Numerical studies of a hyperbolic formulation of the constraint equations

In the framework of General Relativity, the formulation of the initial value problem is given in terms of two systems of PDEs, namely evolution and constraints equations, which have to be solved simultaneously during the whole evolution. Initial data sets are given as solutions of the second system. However, because of the non-linear nature of this set of PDEs, their numerical and analytical treatment is, in general, a complex task.

In this talk we will consider a hyperbolic formulation of the constraints equations. Using a pseudo-spectral approach based on spin-weighted spherical harmonics, we will construct initial data sets which can be interpreted as nonlinear perturbations of a Schwarzschild initial data in Kerr-Schild coordinates. Our numerical results suggest that generic initial data sets obtained with this method may violate fundamental asymptotic conditions.