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
t
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 that
Equation (41) represents the tracefree part of the evolution of the 3-metric,
so (13
) becomes
Using the conformal scalings (22), (35
), and (44
), we obtain
York 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 to
[A2
]
Let us, for convenience, group together all the equations that constitute the conformal thin-sandwich decomposition:
[A3
]
In this decomposition (51), we are free to specify a symmetric tensor
as the conformal 3-metric, a symmetric tracefree tensor
, a scalar function
K,
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.
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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 |