where Z + characterizes the polar perturbations and Z − − the axial perturbations. Here r ∗ ∗ = r + 2 M ln ⁡ ⁡ ( r / 2 M − − 1 ) is the tortoise coordinate (we set G = c = 1 ), r belongs to the Schwarzschild coordinates ( t , r , θ θ , φ φ ) , 2 M is the Schwarzschild radius and σ σ represents the time frequency of the perturbations appearing in the form e i σ σ t . The Regge–Wheeler potential and Zerilli potential are respectively given by