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.
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