  Approximate gravitational potentials are often used to describe analytically
the motion of particles near black holes (BHs), as well as to study the
structure of an accretion disk. Such 'pseudo-Newtonian' potentials are used
with the flat-metric equations. Here we consider the motion of a free particle
near a non-rotating BH in the context of an exact `logarithmic' gravitational
potential. We show how the logarithmic potential gives an exact solution for a
mechanical problem and present the relativistic Bernoulli equation for the
fluid in the Schwarzschild metric.
