This process, sometimes referred to as ``hyperbolic reduction'' consists of several steps. First, one needs to break the invariance of the equations. By imposing suitable gauge conditions one can specify a coordinate system, a linear reference frame and a conformal factor. Then the equations can be written as equations for the components of the geometric quantities with respect to the chosen frame in the chosen conformal gauge and as functions of the chosen coordinates. In the next step, one needs to extract from the equations a subsystem of propagation equations which is hyperbolic so that it has a well-posed initial value problem. It is often referred to as the ``reduced equations''. Finally, one has to make sure that solutions of the reduced system give rise to solutions of the full system. This step may involve the verification that the gauge conditions imposed are compatible with the propagation equations, or that other equations (constraints) not included in the reduced system are preserved under the propagation. The first two steps, choice of gauge and extraction of the reduced system, are very much related. Gauge conditions should be imposed such that they lead to a hyperbolic reduced system. Furthermore, the gauge conditions should be such that they can be imposed locally without loss of generality.
The gauge freedom present in the conformal field equations can easily be determined. The freedom to choose the coordinates amounts to four scalar functions while the linear reference frame, which we take to be orthogonal, can be specified by a Lorentz rotation, which amounts to six free functions. Finally, the choice of a conformal factor contributes another free function. Altogether, there are eleven functions which can be chosen at will.
Once the geometric equations have been transformed into equations for components, the next step is to extract the reduced system. These are equations for the components of the geometric quantities defined above as well as for gauge-dependent quantities: the components of the frame with respect to the coordinate basis, the components of the connection with respect to the given frame and the conformal factor.
There are several well-known choices for coordinates (harmonic, Gauß, Bondi, etc.), as well as for frames (Fermi-Walker transport, Newman-Penrose, etc.). These are usually ``hard-wired'' into the equations and one has no further control on the properties of the gauge. Gauß coordinates for instance have the tendency to become singular when the geodesic congruence which is used for their definition starts to self-intersect. Similarly, Bondi coordinates are attached to null-hypersurfaces which have the tendency to self-intersect thus destroying the coordinate system. In the context of existence proofs and the numerical evolution of the equations it is of considerable interest to have additional flexibility in order to prevent the coordinates or the frame from becoming singular. The goal is to ``fix the gauge'' in as flexible a manner as possible and to obtain reduced equations which still have useful properties.
A scheme to obtain the reduced equations in symmetric
hyperbolic form while still allowing for arbitrary gauges has
been devised by Friedrich [48] (see also [55
] for various examples). The idea is based on the following
observation. Cartan's structure equations which express the
torsion and curvature tensors in terms of tetrad and connection
coefficients are two-form equations: They are skew on two indices
and the information contained in the equations is not enough to
fix the tetrad and the connection by specifying the torsion and
the curvature. The additional information is provided by fixing a
gauge. Normally, this is achieved by reducing the number of
variables, in this case the number of tetrad components and
connection coefficients. However, one can just as well add
appropriate further equations to have enough equations for all
unknowns. The additional equations should be chosen so that the
ensuing system has ``nice'' properties.
We illustrate this procedure by a somewhat trivial example.
Consider, in flat space with coordinates
, a one-form
which we require to be closed:
From this equation we can extract three evolution equations, namely
Obviously, these three equations are not sufficient to
reconstruct
from appropriate initial data. One possibility to proceed from
here is to specify one component of
freely and then obtain equations for the other three. However,
it is easily seen that only by specifying
we can achieve a pure evolution system. Otherwise, we get
mixtures of evolution and constraint equations. So we may note
that proceeding in this way leads to a restriction of
possibilities as to which components should be specified freely
and, in general, it also entails that derivatives of the
specified component appear.
Another possible procedure is to enlarge the system by adding
an equation for the time derivative of
. Doing this covariantly implies that we should add an equation
in the form of a divergence
where F is an arbitrary function. This results in the system
which is symmetric hyperbolic for any choice of
F
. Note also that
F
appears as a source term and only in undifferentiated form.
Clearly, our influence on the component
is now very indirect via the solution of the system, while
before we could specify it directly.
In a similar way, one proceeds in the present case of the
conformal field equations. Note, however, that this way of fixing
a gauge is not at all specific to these equations. Since it
depends essentially only on the form of Cartan's structure
equations it is applicable in all cases where these are part of
the first order system. The Cartan equations can be regarded as
exterior equations for the one-forms
dual to a tetrad
and the connection one-forms
. Similar to the system above, the equations involve only the
exterior derivative of the one-forms and so we expect that we
should add equations in divergence form, namely
with arbitrary gauge source functions
for fixing coordinates and
for choosing a tetrad. Note that
implies
.
In a given gauge (i.e., coordinates and frame field are specified) the gauge sources can be determined from
In fact, these equations are exactly the same equations
as (23) except that they are written in a more invariant form. Now it
is obvious that the gauge sources contain information about the
coordinates and the frame used. What needs to be shown is that
any specification of the gauge sources fixes a gauge. In fact,
suppose we are given functions
and
on
then there exist (locally) coordinates
and a frame
so that in that coordinate system the gauge sources are just the
prescribed functions
and
. This follows from the equations
These are semi-linear wave equations which determine a unique
solution from suitably given initial data close to the initial
surface. Note that on the right hand side of (26) there is a function of the
, and not a source term. The equations can be solved in steps.
Once the coordinates
have been determined from (26
), the right hand side of (27
) can be considered as a source term.
