3.5 Space-like infinity3 The Regular Conformal Field 3.3 Initial value problems

3.4 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 hyperboloidal 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 tex2html_wrap_inline4743 be a hyperboloidal hypersurface in an asymptotically flat vacuum space-time which extends out to tex2html_wrap_inline3905, touching it in a two-surface tex2html_wrap_inline4747 which is topologically a two-sphere. The assumptions on tex2html_wrap_inline4743 are equivalent to the fact that tex2html_wrap_inline4751 is a smooth Riemannian manifold with boundary, which carries a smooth metric tex2html_wrap_inline4612 and a smooth function tex2html_wrap_inline3721, obtained by restriction of the unphysical metric and the conformal factor. The conformal factor is a defining function for the boundary tex2html_wrap_inline4747 (i.e. it vanishes only on tex2html_wrap_inline4747 with non-vanishing gradient), and together with the metric it satisfies


on tex2html_wrap_inline4743, where tex2html_wrap_inline4763 is the metric induced on tex2html_wrap_inline4743 by the physical metric. Furthermore, let tex2html_wrap_inline4767 be the extrinsic curvature of tex2html_wrap_inline4743 in the physical space-time. Together with tex2html_wrap_inline4763 it satisfies the vacuum constraint equations,



where tex2html_wrap_inline4773 is the Levi-Civita connection of tex2html_wrap_inline4763 and tex2html_wrap_inline4777 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,


The momentum constraint (31Popup Equation) implies that c is constant while the hyperboloidal character of tex2html_wrap_inline4743 implies that tex2html_wrap_inline4785 . With these simplifications and a rescaling of tex2html_wrap_inline4763 with a constant factor, the Hamiltonian constraint (30Popup Equation) takes the form


A further consequence of the condition (29Popup Equation) is the vanishing of the magnetic part tex2html_wrap_inline4789 of the Weyl tensor. For any defining function tex2html_wrap_inline4480 of the boundary, the conformal factor has the form tex2html_wrap_inline4793 . Expressing Equation (32Popup Equation) in terms of the unphysical quantities tex2html_wrap_inline4612 and tex2html_wrap_inline3721 yields the single second-order equation


This equation is a special case of the Lichnérowicz equation and is sometimes also referred to as the Yamabe equation. For a given metric tex2html_wrap_inline4612 and boundary defining function tex2html_wrap_inline4480 it is a second-order, non-linear equation for the function tex2html_wrap_inline4803 . Note that the principal part of the equation degenerates on the boundary. Therefore, on the boundary, the Yamabe equation degenerates to the relation


Note also that tex2html_wrap_inline4807 is a solution of (33Popup Equation) which, however, is not useful for our purposes because it would correspond to a conformal factor with vanishing first derivative on tex2html_wrap_inline3905 . Therefore, we require that tex2html_wrap_inline4803 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 tex2html_wrap_inline4803 in terms of the function tex2html_wrap_inline4480 . Taking derivatives of Equation (33Popup Equation), one finds that also the normal derivative of tex2html_wrap_inline4803 is fixed on the boundary in terms of the second derivative of tex2html_wrap_inline4480 .

A given metric tex2html_wrap_inline4763 does not fix a unique pair tex2html_wrap_inline4823 . Therefore, Equation (33Popup Equation) has the property that, for fixed tex2html_wrap_inline4480, rescaling the metric tex2html_wrap_inline4612 with an arbitrary smooth non-vanishing function tex2html_wrap_inline3875 on tex2html_wrap_inline4831 according to tex2html_wrap_inline4833 results in a rescaling of the solution tex2html_wrap_inline4803 of (33Popup Equation) according to tex2html_wrap_inline4837 and, hence, a change in the conformal factor tex2html_wrap_inline4839 .

Now we define the trace-free part tex2html_wrap_inline4841 of the projection tex2html_wrap_inline4843 of the trace-free part of the unphysical Ricci tensor onto tex2html_wrap_inline4731 and consider the equations


which follow from the Equations (20Popup Equation) and (16Popup Equation), respectively. Together with the fields tex2html_wrap_inline4612, tex2html_wrap_inline3721, tex2html_wrap_inline4851 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 tex2html_wrap_inline4747 . This question was answered in [4], where the following theorem was proved:

Theorem 7:  Suppose tex2html_wrap_inline4855 is a three-dimensional, orientable, compact, smooth Riemannian manifold with boundary tex2html_wrap_inline4747 . Then there exists a unique solution tex2html_wrap_inline4803 of (33Popup Equation), and the following conditions are equivalent:

  1. The function tex2html_wrap_inline4803 as well as the tensor fields tex2html_wrap_inline4863 and tex2html_wrap_inline4865 determined on the interior tex2html_wrap_inline4743 from h and tex2html_wrap_inline4793 extend smoothly to all of tex2html_wrap_inline4731 .
  2. The conformal Weyl tensor computed from the data vanishes on tex2html_wrap_inline4747 .
  3. The conformal class of h is such that the extrinsic curvature of tex2html_wrap_inline4747 with respect to its embedding in tex2html_wrap_inline4855 is pure trace.  

Condition ( 3 ) is a weak restriction of the conformal class of the metric h on tex2html_wrap_inline4731, 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 tex2html_wrap_inline3905 is shear-free. Interestingly, the theorem only requires tex2html_wrap_inline4731 to be orientable and does not restrict the topology of tex2html_wrap_inline4731 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 [2Jump To The Next Citation Point In The Article, 3] the assumption (29Popup Equation) 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 (33Popup Equation) has been solved. The data are given by expressions which are formally singular at the boundary because of the division by the conformal factor tex2html_wrap_inline3721 . 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 (79Popup Equation, 80Popup Equation, 81Popup Equation, 82Popup Equation, 83Popup Equation, 84Popup Equation, 85Popup Equation, 86Popup Equation, 87Popup Equation, 88Popup Equation, 89Popup Equation) 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 tex2html_wrap_inline4895 . 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 tex2html_wrap_inline3905 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 tex2html_wrap_inline3905, 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 tex2html_wrap_inline3847 and tex2html_wrap_inline3849 across tex2html_wrap_inline3683 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 [20Jump To The Next Citation Point In The Article]. 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 tex2html_wrap_inline3683, 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-Nordström solutions near tex2html_wrap_inline3683 . So they are not general enough to draw any conclusions about the generic behaviour of asymptotically flat space-times near tex2html_wrap_inline3683 .

3.5 Space-like infinity3 The Regular Conformal Field 3.3 Initial value problems

image Conformal Infinity
Jörg Frauendiener
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