THE INITIAL VALUE PROBLEM IN NUMERICAL RELATIVITY

HARALD P. PFEIFFER
2005 Journal of Hyperbolic Differential Equations
The conformal method for constructing initial data for Einstein's equations is presented in both the Hamiltonian and Lagrangian picture (extrinsic curvature decomposition and conformal thin sandwich formalism, respectively), and advantages due to the recent introduction of a weight-function in the extrinsic curvature decomposition are discussed. I then describe recent progress in numerical techniques to solve the resulting elliptic equations, and explore innovative approaches toward the
more » ... toward the construction of astrophysically realistic initial data for binary black hole simulations. Keywords: Einsteins equations; initial value problem; numerical relativity. Here, ∇ i and R are the covariant derivative and the scalar curvature (trace of the Ricci tensor) of g ij , respectively, and K = K ij g ij denotes the mean curvature. Furthermore, G stands for Newton's constant, ρ and S ij are matter density and stress tensor, respectively, and S = S ij g ij denotes the trace of S ij . The constraint equations are conditions within each hypersurface alone, ensuring that the three-dimensional surface can be embedded into the four-dimensional December 1, 2004 21:46 WSPC/INSTRUCTION FILE Pfeiffer The initial value problem in numerical relativity 3 spacetime: with j i denoting the matter momentum density. Equation (2.4) is called the Hamiltonian constraint, and Eq. (2.5) is the momentum constraint. Cauchy initial data for Einstein's equations consists of (g ij , K ij ) on one hypersurface satisfying the constraint equations (2.4) and (2.5). After choosing lapse and shift (which are arbitrary and merely choose a specific coordinate system), Eqs. (2.2) and (2.3) determine (g ij , K ij ) at later times. Analytically, the constraints equations are preserved under the evolution. In practice, however, during numerical evolution of Eqs. (2.2) and (2.3) or any other formulation of Einstein's equations, many problems arise. The constraints (2.4) and (2.5) restrict four of the twelve degrees of freedom of (g ij , K ij ). As these equations are not of any standard mathematical form, it is not obvious which four degrees of freedom are restricted. Hence, finding any solutions is not trivial, and it is even harder to construct specific solutions that represent certain astrophysically relevant situations like a binary black hole. Preliminaries Both Hamiltonian and Lagrangian viewpoints use a conformal transformation on the spatial metric,