  With the Boltzmann-radiation-hydrodynamics code, which we have developed to
solve numerically the Boltzmann equations for neutrino transfer, the Newtonian
hydrodynamics equations, and the Newtonian self-gravity simultaneously and
consistently, we simulate the collapse of a rotating core of the progenitor
with a zero-age-main-sequence mass of $11.2\,M_\odot$ and a shelluler rotation
of $1\,{\rm rad\,s^{-1}}$ at the center. We pay particular attention in this
paper to the neutrino distribution in phase space, which is affected by the
rotation. By solving the Boltzmann equations directly, we can assess the
rotation-induced distortion of the angular distribution in momentum space,
which gives rise to the rotational component of the neutrino flux. We compare
the Eddington tensors calculated both from the raw data and from the M1-closure
approximation. We demonstrate that the Eddington tensor is determined by
complicated interplays of the fluid velocity and the neutrino interactions and
that the M1-closure, which assumes that the Eddington factor is determined by
the flux factor, fails to fully capture this aspect, especially in the vicinity
of the shock. We find that the error in the Eddington factor reaches $\sim
20\%$ in our simulation. This is due not to the resolution but to the different
dependence of the Eddington and flux factors on the angular profile of the
neutrino distribution function, and hence modification to the closure relation
is needed.
