An implementation of the microphysics in full general relativity : General relativistic neutrino leakage scheme

Performing fully general relativistic simulations taking account of microphysical processes is one of long standing problems in numerical relativity. One of main difficulties in implementation of weak interactions in the general relativistic framework lies on the fact that the characteristic timescale of weak interaction processes (the WP timescale) in hot dense matters is much shorter than the dynamical timescale. Numerically this means that stiff source terms appears in the equations so that an implicit scheme is in general necessary to stably solve the relevant equations. Otherwise a very short timestep will be required to solve them explicitly. Furthermore, in the relativistic framework, the Lorentz factor is coupled with the rest mass density and the energy density. The specific enthalpy is also coupled with the momentum. Due to these couplings, it is very complicated to recover the primitive variables and the Lorentz factor from conserved quantities. At the current status, no implicit procedure have been proposed except for the case of the spherical symmetry. Therefore, an approximate, explicit procedure is developed in the fully general relativistic framework in this paper as an first implementation of the microphysics toward a more realistic sophisticated model. The procedure is based on the so-called neutrino leakage schemes which is based on the property that the characteristic timescale in which neutrinos leak out of the system (the leakage timescale) is much longer than the WP timescale. In this paper, I present a detailed neutrino leakage scheme and a simple and stable method for solving the equations explicitly in the fully general relativistic framework. I also perform a test simulation to check the validity of the present method, showing that it works fairly well.