La salida ahora para un tensor dos veces contravariante en la base ortonormal queda:

CovDerTen2BiSphCom1,

Para primera ecuación:

\boxed{\Delta \Theta_\beta = \frac{3}{2} \mathcal{D}_i \mathcal{D}_j (\alpha \psi^{-6} \hat{A}^{ij}) },

definimos como antes

V^i := \mathcal{D}_j \alpha \psi^{-6} \hat{A}^{ij},

de manera que la ecuación original la reescribimos como

\Delta \Theta_\beta = \frac{3}{2} \mathcal{D}_i V^i,

De esta manera, en nuestras coordenadas obtenemos:

\Delta \Theta_\beta = \frac{3}{2} \mathcal{D}_i V^i = \frac{3}{2} (\mathcal{D}_{\xi} V^{\xi} + \mathcal{D}_{\bar{\eta}} V^{\bar{\eta}} + \mathcal{D}_{\varphi} V^{\varphi}) =

div_biSphComNor1

donde

V^{\xi} = \mathcal{D}_{\xi} (\alpha \psi^{-6} \hat{A}^{\xi \xi}) + \mathcal{D}_{\bar{\eta}} ( \alpha \psi^{-6} \hat{A}^{\xi \bar{\eta}} ) + \mathcal{D}_{\varphi} ( \alpha \psi^{-6} \hat{A}^{\xi \varphi} ),

V^{\bar{\eta}} = \mathcal{D}_{\xi} (\alpha \psi^{-6} \hat{A}^{\bar{\eta} \xi}) + \mathcal{D}_{\bar{\eta}} ( \alpha \psi^{-6} \hat{A}^{\bar{\eta} \bar{\eta}} ) + \mathcal{D}_{\varphi} ( \alpha \psi^{-6} \hat{A}^{\bar{\eta} \varphi} ),

V^{\varphi} = \mathcal{D}_{\xi} (\alpha \psi^{-6} \hat{A}^{\varphi \xi}) + \mathcal{D}_{\bar{\eta}} ( \alpha \psi^{-6} \hat{A}^{\varphi \bar{\eta}} ) + \mathcal{D}_{\varphi} ( \alpha \psi^{-6} \hat{A}^{\varphi \varphi} ),

que desarrollando las covariantes quedan:

V^{\xi} = \frac{(1 - \bar{r})^2}{a} \partial_{\bar{r}} (\alpha \psi^{-6} \hat{A}^{\bar{r} \bar{r}}) +

+ \frac{1 - \bar{r}}{a \bar{r}} [ \partial_{\theta} ( \alpha \psi^{-6} \hat{A}^{\bar{r} \theta}) + \alpha \psi^{-6} \hat{A}^{\bar{r} \bar{r}} - \alpha \psi^{-6} \hat{A}^{\theta \theta} ] +

+ \frac{1 - \bar{r}}{a \bar{r}} [ \csc \theta \partial_{\varphi} ( \alpha \psi^{-6} \hat{A}^{\bar{r} \varphi}) + \alpha \psi^{-6} \hat{A}^{\bar{r} \bar{r}} + \cot \theta \alpha \psi^{-6} \hat{A}^{\bar{r} \theta} - \alpha \psi^{-6} \hat{A}^{\varphi \varphi}] ),

V^{\bar{\eta}} = \frac{(1 - \bar{r})^2}{a} \partial_{\bar{r}} (\alpha \psi^{-6} \hat{A}^{\theta \bar{r}}) +

+ \frac{1 - \bar{r}}{a \bar{r}} [ \partial_{\theta} ( \alpha \psi^{-6} \hat{A}^{\theta \theta}) + 2 \alpha \psi^{-6} \hat{A}^{\bar{r} \theta} ] +

+ \frac{1 - \bar{r}}{a \bar{r}} [ \csc \theta \partial_{\varphi} ( \alpha \psi^{-6} \hat{A}^{\theta \varphi}) + \alpha \psi^{-6} \hat{A}^{\bar{r} \theta} + \cot \theta \alpha \psi^{-6} \hat{A}^{\theta \theta} - \cot \theta \alpha \psi^{-6} \hat{A}^{\varphi \varphi} ] ),

V^{\varphi} = \frac{(1 - \bar{r})^2}{a} \partial_{\bar{r}} (\alpha \psi^{-6} \hat{A}^{\bar{r} \bar{r}}) +

+ \frac{1 - \bar{r}}{a \bar{r}} [ \partial_{\theta} ( \alpha \psi^{-6} \hat{A}^{\bar{r} \theta}) + \alpha \psi^{-6} \hat{A}^{\bar{r} \bar{r}} - \alpha \psi^{-6} \hat{A}^{\theta \theta} ] +

