(ed.). Gravitational waves (IOP, 2001)(422s).

366 Numerical relativityHere we have introduced the 3D spatial covariant derivative operator D aassociated with the 3-metric γ ab (i.e. D a γ bc = 0), and the 3D scalar curvatureR computed from γ ab . These four constraint equations can be used to determineinitial data for γ ab and K ab , which are to be evolved with the evolution equationsdiscussed below. These equations (18.2) and (18.3) are referred to as constraintsbecause, as in the case of electrodynamics, they contain no time derivatives of thefundamental fields γ ab and K ab , and hence do not propagate the solution in time.Next, we will sketch the standard method for obtaining a solution to theseconstraint equations. We follow York and coworkers (e.g., [20]) by writing the3-metric and extrinsic curvature in ‘conformal form’, and also make use of thesimplifying assumption tr K = 0 which causes the Hamiltonian and momentumconstraints to completely decouple (note that actually the equations decoupledwith tr K = constant but we will discuss only the simplest case here). We writeγ ab = 4 ˆγ ab , K ab = −2 ˆK ab , (18.5)where ˆγ ab and the transverse-tracefree part of ˆK ab is regarded as given, i.e. chosento represent the physical system that we want to study. Under the conformaltransformation, with tr K = 0wefind that the momentum constraint becomesˆD b ˆK ab = 0, (18.6)where ˆD a is the 3D covariant derivative associated with ˆγ ab (i.e. ˆD a ˆγ ab = 0).In vacuum, black hole spacetimes ˆK ab can often be solved analytically. Formore details on how to solve the momentum constraints in complicated situations,see [10, 21, 22].The remaining unknown function , must satisfy the Hamiltonianconstraint. The conformal transformation of the scalar curvature isR = −4 ˆR − 8 −5 ˆ, (18.7)where ˆ = ˆγ ab ˆD a ˆD b and ˆR is the scalar curvature of the known metric ˆγ ab .Plugging this back into the Hamiltonian constraint and dividing through by−8 −5 , we obtainˆ − 1 8 ˆR + 1 8 −7 ( ˆK ab ˆK ab ) = 0, (18.8)an elliptic equation for the conformal factor .To summarize, one first specifies ˆγ ab and the transverse-tracefree part ofˆK ab ‘at will’, choosing them to be something ‘closest’ to the spacetime onewants to study. Then one solves (18.6) for the conformal extrinsic curvatureˆK ab . Finally, (18.8) is solved for the conformal factor , so the full solution γ aband K ab can be reconstructed. In this process the elliptic equations are solvedby standard techniques, for example, the conjugate gradient [23] or multigridmethods [24]. In situations where there is a black hole singularity, there couldbe added complications in solving the elliptic equations, and special treatments