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Christoffel Symbols for Schwarzschild Metric (?)

  1. Jul 24, 2009 #1
    ROUGH DRAFT

    I have a beginner's basic question:

    1. Schwarzschild Metric components

    Let [tex]\epsilon[/tex] = rs / r, where rs is the Schwarzschild Radius. Then, as is is well-known:

    [tex]g_{00} = 1 - \epsilon[/tex]
    [tex]g_{11} = - \left( 1 - \epsilon \right)^{-1}[/tex]
    [tex]g_{22} = - r^{2}[/tex]
    [tex]g_{33} = - r^{2} \; sin^{2}(\theta)[/tex]​

    B/c this Schwarzschild Metric Tensor gij is Diagonal, its Inverse gij is also Diagonal, w/ components equal to "one over" those above.


    2. Christoffel Symbol components

    As is well-known:

    [tex]\Gamma^{i}_{k\ell} = {1 \over 2} g^{im} (g_{mk,\ell} + g_{m\ell,k} - g_{k\ell,m})[/tex]​

    But, since the Schwarzschild Metric Tensor is diagonal, [tex]g^{im} = \delta^{im} \; g^{ii}[/tex]. So:

    [tex]\Gamma^{i}_{k\ell} = {1 \over 2} g^{ii} (g_{ik,\ell} + g_{i\ell,k} - g_{k\ell,i}) + 0[/tex]​

    Thus, in this Schwarzschild Polar Coordinate System, w.h.t.:

    [tex]\Gamma^{0}_{k\ell} = {1 \over 2} g^{00} \[ \left( \begin{array}{cccc}
    0 & g_{00,1} & 0 & 0 \\
    g_{00,1} & 0 & 0 & 0 \\
    0 & 0 & 0 & 0 \\
    0 & 0 & 0 & 0 \end{array} \right)\][/tex]

    [tex]\Gamma^{1}_{k\ell} = {1 \over 2} g^{11} \[ \left( \begin{array}{cccc}
    -g_{00,1} & 0 & 0 & 0 \\
    0 & g_{11,1} & 0 & 0 \\
    0 & 0 & -g_{22,1} & 0 \\
    0 & 0 & 0 & -g_{33,1} \end{array} \right)\][/tex]

    [tex]\Gamma^{2}_{k\ell} = {1 \over 2} g^{22} \[ \left( \begin{array}{cccc}
    0 & 0 & 0 & 0 \\
    0 & 0 & g_{22,1} & 0 \\
    0 & g_{22,1} & 0 & 0 \\
    0 & 0 & 0 & -g_{33,2} \end{array} \right)\][/tex]

    [tex]\Gamma^{3}_{k\ell} = {1 \over 2} g^{33} \[ \left( \begin{array}{cccc}
    0 & 0 & 0 & 0 \\
    0 & 0 & 0 & g_{33,1} \\
    0 & 0 & 0 & g_{33,2} \\
    0 & g_{33,1} & g_{33,2} & 0 \end{array} \right)\][/tex]​

    Or, noting that [tex]\partial \epsilon / \partial r = - \epsilon / r[/tex], w.h.t.:


    [tex]\Gamma^{0}_{k\ell} = {1 \over 2} (1 - \epsilon)^{-1} \[ \left( \begin{array}{cccc}
    0 & \epsilon / r & 0 & 0 \\
    \epsilon / r & 0 & 0 & 0 \\
    0 & 0 & 0 & 0 \\
    0 & 0 & 0 & 0 \end{array} \right)\][/tex]

    [tex]\Gamma^{1}_{k\ell} = -{1 \over 2} \left( 1 - \epsilon \right) \[ \left( \begin{array}{cccc}
    -\epsilon / r & 0 & 0 & 0 \\
    0 & \left( 1 - \epsilon \right)^{-2}(\epsilon / r) & 0 & 0 \\
    0 & 0 & 2 r & 0 \\
    0 & 0 & 0 & 2 r \; sin^{2}(\theta) \end{array} \right)\][/tex]

    [tex]\Gamma^{2}_{k\ell} = -{1 \over 2} r^{-2} \[ \left( \begin{array}{cccc}
    0 & 0 & 0 & 0 \\
    0 & 0 & -2 r & 0 \\
    0 & -2 r & 0 & 0 \\
    0 & 0 & 0 & r^{2} \; sin(2 \theta) \end{array} \right)\][/tex]

    [tex]\Gamma^{3}_{k\ell} = -{1 \over 2} r^{-2} \; sin^{-2}(\theta) \[ \left( \begin{array}{cccc}
    0 & 0 & 0 & 0 \\
    0 & 0 & 0 & -2 r \; sin^{2}(\theta) \\
    0 & 0 & 0 & -r^{2} \; sin(2 \theta) \\
    0 & -2 r \; sin^{2}(\theta) & -r^{2} \; sin(2 \theta) & 0 \end{array} \right)\][/tex]​


