1. Initial Data Set formulation The initial data set is given by (gij, pij) on a three dimensional manifold where gij is asymptotically euclidean (so that asymptotic quantities can be defined) and pij falls off quadratically. We also study the case when g^ is asymptotically hyperbolic and the three manifold is asymptotically null. Several important questions remain unsolved for the initial data set and its Cauchy development.
Gravitational waves are predicted by the general theory of relativity. In  D. Christodoulou showed that gravitational waves have a nonlinear memory. We proved in  that the electromagnetic field contributes at highest order to the nonlinear memory effect of gravitational waves. In the present paper, we study this electromagnetic Christodoulou memory effect and compute it for binary neutron star mergers. These are typical sources of gravitational radiation. During these processes, not only mass and momenta are radiated away in form of gravitational waves, but also very strong magnetic fields are produced and radiated away. Thus the observed effect on test masses of a laser interferometer gravitational wave detector will be enlarged by the contribution of the electromagnetic field. Therefore, the present results are important for the planned experiments. Looking at the null asymptotics of spacetimes, which are solutions of the Einstein-Maxwell (EM) equations, we derived in  the electromagnetic Christodoulou memory effect. Moreover, our results allow to answer astro-physical questions, as the knowledge about the amount of energy radiated away in a neutron star binary merger enables us to gain information about the source of the gravitational waves.