  Pseudo-Newtonian potentials are a tool often used in theoretical astrophysics
to capture some key features of a black-hole space-time in a Newtonian
framework. As a result, one can use Newtonian numerical codes, and Newtonian
formalism in general, in an effective description of important astrophysical
processes such as accretion onto black holes.
  In this paper we develop a general pseudo-Newtonian formalism which pertains
to the motion of particles, light, and fluids in stationary space-times. In
return, we are able to assess the applicability of the pseudo-Newtonian scheme.
The simplest and most elegant formulas are obtained in space-times without
gravitomagnetic effects, such as the Schwarzschild rather than the Kerr
space-time; the quantitative errors are smallest for motion with low binding
energy. Included is a ready-to-use set of fluid equations in Schwarzschild
space-time in Cartesian and radial coordinates.
