Para una tríada (tétrada) \{ \hat{e}_i \}, podemos calcular sus coeficientes de estructura c_{ij}^{\phantom{ij}k}: escribimos su corchete de Lie en función de los elementos de la tríada (tétrada):

[\hat{e}_i, \hat{e}_j] = c_{ij}^{\phantom{ij}k} \hat{e}_k.

En el caso de la tríada (tétrada) correspondiente a las esféricas normalizadas:

\{ \frac{(a-\bar{r})^2}{a^2} \partial_{\bar{r}}, \frac{a-\bar{r}}{a \bar{r}} \partial_{\theta}, \frac{a-\bar{r}}{a \bar{r}} \csc \theta \partial_{\varphi} \},

tenemos:

[\frac{(a-\bar{r})^2}{a^2} \partial_{\bar{r}} , \frac{a-\bar{r}}{a \bar{r}} \partial_{\theta}] = -[\frac{a-\bar{r}}{a \bar{r}} \partial_{\theta}, \frac{(a-\bar{r})^2}{a^2} \partial_{\bar{r}}] =

= \frac{(a-\bar{r})^2}{a^2} \partial_{\bar{r}} (\frac{a-\bar{r}}{a \bar{r}}) \partial_{\theta} - \frac{a-\bar{r}}{a \bar{r}} \partial_{\theta} (\frac{(a-\bar{r})^2}{a^2}) \partial_{\bar{r}} + \frac{(a-\bar{r})^2}{a^2} \frac{a-\bar{r}}{a \bar{r}} [\partial_{\bar{r}},\partial_{\theta}] =

-\frac{a-\bar{r}}{a \bar{r}} \frac{a-\bar{r}}{a \bar{r}} \partial_{\theta} = -\frac{a-\bar{r}}{a \bar{r}} \hat{e}_{\theta},

[\frac{(a-\bar{r})^2}{a^2} \partial_{\bar{r}} , \frac{a-\bar{r}}{a \bar{r}} \csc \theta \partial_{\varphi}] = - [\frac{a-\bar{r}}{a \bar{r}} \csc \theta \partial_{\varphi}, \frac{(a-\bar{r})^2}{a^2} \partial_{\bar{r}}] =

= \frac{(a-\bar{r})^2}{a^2} \partial_{\bar{r}} (\frac{a-\bar{r}}{a \bar{r}} \csc \theta) \partial_{\varphi} - \frac{a-\bar{r}}{a \bar{r}} \csc \theta \partial_{\varphi} (\frac{(a-\bar{r})^2}{a^2}) \partial_{\bar{r}} + \frac{(a-\bar{r})^2}{a^2} \frac{a-\bar{r}}{a \bar{r}} \csc \theta [\partial_{\bar{r}},\partial_{\theta}] =

= -\frac{a-\bar{r}}{a \bar{r}} \frac{a-\bar{r}}{a \bar{r}} \csc \theta \partial_{\varphi} = -\frac{a-\bar{r}}{a \bar{r}} \hat{e}_{\varphi},

[\frac{a-\bar{r}}{a \bar{r}} \partial_{\theta} , \frac{a-\bar{r}}{a \bar{r}} \csc \theta \partial_{\varphi}] = - [\frac{a-\bar{r}}{a \bar{r}} \csc \theta \partial_{\varphi}, \frac{a-\bar{r}}{a \bar{r}} \partial_{\theta}] =

= \frac{a-\bar{r}}{a \bar{r}} \partial_{\theta} ( \frac{a-\bar{r}}{a \bar{r}} \csc \theta ) \partial_{\varphi} - \frac{a-\bar{r}}{a \bar{r}} \csc \theta \partial_{\varphi} (\frac{a-\bar{r}}{a \bar{r}} ) \partial_{\theta} + \frac{a-\bar{r}}{a \bar{r}} \frac{a-\bar{r}}{a \bar{r}} \csc \theta [\partial_{\theta} ,\partial_{\varphi}] =

= -\frac{a-\bar{r}}{a \bar{r}} \cot \theta \frac{a-\bar{r}}{a \bar{r}} \csc \theta \partial_{\varphi} = -\frac{a-\bar{r}}{a \bar{r}} \cot \theta \hat{e}_{\varphi}.

De esta manera, tenemos:

[\hat{e}_{\bar{r}}, \hat{e}_{\theta}] = -\frac{a-\bar{r}}{a \bar{r}} \hat{e}_{\theta} = c_{\hat{\bar{r}}\hat{\theta}}^{\phantom{r \theta} \hat{\theta}} = -c_{\hat{\theta} \hat{\bar{r}}}^{\phantom{\theta r} \hat{\theta}},

[\hat{e}_{\bar{r}}, \hat{e}_{\varphi}] = -\frac{a-\bar{r}}{a \bar{r}} \hat{e}_{\varphi} = c_{\hat{\bar{r}}\hat{\varphi}}^{\phantom{r \varphi} \hat{\varphi}} = -c_{\hat{\varphi} \hat{\bar{r}}}^{\phantom{\varphi r} \hat{\varphi}},

[\hat{e}_{\theta}, \hat{e}_{\varphi}] = -\frac{a-\bar{r}}{a \bar{r}} \cot \theta \hat{e}_{\varphi} = c_{\hat{\theta}\hat{\varphi}}^{\phantom{\theta \varphi} \hat{\varphi}} = -c_{\hat{\varphi} \hat{\theta}}^{\phantom{\varphi \theta} \hat{\varphi}},

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