interesting scenario is the following one: A gravitating material system (like e.g. several extended bodies) evolves from a given initial state, possibly interacting with incoming gravitational radiation and emitting outgoing gravitational radiation until it reaches a final state. This situation is sketched in Figure 6.
In more mathematical terms, this requires the solution of an initial value problem: We provide appropriate initial data, describing the initial configuration of the matter and the geometry, on a hyperboloidal hypersurface Σ0, and appropriate boundary data, describing the incoming gravitational radiation, on the piece of ℐ- which is in the future of Σ0. Then we have to show that there is a unique solution of the conformal field equations coupled to the matter equations which exists for some time. If the situation is “close enough” to a Newtonian situation, i.e., the gravitational waves are weak and the matter itself is rather “tame”, then one would expect that there is a solution, i.e. a space-time, which is regular on arbitrary hyperboloidal hypersurfaces intersecting ℐ+. In general, however, we cannot expect to have a regular point i+ representing time-like infinity.
So far, results of this kind are out of reach. The reason is not so much the incorporation of matter into the conformal field equations but a more fundamental one. Space-like infinity i0 is a singularity for the conformal structure of any space-time, which has a non-vanishing ADM-mass. Without the proper understanding of i0 there will be no way to bridge the gap between past and future null-infinity because i0 provides the link between the incoming and the outgoing radiation fields.
The results obtained so far are concerned only with the pure radiation problem, i.e. the vacuum case. In [29] Christodoulou and Klainerman prove the global non-linear stability of Minkowski space, i.e. the existence of global solutions of the Einstein vacuum equations for “small enough” Cauchy data which satisfy certain fall-off conditions at space-like infinity. Their result qualitatively confirms the expectations based on the concept of asymptotic flatness. However, they do not recover the peeling property for the Weyl tensor but a weaker fall-off, which implies that in this class of solutions the conformal compactification would not be as smooth as it was expected to be. This raises the question whether their results are sharp, i.e., whether there are solutions in this class which indeed have their fall-off behaviour. In that case, one would probably have to strengthen the fall-off conditions of the initial data at space-like infinity in order to establish the correct peeling of the Weyl tensor. Then an interesting question arises as to what the physical meaning of these stronger fall-off conditions is. An indication that maybe more restrictive conditions are needed is provided by the analysis of the initial data on hyperboloidal hypersurfaces (see below).
The first result [46] obtained with the conformal field equations is concerned with the asymptotic characteristic initial value problem (see Figure 7) in the analytic case. It was later generalized to the C∞ case.
In this kind of initial value problem, one specifies data on an ingoing null hypersurface \({\mathcal N}\) and that part of ℐ which is in the future of \({\mathcal N}\). The data which have to be prescribed are essentially the so-called null data on \({\mathcal N}\) and ℐ, i.e. those parts of the rescaled Weyl tensor which are entirely intrinsic to the respective null hypersurfaces. In the case of ℐ the null datum is exactly the radiation field.
Theorem 3 [Kánnár [88]] For given smooth null data on an ingoing null hypersurface \({\mathcal N}\) and a smooth radiation field on the part \({\mathcal I}\) of ℐ- which is to the future of the intersection S of \({\mathcal N}\) with ℐ- and certain data on S, there exists a smooth solution of Einstein’s vacuum equations in the future of \({\mathcal N} \cup {\mathcal I}\) which implies the given data on \({\mathcal N} \cup {\mathcal I}\).
The result is in complete agreement with Sachs’ earlier analysis of the asymptotic characteristic initial value problem based on formal expansion methods [128].
Another case is concerned with the existence of solutions representing pure radiation. These are vacuum solutions characterized by the fact that they are smoothly extensible through past time-like infinity, i.e. by the regularity of the point i-. This case has been treated in [49, 51]. A solution of this kind is uniquely characterized by its radiation field, i.e. the intrinsic components of the rescaled Weyl tensor on ℐ-. In the analytic case, a formal expansion of the solution at i- can be derived, and growth conditions on the coefficients can be given to ensure convergence of the formal expansion near i-. Furthermore, there exists a surprising relation between this type of solutions and static solutions, summarized inTheorem 4 [Friedrich] With each asymptotically flat static solution of Einstein’s vacuum field equations can be associated another solution of these equations which has a smooth conformal boundary ℐ- and for which the point i- is regular.
