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

364 Numerical relativityterms, of mixed hyperbolic-elliptic type, and even undefined types, depending oncoordinate conditions. This rich and general structure of the equations implies thatthe techniques developed to solve our problems will be immediately applicable toa large family of diverse scientific applications.The system of equations breaks up naturally into a set of constraintequations, which are elliptic in nature, evolution equations, which are‘hyperbolic’ in nature (more on this below), and gauge equations, which canbe chosen arbitrarily (often leading to more elliptic equations). The evolutionequations guarantee (mathematically) that the elliptic constraints are satisfied atall times provided the initial data satisfied them. This implies that the initialdata are not freely specifiable. Moreover, although the constraints are satisfiedmathematically during evolution, it will not be so numerically. These problemsare discussed in turn below. First, however, we point out that a much simplertheory, familiar to many, has all of these same features. Maxwell’s equationsdescribing electromagnetic radiation have: (a) elliptic constraint equations,demanding that in vacuum the divergence of the electric and magnetic fieldsvanish at all times; (b) evolution equations, determining the time development ofthese fields, given suitable initial data satisfying the elliptic constraint equations;and (c) gauge conditions that can be applied freely to certain variables in thetheory, such as some components of the vector potential. Some choices of vectorpotential lead to hyperbolic evolution equations for the system, and some do not.We will find all of these features present in the much more complicated Einsteinequations, so it is useful to keep Maxwell’s equations in mind when reading thenext sections.In the standard 3 + 1 ADM approach to general relativity [10], the basicbuilding block of the theory—the spacetime metric—is written in the formds 2 =−(α 2 − β a β a ) dt 2 + 2β a dx a dt + γ ab dx a dx b , (18.1)using geometrized units such that the gravitational constant G and the speed oflight c are both equal to unity. Throughout this paper, we use Latin indices tolabel spatial coordinates, running from 1 to 3. The ten functions (α, β a ,γ ab )are functions of the spatial coordinates x a and time t. Indices are raised andlowered by the ‘spatial 3-metric’ γ ab . Notice that the geometry on a 3D spacelikehypersurface of constant time (i.e. dt = 0) is determined by γ ab . As we will seebelow, the Einstein equations control the evolution in time of this 3D geometrydescribed by γ ab , given appropriate initial conditions. The lapse function αand the shift vector β a determine how the slices are threaded by the spatialcoordinates. Together, α and β a represent the coordinate degrees of freedominherent in the covariant formulation of Einstein’s equations, and can therefore bechosen, in some sense, ‘freely’, as discussed below.This formulation of the equations assumes that one begins with aneverywhere spacelike slice of spacetime, that should be evolved forward in time.Due to limited space, we will not discuss promising alternate treatments, based