3.6 Going further3 The Regular Conformal Field 3.4 Hyperboloidal initial data

3.5 Space-like infinity 

As indicated above, the problem at tex2html_wrap_inline3683 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 [57Jump To The Next Citation Point In The Article] 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 tex2html_wrap_inline3683 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 [57Jump To The Next Citation Point In The Article] 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 tex2html_wrap_inline4743 is made. Since the focus is on the behaviour of the fields near space-like infinity, the topology of tex2html_wrap_inline4743 is taken to be tex2html_wrap_inline4922 . It is assumed that the data are time-symmetric (tex2html_wrap_inline4924) and that on tex2html_wrap_inline4743 a (negative definite) metric tex2html_wrap_inline4763 with vanishing scalar curvature is given. Let tex2html_wrap_inline4731 be the conformal completion of tex2html_wrap_inline4932 which is topologically tex2html_wrap_inline4934, obtained by attaching a point i to tex2html_wrap_inline4743, and assume furthermore that there exists a smooth positive function tex2html_wrap_inline3721 on tex2html_wrap_inline4731 with tex2html_wrap_inline4944, tex2html_wrap_inline4946 and tex2html_wrap_inline4948 negative definite. Furthermore, the metric tex2html_wrap_inline4950 extends to a smooth metric on S . Thus, the three-dimensional conformal structure defined by tex2html_wrap_inline4763 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 tex2html_wrap_inline4970, 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 tex2html_wrap_inline4982 in a normal coordinate system tex2html_wrap_inline4984 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 tex2html_wrap_inline3691 essentially by replacing it with the sphere of unit vectors at i . Roughly speaking, this process yields a covering space tex2html_wrap_inline4994 of (a suitable neighbourhood of i in) tex2html_wrap_inline4731 projecting down to tex2html_wrap_inline4731 which has the following properties: The pre-image tex2html_wrap_inline3691 of i is an entire sphere while any other point on tex2html_wrap_inline4731 has exactly two pre-image points. There exists a coordinate r on tex2html_wrap_inline4994 which vanishes on tex2html_wrap_inline3691 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 tex2html_wrap_inline4099 which is compatible with the conformal structure, i.e. which satisfies the condition

displaymath5020

for some one-form tex2html_wrap_inline5022 . If tex2html_wrap_inline5022 is exact, then one can find a metric in the conformal class for which tex2html_wrap_inline4099 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 tex2html_wrap_inline5028 to distinguish it from the conformal factor tex2html_wrap_inline3721 given on the initial surface tex2html_wrap_inline4731, 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 tex2html_wrap_inline3683 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 tex2html_wrap_inline5028 which rescales it to the physical metric. The one-form determined by the conformal geodesics defines a conformal connection tex2html_wrap_inline4099, 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 tex2html_wrap_inline5028 along the conformal geodesics: It is a quadratic function of the natural parameter tex2html_wrap_inline4156 along the curves, vanishing at exactly two points if the initial conditions for the curves are chosen appropriately. The vanishing of tex2html_wrap_inline5028 indicates the intersection of the curves with tex2html_wrap_inline3867 . The intersection points are separated by a finite distance in the parameter tex2html_wrap_inline4156 .

  

Click on thumbnail to view image

Figure 8: The geometry near space-like infinity: The ``point'' tex2html_wrap_inline3683 has been blown up to a cylinder which is attached to tex2html_wrap_inline3849 and tex2html_wrap_inline3847 . The physical space-time is the exterior part of this ``stovepipe''. The ``spheres'' tex2html_wrap_inline3689 and tex2html_wrap_inline3691 are shown in blue and light green, respectively. The brown struts symbolize the conformal geodesics used to set up the construction. Note that they intersect tex2html_wrap_inline3905 and continue into the unphysical part.

Now one fixes an initial surface tex2html_wrap_inline4731 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 tex2html_wrap_inline4731, 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 tex2html_wrap_inline3691 which is carried along the conformal geodesics to form a finite cylinder I . The surfaces tex2html_wrap_inline3867 are the surfaces on which the conformal factor tex2html_wrap_inline5028 vanishes. They touch the cylinder in the two spheres tex2html_wrap_inline3689, respectively. The conformal factor tex2html_wrap_inline5028 vanishes with non-vanishing gradient on I and on tex2html_wrap_inline3867 while on the spheres tex2html_wrap_inline3689 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 tex2html_wrap_inline3689 are the places where `` tex2html_wrap_inline3905 touches tex2html_wrap_inline3683 '' 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 tex2html_wrap_inline3867 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 tex2html_wrap_inline5100 . 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

  equation823

with symmetric matrices tex2html_wrap_inline5102, tex2html_wrap_inline5104, tex2html_wrap_inline5106, tex2html_wrap_inline5108, and tex2html_wrap_inline5110 which depend on the unknown tex2html_wrap_inline4378 and the coordinates tex2html_wrap_inline5114 . 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 tex2html_wrap_inline3691, the initial data coincide necessarily with Minkowski data, which together with the vanishing of tex2html_wrap_inline5028 implies that on the entire cylinder I the coefficient matrix tex2html_wrap_inline5104 vanishes. Thus, the system (35Popup Equation) 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 tex2html_wrap_inline3867 are also characteristics, and at the intersections tex2html_wrap_inline3689 between them and I the system degenerates: The coefficient matrix tex2html_wrap_inline5102 which is positive definite on I and tex2html_wrap_inline3867 looses rank on tex2html_wrap_inline3689 .

The fact that I is a total characteristic implies that one can determine all fields on I from data given on tex2html_wrap_inline3691 . I is not a boundary on which one could specify in- or outgoing fields. This is no surprise, because the system (35Popup Equation) yields an entirely structural transport process which picks up data delivered from tex2html_wrap_inline3849 via tex2html_wrap_inline5152 and moves them to tex2html_wrap_inline3847 via tex2html_wrap_inline5156 . 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 tex2html_wrap_inline3689 means that one has to take special precautions to make sure that the transitions from and to tex2html_wrap_inline3867 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 tex2html_wrap_inline3689 . 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 space-like 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 tex2html_wrap_inline3847 to initial data on tex2html_wrap_inline4731 . The cut of tex2html_wrap_inline3847 is pushed down towards tex2html_wrap_inline5156 where it is picked up by the transport equations of system (35Popup Equation). In a similar way, one can relate the Bondi- and ADM-masses of a space-time.



3.6 Going further3 The Regular Conformal Field 3.4 Hyperboloidal initial data

image Conformal Infinity
Jörg Frauendiener
http://www.livingreviews.org/lrr-2000-4
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de