Stability considerations of a Schwarzschild black hole with an r dependent mass-function

The axial and polar modes for the ring down of a Schwarzschild black hole are calculated using an r-dependent mass-function and finally applying the asymptotic iteration method (AIM). We were able to reach up to 500 iterations, obtaining for the first time convergence for a wide range of large damping modes. The general relativity (GR) and an extended model with an r-dependent mass-function are compared. This mass-function allows an analytical solution for the Tortoise coordinate. The example of the mass-function corresponds to the leading correction for possible extended theories and serves as a starting point to treat other r-dependent parameter mass-functions.