second-order ODE for $X$: where

Boundary Conditions {#sec:BoundaryConditions}
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In the arguments regarding black hole uniqueness, the boundary conditions at the horizon and at infinity play a crucial role. Obviously, if we impose boundary conditions that are too weak, the uniqueness will always be violated. On the other hand, the uniqueness theorem obtained under conditions that are too strong will not be sufficiently powerful in general.

Regularity condition at the horizon
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Because we are considering only regular black holes, the spacetime metric describing a black hole must be regular at the horizon. Hence, it is natural to require that a metric perturbation be regular at the horizon with respect to a coordinate system in which the background metric is regular. For the background metric with , such a coordinate system is given by the Szekeres-type coordinates $U,V$ and $z^i$ that satisfy where $\kappa=f'(r_h)/2$ is the surface gravity of the black hole, and $r_*$ is the coordinate defined by Because the black hole horizon is non-degenerate, we have $\kappa\not=0$, and $UV$ can be expressed near the horizon in terms of a regular positive function $g(r)$ as In the present paper, we require that the components of perturbations of the metric 