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Two questions in Sean Carroll's spacetime and geometry, 4.3

  1. Jul 31, 2010 #1
    A question in Sean Carroll's spacetime and geometry, 4.3

    (I have solved and removed the first question posted before, only the second left)

    1. The problem statement, all variables and given/known data

    Hi, my questions is , how to derive eqn (4.64)
    \delta \Gamma_{\mu \nu}^{\sigma} = - \frac{1}{2} [ g_{\lambda \mu} \nabla_{\nu} ( \delta g^{\lambda \sigma} ) + g_{\lambda \nu} \nabla_{\mu} ( \delta g^{\lambda \sigma}) - g_{\mu \alpha} g_{\nu \beta} \nabla^{\sigma} ( \delta g^{\alpha \beta}) ]

    in Sean Carroll's spacetime and geometry, section 4.3

    2. Relevant equations

    3. The attempt at a solution

    For eqn(4.64), if I use the connection equation (3.27)
    \Gamma_{\mu \nu}^{\sigma} = \frac{1}{2} g^{\sigma \rho} ( \partial_{\mu} g_{\nu \rho } + \partial_{\nu} g_{\rho \mu} - \partial_{\rho} g_{\mu \nu} )

    Try to convert the partial derivative into the covariant derivative as eqn (3.17)

    \Gamma_{\mu \nu}^{\sigma} &=& \frac{1}{2} g^{\sigma \rho} ( \nabla_{\mu} g_{\nu \rho } + \Gamma_{\mu \nu}^{\lambda} g_{\lambda \rho} + \Gamma_{\mu \rho}^{\lambda} g_{\lambda \nu} + \nabla_{\nu} g_{\rho \mu} + \Gamma_{\nu \rho}^{\lambda} g_{\lambda \mu} + \Gamma_{\nu \mu}^{\lambda} g_{\lambda \rho} - \nabla_{\rho} g_{\mu \nu} - \Gamma_{\rho \mu}^{\lambda} g_{\lambda \nu} - \Gamma_{\rho \nu}^{\lambda} g_{\lambda \rho} )

    = \frac{1}{2} g^{\sigma \rho} ( \nabla_{\mu} g_{\nu \rho } + \Gamma_{\mu \nu}^{\lambda} g_{\lambda \rho} + \nabla_{\nu} g_{\rho \mu} + \Gamma_{\nu \mu}^{\lambda} g_{\lambda \rho} - \nabla_{\rho} g_{\mu \nu} )

    How to proceed to the right-hand-side of Sean Carroll's (4.64)?

    Thank you very much!!
    Last edited: Jul 31, 2010
  2. jcsd
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