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

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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. Homework Statement


Hi, my questions is , how to derive eqn (4.64)
[tex]
\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}) ]
[/tex]

in Sean Carroll's spacetime and geometry, section 4.3

Homework Equations





The Attempt at a Solution



For eqn(4.64), if I use the connection equation (3.27)
[tex]
\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} )
[/tex]

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

[tex]
\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} )
[/tex]

[tex]
= \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} )
[/tex]

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

Thank you very much!!
 
Last edited:

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