and the regularity of $h_{ij}$, we obtain the conditions at the horizon. Under these conditions, the coefficients of $\SHB_{ij}$ in $\Lie_\eta C_{aibj}$ given in are bounded at the horizon. Therefore, the corresponding coefficients in $[\delta C_{aibj}]$ should be bounded in the $(U,V,z^i)$ coordinates. Taking account of and the equation $F^a_a=-2(n-2)F$, this leads to the conditions at the horizon. From this and the definitions of $F^a_b$, it follows that at the horizon. Using , we can easily check that all components of $\Lie_\eta C_{****}$ in the $(U,V,z^i)$ coordinates are bounded at the horizon under these boundary conditions. This implies that the gauge-invariant combinations $[\delta C_{****}]$ in the same coordinates should be bounded at the horizon. This requirement yields the additional conditions at the horizon.

In terms of the variables $X$ and $Y$, the boundary conditions at the horizon obtained by the above argument are expressed simply as Note that the conditions on the second derivatives follow from these conditions, with the help of the Einstein equations . Note also that in the case $\Lambda>0$, the same conditions should be imposed at the cosmological horizon.

Asymptotic condition at infinity
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In the cases $\Lambda\le0$, the region outside the horizon extends to $r=\infty$. 