2.4 Stationary Solutions2 The Initial-Value Equations2.2 York-Lichnerowicz Conformal Decompositions

2.3 Conformal Thin-Sandwich Decomposition 

The two general initial-value decompositions outlined in Section  2.2.1 and Section  2.2.2 require identical freely specified data (tex2html_wrap_inline2989, tex2html_wrap_inline3153, and K), yet they usually produce different physical initial data. One shortcoming of these approaches is that they provide no direct insight into how to choose the freely specifiable data. All of the data are determined by schemes that involve only a single spacelike hypersurface. The resulting constraint equations are independent of the kinematical variables tex2html_wrap_inline3011 and tex2html_wrap_inline3031 that govern how the coordinates move through spacetime, and thus there is no connection to dynamics. York's conformal thin-sandwich decomposition [115] takes a different approach by considering the evolution of the metric between two neighboring hypersurfaces (the thin sandwich). This decomposition is very similar to an approach originated by Wilson [104Jump To The Next Citation Point In The Article, 46Jump To The Next Citation Point In The Article], but is somewhat more general. Perhaps the most attractive feature of this decomposition is the insight it yields into the choice of the freely specifiable data.

The decomposition begins with the standard conformal decomposition of the 3-metric (16Popup Equation). However, we next make use of the evolution equation for the metric (13Popup Equation) in order to connect the 3-metrics on the two neighboring hypersurfaces. Label the two slices by t and tex2html_wrap_inline3257, with tex2html_wrap_inline3259, then tex2html_wrap_inline3261 . 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:

  equation593

  equation600

and

  equation606

The latter definition is made for convenience, so that we can treat tex2html_wrap_inline3107, tex2html_wrap_inline2989, and tex2html_wrap_inline3267 as regular scalars and tensors instead of as scalar- and tensor-densities within this thin-sandwich formalism.

The conformal scaling of tex2html_wrap_inline3269 follows directly from (16Popup Equation), (41Popup Equation), (42Popup Equation), (43Popup Equation), and the identity that, for any small perturbation, tex2html_wrap_inline3271 . The result is

  equation622

which relies on the useful intermediate result that Popup Footnote

  equation631

Equation (41Popup Equation) represents the tracefree part of the evolution of the 3-metric, so (13Popup Equation) becomes

  equation636

Using the conformal scalings (22Popup Equation), (35Popup Equation), and (44Popup Equation), we obtain

  equation646

York has pointed out that it is natural to use the following conformal rescaling of the lapse:

  equation657

This rescaling follows naturally from the ``slicing function'' that replaces the usual lapse (tex2html_wrap_inline3279) which has been critical in solving several problems [4]. It also results in the natural conformal scaling (22Popup Equation) postulated for the tracefree part of the extrinsic curvature. Substituting (48Popup Equation) into (47Popup Equation) yields what is taken as the definition of the tracefree part of the conformal extrinsic curvature,

  equation664

Because the tracefree extrinsic curvature satisfies the normal conformal scaling, the Hamiltonian constraint will take on the same form as in (32Popup Equation). However, the momentum constraint will have a very different form. Combining equations (16Popup Equation), (19Popup Equation), (21Popup Equation), (22Popup Equation), and (49Popup Equation) with the momentum constraint (15Popup Equation), we find that it simplifies to

  equation681   [A2Jump To The Next Amendment]

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

  eqnarray694   [A3Jump To The Next Amendment]

In this decomposition (51Popup Equation), we are free to specify a symmetric tensor tex2html_wrap_inline2989 as the conformal 3-metric, a symmetric tracefree tensor tex2html_wrap_inline3283, a scalar function K, and the scalar function tex2html_wrap_inline3287 . Solving this set of equations with appropriate boundary conditions yields initial data tex2html_wrap_inline2979 and tex2html_wrap_inline3037 on a single hypersurface. However, we also know the following: If we chose to use the shift vector obtained from solving (50Popup Equation) and the lapse from (48Popup Equation) via our choice of tex2html_wrap_inline3287 and our solution to the Hamiltonian constraint, then the rate of change of the physical 3-metric is given by

  equation741

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.



2.4 Stationary Solutions2 The Initial-Value Equations2.2 York-Lichnerowicz Conformal Decompositions

image Initial Data for Numerical Relativity
Gregory B. Cook
http://www.livingreviews.org/lrr-2000-5
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