    3. Geodesic Equation (?!)

    [tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
    c t \\
    r \\
    \theta \\
    \phi \end{array} \right) + \left( \begin{array}{c}
    (1 - \epsilon)^{-1} {\epsilon \over r} {\partial (ct) \over \partial s} {\partial r \over \partial s} \\
    {1 \over 2} \left( 1 - \epsilon \right) \left( {\epsilon \over r}{\partial (ct) \over \partial s}^{2} - \left( 1 - \epsilon \right)^{-2}{\epsilon \over r}{\partial r \over \partial s}^{2} - 2 r {\partial \theta \over \partial s}^{2} - 2 r \; sin^{2}(\theta) {\partial \phi \over \partial s}^{2} \right) \\
    {2 \over r}{\partial r \over \partial s} {\partial \theta \over \partial s} - {sin(2 \theta) \over 2}{\partial (\phi) \over \partial s}^{2} \\
    {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} + 2 cot(\theta){\partial \theta \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]


    4. Zero-Gravity limit (??)

    If [tex]\epsilon = 0[/tex], w.h.t.:

    [tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
    c t \\
    r \\
    \theta \\
    \phi \end{array} \right) + \left( \begin{array}{c}
    0 \\
    - r {\partial (\theta) \over \partial s}^{2} - r \; sin^{2}(\theta) {\partial (\phi) \over \partial s}^{2} \\
    {2 \over r}{\partial r \over \partial s} {\partial \theta \over \partial s} + {sin(2 \theta) \over 2}{\partial (\phi) \over \partial s}^{2} \\
    {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} + 2 cot(\theta){\partial \theta \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]

    Further restricting [tex]\theta = {\pi \over 2}[/tex], w.h.t.:

    [tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
    c t \\
    r \\
    \theta \\
    \phi \end{array} \right) + \left( \begin{array}{c}
    0 \\
    - r {\partial (\phi) \over \partial s}^{2} \\
    0 \\
    {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]

    Is this the equation of a straight line in Polar Coordinates ?


    5. Weak-Gravity limit (??)

    If [tex]\epsilon << 1[/tex], w.h.t.:

    [tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
    c t \\
    r \\
    \theta \\
    \phi \end{array} \right) + \left( \begin{array}{c}
    {\epsilon \over r} {\partial (ct) \over \partial s} {\partial r \over \partial s} \\
    {1 \over 2} \left( {\epsilon \over r}{\partial (ct) \over \partial s}^{2} + {\epsilon \over r}{\partial (r) \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r {\partial (\theta) \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r \; sin^{2}(\theta) {\partial (\phi) \over \partial s}^{2} \right) \\
    {2 \over r}{\partial r \over \partial s} {\partial \theta \over \partial s} + {sin(2 \theta) \over 2}{\partial (\phi) \over \partial s}^{2} \\
    {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} + 2 cot(\theta){\partial \theta \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]

    Further restricting [tex]\theta = {\pi \over 2}[/tex], w.h.t.:

    [tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
    c t \\
    r \\
    \theta \\
    \phi \end{array} \right) + \left( \begin{array}{c}
    {\epsilon \over r} {\partial (ct) \over \partial s} {\partial r \over \partial s} \\
    {1 \over 2} \left( {\epsilon \over r}{\partial (ct) \over \partial s}^{2} + {\epsilon \over r}{\partial (r) \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r {\partial (\phi) \over \partial s}^{2} \right) \\
    0 \\
    {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]

    Does this reduce to Newton's equations ?
     
    Last edited: Jul 24, 2009
  2. jcsd
  3. Jul 24, 2009 #2
  4. Jul 24, 2009 #3
    Verifying Geodesic Equation by applying Euler-Lagrange Equation to Scwarzschild Metric-derived Lagrangian

    We apply the Euler-Lagrange Equation to the Scwarzschild Metric-derived Lagrangian:

    [tex]L \equiv g_{\mu \nu} {d x^{\mu} \over ds} {d x^{\nu} \over ds} = 1[/tex]​

    Explicitly, w.h.t.:

    [tex]L \equiv (1 - \epsilon) c^{2} {dt \over ds}^{2} - (1 - \epsilon)^{-1} {dr \over ds}^{2} - r^{2} {d \theta \over ds}^{2} - r^{2} sin^{2}(\theta) {d \phi \over ds}^{2}[/tex]​

    Applying the Euler-Lagrange Equation:

