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Christoffel Symbol of the Second Kind

The expression for the covariant derivative of the metric tensor can be solved for the Affine Connections. To do this, takes some manipulation of indices. Start with the expression that we have already proved:

$\displaystyle g_{\alpha \beta,\sigma}=g_{\rho \beta}\Gamma_{\alpha \sigma}^{\rho} +g_{\alpha \rho} \Gamma_{\beta \sigma}^{\rho}$ (21)

Now write this down again but make the replacements

$\displaystyle \alpha \rightarrow \beta \rightarrow \sigma \rightarrow \alpha.$ (22)

That is, make a cyclic permutation of indices. The result is:

$\displaystyle g_{\beta \sigma,\alpha}=g_{\rho \sigma}\Gamma_{\beta \alpha}^{\rho} +g_{\beta \rho} \Gamma_{\sigma \alpha}^{\rho}$ (23)

Do it once more, but also multiply by -1:

$\displaystyle -g_{\sigma \alpha, \beta}=-g_{\rho \alpha}\Gamma_{\sigma \beta}^{\rho} -g_{\sigma \rho} \Gamma_{\alphai \beta}^{\rho}$ (24)

Now add corresponding sides of the above three equations together, taking account of the fact that the metric tensor is symmetric and the Affine connections are symmetric. Four of the terms on the right sides cancel out, and the remaining ones add up. The result is:

$\displaystyle 2\Gamma_{\sigma \alpha}^{\rho}g_{\beta \rho} = g_{\alpha \beta , \sigma}+ g_{\beta \sigma , \alpha}- g_{\sigma \alpha , \beta}.$ (25)

Lastly, multiply the result by $ g^{\beta \gamma}$ and sum over $ \beta$. The result is the expression for the affine connection in terms of what is called the Christoffel Symbol of the second kind:

$\displaystyle \Gamma_{\sigma \alpha}^{\gamma} = {1 \over 2} g^{\gamma \beta}\le...
... \sigma, \alpha}+ g_{\beta \alpha , \sigma}- g_{\sigma \alpha , \beta} \right].$ (26)

This clearly shows that if all the gradients of the metric tensor are zero then all of the Christoffel symbols of the second kind are zero, and vice versa. In sum, The Principle of Equivalence is incorporated into the formalism by the simple requirement that:

$\displaystyle \Gamma_{\alpha \beta}^{\mu}=\Gamma_{\beta \alpha}^{\mu}.$ (27)


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Next: About this document ... Up: Second Lecture on General Previous: The Equivalence Principle
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2002-12-02