This result establishes the existence of a large class of purely radiative solutions.
For applications, however, the most important type of initial value problem so far, in the sense that the asymptotic behaviour can be controlled, has been the hyperboloidal initial value problem where data are prescribed on a hyperboloidal hypersurface. This is a space-like hypersurface whose induced physical metric behaves asymptotically like a surface of constant negative curvature (see Section 2.4). In the conformal picture, a hyperboloidal hypersurface is characterized simply by the geometric fact that it intersects ℐ transversely in a two-dimensional space-like surface. Prototypes of such hypersurfaces are the space-like hyperboloids in Minkowski space-time. In the Minkowski picture they can be seen to become asymptotic to null cones which suggests that they reach null-infinity. However, the picture is deceiving: The conformal structure is such that the hyperboloids always remain space-like, the null-cones and the hyperboloids never become tangent. The intersection is a two-dimensional surface S, a “cut” of ℐ. The data implied by the conformal fields on such a hypersurface are called hyperboloidal initial data. The first result obtained for the hyperboloidal initial value problem states that if the space-time admits a hypersurface which extends smoothly across ℐ+ with certain smooth data given on it, then the smoothness of ℐ+ will be guaranteed at least for some time into the future. This is contained inTheorem 5 [Friedrich [47]] Smooth hyperboloidal initial data on a hyperboloidal hypersurface Σ determine a unique solution of Einstein’s vacuum field equations which admits a smooth conformal boundary at null-infinity in the future of Σ.
There exists also a stability result which states that there are solutions which behave exactly like Minkowski space near future time-like infinity:Theorem 6 [Friedrich [50]] If the hyperboloidal initial data are in a sense sufficiently close to Minkowskian hyperboloidal data, then there exists a conformal extension of the corresponding solution which contains a point i+ such that ℐ+ is the past null cone of that point.
It should be emphasized that this result implies that the physical metric of the corresponding solution is regular for all future times. Thus, the theorem constitutes a (semi-)global existence result for the Einstein vacuum equations.
Hyperboloidal initial data
Now the obvious problem is to determine hyperboloidal initial data. That such data exist follows already from Theorem 4 because one can construct hyper-boloidal hypersurfaces together with data on them in any of the radiative solutions whose existence is guaranteed by that theorem. However, one can also construct such data sets in a similar way to the construction of Cauchy data on an asymptotically Euclidean hypersurface by solving the constraint equations implied on the Cauchy surface. Let ̃Σ be a hyperboloidal hypersurface in an asymptotically flat vacuum space-time which extends out to ℐ, touching it in a two-surface ∂Σ which is topologically a two-sphere. The assumptions on ̃Σ are equivalent to the fact that on ̃Σ ≔ Σ ∪ S is a smooth Riemannian manifold with boundary, which carries a smooth metric hab and a smooth function Ω, obtained by restriction of the unphysical metric and the conformal factor. The conformal factor is a defining function for the boundary ∂Σ (i.e. it vanishes only on ∂Σ with non-vanishing gradient), and together with the metric it satisfies
$${\tilde h_{ab}} = {\Omega ^{ - 2}}{h_{ab}}$$
((29))
on ̃Σ, where ̃hab is the metric induced on ̃Σ by the physical metric. Furthermore, let ̃χab be the extrinsic curvature of ̃Σ in the physical space-time. Together with ̃hab it satisfies the vacuum constraint equations,
$$\tilde R - {\tilde \chi _{ab}}{\tilde \chi ^{ab}} + {({\tilde \chi ^a}_{\;\;a})^2} = 0,$$
((30))
$${\tilde \partial _a}{\tilde \chi ^a}_{\;\;b} - {\tilde \partial _b}\tilde \chi _{\;\;a}^a = 0,$$
((31))
where ̃∂a is the Levi-Civita connection of ̃hab and ̃R is its scalar curvature.
