Equations (63) and (64
) form a coupled nonlinear set of elliptic equations which must
be solved iteratively, in general. The two equations can,
however, be decoupled if a mean curvature slicing (
) is assumed. Given the free data
,
,
and
, the constraints are solved for
,
and
. The actual metric
and curvature
are then reconstructed by the corresponding conformal
transformations to provide the complete initial data.
Anninos
[7]
describes a procedure using York’s formalism to construct
parametrized inhomogeneous initial data in freely specifiable
background spacetimes with matter sources. The procedure is
general enough to allow gravitational wave and Coulomb
nonlinearities in the metric, longitudinal momentum fluctuations,
isotropic and anisotropic background spacetimes, and can
accommodate the conformal-Newtonian gauge to set up gauge
invariant cosmological perturbation solutions as free data.
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