Analytic solutions to the accretion of a rotating finite cloud towards a central object - II. Schwarzschild space-time

We construct a general relativistic model for the accretion flow of a rotating finite cloud of non-interacting particles infalling on to a Schwarzschild black hole. The streamlines start at a spherical shell, where boundary conditions are fixed with wide flexibility, and are followed down to the point at which they either cross the black hole horizon or become incorporated into an equatorial thin disc. Analytic expressions for the streamlines and the velocity field are given, in terms of Jacobi elliptic functions, under the assumptions of stationarity and ballistic motion. A novel approach allows us to describe all of the possible types of orbit with a single formula. A simple numerical scheme is presented for calculating the density field. This model is the relativistic generalization of the Newtonian one developed by Mendoza, Tejeda & Nagel, and, due to its analytic nature, it can be useful in providing a benchmark for general relativistic hydrodynamical codes and for exploring the parameter space in applications involving accretion on to black holes when the approximations of steady state and ballistic motion are reasonable ones.