Several results of increasing generality have been obtained. We discuss only the simplest case here, referring to the literature for the more general results. Assume that the extrinsic curvature is a pure trace term,
$${\tilde \chi _{ab}} = \frac{1}{3}c{\tilde h_{ab}}.$$
The momentum constraint (31) implies that c is constant while the hyperboloidal character of ̃Σ implies that c ≠ 0. With these simplifications and a rescaling of ̃hab with a constant factor, the Hamiltonian constraint (30) takes the form
$$\tilde R = - 6.$$
((32))
A further consequence of the condition (29) is the vanishing of the magnetic part Bab of the Weyl tensor. For any defining function ω of the boundary, the conformal factor has the form Ω = ϕ-2ω. Expressing Equation (32) in terms of the unphysical quantities hab and Ω yields the single second-order equation
$$8{\omega ^2}\Delta \phi - 8\omega {\partial ^a}\omega {\partial _a}\phi - [{\omega ^2}R + 4\omega \Delta \omega - 6{\partial ^a}\omega {\partial _a}\omega ]\phi = 6{\phi ^5}.$$
((33))
This equation is a special case of the Lichnérowicz equation and is sometimes also referred to as the Yamabe equation. For a given metric hab and boundary defining function ω it is a second-order, non-linear equation for the function ϕ. Note that the principal part of the equation degenerates on the boundary. Therefore, on the boundary, the Yamabe equation degenerates to the relation
$$\phi {\partial ^a}\omega {\partial _a}\omega = {\phi ^5}.$$
Note also that ϕ = 0 is a solution of (33) which, however, is not useful for our purposes because it would correspond to a conformal factor with vanishing first derivative on ℐ. Therefore, we require that ϕ be non-vanishing on the boundary, i.e. bounded from below by a strictly positive constant. Then the relation above determines the boundary values of ϕ in terms of the function ω . Taking derivatives of Equation (33), one finds that also the normal derivative of ϕ is fixed on the boundary in terms of the second derivative of ω.
A given metric ̃hab does not fix a unique pair (hab, Ω). Therefore, Equation (33) has the property that, for fixed ω, rescaling the metric hab with an arbitrary smooth non-vanishing function θ on Σ ∪ S according to hab ↦ θ-4hab results in a rescaling of the solution ϕ of (33) according to ϕ ↦ θϕ and, hence, a change in the conformal factor Ω ↦ θ-2Ω.
Now we define the trace-free part sab = ϕab - 1/3habϕcc of the projection ϕab of the trace-free part of the unphysical Ricci tensor onto Σ and consider the equations
$$\begin{array}{l}
{s_{ab}} = - {\Omega ^{ - 1}}({\partial _a}{\partial _b}\Omega - \frac{1}{3}{h_{ab}}\Delta \Omega )\:,\\
{E_{ab}} = - {\Omega ^{ - 1}}({R_{ab}} - \frac{1}{3}{h_{ab}}R - {s_{ab}})\:,
\end{array}$$
((1))
which follow from the Equations (20) and (16), respectively. Together with the fields hab, Ω, s = 1/3ΔΩ they provide initial data for all the quantities appearing in the evolution equations under the given assumptions. As they stand, these expressions are formally singular at the boundary and one needs to worry about the possibility of a smooth extension of the field to ∂Σ. This question was answered in [4], where the following theorem was proved:Theorem 7 Suppose (Σ, h) is a three-dimensional, orientable, compact, smooth Riemannian manifold with boundary ∂Σ. Then there exists a unique solution ϕ of (33), and the following conditions are equivalent:
-
1.
The function ϕ as well as the tensor fields sab and Eab determined on the interior ̃Σ from h and Ω = ϕ-2ω extend smoothly to all of Σ.
-
2.
The conformal Weyl tensor computed from the data vanishes on ∂Σ.
-
3.
The conformal class of h is such that the extrinsic curvature of •Σ with respect to its embedding in (Σ, h) is pure trace.
Condition (3) is a weak restriction of the conformal class of the metric h on Σ, since it is only effective on the boundary. It is equivalent to the fact that in the space-time which evolves from the hyperboloidal data, null-infinity ℐ is shear-free. Interestingly, the theorem only requires Σ to be orientable and does not restrict the topology of Σ any further.
