on , where is the metric induced on by the physical metric. Furthermore, let be the extrinsic curvature of in the physical space-time. Together with it satisfies the vacuum constraint equations,
where is the Levi-Civita connection of and 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 (31) implies that c is constant while the hyperboloidal character of implies that . With these simplifications and a rescaling of with a constant factor, the Hamiltonian constraint (30) takes the form
A further consequence of the condition (29) is the vanishing of the magnetic part of the Weyl tensor. For any defining function of the boundary, the conformal factor has the form . Expressing Equation (32) in terms of the unphysical quantities and 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 and boundary defining function it is a second-order, non-linear equation for the function . 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 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 . Therefore, we require that 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 in terms of the function . Taking derivatives of Equation (33), one finds that also the normal derivative of is fixed on the boundary in terms of the second derivative of .
A given metric does not fix a unique pair . Therefore, Equation (33) has the property that, for fixed , rescaling the metric with an arbitrary smooth non-vanishing function on according to results in a rescaling of the solution of (33) according to and, hence, a change in the conformal factor .
Now we define the trace-free part of the projection of the trace-free part of the unphysical Ricci tensor onto and consider the equations
which follow from the Equations (20) and (16), respectively. Together with the fields , , 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 . This question was answered in [4], where the following theorem was proved:
Theorem 7: Suppose is a three-dimensional, orientable, compact, smooth Riemannian manifold with boundary . Then there exists a unique solution of (33), and the following conditions are equivalent:
Condition ( 3 ) is a weak restriction of the conformal class of the metric h on , 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 is shear-free. Interestingly, the theorem only requires to be orientable and does not restrict the topology of 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 . 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 . 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 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 , 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 and across 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 , 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 . So they are not general enough to draw any conclusions about the generic behaviour of asymptotically flat space-times near .
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 |