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

CovDerTen2SphCom1,

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}_{\bar{r}} V^{\bar{r}} + \mathcal{D}_{\theta} V^{\theta} + \mathcal{D}_{\varphi} V^{\varphi}) =

= \frac{3 - 3\bar{r} }{2a \bar{r}} [ (\bar{r}-\bar{r}^2) \partial_{\bar{r}} V^{\bar{r}} + 2 V^{\bar{r}} + \partial_{\theta} V^{\theta} + \cot \theta V^{\theta} + \csc \theta \partial_{\varphi} V^{\varphi} ],

donde

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

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

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

que desarrollando las covariantes quedan:

V^{\bar{r}} = \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^{\theta} = \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 (solo escribimos como empezaría debido a la longitud de la ecuación):

\Delta \Theta_\beta = \frac{3 - 3\bar{r} }{2a \bar{r}} \Big [ (\bar{r}-\bar{r}^2) \partial_{\bar{r}} [\frac{(1 - \bar{r})^2}{a} \partial_{\bar{r}} (\alpha \psi^{-6} \hat{A}^{\bar{r} \bar{r}}) + \ldots ] + \ldots \Big ]

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

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

S^{\varphi}_\beta = 2 \big [ \mathcal{D}_{\bar{x}} (\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}_{\bar{y}} \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^{\bar{r}} = 2 V^{\bar{r}} - \frac{(1-\bar{r})^2}{3a} \partial_{\bar{r}} \Theta_{\beta},

S^{\theta} = 2 V^{\theta} - \frac{1-\bar{r}}{3a\bar{r}} \partial_{\theta} \Theta_{\beta},

S^{\varphi} = 2 V^{\varphi} - \frac{1-\bar{r}}{3a\bar{r}} \csc \theta \partial_{\varphi} \Theta_{\beta}.

Hacemos a continuación el cambio:

[S^{\bar{r}}(\bar{r},\theta,\varphi),S^{\theta}(\bar{r},\theta,\varphi),S^{\varphi}(\bar{r},\theta,\varphi)] \rightarrow

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

y resolvemos:

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

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

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

deshaciendo el cambio:

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

\rightarrow [\beta^{\bar{r}}(\bar{r},\theta,\varphi),\beta^{\theta}(\bar{r},\theta,\varphi),\beta^{\varphi}(\bar{r},\theta,\varphi)]

para terminar.

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

CovDerTen2CarCom2,

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}_{\bar{x}} V^{\bar{x}} + \mathcal{D}_{\bar{y}} V^{\bar{y}} + \mathcal{D}_{\bar{z}} V^{\bar{z}}) =

= \frac{3}{2} (\frac{1 + \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} V^{\bar{x}}+ \frac{1 + \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} V^{\bar{y}} + \frac{1 + \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} V^{\bar{z}}),

donde

V^{\bar{x}} = \mathcal{D}_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{x}}) + \mathcal{D}_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{y}} ) + \mathcal{D}_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{z}} ),

V^{\bar{y}} = \mathcal{D}_{\bar{x}} (\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}} ),

V^{\bar{z}} = \mathcal{D}_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{x}}) + \mathcal{D}_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{y}} ) + \mathcal{D}_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{z}} ),

que desarrollando las covariantes quedan:

V^{\bar{x}} = \frac{1 + \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{x}}) +

+ \frac{1 + \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{y}} ) + \frac{1 + \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{z}} ),

V^{\bar{y}} = \frac{1 + \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{x}}) +

+ \frac{1 + \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{y} \bar{y}} ) + \frac{1 + \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{y} \bar{z}} ),

V^{\bar{z}} = \frac{1 + \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{x}}) +

+ \frac{1 + \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{y}} ) + \frac{1 + \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{z}} ).

