Therefore, we must impose some asymptotic condition. In the present paper, we require that all components of the metric perturbation $h_{\mu\nu}$ in the natural orthonormal basis of the background metric fall off at infinity, i.e., First, for a tensor perturbation, this requirement gives the single condition Next, for a stationary vector perturbation, these conditions can be expressed in terms of the gauge-invariant variables $F_t=f_t$ and $F^{(1)}=-r(f_t/r)'$ as Note that we do not need an asymptotic condition on $F_r$, because $F_r$ vanishes for a stationary perturbation.

Finally, for a stationary scalar perturbation, the Einstein equation $F^a_a=-2(n-2)F$ yields From this, we obtain the asymptotic condition Hence, the above asymptotic conditions on the metric perturbations can be expressed in terms of the gauge-invariant variables $X$ and $Y$ as

Exceptional vector perturbations {#subsec:ExceptionalMode2}
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As shown in §\[subsec:BasicEqs:Vector\], we have an exact general solution for the exceptional vector perturbation with $k_V^2=(n-1)K$ for $K\ge0$. There is a subtlety concerning the regularity of this solution. In the case $\Lambda\le0$, the above asymptotic condition is satisfied by only for the choice $C=0$. The solution corresponding to this choice is obtained when we treat, for example, the Kerr metric with small angular momentum as a perturbed form of 