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

Einstein equations for relativity 367would have to be introduced, for example, the ‘puncture’ treatment of [25], oremploying an ‘isometry’ operation to provide boundary conditions on black holethroats, ensuring identical spatial geometries inside and outside the throat (see,e.g., [21, 26], or [27] for more details).While this is a well-established process for generating an initial data setfor numerical study, there is a fundamental difficulty in using this approach togenerate initial data corresponding to a physical system one wants to evolve, forexample, a coalescing binary system. It is not clear how to choose the ‘closest’ˆγ ab , and the corresponding free components in ˆK ab , so that the resulting γ ab andK ab represent the inspiralling system at its late stage of inspiral. This late stage isthe so-called ‘intermediate challenge problem’ of binary black holes [28], an areaof much current interest.18.2.2 Evolution equations18.2.2.1 The standard evolution systemWith the initial data γ ab and K ab specified, we now consider their evolution intime. There are six evolution equations for the 3-metric γ ab that are secondorder in time, resulting from projections of the full 4D Einstein equations ontothe 3D spacelike slice [10]. These are most often written as a first-order in-timesystem of twelve evolution equations, usually referred to as the ‘ADM’ evolutionsystem [10, 29]:∂ t γ ab =−2αK ab + D a β b + D b β a (18.9)∂ t K ab =−D a D b α + α[R ab + (tr K )K ab − 2K ac K c b]+ β c D c K ab + K ac D b β c + K cb D a β c . (18.10)Here R ab is the Ricci tensor of the 3D spacelike slice labelled by a constant valueof t. Note that these are quantities defined only on a t = constant hypersurface,and require only the 3-metric γ ab in their construction. Do not confuse them withthe conventional 4D objects! The complete set of Einstein equations are containedin constraint equations (18.2), (18.3) and the evolution equations (18.10), (18.9).Note that (18.9) is simply the definition of the extrinsic curvature K ab (18.4).These equations are analogous to the evolution equations for the electric andmagnetic fields of electrodynamics. Given the ‘lapse’ α and ‘shift’ β a , discussedbelow, they allow one to advance the system forward in time.18.2.2.2 Hyperbolic evolution systemsThe evolution equations (18.10) and (18.9) have been presented in the ‘standardADM form’, which has served numerical relativity well over the last fewdecades. However, the equations are enormously complicated; the complicationis hidden in the definition of the curvature tensor R ab and the covariantdifferentiation operator D a . In particular, although they describe physical