I am fully aware of the fact that the gravitational potential is GM/R_s, where R_s is the Schwarzschild radius. But I was thinking of deriving that with a different thought.

Suppose an electron of mass m is Z' distance away from the even horizon. The gravitational force experienced by the electron will be GMm/Z'^2. So the potential experienced by the particle will be -GMm/Z' at the point Z'. Now to figure out the the potential across the Z axis I need to integrate this expression w.r.t Z to get GMm*Ln|Z|. This is the potential of the black hole over the Z axis? Is that right?

M: mass of the black hole