Constrained nonlinear initial data

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 $γ = ψ4 ˆγ ij ij$ , the trace-free momentum components $ij ij ij −10 ˆij A = K − γ K ∕3 = ψ A$ , and matter source terms $i − 10i s = ψ sˆ$ and $− n ρH = ψ ˆρH$ , where $n > 5$ 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 $Aˆij = Aˆij + (ˆlw)ij ∗$ , the Hamiltonian and momentum constraints are written as

ˆ ˆ i ˆR- 1-ˆ ˆij − 7 1-- 2 5 5−n ∇i ∇ ψ − 8 ψ + 8 AijA ψ − 12K ψ + 2πG ρˆψ = 0, (63 ) j i 1 i( j) i j 2 6 i i (ˆ∇j ˆ∇ w ) + --ˆ∇ ˆ∇jw + Rˆjw − -ψ ˆ∇ K − 8πG ˆs = 0, (64 ) 3 3

where the longitudinal part of $Aˆij$ is reconstructed from the solutions by

The transverse part of $ˆ ij A$ is constrained to satisfy $ˆ ˆij ˆ j ∇j A∗ = A∗j = 0$ .

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 ( $K = K (t)$ ) is assumed. Given the free data $K$ , $ˆγij$ , $i ˆs$ and $ρˆ$ , the constraints are solved for $Aˆij∗$ , $(ˆlw)ij$ and $ψ$ . The actual metric $γij$ and curvature $Kij$ 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.