2.3 Conformal Thin-Sandwich Decomposition2 The Initial-Value Equations2.1 Initial Data

2.2 York-Lichnerowicz Conformal Decompositions 

For general initial-data configurations, the most widely used class of constraint decompositions are the York-Lichnerowicz conformal decompositions. At their heart are a conformal decomposition of the metric and certain components of the extrinsic curvature, together with a transverse-traceless decomposition of the extrinsic curvature.

First, the metric is decomposed into a conformal factor tex2html_wrap_inline3107 multiplying an auxiliary 3-metric [68, 107, 108]:

  equation222

The auxiliary 3-metric tex2html_wrap_inline2989 is often called the conformal or background 3-metric, and it carries five degrees of freedom. Its natural definition is given by

  equation228

leaving tex2html_wrap_inline3111, but we are free to choose any normalization for tex2html_wrap_inline3113 . Using (16Popup Equation), we can rewrite the Hamiltonian constraint (14Popup Equation) as

  equation236

where tex2html_wrap_inline3115 is the scalar Laplace operator, and tex2html_wrap_inline3117 and tex2html_wrap_inline3119 are the covariant derivative and Ricci scalar associated with tex2html_wrap_inline2989 . Equation (18Popup Equation) is a quasilinear elliptic equation for the conformal factor tex2html_wrap_inline3107, and we see that the Hamiltonian constraint naturally constrains the 3-metric.

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,

  equation249

The decomposition proceeds by using the fact that we can covariantly split any symmetric tracefree tensor as follows:

  equation256

Here, tex2html_wrap_inline3125 is a symmetric, transverse-traceless tensor (i. e., tex2html_wrap_inline3127 and tex2html_wrap_inline3129) and

  equation266

After separating out the transverse-traceless portion of tex2html_wrap_inline3131, what remains, tex2html_wrap_inline3133, is referred to as its ``longitudinal'' part. We now want to apply this transverse-traceless decomposition to the tracefree part of the extrinsic curvature tex2html_wrap_inline3135 . However, the conformal decomposition of the metric leaves us with at least two ways to proceed.

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 (21Popup Equation) 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.

2.2.1 Conformal Transverse-Traceless Decomposition 

Let us first consider decomposing tex2html_wrap_inline3137 with respect to the conformal 3-metric [111, 116Jump To The Next Citation Point In The Article]. 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 tex2html_wrap_inline3139 by

  equation287

Next, the transverse-traceless decomposition is applied to the conformal extrinsic curvature,

  equation299

Note that the longitudinal operator tex2html_wrap_inline3141 and the symmetric, transverse-tracefree tensor tex2html_wrap_inline3143 are both defined with respect to covariant derivatives compatible with tex2html_wrap_inline2989 .

Applying equations (16Popup Equation), (19Popup Equation), (21Popup Equation), (22Popup Equation), and (23Popup Equation) to the momentum constraints (15Popup Equation), we find that they simplify to

  equation318

where

  equation324   [A1Jump To The Next Amendment]

and we have used the fact that

  equation332

for any symmetric tracefree tensor tex2html_wrap_inline3131 .

In deriving equation (24Popup Equation), we have also used the fact that tex2html_wrap_inline3143 is transverse (i. e. tex2html_wrap_inline3151). However, in general, we will not know if a given symmetric tracefree tensor, say tex2html_wrap_inline3153, is transverse. By using (20Popup Equation) we can obtain its transverse-traceless part tex2html_wrap_inline3143 via

  equation353

and using the fact that if tex2html_wrap_inline3143 is transverse, we find

  equation364

Thus, Eqs. (27Popup Equation) and (28Popup Equation) give us a general way of constructing the required symmetric transverse-traceless tensor from a general symmetric traceless tensor.

Using the linearity of tex2html_wrap_inline3141, we can rewrite (23Popup Equation) as

  equation377

where

  equation386

Similarly, using the linearity of tex2html_wrap_inline3161, we can rewrite (24Popup Equation) as

  equation391

By solving directly for tex2html_wrap_inline3163, we can combine the steps of decomposing tex2html_wrap_inline3153 with that of solving the momentum constraints.