This theorem gives the answer in a highly simplified case because the freedom in the extrinsic curvature has been suppressed. But there are also several other, less restrictive, treatments in the literature. In [2, 3] the assumption (29) is dropped allowing for an extrinsic curvature which is almost general apart from the fact that the mean curvature is required to be constant. In [85] also this requirement is dropped (but, in contrast to the other works, there is no discussion of smoothness of the implied conformal initial data), and in [87] the existence of hyperboloidal initial data is discussed for situations with a non-vanishing cosmological constant.
The theorem states that one can construct the essential initial data for the evolution once Equation (33) has been solved. The data are given by expressions which are formally singular at the boundary because of the division by the conformal factor Ω. This is of no consequences for the analytical considerations if Condition (3) in the theorem is satisfied. However, even then it is a problem for the numerical treatments because one has to perform a limit process to get to the values of the fields on the boundary. This is numerically difficult. Therefore, it would be desirable to solve the conformal constraints directly. It is clear from Equations (79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89) that the conformal constraints are regular as well. Some of the equations are rather simple but the overall dependencies and interrelations between the equations are very complicated. At the moment there exists no clear analytical method (or even strategy) for solving this system. An interesting feature appears in connection with Condition (3) of the theorem and analogous conditions in the more general cases. The necessity of having to impose this condition seems to indicate that the development of hyperboloidal data is not smooth but in general at most \({{\mathcal C}^2}\). If the condition were not imposed then logarithms appear in an expansion of the solution of the Yamabe equation near the boundary, and it is rather likely that these logarithmic terms will be carried along with the time evolution, so that the developing null-infinity looses differentiability. Thus, the conformal boundary is not smooth enough and, consequently, the Weyl tensor need not vanish on ℐ which, in addition, is not necessarily shear-free. The Sachs peeling property is not completely realized in these situations. One can show [2] that generically hyperboloidal data fall into the class of “poly-homogeneous” functions which are (roughly) characterized by the fact that they allow for asymptotic expansions including logarithmic terms. This behaviour is in accordance with other work [148] on the smoothness on ℐ, in particular with the Bondi-Sachs type expansions which were restricted by the condition of analyticity (i.e. no appearance of logarithmic terms). It is also consistent with the work of Christodoulou and Klainerman.
Solutions of the hyperboloidal initial value problem provide pieces of space-times which are semi-global in the sense that their future (or past) development is determined. However, the domain of dependence of a hyperboloidal initial surface does not include space-like infinity and one may wonder whether this fact is the reason for the apparent generic non-smoothness of null-infinity. Is it not conceivable that the possibility of making a connection between ℐ+ and ℐ- across i0 to build up a global space-time automatically excludes the non-smooth data? If we let the hyperboloidal initial surface approach space-like infinity it might well be that Condition (3) imposes additional conditions on asymptotically flat Cauchy data at spatial infinity. These conditions would make sure that the development of such Cauchy data is an asymptotically flat space-time, in particular that it has a smooth conformal extension at null-infinity.
These questions give some indications about the importance of gaining a detailed understanding of the structure of gravitational fields near space-like infinity. One of the difficulties in obtaining more information about the structure at space-like infinity is the lack of examples which are general enough. There exist exact radiative solutions with boost-rotation symmetry [20]. They possess a part of a smooth null-infinity which, however, is incomplete. This is a general problem because the existence of a complete null-infinity with non-vanishing radiation restricts the possible isometry group of a space-time to be at most one-dimensional with space-like orbits [13]. Some of the boost-rotation symmetric space-times even have a regular i0, thus they have a vanishing ADM-mass. Other examples exist of space-times which are solutions of the Einstein-Maxwell [32] or Einstein-Yang-Mills [14] equations. They have smooth and complete null-infinities. However, they were constructed in a way which enforces the field to coincide with the Schwarzschild or the Reissner-Nordstrom solutions near i0. So they are not general enough to draw any conclusions about the generic behaviour of asymptotically flat space-times near i0.
