### 6.3 Constrained nonlinear initial data

One cannot take arbitrary data to initiate an evolution of the Einstein equations. The data must satisfy
the constraint equations (9) and (10). York [163] 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
for uniqueness of solutions to the elliptic equation (63) 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 (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.