  We describe in detail the implementation of a simplified approach to
radiative transfer in general relativity by means of the well-known neutrino
leakage scheme (NLS). In particular, we carry out an extensive investigation of
the properties and limitations of the NLS for isolated relativistic stars to a
level of detail that has not been discussed before in a general-relativistic
context. Although the numerous tests considered here are rather idealized, they
provide a well-controlled environment in which to understand the relationship
between the matter dynamics and the neutrino emission, which is important in
order to model the neutrino signals from more complicated scenarios, such as
binary neutron-star mergers. When considering nonrotating hot neutron stars we
confirm earlier results of one-dimensional simulations, but also present novel
results about the equilibrium properties and on how the cooling affects the
stability of these configurations. In our idealized but controlled setup, we
can then show that deviations from the thermal and weak-interaction equilibrium
affect the stability of these models to radial perturbations, leading models
that are stable in the absence of radiative losses, to a gravitational collapse
to a black hole when neutrinos are instead radiated.