Por tanto, combiando todo, tenemos:

\Delta \Theta_\beta =

= \frac{3 + 3cos(\pi \bar{x})}{2 a \pi} \partial_{\bar{x}} \big [ \frac{1 + \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{x}}) +

+ \frac{1 + \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{y}} ) +

+ \frac{1 + \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{z}} ) \big ] +

+ \frac{3 + 3 \cos (\pi \bar{y})}{2b \pi} \partial_{\bar{y}} \big [ \frac{1 + \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{x}}) +

+ \frac{1 + \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{y} \bar{y}} ) +

+ \frac{1 + \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{y} \bar{z}} ) \big ] +

+ \frac{3 + 3 \cos(\pi \bar{z})}{2c \pi} \partial_{\bar{z}} \big [ \frac{1 + \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{x}}) +

\frac{1 + \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{y}} ) +

+ \frac{1 + \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{z}} ) \big ],

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} }.

son:

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

\Delta \beta^{\bar{y}} = 2 \big [ \mathcal{D}_{\bar{x}} (\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}_{\bar{y}} \Theta_\beta,

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

y al sustituir las derivadas covariantes:

\Delta \beta^{\bar{x}} = \frac{2 + 2 \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{x}}) +

+ \frac{2 + 2 \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{y}}) +

+ \frac{2 + 2 \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{z}}) - \frac{1 + \cos(\pi \bar{x})}{3a \pi} \partial_{\bar{x}} \Theta_\beta,

\Delta \beta^{\bar{y}} = \frac{2 + 2 \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{x}}) +

+ \frac{2 + 2 \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{y}}) +

+ \frac{2 + 2 \cos(\pi \bar{z})}{c} \partial_{\bar{z}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{z}}) - \frac{1 + \cos(\pi \bar{y})}{3b \pi} \partial_{\bar{y}} \Theta_\beta,

\Delta \beta^{\bar{z}} = \frac{2 + 2 \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{x}}) +

+ \frac{2 + 2 \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{y}}) +

+ \frac{2 + 2 \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{z}}) - \frac{1 + \cos(\pi \bar{z})}{3c \pi} \partial_{\bar{z}} \Theta_\beta.

Copio a continuación la salida generada por nuestra función en Mathematica que nos calcula todas las derivadas covariantes de tensores con dos índices (aunque en este caso particular no es excesivamente laborioso, si lo es para el resto de entradas, por lo que evitaremos morir en el intento de pasarlas a latex 😉 ) . En particular, aquí lo hacemos para un vector dos veces contravariante y para la base ortonormal:

CovDerTen2CarCom1,

donde primer término corresponde al factor que acompaña a la derivada parcial y la matriz contiene los factores que acopañan a cada par de valores de los índices.

Vamos a ver ahora, en este caso, como quedan las ecuaciones del shift. Para la primera:

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

como contraemos el índice j quedando libre únicamente el i, definimos

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,

que nos ayudará a no liarnos, ya que ésta última queda como una derivada covariante de un vector donde éste, a su vez, lo calcularemos a parte como la derivada covariante de un tensor dos veces contravariante.

De esta manera, en nuestras coordenadas tenemos:

\Delta \Theta_\beta = \frac{3}{2} \mathcal{D}_i V^i = \frac{3}{2} (\mathcal{D}_{\bar{x}} V^{\bar{x}} + \mathcal{D}_{\bar{y}} V^{\bar{y}} + \mathcal{D}_{\bar{z}} V^{\bar{z}}) =

= \frac{3}{2} (\frac{|\bar{x}^2-1|}{a} \partial_{\bar{x}} V^{\bar{x}}+ \frac{|\bar{y}^2-1|}{b} \partial_{\bar{y}} V^{\bar{y}} + \frac{|\bar{z}^2-1|}{c} \partial_{\bar{z}} V^{\bar{z}}),

donde

V^{\bar{x}} = \mathcal{D}_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{x}}) + \mathcal{D}_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{y}} ) + \mathcal{D}_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{z}} ),

V^{\bar{y}} = \mathcal{D}_{\bar{x}} (\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}} ),

V^{\bar{z}} = \mathcal{D}_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{x}}) + \mathcal{D}_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{y}} ) + \mathcal{D}_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{z}} ),

que desarrollando las covariantes según lo encontrado al principio del post, quedan:

V^{\bar{x}} = \frac{|\bar{x}^2-1|}{a} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{x}}) + \frac{|\bar{y}^2-1|}{b} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{y}} ) + \frac{|\bar{z}^2-1|}{c} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{z}} ),

V^{\bar{y}} = \frac{|\bar{x}^2-1|}{a} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{x}}) + \frac{|\bar{y}^2-1|}{b} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{y} \bar{y}} ) + \frac{|\bar{z}^2-1|}{c} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{y} \bar{z}} ),

V^{\bar{z}} = \frac{|\bar{x}^2-1|}{a} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{x}}) + \frac{|\bar{y}^2-1|}{b} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{y}} ) + \frac{|\bar{z}^2-1|}{c} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{z}} ).

