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  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
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