After applying (19Popup Equation) and (22Popup Equation) to the Hamiltonian constraint (18Popup Equation), we obtain the following full decomposition, which I will list together here for convenience:

  eqnarray406

In the decomposition given by (32Popup Equation), we are free to specify a symmetric tensor tex2html_wrap_inline2989 as the conformal 3-metric, a symmetric tracefree tensor tex2html_wrap_inline3153, and a scalar function K . Then, with given matter energy and momentum densities, tex2html_wrap_inline3067 and tex2html_wrap_inline3093, and appropriate boundary conditions, the coupled set of constraint equations for tex2html_wrap_inline3107 and tex2html_wrap_inline3163 are solved. Finally, given the solutions, we can construct the physical initial data, tex2html_wrap_inline2979 and tex2html_wrap_inline3183 .

The decomposition outlined above has the interesting property that if we choose K to be constant and if the momentum density vanishes Popup Footnote, then the momentum constraint equations fully decouple from the Hamiltonian constraint. As we will see later, this simplification has proven to be useful.

2.2.2 Physical Transverse-Traceless Decomposition 

Alternatively, we can decompose tex2html_wrap_inline3137 with respect to the physical 3-metric [81, 82, 83]. We decompose the extrinsic curvature as

  equation451

In this case, the longitudinal operator tex2html_wrap_inline3193 and the symmetric transverse-tracefree tensor tex2html_wrap_inline3195 are both defined with respect to covariant derivatives compatible with tex2html_wrap_inline2979 .

Applying equations (16Popup Equation), (19Popup Equation), (21Popup Equation), (33Popup Equation), and (26Popup Equation) to the momentum constraint (15Popup Equation), we find that it simplifies to

  equation467

where we have used the fact that

  equation474

As in the previous section, we will obtain the symmetric transverse-traceless tensor tex2html_wrap_inline3195 from a general symmetric tracefree tensor tex2html_wrap_inline3153 by using (20Popup Equation). In this case, we take

  equation486

and use the fact that tex2html_wrap_inline3195 is transverse, to obtain

  equation496

Again, we can define

  equation505

and use the linearity of tex2html_wrap_inline3141 and tex2html_wrap_inline3161 to combine the process of obtaining the transverse-traceless part of tex2html_wrap_inline3209 and solving the momentum constraints. We obtain the following full decomposition, which I will list together here for convenience Popup Footnote :

  eqnarray516

In the decomposition given by (39Popup Equation), we are again free to specify a symmetric tensor tex2html_wrap_inline2989 as the conformal 3-metric, a symmetric tracefree tensor tex2html_wrap_inline3153, and a scalar function K . Then, with given matter energy and momentum densities, tex2html_wrap_inline3067 and tex2html_wrap_inline3093, and appropriate boundary conditions, the coupled set of constraint equations for tex2html_wrap_inline3107 and tex2html_wrap_inline3163 are solved. Finally, given the solutions, we can construct the physical initial data, tex2html_wrap_inline2979 and tex2html_wrap_inline3183 .

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 tex2html_wrap_inline2989, tex2html_wrap_inline3153, and K 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 tex2html_wrap_inline2979 and tex2html_wrap_inline3037, which satisfies the constraint equations (14Popup Equation) and (15Popup Equation). For any everywhere-positive function tex2html_wrap_inline3239, we define our freely specifiable data as follows:

  eqnarray565

Then the solution to both sets of equations, assuming we use correct boundary conditions, will be tex2html_wrap_inline3241 and tex2html_wrap_inline3243, which yields the original data as the solution for each decomposition.



2.3 Conformal Thin-Sandwich Decomposition2 The Initial-Value Equations2.1 Initial Data

image Initial Data for Numerical Relativity
Gregory B. Cook
http://www.livingreviews.org/lrr-2000-5
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de