6.5 Newtonian limit 6 Appendix: Basic Equations and 6.3 Sources of matter

6.4 Constrained nonlinear initial data 

One cannot take arbitrary data to initiate an evolution of the Einstein equations. The data must satisfy the constraint equations (11Popup Equation) and (12Popup Equation
). York [127] developed a procedure to generate proper initial data by introducing conformal transformations of the 3-metric tex2html_wrap_inline3179 , the trace-free momentum components tex2html_wrap_inline3181 , and matter source terms tex2html_wrap_inline3183 and tex2html_wrap_inline3185 , where n>5 for uniqueness of solutions to the elliptic equation (53Popup Equation) below. In this procedure, the conformal (or ``hatted'') variables are freely specifiable. Further decomposing the free momentum variables into transverse and longitudinal components tex2html_wrap_inline3189 , the Hamiltonian and momentum constraints are written as



where the longitudinal part of tex2html_wrap_inline3191 is reconstructed from the solutions by


The transverse part of tex2html_wrap_inline3191 is constrained to satisfy tex2html_wrap_inline3195 .

Equations (53Popup E
quation) and (54Popup Equation) 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, tex2html_wrap_inline3201 , tex2html_wrap_inline3203 and tex2html_wrap_inline3205 , the constraints are solved for tex2html_wrap_inline3207 , tex2html_wrap_inline3209 and tex2html_wrap_inlin
e3017 . The actual metric tex2html_wrap_inline2983 and curvature tex2html_wrap_inline2995 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.

6.5 Newtonian limit 6 Appendix: Basic Equations and 6.3 Sources of matter

Computational Cosmology: from the Early Universe to the Large Scale Structure
Peter Anninos
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de