2.2 York-Lichnerowicz Conformal Decompositions2 The Initial-Value Equations2 The Initial-Value Equations

2.1 Initial Data 

In the Cauchy formulation of Einstein's equations, we begin by foliating the 4-dimensional manifold as a set of spacelike, 3-dimensional hypersurfaces (or slices) tex2html_wrap_inline3001 . These slices are labeled by a parameter t or, more simply, each slice of the 4-dimensional manifold is a tex2html_wrap_inline3005 hypersurface. Following the standard tex2html_wrap_inline2971 decomposition [6, 111Jump To The Next Citation Point In The Article, 37], we let tex2html_wrap_inline3009 be the future-pointing timelike unit normal to the slice, with

  equation94

Here, tex2html_wrap_inline3011 is called the lapse function (frequently denoted N in the literature). The scalar lapse function sets the proper interval measured by observers as they move between slices on a path that is normal to the hypersurface (so-called normal observers):

  equation98

Of course, there is no reason that observers must move along a path normal to the hypersurface. In general, we can define the time vector as

  equation104

where

  equation107

Here, tex2html_wrap_inline3015 is called the shift vector (frequently denoted tex2html_wrap_inline3017 in the literature).

Because of (9Popup Equation), tex2html_wrap_inline3015 has only three independent components and is a spatial vector, tangent to the hypersurface on which it resides. At this point, it is convenient to introduce a coordinate system adapted to the foliation tex2html_wrap_inline3001 . Let tex2html_wrap_inline3023 be the spatial coordinates in the slice. The fourth coordinate, t, is the parameter labeling each slice. With this adapted coordinate system, we find that 3-dimensional coordinate values remain constant as we move between slices along the tex2html_wrap_inline3027 direction (8Popup Equation). The four parameters, tex2html_wrap_inline3011 and tex2html_wrap_inline3031, are a manifestation of the 4-dimensional coordinate invariance, or gauge freedom, in Einstein's theory. If we let tex2html_wrap_inline2979 represent the metric of the spacelike hypersurfaces, then we can rewrite the interval (5Popup Equation) as

  equation116

In the Cauchy formulation of Einstein's equations, tex2html_wrap_inline2979 is regarded as the fundamental variable and values for its components must be given as part of a well-posed initial-value problem. Since Einstein's equations are second order, we must also specify something like a time derivative of the metric. For this, we use the second fundamental form, or extrinsic curvature, of the slice, tex2html_wrap_inline3037, defined Popup Footnote by

  equation129

where tex2html_wrap_inline3039 denotes the Lie derivative along the tex2html_wrap_inline3009 direction.

Together, tex2html_wrap_inline2979 and tex2html_wrap_inline3037 are the minimal set of initial data that must be specified for a Cauchy evolution of Einstein's equations. The metric tex2html_wrap_inline2979 on a hypersurface is induced on that surface by the 4-metric tex2html_wrap_inline2975 . This means that the values tex2html_wrap_inline2979 receives depend on how tex2html_wrap_inline3053 is embedded in the full spacetime. In order for the foliation of slices tex2html_wrap_inline3001 to fit into the higher-dimensional space, they must satisfy the Gauss-Codazzi-Ricci conditions. Combining these conditions with Einstein's equations, and using (10Popup Equation), the six evolution equations become

  equation143

Here, tex2html_wrap_inline3057 is the spatial covariant derivative compatible with tex2html_wrap_inline2979, tex2html_wrap_inline2985 is the Ricci tensor associated with tex2html_wrap_inline2979, tex2html_wrap_inline3065, tex2html_wrap_inline3067 is the matter energy density, tex2html_wrap_inline3069 is the matter stress tensor, and tex2html_wrap_inline3071 . Popup Footnote We have also used the fact that in our adapted coordinate system, tex2html_wrap_inline3085 . The set of second-order evolution equations is completed by rewriting the definition of the extrinsic curvature (11Popup Equation) as

  equation177

Equations (12Popup Equation) and (13Popup Equation) are a first-order representation of a complete set of evolution equations for given initial data tex2html_wrap_inline2979 and tex2html_wrap_inline3037 . However, the data cannot be freely specified in their entirety. The four constraint equations, following the same procedure outlined above for the evolution equations, become

  equation186

and

  equation192

Here, tex2html_wrap_inline3091 and tex2html_wrap_inline3093 is the matter momentum density. Equation (14Popup Equation) is referred to as the Hamiltonian or scalar constraint, while (15Popup Equation) are referred to as the momentum or vector constraints. Valid initial data for the evolution equations (12Popup Equation) and (13Popup Equation) must satisfy this set of constraints. And, as mentioned earlier, the Bianchi identities (4Popup Equation) guarantee that the evolution equations will preserve the constraints on future slices of the evolution.

As we will see below, the Hamiltonian constraint (14Popup Equation) most naturally constrains the 3-metric tex2html_wrap_inline2979, while the momentum constraints (15Popup Equation) naturally constrain the extrinsic curvature tex2html_wrap_inline3037 . Taking the constraints into consideration, it seems that the 3-metric has five degrees of freedom remaining, while the extrinsic curvature has three. But we know that the gravitational field in Einstein's theory has two dynamical degrees of freedom, so we expect that both tex2html_wrap_inline2979 and tex2html_wrap_inline3037 should each have only two free components. The answer to this problem is, once again, the coordinate invariance of Einstein's theory. This seems strange at first, because we have already used the coordinate invariance of the theory to narrow our scope from the ten components of the 4-metric to the six components of the 3-metric. However, the lapse and shift do not completely specify the coordinate gauge. Rather, they specify how an initial choice of gauge will evolve with the foliation. The metric on a given hypersurface retains full 3-dimensional coordinate invariance, reducing the number of freely specifiable components to two [107Jump To The Next Citation Point In The Article, 108Jump To The Next Citation Point In The Article]. There also remains one degree of gauge freedom associated with the time coordinate which must be fixed. Each hypersurface represents a tex2html_wrap_inline3103 slice of the spacetime, so how the initial hypersurface is embedded in the full 4-dimensional manifold represents our temporal gauge choice. There is no unique way to specify this choice, but it is often convenient to let the trace of the extrinsic curvature K represent this temporal gauge choice [108Jump To The Next Citation Point In The Article]. Thus, we find that we are allowed to choose freely five components of the 3-metric and three components of the extrinsic curvature. However, only two of the components for each field represent dynamical degrees of freedom, the remainder are gauge degrees of freedom.

The four constraint equations, (14Popup Equation) and (15Popup Equation), represent conditions which the 3-metric and extrinsic curvature must satisfy. But, they do not specify which components (or combination of components) are constrained and which are freely specifiable. In the weak field limit where Einstein's equations can be linearized, there are clear ways to determine which components are dynamic, which are constrained, and which are gauge. However, in the full nonlinear theory, there is no unique decomposition. In this case, one must choose a method for decomposing the constraint equations. The goal is to transform the equations into standard elliptic forms which can be solved given appropriate boundary conditions [80, 82Jump To The Next Citation Point In The Article, 111Jump To The Next Citation Point In The Article]. Each different decomposition yields a unique set of elliptic equations to be solved and a unique set of freely specifiable parameters which must be fixed somehow. Seemingly similar sets of assumptions applied to different decompositions can lead to physically different initial conditions.



2.2 York-Lichnerowicz Conformal Decompositions2 The Initial-Value Equations2 The Initial-Value Equations

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