    [tex]{\partial L \over \partial x^{\mu}} - {d \over ds}{\partial L \over \partial ({\partial x^{\mu} \over \partial s})} = 0 [/tex]​

    w.h.t.:

    [tex]\left( \begin{array}{c}
    0 \\
    c^{2} {\epsilon \over r}{\partial t \over \partial s}^{2} + (1 - \epsilon)^{-2} {\epsilon \over r}{\partial r \over \partial s}^{2} - 2 r {\partial \theta \over \partial s}^{2} - 2 r sin^{2}(\theta) {\partial \phi \over \partial s}^{2}\\
    - r^{2} sin(2 \theta) {\partial \phi \over \partial s}^{2} \\
    0 \end{array} \right) - \left( \begin{array}{c}
    2 c^{2} {\epsilon \over r} {\partial r \over \partial s}{\partial t \over \partial s} + 2 c^{2} (1 - \epsilon) {\partial^{2} t \over \partial s^{2}} \\
    2 (1 - \epsilon)^{-2} {\epsilon \over r}{\partial r \over \partial s}^{2} - 2 (1 - \epsilon)^{-1}{\partial^{2} r \over \partial s^{2}} \\
    -4 r {\partial r \over \partial s}{\partial \theta \over \partial s} - 2 r^{2}{\partial^{2} \theta \over \partial s^{2}} \\
    -2 r^{2} sin^{2}(\theta){\partial^{2} \phi \over \partial s^{2}} - 2 r^{2} sin(2 \theta) {\partial \theta \over \partial s}{\partial \phi \over \partial s} - 4 r {\partial r \over \partial s}{\partial \phi \over \partial s} \end{array} \right) = 0[/tex]​

    This seems to be substantially similar to the above-derived Geodesic Equation (so far).
     
    Last edited: Jul 24, 2009
  5. Jul 25, 2009 #4
    UPDATE

    4. Zero-Gravity limit (??)

    If [tex]\epsilon = 0[/tex], w.h.t.:

    [tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
    c t \\
    r \\
    \theta \\
    \phi \end{array} \right) + \left( \begin{array}{c}
    0\\
    - r {\partial \theta \over \partial s}^{2} - r \; sin^{2}(\theta) {\partial \phi \over \partial s}^{2} \\
    {2 \over r}{\partial r \over \partial s} {\partial \theta \over \partial s} - {sin(2 \theta) \over 2} {\partial (\phi) \over \partial s}^{2} \\
    {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} + 2 cot(\theta){\partial \theta \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]

    Further restricting [tex]\theta = {\pi \over 2}[/tex], w.h.t.:

    [tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
    c t \\
    r \\
    \theta \\
    \phi \end{array} \right) + \left( \begin{array}{c}
    0 \\
    - r {\partial (\phi) \over \partial s}^{2} \\
    0 \\
    {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]

    Is this the equation of a straight line in Polar Coordinates ?


    5. Weak-Gravity limit (??)

    If [tex]\epsilon << 1[/tex], w.h.t.:

    [tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
    c t \\
    r \\
    \theta \\
    \phi \end{array} \right) + \left( \begin{array}{c}
    {\epsilon \over r} {\partial (ct) \over \partial s} {\partial r \over \partial s} \\
    {1 \over 2} \left( {\epsilon \over r}{\partial (ct) \over \partial s}^{2} - {\epsilon \over r}{\partial r \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r {\partial \theta \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r \; sin^{2}(\theta) {\partial \phi \over \partial s}^{2} \right) \\
    {2 \over r}{\partial r \over \partial s} {\partial \theta \over \partial s} + {sin(2 \theta) \over 2}{\partial \phi \over \partial s}^{2} \\
    {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} + 2 cot(\theta){\partial \theta \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]

    Further restricting [tex]\theta = {\pi \over 2}[/tex], w.h.t.:


    [tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
    c t \\
    r \\
    \theta \\
    \phi \end{array} \right) + \left( \begin{array}{c}
    {\epsilon \over r} {\partial (ct) \over \partial s} {\partial r \over \partial s} \\
    {1 \over 2} \left( {\epsilon \over r}{\partial (ct) \over \partial s}^{2} - {\epsilon \over r}{\partial r \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r {\partial \phi \over \partial s}^{2} \right) \\
    0 \\
    {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]

    Does this reduce to Newton's equations ?
     
  6. Jul 26, 2009 #5

    cristo

    User Avatar
    Staff Emeritus
    Science Advisor

    There doesn't really appear to be a question here, which is probably why you haven't received any replies. If I were you, I would re-check your first calculation, and compare to well known results. (See, for example, http://arxiv.org/abs/0904.4184 for a useful catalogue of spacetimes).
     
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