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Calculate Christoffel Symbols

  1. Jun 13, 2013 #1
    Problem with Metric Ansatz

    Metric ansatz:
    [itex]
    ds^{2} = e^{\tilde{A}(\tilde{\tau})} d\tilde{t} - d\tilde{r} - e^{\tilde{C}(\tilde{\tau})} dΩ
    [/itex]

    where: [itex]d\tilde{r} = e^{\frac{B}{2}} dr[/itex]

    Metric tensor:

    [itex]g_{μv}=diag (e^{\tilde{A}(\tilde{\tau})}, -1,- e^{\tilde{C}(\tilde{\tau})}, - e^{\tilde{C}(\tilde{\tau})} sin^2 θ ) [/itex] correct?

    Christoffel Symbols:

    [itex]\Gamma^{μ}_{vρ}=\frac{1}{2}g^{μλ}(∂_{v}g_{ρλ}+∂_{ρ}g_{μλ}-∂_{λ}g_{vρ})[/itex]
    [itex]\Gamma^{0}_{00}=\frac{1}{2}g^{00}(∂_{0}g_{00}+∂_{0}g_{00}-∂_{0}g_{00})[/itex]
    [itex]\Gamma^{0}_{00}=\frac{1}{2}g^{00}(∂_{0}g_{00})[/itex]
    [itex]\Gamma^{0}_{00}=\frac{1}{2}e^{-\tilde{A}(\tilde{\tau})}(\frac{∂}{∂\tilde{t}}e^{\tilde{A}(\tilde{\tau})})[/itex]
    [itex]\Gamma^{0}_{00}= 0[/itex] correct?
    [itex]\vdots[/itex]
    etc

    And:

    [itex]\frac{∂}{∂\tilde{r}}e^{\tilde{A}(\tilde{\tau})}= 0 [/itex] correct?

    What is the difference between:

    [itex]\tilde{t}[/itex] and [itex]\tilde{\tau}[/itex] ?
    [itex]{r}[/itex] and [itex]\tilde{r}[/itex] ?
     
    Last edited: Jun 13, 2013
  2. jcsd
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