Space-like infinity
As indicated above, the problem at i0 is one of the most urgent and important ones related to the conformal properties of isolated systems. However, it is also one of the most complicated ones and a thorough discussion of all its aspects is not possible here. We will try to explain in rough terms what the new developments are, but we have to refer to [57] for the rigorous statements and all the details.
There have been several approaches over the years towards a treatment of space-like infinity. Geroch [66] gave a geometric characterization along the same lines as for null-infinity based on the conformal structure of Cauchy surfaces. He used his construction to define multipole moments for static space-times [63, 64], later to be generalized to stationary space-times by Hansen [74]. It was shown by Beig and Simon [19, 136] that the multipole moments uniquely determine a stationary space-time and vice versa.
Different geometric characterizations of spatial infinity in terms of the four-dimensional geometry were given by Sommers [137], Ashtekar and Hansen [10, 6], and by Ashtekar and Romano [11]. The difficulties in all approaches which try to characterize the structure of gravitational fields at space-like infinity in terms of the four-dimensional geometry arise from the lack of general results about the evolution of data near spatial infinity. Since there are no radiating solutions which are general enough at spatial infinity to provide hints, one is limited more or less to one’s intuition. So all these constructions essentially impose “reasonable” asymptotic conditions on the gravitational field at i0 and from them derive certain nice properties of space-times which satisfy these conditions. But there is no guarantee that there are indeed solutions of the Einstein equations which exhibit the claimed asymptotic behaviour. In a sense, all these characterisations are implicit definitions of certain classes of space-times (namely those which satisfy the imposed asymptotic conditions). What is needed is an analysis at space-like infinity which is not only guided by the geometry but which also takes the field equations into account (see e.g. [18, 17] for such attempts using formal power series).
Recently, Friedrich [57] has given such an analysis of space-like infinity which is based exclusively on the initial data, the field equations and the conformal structure of the space-time. In this representation several new aspects come together. First, in order to simplify the analysis, an assumption on the initial data (metric and extrinsic curvature) on an asymptotically Euclidean hypersurface ̃Σ is made. Since the focus is on the behaviour of the fields near space-like infinity, the topology of ̃Σ is taken to be ℝ3. It is assumed that the data are time-symmetric (̃χab = 0) and that on ̃Σ a (negative definite) metric ̃hab with vanishing scalar curvature is given. Let Σ be the conformal completion of (̃Σ, ̃hab) which is topologically S3, obtained by attaching a point i to ̃Σ, and assume furthermore that there exists a smooth positive function Ω on Σ with Ω(i) = 0, dΩ(i) = 0 and Hess Ω(i) negative definite. Furthermore, the metric hab = Ω2̃hab extends to a smooth metric on S. Thus, the three-dimensional conformal structure defined by ̃hab is required to be smoothly extensible to the point i.
From these assumptions follows that the conformal factor near i is determined by two smooth functions U and W, where U is characterized by the geometry near i while W collects global information because \(W(i) = {\textstyle{1 \over 2}}{m_{{\rm{ADM}}}}\), while U(i) = 1. With this information the rescaled Weyl tensor, the most important piece of initial data for the conformal field equations, near i is found to consist of two parts, a “massive” and a “mass-less” part. Under suitable conditions, the mass-less part, determined entirely by the local geometry near i, can be extended in a regular way to i, while the massive part always diverges at i as 1/|x|3 in a normal coordinate system (x1, x2, x3) at i unless the ADM-mass vanishes.
In order to analyze the singular behaviour of the initial data in more detail, the point i is blown up to a spherical set I0 essentially by replacing it with the sphere of unit vectors at i. Roughly speaking, this process yields a covering space \({\mathcal C}\) of (a suitable neighbourhood of i in) Σ projecting down to Σ which has the following properties: The pre-image I0 of i is an entire sphere while any other point on Σ has exactly two pre-image points. There exists a coordinate r on \({\mathcal C}\) which vanishes on I0 and which is such that on each pair of pre-image points it takes values r and -r, respectively. The actual blowup procedure involves a rather involved bundle construction which also takes into account the tensorial (respectively spinorial) nature of the quantities in question. The reader is referred to [57] for details.
