Tenemos:

1. $\bar{r} := \frac{r}{r+a}$
2. $\Delta := \frac{(1-\bar{r})^4}{a^2} \partial_{\bar{r}\bar{r}} + \frac{(1-\bar{r})^4}{a^2}\frac{2}{\bar{r}} \partial_{\bar{r}}$
3. $\Delta \Theta_X = 6 \pi [ \frac{(1-\bar{r})^2}{a} \partial_{\bar{r}} S^*_{\bar{r}} + \frac{1 - \bar{r}}{a} \frac{2}{\bar{r}} S^*_{\bar{r}} + \frac{1-\bar{r}}{a} \frac{\cot \theta}{\bar{r}} S^*_{\theta}]$
4. $\Delta X^{\bar{r}} = 8 \pi S^*_{\bar{r}} - \frac{1}{3} \frac{(1-\bar{r})^2}{a} \partial_{\bar{r}} \Theta_X$
5. $\hat{A}^{\bar{r}\bar{r}} = \frac{4}{3}\frac{(1-\bar{r})^2}{a} \partial_{\bar{r}} X^{\bar{r}} - \frac{2}{3}2\frac{1-\bar{r}}{a}\frac{1}{\bar{r}} X^{\bar{r}}$
6. $\Delta \Theta_\beta = \frac{3}{2}[\frac{(1-\bar{r})^4}{a^2} \partial_{\bar{r}\bar{r}}u + \frac{(1-\bar{r})^3(2-\bar{r})}{2a^2} \frac{4}{\bar{r}}u + \frac{(1-\bar{r})^2}{a^2} \frac{2}{\bar{r}^2}u]$

con:

$u:=\alpha \psi^{-6} \hat{A}^{\bar{r}\bar{r}}$

y:

$\{\frac{2 (\bar{r}_i - 1)^4 (\bar{r_i} - (\bar{r}_{i+1} - \bar{r}_i))}{a^2 (\bar{r}_i - \bar{r}_{i-1}) \bar{r_i} (\bar{r}_{i+1} - \bar{r}_{i-1} )},$

$\frac{(\bar{r}_i - 1)^2 (\frac{-2}{h_\theta^2 \bar{r}_i^2} + \frac{(\bar{r}_i - 1)^2 ((r_{i+1}-r_i)-(r_i-r_{i-1})-2)}{(r_{i+1}-r_i)(r_i - r_{i-1})} + \frac{-2 \csc^2 \theta_i}{h_\varphi^2 \bar{r}_i^2})}{a^2},$

$\frac{2 (\bar{r}_i - 1)^4 ((\bar{r}_{i} - \bar{r}_{i-1}) + \bar{r_i})}{a^2 (\bar{r}_{i+1} - \bar{r}_i) \bar{r}_i (\bar{r}_{i+1} - \bar{r}_{i-1} )} \}$