  We develop a neutrino transfer code for core-collapse simulations, that
directly solves the multidimensional Boltzmann equations in full general
relativity. We employ the discrete ordinate method, which discretizes the six
dimensional phase space. The code is an extension of our special relativistic
code coupled to a Newtonian hydrodynamics code, which is currently employed for
core-collapse supernova simulations. In order to demonstrate our code's
capability to treat general relativistic effects, we conduct some tests: we
first compute the free streaming of neutrinos in the Schwarzschild and Kerr
spacetimes and compare the results with the geodesic curves; in the
Schwarzschild case we deploy not only a 1-dimensional grid in space under
spherical symmetry but also a 2-dimensional spatial mesh under axisymmetry in
order to assess the capability of the code to compute the spatial advection of
neutrinos; secondly, we calculate the neutrino transport in a fixed matter
background, which is taken from a core-collapse supernova simulation with our
general relativistic but spherically symmetric Boltzmann-hydrodynamics code, to
obtain a steady neutrino distribution; the results are compared with those
given by the latter code.
