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
which 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
D
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 (45) and (46
) is completely equivalent to the Cartan equations for
which define the curvature and torsion tensors.
This completes the preliminaries and we can now go on to
perform the splitting of the equations. Out 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 (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 are rather straightforward. We
obtain
while Equation (20) yields four equations:
Finally, the equation (21) for
S
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
i
=1,2,3. 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 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 tensorfields 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 (41) and (43
) 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 (45
) and (46
), which come from acting on
, this procedure yields
Now we have collected all the equations which can be extracted
from the conformal field equations and Cartan's structure
equations. What remains to do 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 tensorfields
and
whose components happen to agree with them in the specified
basis. Thus,
Computing these tensorfields from (76) gives
Now we are ready to collect the constraints:
Finally, we collect the evolution equations:
This is the complete system of evolution equation which 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
. Altogether these are 65 equations.
![]() |
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 |