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The general relativistic effective potential in Schwarzschild geometry (in units $G=M=c=1$) is expressed as $$V_{eff}=\sqrt{\left(1-\frac{2}{r}\right)\left(1+\frac{l^2}{r^2}\right)}$$ where $r$ is the radial coordinate and $l$ is the cinserved angular momentum.

However to model accretion disks around Schwarzschild black holes, pseudo-Newtonian potentials are used which agrees well with the general relativistic result. The pseudo-Newtonian potential widely used is the Paczynski-Wiita potential which can be expressed (including the centrifugal term) as $$V_{PW}=-\frac{1}{r-2}+\frac{l^2}{2r^2}$$

My questions are as follows:

  1. Why are we using pseudo-Newtonian potentials when we already know the exact general relativistic form of the potential? Can't we use the exact potential $V_{eff}$ to model accretion disks?
  2. The exact potential $V_{eff}$ is obtained from particle dynamics in Schwarzschild geometry whereas in accretion disks we are mostly concerned about fluid flows. So, would it be convenient if any exact potential is obtained by solving the equations of hydrodynamics in Schwarzschild geometry? Would such an exact hydrodynamic potential more useful than pseudo-Newtonian potentials?
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