\begin{align}
	&[g^{\tau \tau'}W_{,\tau\tau'}]=-\frac{1}{4\pi^{2}}\sum_{l=0}^{\infty}(2l+1)\sum_{n=1}^{\infty}\frac{n^{2}\kappa_+^{2}}{f(r)}\mathsf{g}_{nl}(r)\nonumber\\
	&\qquad-\frac{1}{4\pi^{2}}\left\{\mathcal{T}_{10}^{(p)}+\mathcal{D}_{22}^{(-)}+\frac{1}{f(r)}(\mathcal{T}_{11}^{(p)}+\mathcal{D}_{21}^{(-)})\right\},\\
	&[g^{\phi \phi'}W_{,\phi\phi'}]=\frac{1}{16\pi^{2}r^{2}}\sum_{l=0}^{\infty}(2l+1)l(l+1)\sum_{n=0}^{\infty}(2-\delta^{n}_{0})\mathsf{g}_{nl}(r)\nonumber\\
	&\qquad+\frac{1}{4\pi^{2}}\left\{\mathcal{T}_{10}^{(p)}+\mathcal{D}_{22}^{(-)}\right\},
	\end{align}
