Initial data for N black holes

The N-body problem in general relatively is of enormous difficulty, especially in the nonlinear regime, where radiation is important. It is now possible to study this problem by treating it as a Cauchy problem and by using large-scale computers to develop numerical models. With this motivation, the Cauchy formulation of Einstein equations is described. It consists of the Hamiltonian and momentum constraint equations, and the evolution equations. The constraints are analyzed using the conformal technique. The first step is this approach is to set up initial data compatible with the constraints. The N-body data, in general, will depend on the stress-energy tensor of the bodies. Hence this issue is bypassed by considering the matter free representation of particles in terms of the geometries of certain non-Euclidean manifolds. Problems such as the dynamics of a binary system of black holes are more interesting. They required data representing holes with nonzero momenta. Hence the extrinsic curvature of the initial hypersurface cannot be taken to be zero