

6 Appendix: Reduction of the
Conformal Field Equations
In this appendix we show how to perform the reduction process for
the conformal field equations to obtain the symmetric hyperbolic
system of evolution equations and the constraints. We assume that
we are given a time-like unit vector
with respect to
which the reduction is done. Any vector
may be decomposed
into parts perpendicular and parallel to
,
and similarly for one-forms
. We call a vector spatial with respect to
if it is orthogonal to
. In particular, the
metric
gives rise to a spatial metric
by the decomposition
The volume four-form
that is defined by the
metric also gives rise to a decomposition as follows:
The covariant derivative operator
is written
analogously,
thus defining two derivative operators:
with
and
. The covariant derivative of
itself is an important field. It gives rise to two
component fields defined by
Note that
is spatial in both its indices and that
there is no symmetry implied between the two indices. Similarly,
is automatically spatial.
It is useful to define two new derivative
operators
and
by the following relations:
These operators have the property that they are compatible with the
spatial metric
and that they annihilate
and
. If
is the unit normal of a
hypersurface, i.e. if
is hypersurface orthogonal,
then
is symmetric,
is the induced (negative
definite) metric on the hypersurface, and
is its Levi-Civita connection. In general this is
not the case and so the operator
possesses torsion.
In particular, we obtain the following commutators (acting on
scalars and spatial vectors):
These commutators are obtained from the commutators between the
derivative operators
and
by expressing them in terms
of
and
on the one hand, and by the
four-dimensional connection
on the other hand. This
procedure yields two equations for the derivatives of
,
The information contained in the commutator relations and in the
Equations (47) and (48) is completely
equivalent to the Cartan equations for
that define the
curvature and torsion tensors.
This completes the preliminaries and we can now
go on to perform the splitting of the equations. Our intention is
to end up with a system of equations for all the spatial parts of
the fields. In order not to introduce too many different kinds of
indices, all indices refer to the four-dimensional space-time, but
they are all spatial, i.e. any transvection with
and
vanishes. If we introduce hypersurfaces
with normal vector
, then there exists an isomorphism
between the tensor algebra on the hypersurfaces and the subalgebra
of spatial four-dimensional tensors.
We start with the tensorial part of the
equations. To this end we decompose the fields into various spatial
parts and insert these decompositions into the conformal field
equations defined by Equations (17, 18, 19, 20, 21). The fields are
decomposed as follows:
The function
is fixed in terms of
because
is trace-free.
Inserting the decomposition of
into Equation (18), decomposing the
equations into various spatial parts, and expressing derivatives in
terms of the operators
and
yields four equations:
Here we have defined
. Treating the other fields
and equations in a similar way, we obtain Equation (17) in the form of four
equations:
The equation (19) for the conformal
factor is rather straightforward. We obtain
while Equation (20) yields four
equations:
Finally, the equation (21) for
gives two equations
This completes the gauge-independent part of the equations. In
order to deal with the gauges we now have to introduce an arbitrary
tetrad and arbitrary coordinates. We extend the time-like unit
vector to a complete tetrad
with
for
. Let
with
be four arbitrary functions,
which we use as coordinates. Application of
and
to the coordinates yields
The four functions
and the four one-forms
may be regarded as the 16 expansion coefficients of
the tetrad vectors in terms of the coordinate basis
because of the identity
In a similar spirit, we apply the derivative
operators to the tetrad and obtain
Again, transvection with
on any index of
and
vanishes. Furthermore, both
and
are antisymmetric in their
(last two) indices. Together with the 12 components of
and
these fields provide an additional 12
components, which account for the 24 connection coefficients of the
four-dimensional connection
with respect to the chosen
tetrad.
Note that these fields are not tensor fields.
They do not transform as tensors under the change of tetrad. Since
we will keep the tetrad fixed here, we may, however, regard them as
defining tensor fields whose components happen to coincide with
them in the specified tetrad.
In order to extract the contents of the first of
Cartan’s structure equations, one needs to apply the
commutators (43) and (45) to the coordinates to
obtain
Similarly, the second of Cartan’s structure equations is exploited
by applying the commutators to the tetrad vectors.
Equation (16) is then used to
substitute for the Riemann tensor in terms of the gravitational
field, the trace-free part of the Ricci tensor, and the scalar
curvature. Apart from the Equations (47) and (48), which come from
acting on
, this procedure yields
Now we have collected all the equations that can be extracted from
the conformal field equations and Cartan’s structure equations.
What remains to be done is to separate them into constraints and
evolution equations. Before doing so, we notice that we do not have
enough evolution equations for the tetrad components and the
connection coefficients. The remedy to this situation is explained
in Section 3.2. It amounts to adding
appropriate “divergence equations”. We obtain these by computing
the “gauge source functions”. The missing equations for the
coordinates are obtained by applying the d’Alembert operator to the
coordinates. Expressing the wave operator in terms of
and
yields the additional equations
In order to find the missing equations for the tetrad we need to
compute the gauge source functions
In a similar way as explained above, we may regard these functions
as components of tensor fields
and
whose components happen to agree with them in the
specified basis. Thus,
Computing these tensor fields from (78, 79) gives
Now we are ready to collect the constraints:
Finally, we collect the evolution equations:
This is the complete system of evolution equations that can be
extracted from the conformal field equations. As it is written,
this system is symmetric hyperbolic. This is not entirely obvious
but rather straightforward to verify. It is important to keep in
mind that with our conventions the spatial metric
is negative definite.
Altogether these are 65 equations.

