The decomposition begins with the standard conformal decomposition of the 3-metric (16). However, we next make use of the evolution equation for the metric (13) in order to connect the 3-metrics on the two neighboring hypersurfaces. Label the two slices by and , with , then . We would like to specify how the 3-metric evolves, but we do not have full freedom to do this. We know we can freely specify only the conformal 3-metric, and similarly, we are free to specify only the evolution of the conformal 3-metric. We make the following definitions:

and The latter definition is made for convenience, so that we can treat , , and as regular scalars and tensors instead of as scalar- and tensor-densities within this thin-sandwich formalism.The conformal scaling of follows directly from (16), (41), (42), (43), and the identity that, for any small perturbation, . The result is

which relies on the useful intermediate result thatEquation (41) represents the tracefree part of the evolution of the 3-metric, so (13) becomes

Using the conformal scalings (22), (35), and (44), we obtainYork has pointed out that it is natural to use the following conformal rescaling of the lapse:

This rescaling follows naturally from the “slicing function” that replaces the usual lapse () which has been critical in solving several problems [4]. It also results in the natural conformal scaling (22) postulated for the tracefree part of the extrinsic curvature. Substituting (48) into (47) yields what is taken as the definition of the tracefree part of the conformal extrinsic curvature,Because the tracefree extrinsic curvature satisfies the normal conformal scaling, the Hamiltonian constraint will take on the same form as in (32). However, the momentum constraint will have a very different form. Combining equations (16), (19), (21), (22), and (49) with the momentum constraint (15), we find that it simplifies toUpdate

Let us, for convenience, group together all the equations that constitute the conformal thin-sandwich decomposition:

In this decomposition (51)Update, we are free to specify a symmetric tensor as the conformal 3-metric, a symmetric tracefree tensor , a scalar function , and the scalar function . Solving this set of equations with appropriate boundary conditions yields initial data and on a single hypersurface. However, we also know the following: If we chose to use the shift vector obtained from solving (50) and the lapse from (48) via our choice of and our solution to the Hamiltonian constraint, then the rate of change of the physical 3-metric is given by This direct information about the consequences of our choices for the freely specifiable data is something not present in the previous decompositions. As we will see later, this framework has been used to construct initial data that are in quasiequilibrium.http://www.livingreviews.org/lrr-2000-5 |
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