Por tanto, combiando todo, tenemos:

\Delta \Theta_\beta =

= \frac{3|\bar{x}^2-1|}{2a} \partial_{\bar{x}} \big [ \frac{|\bar{x}^2-1|}{a} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{x}}) +

+ \frac{|\bar{y}^2-1|}{b} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{y}} ) +

+ \frac{|\bar{z}^2-1|}{c} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{z}} ) \big ] +

+ \frac{3|\bar{y}^2-1|}{2b} \partial_{\bar{y}} \big [ \frac{|\bar{x}^2-1|}{a} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{x}}) +

+ \frac{|\bar{y}^2-1|}{b} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{y} \bar{y}} ) +

+ \frac{|\bar{z}^2-1|}{c} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{y} \bar{z}} ) \big ] +

+ \frac{3|\bar{z}^2-1|}{2c} \partial_{\bar{z}} \big [ \frac{|\bar{x}^2-1|}{a} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{x}}) +

\frac{|\bar{y}^2-1|}{b} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{y}} ) +

+ \frac{|\bar{z}^2-1|}{c} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{z}} ) \big ],

Para terminar, nos quedan la ecuaciónes:

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

En primer lugar, bajamos el índice de la derivada contravariante:

\mathcal{D}^{\bar{x}} \Theta_\beta = f^{\bar{x} i} \mathcal{D}_i \Theta_\beta = f^{\bar{x} \bar{x}} \mathcal{D}_{\bar{x}} \Theta_\beta + f^{\bar{x} \bar{y}} \mathcal{D}_{\bar{y}} \Theta_\beta + f^{\bar{x} \bar{z}} \mathcal{D}_{\bar{z}} \Theta_\beta = \mathcal{D}_{\bar{x}} \Theta_\beta,

y de la misma manera:

\mathcal{D}^{\bar{y}} = \mathcal{D}_{\bar{y}} y \mathcal{D}^{\bar{z}} = \mathcal{D}_{\bar{z}}.

Así pues, lo que nos queda es:

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

\Delta \beta^{\bar{y}} = 2 \big [ \mathcal{D}_{\bar{x}} (\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}_{\bar{y}} \Theta_\beta,

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

que al sustituir las derivadas covariantes del tensor dos veces contravariante \hat{A}^{ab} y del escalar \Theta_\beta, por su valor calculado al principio del post, quedan:

\Delta \beta^{\bar{x}} = \frac{2|\bar{x}^2-1|}{a} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{x}}) +

+ \frac{2|\bar{y}^2-1|}{b} \partial_{\bar{y}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{y}}) +

+ \frac{2|\bar{z}^2-1|}{c} \partial_{\bar{z}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{z}}) - \frac{|\bar{x}^2-1|}{3a} \partial_{\bar{x}} \Theta_\beta,

\Delta \beta^{\bar{y}} = \frac{2|\bar{x}^2-1|}{a} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{x}}) +

+ \frac{2|\bar{y}^2-1|}{b} \partial_{\bar{y}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{y}}) +

+ \frac{2|\bar{z}^2-1|}{c} \partial_{\bar{z}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{z}}) - \frac{|\bar{y}^2-1|}{3b} \partial_{\bar{y}} \Theta_\beta,

\Delta \beta^{\bar{z}} = \frac{2|\bar{x}^2-1|}{a} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{x}}) +

+ \frac{2|\bar{y}^2-1|}{b} \partial_{\bar{y}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{y}}) +

+ \frac{2|\bar{z}^2-1|}{c} \partial_{\bar{z}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{z}}) - \frac{|\bar{z}^2-1|}{3c} \partial_{\bar{z}} \Theta_\beta.

octubre 2017
L M X J V S D
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