So far, the discretization of the equations has been rather
straightforward. One of the schemes which have been used is the
higher dimensional Lax-Wendroff scheme, also called the rotated
Richtmyer scheme, a discretization scheme with second order
accuracy. It has been employed alone [38] or together with Strang splitting [82
] to treat the principal part of the equations differently from
the source part. Since a second order scheme requires much more
computing resources compared to higher order methods to achieve
the same accuracy, Hübner started to use the method of
lines [73
] with a fourth order scheme to compute the spatial derivatives
and fourth order Runge-Kutta for the evolution in time. He
reports [82
] that the fourth order method is very much superior to the
second order scheme in terms of efficiency. (The feasibility of
the method of lines in relativity has been studied by
Hungerbühler [84] using pseudo-spectral methods for the spatial derivatives and a
combination of Adams-Bashforth and Adams-Moulton schemes for the
time evolution.)
The conformal field equations and the propagation equations
derived from them are quasi-linear. This implies that the
characteristics of the system depend on the current solution and
this, in turn, means that one has to be able to change the
time-step
between successive time-slices in order to keep a stable
evolution scheme. This is necessary because the schemes are
explicit schemes and, therefore, subject to the
Courant-Friedrichs-Lewy condition which states that the numerical
domain of dependence of a point should always include the
analytical domain of dependence. This requirement already
excludes the popular leapfrog scheme which is nevertheless used
sometimes also for evolving the Einstein equations. A general
criterion for computing the maximal time-step allowed in each
iteration in arbitrary dimension has been derived in [38
].
Another important point in the development of evolution codes is the numerical treatment of the boundaries. As explained already above, it is one of the advantages in the conformal approach that the outer boundary is not as influential as it is in the conventional approach using the standard Cauchy problem. It was also pointed out that in this case it is enough to impose a boundary condition which results in a numerically stable code because the outer boundary is located in the unphysical region and, therefore, cannot influence the physical space-time.
The proper way to treat the boundary is to prescribe
conditions which are compatible with the full conformal field
equations, in particular with their restriction to the boundary
manifold. This has not been done so far. Since the outer boundary
is not important for the physical effects, other ways of dealing
with the boundary have been devised. One way is to forget about
the restrictions of the conformal field equations to the boundary
and to analyze the possible boundary conditions for the
propagation equations. To first order, one can define ingoing and
outgoing fields on the boundary. Then a sufficiently general
boundary condition will be obtained by specifying the ingoing
fields in terms of the outgoing ones. Although this boundary
treatment is not compatible with the restriction of the conformal
field equations to the boundary, it is compatible with the
evolution equations. This means that the evolution can remain
stable although the solution will not satisfy the constraints in
the domain of influence of the boundary, which, however, is
always in the unphysical part of space-time. This method has been
used in [38] with satisfactory results. In particular, the boundary did not
give rise to non-physical modes. These findings are in agreement
with the analysis of numerical boundary conditions by
Trefethen [141].
Another, very clever method for dealing with the boundary has been found by Hübner [81]. He realized that it is sufficient to solve the conformal field equations in the physical space-time only, and not necessary to solve them in the unphysical region as long as the characteristics remain such that the information created in the unphysical part of the computational domain cannot reach the physical part. Consequently, in his treatment the grid is divided into three zones: the inner zone, the outer zone, and a transition zone. The inner zone covers the physical space-time (flagged by a positive conformal factor) and some part of the adjacent unphysical region. On this part of the grid the conformal field equations are solved. In the outer zone, which is located in a neighbourhood of the grid boundary, one solves an advection equation which propagates outwards, off the grid. In the transition zone, a sufficiently smooth interpolation between these two systems of equations is solved. The effect is that the boundary condition which has to be imposed on the grid boundaries is very simple and that the noise which is generated in the transition region is propagated away from the physical region outwards towards the grid boundary.