+ \frac{1 - \bar{r}}{a \bar{r}} [ \csc \theta \partial_{\varphi} ( \alpha \psi^{-6} \hat{A}^{\bar{r} \varphi}) + \alpha \psi^{-6} \hat{A}^{\bar{r} \bar{r}} + \cot \theta \alpha \psi^{-6} \hat{A}^{\bar{r} \theta} - \alpha \psi^{-6} \hat{A}^{\varphi \varphi}] ),

que combinandolo con la anterior, queda:

Finalmente, las ecuaciones:

\boxed{\Delta \beta^i = 2\mathcal{D}_j ( \alpha \psi^{-6} \hat{A}^{ij} ) - \frac{1}{3} \mathcal{D}^i \Theta_{\beta} },

con las que procederemos de manera similar a como hemos hecho con las X^i, es decir, calculando las fuentes en una base, haciendo un cambio de base que las desacople (cartesianas), resolviendolas de manera independiente y volviendo a la base original:

S^i_\beta (\bar{r},\theta,\varphi) := 2\mathcal{D}_j ( \alpha \psi^{-6} \hat{A}^{ij} ) - \frac{1}{3} \mathcal{D}_i \Theta_{\beta},

que quedan:

S^{\xi}_\beta= 2 \big [ \mathcal{D}_{\xi} (\alpha \psi^{-6} \hat{A}^{\xi \xi}) + \mathcal{D}_{\bar{\eta}} (\alpha \psi^{-6} \hat{A}^{\xi \bar{\eta}}) + \mathcal{D}_{\varphi} (\alpha \psi^{-6} \hat{A}^{\xi \varphi}) \big ] - \frac{1}{3} \mathcal{D}_{\xi} \Theta_\beta,

S^{\bar{\eta}}_\beta = 2 \big [ \mathcal{D}_{\xi} (\alpha \psi^{-6} \hat{A}^{\bar{\eta} \xi}) + \mathcal{D}_{\bar{\eta}} (\alpha \psi^{-6} \hat{A}^{\bar{\eta} \bar{\eta}}) + \mathcal{D}_{\varphi} (\alpha \psi^{-6} \hat{A}^{\bar{\eta} \varphi}) \big ] - \frac{1}{3} \mathcal{D}_{\bar{\eta}} \Theta_\beta,

S^{\varphi}_\beta = 2 \big [ \mathcal{D}_{\xi} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{x}}) + \mathcal{D}_{\bar{y}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{y}}) + \mathcal{D}_{\bar{z}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{z}}) \big ] - \frac{1}{3} \mathcal{D}_{\varphi} \Theta_\beta,

donde las derivadas covariantes del tensor dos veces contravariante:

T^{ij}:=\alpha \psi^{-6} \hat{A}^{ij}

son como acabamos de hacer en la ecuación anterior y las del escalar \Theta_\beta es como ya hicimos con las X^i:

S^{\xi} = 2 V^{\xi} - \frac{\mbox{\scriptsize cosh} \frac{b \bar{\eta}}{1 - \bar{\eta}} - \cos \xi}{3a} \partial_{\xi} \Theta_{\beta},

S^{\bar{\eta}} = 2 V^{\bar{\eta}} - \frac{\mbox{\scriptsize cosh} \frac{b \bar{\eta}}{1 - \bar{\eta}} - \cos \xi}{3a} \frac{(\bar{\eta} - 1)^2}{b} \partial_{\bar{\eta}} \Theta_{\beta},

S^{\varphi} = 2 V^{\varphi} - \frac{\mbox{\scriptsize cosh} \frac{b \bar{\eta}}{1 - \bar{\eta}} - \cos \xi}{3a} \csc \xi \partial_{\varphi} \Theta_{\beta}.

Hacemos a continuación el cambio:

[S^{\xi}(\xi,\bar{\eta},\varphi),S^{\bar{\eta}}(\xi,\bar{\eta},\varphi),S^{\varphi}(\xi,\bar{\eta},\varphi)] \rightarrow

\rightarrow [S^x(\xi,\bar{\eta},\varphi), S^y(\xi,\bar{\eta},\varphi), S^z(\xi,\bar{\eta},\varphi)],

y resolvemos:

\Delta \beta^{x} = S^{x}

\Delta \beta^{y} = S^{y}

\Delta \beta^{z} = S^{z},

deshaciendo el cambio:

[\beta^x(\xi,\bar{\eta},\varphi), \beta^y(\xi,\bar{\eta},\varphi), \beta^z(\xi,\bar{\eta},\varphi)] \rightarrow

\rightarrow [\beta^{\xi}(\xi,\bar{\eta},\varphi),\beta^{\bar{\eta}}(\xi,\bar{\eta},\varphi),\beta^{\varphi}(\xi,\bar{\eta},\varphi)]

para terminar.

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