Gravitational waves in dynamical spacetimes with matter content in the fully constrained formulation

The fully constrained formulation (FCF) of general relativity is a framework introduced as an alternative to the hyperbolic formulations traditionally used in numerical relativity. The FCF equations form a hybrid elliptic-hyperbolic system of equations including explicitly the constraints. We present an implicit-explicit numerical algorithm to solve the hyperbolic part, whereas the elliptic sector shares the form and properties with the well-known conformally flat condition approximation. We show the stability and convergence properties of the numerical scheme with numerical simulations of vacuum solutions. We have performed the first numerical evolutions of the coupled system of hydrodynamics and Einstein equations within FCF. As a proof of principle of the viability of the formalism, we present 2D axisymmetric simulations of an oscillating neutron star. In order to simplify the analysis we have neglected the backreaction of the gravitational waves into the dynamics, which is small (< 2%) for the system considered in this work. We use spherical coordinates grids which are well adapted for simulations of stars and allow for extended grids that marginally reach the wave zone. We have extracted the gravitational wave signature and compared it to the Newtonian quadrupole and hexadecapole formulas. Both extraction methods show agreement within the numerical errors and the approximations used (similar to 30%).