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Einstein Equations of this metric

  1. Apr 12, 2015 #1
    1. The problem statement, all variables and given/known data

    (a) Find the christoffel symbols
    (b) Find the einstein equations
    (c) Find A and B
    (d) Comment on this metric

    2014_B5_Q2.png

    2. Relevant 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]

    3. The attempt at a solution

    Part(a)

    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..
     
    Last edited: Apr 12, 2015
  2. jcsd
  3. Apr 16, 2015 #2
  4. Apr 17, 2015 #3
    bumpp
     
  5. Apr 18, 2015 #4
  6. Apr 19, 2015 #5
    bumpp
     
  7. Apr 20, 2015 #6
  8. Apr 22, 2015 #7
  9. Apr 23, 2015 #8
  10. Apr 24, 2015 #9
    bumpp on part (b)/(c)
     
  11. Apr 25, 2015 #10
    Would appreciate help with my "ricci-nightmare"
     
  12. Apr 26, 2015 #11
    Anyone managed to get a different result for the ricci tensors yet?
     
  13. Apr 30, 2015 #12
    anyone else had a go with the ricci tensors?
     
  14. May 4, 2015 #13
    tried again, still didn't get the required ricci tensors.
     
  15. May 4, 2015 #14

    thierrykauf

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    Here is what I find for one term for Ricci
     
  16. May 7, 2015 #15
    I think the term is not appearing, do you mind posting it again?
     
  17. May 10, 2015 #16
    bumpp ricci
     
  18. May 10, 2015 #17

    thierrykauf

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    Gold Member

    Hold on. Here is what I find
     
  19. May 10, 2015 #18

    thierrykauf

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    Here is what I find [tex]R_{00}=\Gamma^x_{x0}\Gamma^x_{x0} + \Gamma^x_{x0}\Gamma^y_{y0} + \Gamma^z_{z0}\Gamma^y_{y0}[/tex]
     
  20. May 14, 2015 #19
    Thanks alot for replying. I'll give it a go later today and post my updated work.
     
  21. May 14, 2015 #20

    thierrykauf

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    Gold Member

    Please do. And let's see what you have.
     
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