space $\K^n$ whose Ricci curvature $\hat R_{ij}$ satisfies the condition with $K=0,\pm1$. Although perturbative uniqueness can be proved only in the case in which $\K^n$ has a constant curvature, most analysis in the present paper is done without assuming this condition, in order to see what happens when $\K^n$ does not have a constant curvature.

We assume that this metric satisfies the vacuum Einstein equations for $(n+2)$-dimensional spacetimes and represents a black hole. Then, the two-dimensional metric $g_{ab}$ is given by where This metric describes a regular black hole spacetime only when $\lambda$ satisfies the condition[^2] Under this condition, in general, the region outside the black hole corresponds to the range $r_h<r<\infty$, where $r_h$ is the horizon radius, which is given by the (smallest positive) solution to When we study the behaviour of perturbations, we consider this entire range for $\lambda\le0$. For $\lambda>0$, however, we consider only the region $r_h<r<r_c$, where $r_c$ is the radius of the cosmological horizon given by the larger positive root of $f(r)=0$, because the perturbative uniqueness of asymptotically de Sitter black holes can be proved only by looking at the behaviour of perturbations in this region.

In this black hole background, the linearised Einstein equations 