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

[Comanche]

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)
$$\delta \Gamma_{\mu \nu}^{\sigma} = - \frac{1}{2} [ g_{\lambda \mu} \nabla_{\nu} ( \delta g^{\lambda \sigma} ) + g_{\lambda \nu} \nabla_{\mu} ( \delta g^{\lambda \sigma}) - g_{\mu \alpha} g_{\nu \beta} \nabla^{\sigma} ( \delta g^{\alpha \beta}) ]$$

in Sean Carroll's spacetime and geometry, section 4.3

Homework Equations

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!

 

[mark726]

Thank you for your question. In order to derive equation (4.64), we can start by using the connection equation (3.27) as you have done. However, we also need to use the definition of the covariant derivative, which is given by:

\nabla_{\mu} T^{\sigma} = \partial_{\mu} T^{\sigma} + \Gamma_{\mu \nu}^{\sigma} T^{\nu}

where T^{\sigma} is any tensor. Now, let us apply this to the term \delta g^{\lambda \sigma} in your expression:

\nabla_{\nu} ( \delta g^{\lambda \sigma}) = \partial_{\nu} ( \delta g^{\lambda \sigma}) + \Gamma_{\nu \alpha}^{\sigma} ( \delta g^{\lambda \alpha})

Similarly, we can apply the definition to the other terms in your expression. After some algebraic manipulation, we can obtain the right-hand-side of equation (4.64). I hope this helps. Let me know if you need further clarification.