The basic approach for finding stationary solutions begins by
simplifying the metric to take into account the symmetries. Many
different forms have been used for the metric (cf. Refs. [8,
9,
22,
34
,
62
,
21
]). I will use a decomposition that makes comparison with the
previous decompositions straightforward. First, define the
interval as
This form of the metric can describe any stationary spacetime. Notice that the lapse is related to the conformal factor by
and that the shift vector has only one component
I have used the usual conformal decomposition of the 3-metric
(16) and have written the conformal 3-metric with two parameters
as
The four functions
,
,
A, and
B
are functions of
r
and
only.
The equations necessary to solve for these four functions are
derived from the constraint equations (14) and (15
),
and
the evolution equations (12
) and (13
). For the evolution equations, we use the fact that
and
. The metric evolution equation (13
) defines the extrinsic curvature in terms of derivatives of the
shift
With the given metric and shift, we find that K =0 and the divergence of the shift also vanishes. This means we can write the tracefree part of the extrinsic curvature as
We find that the Hamiltonian and momentum constraints take on
the forms given by the conformal thin-sandwich decomposition (51) with
and
. Only one of the momentum constraint equations is non-trivial,
and we find that the constraints yield elliptic equations for
and
. What remains unspecified as yet are
A
and
B
(i. e., the conformal 3-metric).
The conformal 3-metric is determined by the evolution equations for the traceless part of the extrinsic curvature. Of these five equations, one can be written as an elliptic equation for B, and two yield complementary equations that can each be solved by quadrature for A . The remaining equations are redundant as a result of the Bianchi identities.
Of course, the clean separation of the equations I have suggested above is an illusion. All four equations must be solved simultaneously, and clever combinations of the four metric quantities can greatly simplify the task of solving the system of equations. This accounts for the numerous different systems used for solving for stationary solutions.
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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 |