Finally, we need to discuss the gauge freedom in the choice of
the conformal factor
. In many discussions of asymptotic structure the conformal
factor is chosen in such a way that null-infinity is
divergence-free, in addition to the vanishing of its shear, which
is a consequence of the asymptotic vacuum equations. That means
that infinitesimal area elements remain unchanged in size as they
are parallelly transported along the generators of
. Since they also remain unchanged in form due to the vanishing
shear of
, they remain invariant and hence they can be used to define a
unique metric on the space of generators of
. This choice simplifies many calculations on
, still leaving the conformal factor quite arbitrary away from
. Yet, in numerical applications this choice of the conformal
factor may be too rigid and so one needs a flexible method for
fixing the conformal factor.
It turns out that one can introduce a gauge source function
for the conformal gauge as well. Consider the change of the
scalar curvature under the conformal rescaling
,
: It transforms according to
Reading this transformation law as an equation for
we obtain
It follows from this equation that we may regard the scalar
curvature as a gauge source function for the conformal factor:
For, suppose we specify the function
arbitrarily on
, then Equation (28
) is a non-linear wave equation for
which can be solved given suitable initial data. This determines
a unique
, hence a unique
and
such that the scalar curvature of the rescaled metric
has scalar curvature
. Note that these considerations are local. They show that
locally the gauges can be fixed arbitrarily. However, the problem
of identifying and fixing a gauge globally is very difficult but
also very important because only when the gauges are globally
known one can really compare two different space-times.
Having established that the gauge sources do in fact, locally,
fix a unique gauge we can now split the system of conformal field
equations into evolution equations and constraints. The resulting
system of equations is exhibited below. The reduction process is
rather straightforward but tedious. It is sketched in
Appendix
6
. Here, we only describe it very briefly. We introduce an
arbitrary time-like unit vectorfield
which has a priori no relation to the tetrad field used for
framing. We split all the tensorial quantities into the parts
which are parallel and orthogonal to that vector field using the
projector
. The connection coefficients for the four-dimensional
connection
are treated differently. We introduce the covariant derivatives
of the vectorfield
by
They account for half (9+3) of the four-dimensional connection
coefficients. The other half is captured by defining a covariant
derivative
which has the property that it annihilates both
and
and agrees with
when acting on tensors orthogonal to
(see Equations (40
)). Note that we have not required that
be the time-like member of the frame, nor have we assumed that
it be hypersurface orthogonal. In the latter case,
is the extrinsic curvature of the family of hypersurfaces
orthogonal to
and hence it is symmetric. Furthermore, the derivative
agrees with the Levi-Civita connection of the metric
induced on the leaves by the metric
.
We write the equations in terms of the derivative
and the ``time derivative''
which is defined in a way similar to
(see Equation (40
)), because in this form it is quite easy to see the symmetric
hyperbolicity of the equations.
As they stand, the Equations (90,
91
,
92
,
93
,
94
,
95
,
96
,
97
,
98
,
99
,
100
,
101
,
102
,
103
) form a symmetric hyperbolic system of evolution equations for
the collection of 65 unknowns
This property is present irrespective of the particular gauge.
For any choice of the gauge source functions
,
,
and
, the system is symmetric hyperbolic. The fact that the gauge
sources appear only in undifferentiated form implies that one can
specify them not only as functions of the space-time coordinates
but also as functions of the unknown fields. In this way, one can
feed information about the current status of the evolution back
into the system in order to influence the future development.
Other ways of specifying the coordinate gauge, including the
familiar choice of a lapse function and a shift vector, are not
as flexible because then not only these functions themselves
appear in the equations, but also their derivatives. Specifying
them as functions of the unknown fields alters the principal part
of the system and, hence, the propagation properties of the
solution. This may not only corrupt the character of the system
but it may also be disastrous for the numerical applications
because an uncontrolled change of the local propagation speeds
implies that the stability of a numerical scheme can break down
due to violation of the CFL condition (see [38] for a more detailed discussion of these issues). However, due
to the intuitive meaning of lapse and shift they are used (almost
exclusively) in numerical codes.
There are several other ways of writing the equations. Apart
from various possibilities to specify the gauges which result in
different systems with different numbers of unknowns, one can
also set up the equations using spinorial methods. This was the
method of choice in almost all of Friedrich's work (see
e.g. [52] and also [39
]). The ensuing system of equations is analogous to those
obtained here using the tetrad formalism. The main advantages of
using spinors is the fact that the reduction process
automatically leads to a symmetric hyperbolic system, that the
variables are components of irreducible spinors which allows for
the elimination of redundancies, and that variables and equations
become complex and hence easier to handle.
Another possibility is to ignore the tetrad formalism
altogether (or, more correctly, to choose as a basis for the
tangent spaces the natural coordinate frame). This also results
in a symmetric hyperbolic system of equations (see [55,
81]), in which the gauge dependent variables are not the frame
components with their corresponding connection coefficients but
the components of the spatial metric together with the usual
Christoffel symbols and the extrinsic curvature.
The fact that the reduced equations form a symmetric hyperbolic system leads, via standard theorems, to the existence of smooth solutions which evolve uniquely from suitable smooth data given on an initial surface. We have the
Theorem 2:
[Friedrich [48]] For functions
,
,
,
on
and data given on some initial surface let
u
be the solution of the reduced equations. If
u
satisfies the conformal field equations (16
,
17
,
18
,
19
,
20
,
21
) on the initial surface then, in fact, it satisfies them on the
entire domain of dependence of the initial surface in the
space-time defined by
u
.
The proof of this theorem relies on the existence of a
``subsidiary system'' of equations for the zero-quantity
Z
(see Equation (15)), whose vanishing indicates the validity of the conformal field
equations. This system turns out to be linear, symmetric
hyperbolic and
homogeneous
. Thus, one has uniqueness of the solutions so that
Z
vanishes in the domain of dependence of the initial surface if
it vanishes on the surface. Hence, the conformal field equations
hold. It can be shown that solutions obtained from different
gauge source functions are in the same conformal class, so they
lead to the same physical space-time.
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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 |