Pseudo-Newtonian equations for evolution of particles and fluids in   stationary space-times
  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. 
Vojtech Witzany, Claus Laemmerzahl
gr-qc astro-ph.HE