Our next point is concerned with the extraction of the
radiative information from the numerically generated data. This
is the part of the entire numerical process where the superiority
of the conformal approach becomes apparent. How does one
determine the radiative field? First of all, one needs to find
on the current time-slice. Since
is the surface on which the conformal factor
vanishes and since
is explicitly known during the evolution the location of
is a simple task. The next problem is concerned with the
orientation of the tetrad on
. The asymptotic quantities are defined with respect to a
specific geometrically characterized tetrad, a Bondi frame. But,
in general, this tetrad is completely unrelated to the
``computational'' tetrad used for the evolution. Therefore, one
needs to find the transformation from one to the other at each
point of
. Without going into too much details (see [38
,
42
,
41
]) we remark that most asymptotic quantities, in particular the
radiation field, are of a local character so they can be read off
without constructing a Bondi frame. This is rather fortunate
because there are global issues involved in the transformation
from the computational tetrad to the Bondi frame. These have
implications for the determination of global quantities like the
Bondi energy-momentum four-vector, but they have no effect on the
radiation field, which is defined as that (complex) component of
which is entirely intrinsic to
:
Here,
is a null-vector tangent to the generators of
, i.e.,
, and
is any complex space-like null-vector which is orthogonal to
. It is useful to require the space-like vector to be tangent to
the intersection of
with the current time-slice. Augmenting these two vectors by a
further real null-vector
yields a null-tetrad which is fixed up to rotations in the
(two-dimensional) tangent space of that intersection and boosts
in the plane orthogonal to it. The behaviour of
under these transformations is that of a GHP-weighted
quantity [67,
117] with boost weight -2 and spin weight -2. This corresponds to
the quadrupole-like character of the gravitational radiation
field. However,
really depends only on the null-vector
. For, suppose we perform a null-rotation around
, then
transforms into
for some complex valued function
on
. But
is invariant under this transformation. So in order to find
it is only necessary to transform from the given computational
tetrad to the tetrad specified above which is rather
straightforward. In fact, the computation of
involves only the combination of certain components of the
gravitational field with powers of
.
The final step in the correct determination of the radiation
is to find the correct time parameter. Suppose we follow a
specific null generator of
crossing through successive time-slices. On each slice we
compute
on that generator. Then we obtain the radiation emitted by the
source into the direction specified by the generator as a
function of our computational time parameter. Since the time
coordinate is rather arbitrary, this means that the wave form
determined so far has no physical meaning. The problem is already
present in Maxwell's theory: Suppose we have an emitter which
sends out a pure sine wave. A detector far away from the source
cannot determine the absolute frequency of the signal because the
relative velocity of emitter and receiver might be non-zero but
the detector should also find a pure sine signal. However, this
will be true only if the detector records the signal as a
function of proper time. Any other time parameter along the
detector's world-line will not produce a pure sine.
What one needs to do in the general case is to select among all parameters along the generator a specific, geometrically distinguished one, namely a Bondi parameter. A generator and such a parameter along it can be understood as a certain limit of freely falling observers with proper time clocks as they move towards infinity [42]. Bondi parameters are obtained as solutions of an ordinary, linear, second order differential equation, which is conformally invariant.
The computation of the Bondi energy-momentum is a global
procedure, i.e., it depends on properties of the entire cut of
with the current time-slice. There are two steps involved in
this procedure. First, one needs to obtain the asymptotic
translation group (see e.g. [118]) on each cut. This provides four functions on the cut which are
then, in a second step, integrated against the ``mass aspect''
which is another function obtained from the ``Coulomb'' part
of the gravitational field, and the ``news function'' which is a
combination of components of the Ricci tensor and connection
coefficients. The first step, the determination of the
translation group, is the global step because it involves solving
a second order elliptic equation on the cut. These issues are
discussed in more detail in [41].
All these procedures for finding the relevant data on
have been worked out analytically and they have also been tested
(at least in part) numerically [38
]. The tests have been performed under the assumption that
null-infinity admits toroidal cuts, which has the advantage that
one can actually compare the numerical results with analytical
expressions because a whole class of exact solutions [135
,
80
] is known to exist. Admittedly, such space-times are rather
unphysical, but since most of the extraction procedures are local
there is no doubt that they will also work in more realistic
cases.
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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 |