Consider now a four-dimensional neighbourhood of space-like infinity. The next important step is the realization that, in order to take full advantage of the conformal structure of space-time, it is not enough to simply allow for metrics which are conformally equivalent to the physical metric but that one should also allow for more general connections. Instead of using a connection which is compatible with a metric in the conformal class, one may use a connection ∇a which is compatible with the conformal structure, i.e. which satisfies the condition
$${\nabla _c}{\tilde g_{ab}} = 2{\lambda _c}{\tilde g_{ab}}$$
for some one-form λa. If λa is exact, then one can find a metric in the conformal class for which ∇a is the Levi-Civita connection. Generally, however, this will not be the case. This generalization is motivated by the use of conformal geodesics as indicated below, and its effect is to free up the conformal factor, which we call Θ to distinguish it from the conformal factor Ω given on the initial surface Σ, from the connection (recall that two connections which are compatible with metrics in the same conformal class differ only by terms which are linear in the first derivative of the conformal factor relating the metrics). As a consequence, the conformal field equations, when expressed in terms of a generalized connection, do not any longer contain an equation for the conformal factor. It appears, instead, as a gauge source function for the choice of conformal metric. Additionally, a free one-form appears which characterizes the freedom in the choice of the conformal connection.
To fix this freedom, Friedrich uses conformal geodesics [60]. These are curves which generalize the concept of auto-parallel curves. They are given in terms of a system of ordinary differential equations (ODE’s) for their tangent vector together with a one-form along them. In coordinates this corresponds to a third-order ODE for the parameterization of the curve. Their crucial property is that they are defined entirely by the four-dimensional conformal structure with no relation to any specific metric in that conformal structure.
A time-like congruence of such curves is used to set up a “Gauß” coordinate system in a neighbourhood of i0 and to define a conformal frame, a set of four vectorfields which are orthonormal for some metric in the conformal class. This metric in turn defines a conformal factor Θ which rescales it to the physical metric. The one-form determined by the conformal geodesics defines a conformal connection ∇a, thus fixing the freedom in the connection. In this way, the gauge is fixed entirely in terms of the conformal structure.
If the physical space-time is a vacuum solution of the Einstein equations then one can say more about the behaviour of the conformal factor Θ along the conformal geodesics: It is a quadratic function of the natural parameter τ along the curves, vanishing at exactly two points if the initial conditions for the curves are chosen appropriately. The vanishing of Θ indicates the intersection of the curves with ℐ±. The intersection points are separated by a finite distance in the parameter τ.
Now one fixes an initial surface Σ with data as described above, and the conformal geodesics are used to set up the coordinate system and the gauge as above. When the blow-up procedure is performed for Σ, a new finite representation of space-like infinity is obtained which is sketched in Figure 8.
The point i on the initial surface has been replaced by a sphere I0 which is carried along the conformal geodesics to form a finite cylinder ℐ±. The surfaces are the surfaces on which the conformal factor Θ vanishes. They touch the cylinder in the two spheres I±, respectively. The conformal factor Θ vanishes with non-vanishing gradient on I and on ℐα while on the spheres I± its gradient also vanishes.
In this representation there is for the first time a clean separation of the issues which go on at space-like infinity: The spheres I± are the places where “ℐ touches i0” while the finite cylinder I serves two purposes. On the one hand, it represents the endpoints of space-like geodesics approaching from different directions, while, on the other hand, it serves as the link between past and future null-infinity. The part “outside” the cylinder where r is positive between the two null surfaces ℐ; corresponds to the physical space-time, while the part with r negative is not causally related to the physical space-time but constitutes a smooth extension. For easy reference, we call this entire space an extended neighbourhood of space-like infinity.
The conformal field equations, when expressed in the conformal Gauß gauge of this generalized conformal framework, yield a system of equations which has similar properties as the earlier version: It is a system of equations for a frame, the connection coefficients with respect to the frame, and the curvature, split up into the Ricci and the Weyl parts; they allow the extraction of a reduced system which is symmetric hyperbolic and propagates the constraint equations.
