Consider the Schwarzschild blackhole vaccuum solution. Now let a test particle drop from "coordinate rest" at a finite r, and watch it fall in. Is there a coordinate transformation in which we go to a rotating frame where the blackhole is now a Kerr blackhole and the test particle follows the appropriate trajectory for the Kerr blackhole vaccuum solution? If yes, then aren't the solutions related by a diffeomorphism and therefore "physically" the same thing. If no, what prevents us from mapping one static vaccuum solution onto another?