The Paczynski-Wiita potential: A step-by-step "derivation"

Paczynski realized that a properly chosen gravitational potential may accurately model (in a "pseudo Newtonian" theory) general relativistic effects that determine motion of matter near a non-rotating black hole. Paczynski's choice, known today as the "Paczynski-Wiita potential", proved to be very practical. It was used by numerous researchers in the black hole accretion theory, and became a standard tool in relativistic astrophysics. The model is an example of Paczynski's admired ability to invent "out of nowhere" simple ideas that were brilliant, deep and useful. Paczynski has guessed intuitively the form of the potential. However, it could be also derived by a a step-by-step formal procedure. I show the derivation here. My derivation is based on a standard definition of the relativistic "effective potential" in the Schwarzschild spacetime. The relativistic effective potential may be uniquely divided into its "gravitational" and "centrifugal" part. The gravitational part differs from the Paczynski-Wiita potential only by a constant.