Its solutions yield solutions of the vacuum Einstein equations whenever Θ > 0. The Bianchi identities, which form the only sub-system consisting of partial differential equations, again play a key role in the system. Due to the use of the conformal Gauß gauge, all other equations are simply transport equations along the conformal geodesics.
The reduced system is written in symbolic form as
$${{\rm{A}}^\tau }\frac{\partial }{{\partial \tau }}{\rm{u}} + {{\rm{A}}^r}\frac{\partial }{{\partial r}}{\rm{u + }}{{\rm{A}}^\theta }\frac{\partial }{{\partial \theta }}{\rm{u + }}{{\rm{A}}^\phi }\frac{\partial }{{\partial \phi }}{\rm{u = Bu,}}$$
((35))
with symmetric matrices Aτ, Ar, Aθ, Aϕ, and C which depend on the unknown u and the coordinates (τ, r, θ, ϕ). This system as well as initial data for it, first defined only on the original space-time, can be extended in a regular way to an extended neighbourhood of space-like infinity which allows for the setup of a regular initial value problem at space-like infinity. Its properties are most interesting: When restricted to I0, the initial data coincide necessarily with Minkowski data, which together with the vanishing of Θ implies that on the entire cylinder I the coefficient matrix Ar vanishes. Thus, the system (35) degenerates into an interior symmetric hyperbolic system on I. Therefore, the finite cylinder I is a total characteristic of the system. The two null-infinities ℐ± are also characteristics, and at the intersections I± between them and I the system degenerates: The coefficient matrix Aτ which is positive definite on I and ℐα looses rank on I±.
The fact that I is a total characteristic implies that one can determine all fields on I from data given on I0. I is not a boundary on which one could specify in- or outgoing fields. This is no surprise, because the system (35) yields an entirely structural transport process which picks up data delivered from ℐ- via I- and moves them to ℐ+ via I+. It is also consistent with the standard Cauchy problem where it is known that one cannot specify any data “at infinity”.
The degeneracy of the equations at I± means that one has to take special precautions to make sure that the transitions from and to ℐ± are smooth. In fact, not all data “fit through the pipe”: Friedrich has derived restrictions on the initial data of solutions of the finite initial value problem which are necessary for regularity through I±. They are conditions on the conformal class of the initial data, stating that the Cotton tensor and all its symmetrized and trace-removed derivatives should vanish at the point i in the initial surface. If this is not the case, then the solution of the intrinsic system will develop logarithmic singularities which will then probably spread across null-infinity, destroying its smoothness. So here is another concrete indication that initial data have to be restricted albeit in a rather mild way in order for the smooth picture of asymptotic flatness to remain intact. It is not known whether this condition is also sufficient nor what its physical implications are.
Note that the conditions on the Cotton tensor are entirely local-at-infinity. This is the first time that such local conditions have been derived. It is rather surprising that the equations should render this possible.
The setting described in the above paragraphs certainly provides the means to analyze the consequences of the conformal Einstein evolution near spacelike infinity and to understand the properties of gravitational fields in that region. The finite picture allows the discussion of the relation between various concepts which are defined independently at null and space-like infinity. As one application of this kind Friedrich and Kánnár [58] have related the Newman-Penrose constants which are defined by a surface integral over a cut of ℐ+ to initial data on Σ. The cut of ℐ+ is pushed down towards I+ where it is picked up by the transport equations of system (35). In a similar way, one can relate the Bondi- and ADM-masses of a space-time.
Going further
Readers who are interested in obtaining information about the current status of existence theorems in general relativity are referred to Rendall’s recently updated Living Reviews article [121].
Although we have restricted ourselves to the case of a vanishing cosmological constant this does not mean that other cases have not been analyzed. Friedrich [52] has shown the existence of asymptotically simple de Sitter type solutions of the Einstein-Maxwell-Yang-Mills equations and stability of de Sitter space. More recently [54], with the general version of his conformal field equations, he was able to demonstrate the existence of asymptotically simple anti-de Sitter type solutions.
Finally, it should also be mentioned that recently some work has gone into generalizing the Bondi-Sachs calculations to poly-homogeneous space-times [30, 143, 144].