First, the metric is decomposed into a conformal factor multiplying an auxiliary 3-metric [69, 107, 108]:
The conformal decomposition of the Hamiltonian constraint was proposed by Lichnerowicz. But, the key to the full decomposition is the treatment of the extrinsic curvature introduced by York [109, 110]. This begins by splitting the extrinsic curvature into its trace and tracefree parts,
The goal of the decomposition is to produce a coupled set of elliptic equations to be solved with some prescribed boundary conditions. We have already reduced the Hamiltonian constraint to an elliptic equation being solved on a background space in terms of differential operators that are compatible with the conformal 3-metric. In the end, we want to reduce the momentum constraints to a set of elliptic equations based on differential operators that are compatible with the same conformal 3-metric. But, the longitudinal operator (21) can be defined with respect to any metric space. In particular, it is natural to consider decompositions with respect to both the physical and conformal 3-metrics.
Let us first consider decomposing with respect to the conformal 3-metric [111, 116]. As we will see, when certain assumptions are made, this decomposition has the advantage of producing a simpler set of elliptic equations that must be solved. The first step is to define the conformal tracefree extrinsic curvature by
Applying equations (16), (19), (21), (22), and (23) to the momentum constraints (15), we find that they simplify toUpdate
In deriving equation (24), we have also used the fact that is transverse (i.e. ). However, in general, we will not know if a given symmetric tracefree tensor, say , is transverse. By using (20) we can obtain its transverse-traceless part viaconstructing the required symmetric transverse-traceless tensor from a general symmetric traceless tensor.
Using the linearity of , we can rewrite (23) as
After applying (19) and (22) to the Hamiltonian constraint (18), we obtain the following full decomposition, which I will list together here for convenience:
In the decomposition given by (32), we are free to specify a symmetric tensor as the conformal 3-metric, a symmetric tracefree tensor , and a scalar function . Then, with given matter energy and momentum densities, and , and appropriate boundary conditions, the coupled set of constraint equations for and are solved. Finally, given the solutions, we can construct the physical initial data, and .
The decomposition outlined above has the interesting property that if we choose to be constant and if the momentum density vanishes4, then the momentum constraint equations fully decouple from the Hamiltonian constraint. As we will see later, this simplification has proven to be useful.
Alternatively, we can decompose with respect to the physical 3-metric [82, 83, 84]. We decompose the extrinsic curvature as
Applying equations (16), (19), (21), (33), and (26) to the momentum constraint (15), we find that it simplifies to
As in the previous section, we will obtain the symmetric transverse-traceless tensor from a general symmetric tracefree tensor by using (20). In this case, we take5:
In the decomposition given by (39), we are again free to specify a symmetric tensor as the conformal 3-metric, a symmetric tracefree tensor , and a scalar function . Then, with given matter energy and momentum densities, and , and appropriate boundary conditions, the coupled set of constraint equations for and are solved. Finally, given the solutions, we can construct the physical initial data, and .
Notice that, while very similar to the decomposition from Section 2.2.1, the sets of equations are distinctly different. In general, if we make the same choices for the freely specifiable data in both decompositions (i.e., we choose , , and the same), we will produce two different sets of initial data. Both will be equally valid solutions of the constraint equations, but they will have distinct physical properties.
There is at least one exception to this. Assume we have a valid set of initial data and , which satisfies the constraint equations (14) and (15). For any everywhere-positive function , we define our freely specifiable data as follows:
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