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
enclosing the system. Thereby, we effectively replace the outer
part by data on
. The evolution of our system is
determined by those data and initial data on some space-like
hypersurface
. 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
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
?
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 , 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 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
,
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 to measure distance and time, we use a different
metric
, which is “scaled down” with a scale
factor
. If
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
.
We can imagine attaching points to the space-time that are finite
with respect to
but which are at infinity with respect
to
. 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.