)
and (12
). York [127] developed
a procedure to generate proper initial data by introducing
conformal transformations of the 3-metric
, the
trace-free momentum components
,
and matter source terms 
and
, where n>5 for uniqueness
of solutions to the elliptic equation (53
) below.
In this procedure, the conformal (or ``hatted'')
variables are freely specifiable.
Further decomposing the free momentum variables into transverse
and longitudinal components
,
the Hamiltonian and momentum constraints are written as
where the longitudinal part of 
is
reconstructed from the solutions by
The transverse part of is constrained to satisfy
.
Equations (53) and (54) 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 (K=K(t)) is assumed.
Given the free data K, 
, 
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.
Reference [6] 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.
|
Computational Cosmology: from the Early Universe to the Large Scale Structure Peter Anninos http://www.livingreviews.org/lrr-2001-2 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |
