Stellar core collapse in full general relativity with microphysics - Formulation and Spherical collapse test -

One of the longstanding issues in numerical relativity is to enable a simulation taking account of microphysical processes (e.g., weak interactions and neutrino cooling). We develop an approximate and explicit scheme in the fully general relativistic framework as a first implementation of the microphysics toward a more realistic and sophisticated modeling. In this paper, we describe in detail a method for implementation of a realistic equation of state, the electron capture and the neutrino cooling in a multidimensional, fully general relativistic code. The procedure is based on the so-called neutrino leakage scheme. To check the validity of the code, we perform a two dimensional (2D) simulation of spherical stellar core collapse. Until the convective activities set in, our results approximately agree, or at least are consistent, with those in the previous so-called state-of-the-art simulations. In particular, the radial profiles of thermodynamical quantities and the time evolution of the neutrino luminosities agree quantitatively. The convection is driven by negative gradients of the entropy per baryon and the electron fraction as in the previous 2D Newtonian simulations. We clarify which gradient is more responsible for the convection. Gravitational waves from the convection are also calculated. We find that the characteristic frequencies of the gravitational-wave spectra are distributed for higher frequencies than those in Newtonian simulations due to the general relativistic effects.