3+1 covariant suite of numerical relativity evolution systems

A suite of three evolution systems is presented in the framework of the 3+1 formalism. The first one is of second order in space derivatives and has the same causal structure as the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system for a suitable choice of parameters. The second one is the standard first order version of the first one and has the same causal structure as the Bona-Mass\'o system for a given parameter choice. The third one is obtained from the second one by reducing the space of variables in such a way that the only modes that propagate with zero characteristic speed are the trivial ones. This last system has the same structure as the ones recently presented by Kidder, Scheel, and Teukolsky: the correspondence between the two sets of parameters is explicitly given. The fact that the suite started with a system in which all the dynamical variables behave as tensors (contrary to what happens with the BSSN system) allows one to keep the same parametrization when passing from one system to the next in the suite. The direct relationship between each parameter and a particular characteristic speed, which is quite evident in the second and third systems, is a direct consequence of the manifest 3+1 covariance of the approach.