2.1 Isolated systems

The conformal structure of space-times has found a wide range of interesting applications in general relativity with various motivations. Of particular importance for us is the emergence of conformal geometric ideas in connection with isolated systems (see also the related discussions in [79Jump To The Next Citation Point66Jump To The Next Citation Point]). As an illustration, consider a gravitating system (e.g. a binary system or a star) somewhere in our universe, evolving according to its own gravitational interaction, and possibly reacting to gravitational radiation impinging on it from the outside. Thereby, it will also emit gravitational radiation. We are interested in detecting and evaluating these waves because they provide us with important information about the physics governing the system. For several obvious reasons, it is desirable not only to have a description of such situations within the theoretical framework of the theory but, furthermore, to have the ability to simulate them numerically.

Two problems arise: First, we need to idealize the physical situation in an appropriate way, since it is hopeless to try to analyze the behaviour of the system in its interaction with the rest of the universe. We are mainly interested in the behaviour of the system and not so much in other processes taking place at large distances from the system. Since we would like to ignore those regions, we need a way to isolate the system from their influence.

We might want to do this by cutting away the uninteresting parts of the universe along a time-like cylinder T enclosing the system. Thereby, we effectively replace the outer part by data on T. The evolution of our system is determined by those data and initial data on some space-like hypersurface S. But now we are faced with the problem of interpreting the data. It is well known that initial data are obtained from some free data by solving elliptic equations. This is a global procedure. It is very difficult to give a physical meaning to initial data obtained in this way, and it is even more difficult, if not impossible, to specify a system, i.e. to determine initial data, exclusively from (local) physical properties of the constituents of the system like energy-momentum, spin, material properties, and such. In a similar spirit, the data on the time-like boundary T are complicated and only to a rather limited extent do they lend themselves to physical interpretation. For instance, it is not known how to extract from those data any piece that would unambiguously correspond to the radiation emitted by the system. Another problem is related to the arbitrariness in performing the cut. How can we be sure that we capture essentially the same behaviour independently of how we define T?

Thus, we are led to consider a different kind of “isolation procedure”. We imagine the system as being “alone in the universe” in the sense that we assume it being embedded in a space-time manifold that is asymptotically flat. How to formulate this is a priori rather vague. Somehow, we want to express the fact that the space-time “looks like” Minkowski space-time “at large distances” from the source. Certainly, fall-off conditions for the curvature have to be imposed as one recedes from the source and these conditions should be compatible with the Einstein equations. This means that there should exist solutions of the Einstein equations that exhibit these fall-off properties. We would then, on some initial space-like hypersurface S, prescribe initial data which should, on the one hand, satisfy the asymptotic conditions. On the other hand, the initial data should approximate in an appropriate sense the initial conditions that give rise to the real behaviour of the system. Our hope is that the evolution of these data provides a reasonable approximation of the real behaviour. As before, the asymptotic conditions (which, in a sense, replace the influence of the rest of the universe on the system) should not depend on the particular system under consideration. They should provide some universal structure against which we can gauge the information gained. Otherwise, we would not be able to compare different systems. Furthermore, we would hope that the conditions are such that there is a well defined way to allow for radiation to be easily extracted. It turns out that all these desiderata are in fact realized in the final formulation.

These considerations lead us to focus on space-times that are asymptotically flat in the appropriate sense. However, how should this notion be defined? How can we locate “infinity”? How can we express conditions “at infinity”?

This brings us to the second problem mentioned above. Even if we choose the idealization of our system as an asymptotically flat space-time manifold, we are still facing the task of adequately simulating the situation numerically. This is a formidable task, even when we ignore complications arising from difficult matter equations. The simulation of gravitational waves in an otherwise empty space-time coming in from infinity, interacting with themselves, and going out to infinity is a challenging problem. The reason is obvious: Asymptotically flat space-times necessarily have infinite extent, while computing resources are finite.

The conventional way to overcome this apparent contradiction is the introduction of an artificial boundary “far away from the interesting regions”. During the simulation this boundary evolves in time, thus defining a time-like hypersurface in space-time. There one imposes conditions which, it is hoped, approximate the asymptotic conditions. However, introducing the artificial boundary is nothing but the reintroduction of the time-like cylinder T on the numerical level with all its shortcomings. Instead of having a “clean” system that is asymptotically flat and allows well defined asymptotic quantities to be precisely determined, one is now dealing again with data on a time-like boundary whose meaning is unclear. Even if the numerical initial data have been arranged so that the asymptotic conditions are well approximated initially by the boundary conditions on T, there is no guarantee that this will remain so when the system is evolved. Furthermore, the numerical treatment of an initial-boundary value problem is much more complicated than an initial value problem because of instabilities that can easily be generated at the boundary.

What is needed, therefore, is a definition of asymptotically flat space-times that allows one to overcome both the problem of “where infinity is” and the problem of simulating an infinite system with finite resources. The key observation in this context is that “infinity” is far away with respect to the space-time metric. This means that one needs infinitely many “metre sticks” in succession in order to “get to infinity”. But, what if we replaced these metre sticks by ones that grow in length the farther out we go? Then it might be possible that only a finite number of them suffices to cover an infinite range, provided the growth rate is just right. This somewhat naive picture can be made much more precise: Instead of using the physical space-time metric &tidle;g to measure distance and time, we use a different metric g = Ω2&tidle;g, which is “scaled down” with a scale factor Ω. If Ω can be arranged to approach zero at an appropriate rate, then this might result in “bringing infinity in to a finite region” with respect to the unphysical metric g. We can imagine attaching points to the space-time that are finite with respect to g but which are at infinity with respect to &tidle;g. In this way we can construct a boundary consisting of all the end points of the succession of finitely many rescaled metre sticks arranged in all possible directions. This construction works for Minkowski space and so it is reasonable to define asymptotically flat space-times as those for which the scaling-down of the metric is possible.

We arrived at this idea by considering the metric structure only “up to arbitrary scaling”, i.e. by looking at metrics which differ only by a factor. This is the conformal structure of the space-time manifold in question. By considering the space-time only from the point of view of its conformal structure we obtain a picture of the space-time which is essentially finite but which leaves its causal properties, and hence the properties of wave propagation unchanged. This is exactly what is needed for a rigorous treatment of radiation emitted by the system and also for the numerical simulation of such situations.

The way we have presented the emergence of the conformal structure as the essence of asymptotically flat space-times is not how it happened historically. Since it is rather instructive to see how various approaches finally came together in the conformal picture, we will present in the following Section 2.2 a short overview of the history of the subject.

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