Usuário(a):Jcband/cosmologia

$ds^{2}=-dt^{2}+a^{2}(t)[d\chi ^{2}+sen^{2}(\chi )(d\theta ^{2}+sen^{2}(\theta )d\phi ^{2})]$

Variáveis: $X^{0}=t;X^{1}=\chi ;X^{2}=\theta ;X^{3}=\phi$

Lista dos elementos não nulos do tensor métrico:

$g_{00}=-1$; $g_{11}=a^{2}(t)$; $g_{22}=a^{2}(t)sen^{2}(\chi )$; $g_{33}=a^{2}(t)sen^{2}(\chi )sen^{2}(\theta )$

Fórmula geral dos símbolos de Christoffel não nulos para eese caso, onde os termos fora da diagonal do tensor métrico são nulos:

$\Gamma _{bc}^{a}={\frac {1}{2}}g^{aa}({\frac {\partial g^{ab}}{\partial X^{c}}}+{\frac {\partial g^{ac}}{\partial X^{b}}}-{\frac {\partial g^{bc}}{\partial X^{a}}})$;

Símbolos de Christoffel não nulos, considerando que só as derivadas em relação ao tempo, e os termos da diagonal da métrica são diferentes de zero:

$\Gamma _{11}^{0}={\frac {1}{2}}g^{00}({\frac {\partial g^{01}}{\partial X^{1}}}+{\frac {\partial g^{01}}{\partial X^{1}}}-{\frac {\partial g^{11}}{\partial X^{0}}})={\frac {1}{2}}(-1)(-2aa')=aa'$

$\Gamma _{22}^{0}={\frac {1}{2}}g^{00}({\frac {\partial g^{02}}{\partial X^{2}}}+{\frac {\partial g^{02}}{\partial X^{2}}}-{\frac {\partial g^{22}}{\partial X^{0}}})={\frac {1}{2}}(-1)(-2aa'sen^{2}(\chi ))=aa'sen^{2}(\chi )$

$\Gamma _{33}^{0}={\frac {1}{2}}g^{00}({\frac {\partial g^{03}}{\partial X^{3}}}+{\frac {\partial g^{03}}{\partial X^{3}}}-{\frac {\partial g^{33}}{\partial X^{0}}})={\frac {1}{2}}(-1)(-2aa'sen^{2}(\chi )sen^{2}(\theta ))=aa'sen^{2}(\chi )sen^{2}(\theta )$

$\Gamma _{10}^{1}={\frac {1}{2}}g^{11}({\frac {\partial g^{11}}{\partial X^{0}}}+{\frac {\partial g^{10}}{\partial X^{1}}}-{\frac {\partial g^{10}}{\partial X^{1}}})={\frac {1}{2}}({\frac {1}{a^{2}}})(2aa')={\frac {a'}{a}}$

$\Gamma _{01}^{1}=\Gamma _{10}^{1}={\frac {a'}{a}}$

$\Gamma _{20}^{2}={\frac {1}{2}}g^{22}({\frac {\partial g^{22}}{\partial X^{0}}}+{\frac {\partial g^{20}}{\partial X^{2}}}-{\frac {\partial g^{20}}{\partial X^{2}}})={\frac {1}{2}}({\frac {1}{a^{2}sen^{2}(\chi )}})(2aa'sen^{2}(\chi ))={\frac {a'}{a}}$

$\Gamma _{02}^{2}=\Gamma _{20}^{2}={\frac {a'}{a}}$

$\Gamma _{30}^{3}={\frac {1}{2}}g^{33}({\frac {\partial g^{33}}{\partial X^{0}}}+{\frac {\partial g^{30}}{\partial X^{3}}}-{\frac {\partial g^{30}}{\partial X^{3}}})={\frac {1}{2}}({\frac {1}{a^{2}sen^{2}(\chi )sen^{2}(\theta )}})(2aa'sen^{2}(\chi )sen^{2}(\theta ))={\frac {a'}{a}}$

$\Gamma _{03}^{3}=\Gamma _{30}^{3}={\frac {a'}{a}}$

Cálculo do tensor de Ricci, Fórmula geral: $R_{ij}=\sum _{a=0}^{3}{\frac {\partial \Gamma _{ij}^{a}}{\partial x^{a}}}-\sum _{a=0}^{3}{\frac {\partial \Gamma _{ai}^{a}}{\partial x^{j}}}+\sum _{a=0}^{3}\sum _{b=0}^{3}\left(\Gamma _{ab}^{a}\Gamma _{ij}^{b}-\Gamma _{ib}^{a}\Gamma _{aj}^{b}\right)$

$R_{00}=\sum _{a=0}^{3}{\frac {\partial \Gamma _{00}^{a}}{\partial x^{a}}}-\sum _{a=0}^{3}{\frac {\partial \Gamma _{a0}^{a}}{\partial x^{0}}}+\sum _{a=0}^{3}\sum _{b=0}^{3}\left(\Gamma _{ab}^{a}\Gamma _{00}^{b}-\Gamma _{0b}^{a}\Gamma _{a0}^{b}\right)$

$R_{00}=-{\frac {\partial \Gamma _{10}^{1}}{\partial x^{0}}}-{\frac {\partial \Gamma _{20}^{2}}{\partial x^{0}}}-{\frac {\partial \Gamma _{30}^{3}}{\partial x^{0}}}-\left(\Gamma _{01}^{1}\right)^{2}-\left(\Gamma _{02}^{2}\right)^{2}-\left(\Gamma _{03}^{3}\right)^{2}$

$R_{00}=-3{\frac {a''}{a}}$

$R_{11}=\sum _{a=0}^{3}{\frac {\partial \Gamma _{11}^{a}}{\partial x^{a}}}-\sum _{a=0}^{3}{\frac {\partial \Gamma _{a1}^{a}}{\partial x^{1}}}+\sum _{a=0}^{3}\sum _{b=0}^{3}\left(\Gamma _{ab}^{a}\Gamma _{11}^{b}-\Gamma _{1b}^{a}\Gamma _{a1}^{b}\right)$

$R_{11}={\frac {\partial \Gamma _{11}^{0}}{\partial x^{0}}}-\Gamma _{11}^{0}\Gamma _{01}^{1}=a'^{2}+aa''-(aa')({\frac {a'}{a}})=aa''$

$R_{22}={\frac {\partial \Gamma _{22}^{0}}{\partial x^{0}}}-\Gamma _{22}^{0}\Gamma _{02}^{2}=(a'^{2}+aa'')sen^{2}(\chi )-(aa'sen^{2}(\chi )({\frac {a'}{a}}))=aa''sen^{2}(\chi )$

$R_{33}={\frac {\partial \Gamma _{33}^{0}}{\partial x^{0}}}-\Gamma _{33}^{0}\Gamma _{03}^{3}=(a'^{2}+aa'')sen^{2}(\chi )sen^{2}(\theta )-(aa'sen^{2}(\chi )sen^{2}(\theta )({\frac {a'}{a}}))=aa''sen^{2}(\chi )sen^{2}(\theta )$