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Homework Statement
[/B]
(a) Find the christoffel symbols
(b) Find the einstein equations
(c) Find A and B
(d) Comment on this metric
Homework Equations
[tex]\Gamma_{\alpha\beta}^\mu \frac{1}{2} g^{\mu v} \left( \partial_\alpha g_{\beta v} + \partial_\beta g_{\alpha v} - \partial_\mu g_{\alpha \beta} \right) [/tex]
[tex]R_{v \beta} = \partial_\mu \Gamma_{\beta v}^\mu - \partial_\beta \Gamma_{\mu v}^\mu + \Gamma_{\mu \epsilon}^\mu \Gamma_{v \beta}^\epsilon - \Gamma_{\epsilon \beta}^\mu \Gamma_{v \mu}^\epsilon [/tex]
The Attempt at a Solution
Part(a)[/B]
After some math, I found the christoffel symbols to be:
##\Gamma_{11}^0 = \frac{A A^{'}}{c^2}##
##\Gamma_{22}^0 = \frac{B B^{'}}{c^2}##
##\Gamma_{33}^0 = \frac{B B^{'}}{c^2}##
##\Gamma_{01}^1 = \frac{A^{'}}{A}##
##\Gamma_{02}^2 = \frac{B^{'}}{B}##
##\Gamma_{03}^3 = \frac{B^{'}}{B}##
Part (b)
Now brace yourselves for the ricci tensors...
[tex]R_{00} = -\partial_0 \left( \Gamma_{01}^1 + \Gamma_{02}^2 + \Gamma_{03}^3 \right) - \Gamma_{10}^1 \Gamma_{01}^1 - 2\Gamma_{20}^2 \Gamma_{02}^2 [/tex]
[tex]R_{00} = -\frac{A^{''}}{A} - 2 \frac{B^{''}}{B}[/tex]
By symmetry, ##R_{01} = R_{02} = R_{03} = R_{12} = R_{13} = R_{23} = 0##.
Now to find the ##11## component:
[tex]R_{11} = \partial_0 \Gamma_{11}^0 + \Gamma_{11}^0 \left( \Gamma_{10}^1 + \Gamma_{20}^2 + \Gamma_{30}^3 \right) - \Gamma_{11}^0 \Gamma_{10}^1 - \Gamma_{01}^1 \Gamma_{11}^0 [/tex]
[tex] = \partial_0 \Gamma_{11}^0 + 2 \Gamma_{11}^0 \Gamma_{20}^2 - \Gamma_{11}^0 \Gamma_{10}^1 [/tex]
[tex] R_{11} = \frac{A A^{''}}{c^2} + 2 \left( \frac{A}{B} \right) \frac{A^{'} B^{'}}{c^2} [/tex]
By symmetry, to find ##22## and ##33## components, we swap ##A## with ##B##:
[tex]R_{22} = R_{33} = \frac{B B^{''}}{c^2} + 2 \left( \frac{B}{A} \right) \frac{A^{'} B^{'}}{c^2}[/tex]The einstein field equations are given by:
[tex]G^{\alpha \beta} = \frac{8 \pi G}{c^4} T^{\alpha \beta} - \Lambda g^{\alpha \beta} [/tex]
Thus, the simultaneous equations we seek are:
[tex] G^{00} = \frac{8 \pi G}{c^4} T^{00} [/tex]
For ##\mu, v \neq 0## we have
[tex] R_{\mu v} = 0[/tex]
So we simply equate ##R_11 = 0##, ##R_22 = R_{33} = 0##.However, the equations don't match..
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