Initial value problems

3.3 Initial value problems

From the physical point of view the most interesting scenario is the following one: A gravitating material system (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

Figure 6: The physical scenario: The figure describes the geometry of an isolated system. Initial data are prescribed on the blue parts, i.e. on a hyperboloidal hypersurface and the part of $− ℐ$ which is in its future. Note that the two cones $ℐ +$ and $ℐ −$ are separated to indicate the non-trivial transition between them.

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 $ℐ −$ that 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 that 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, that 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 that 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 $0 i$ 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 [31 ] 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 that 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 that 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 [56 ] 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.

Figure 7: The geometry of the asymptotic characteristic initial value problem: Characteristic data are given on the blue parts, i.e. an ingoing null surface and the part of $− ℐ$ that is in its future. Note that the ingoing surface may develop self-intersections and caustics.

In this kind of initial value problem, one specifies data on an ingoing null hypersurface $𝒩$ and that part of $ℐ$ that is in the future of $𝒩$ . The data that have to be prescribed are essentially the so-called null data on $𝒩$ and $ℐ$ , i.e. those parts of the rescaled Weyl tensor that are entirely intrinsic to the respective null hypersurfaces. In the case of $ℐ$ , the null datum is exactly the radiation field.

Theorem 3 (Kánnár [104 ]): For given smooth null data on an ingoing null hypersurface $𝒩$ and a smooth radiation field on the part $ℐ$ of $ℐ −$ that is to the future of the intersection $S$ of $𝒩$ with $ℐ −$ and certain data on $S$ , there exists a smooth solution of Einstein’s vacuum equations in the future of $𝒩 ∪ ℐ$ that implies the given data on $𝒩 ∪ ℐ$ .

The result is in complete agreement with Sachs’ earlier analysis of the asymptotic characteristic initial value problem based on formal expansion methods [145 ].

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 [59 , 61 ]. 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 in

Theorem 4 (Friedrich): With each asymptotically flat static solution of Einstein’s vacuum field equations can be associated another solution of these equations that 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 that 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 in

Theorem 5 (Friedrich [57 ]): Smooth hyperboloidal initial data on a hyperboloidal hypersurface $Σ$ determine a unique solution of Einstein’s vacuum field equations that admits a smooth conformal boundary at null-infinity in the future of $Σ$ .

There exists also a stability result that states that there are solutions that behave exactly like Minkowski space near future time-like infinity:

Theorem 6 (Friedrich [60 ]): 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.