the Schwarzschild metric in the standard $(t,r,\theta,\phi)$ coordinates. However, under this gauge condition, the regularity condition at the horizon, , is not satisfied for $J\not=0$. Nevertheless, this solution is regular at the horizon, because we can set $C=-J/(n+1)/r_h^{n+1}$ through a gauge transformation, for which the solution satisfies . The apparent singular behaviour of the perturbation for the former gauge arises because the Killing vector $\partial_t$ vanishes at the bifurcation sphere of the horizons for the Schwarzschild metric, while it becomes space-like and does not vanish there for the Kerr metric. That is, the mapping used to compare the two metrics is singular at the bifurcation sphere. In contrast, for the latter gauge, which corresponds to the redefinition of the angular coordinate from $\phi$ to $\tilde\phi=\phi
 -\Omega_h t$, with $\Omega_h$ being the angular velocity of the horizon, $\partial_t$ becomes parallel to null generators of the horizon and vanishes at the bifurcation sphere. Hence, the mapping between two spacetimes is regular at the horizon. The situation in the $\Lambda>0$ case is similar. In this case, to prove the regularity of the solution, we have to employ different choices of $C$ at the black hole horizon and at the cosmological horizon.

In connection 