Kaluza-Klein Christoffel Symbols

Question

I have a question regarding the connection coefficients as they pertain to the following paper: http://www.weylmann.com/kaluza.pdf . When I try to calculate the 4D Christoffel symbols from the 4D part of the 5D metric $\tilde{g}_{\mu \nu}=g_{\mu \nu}+kA_\mu A_\nu$ I get an extra term that shouldn't be there. I get:

$$\tilde{\Gamma}^\lambda_{\mu \nu }=\frac{1}{2}\tilde{g}^{\lambda \sigma}(\partial_\mu \tilde{g}_{\nu \sigma}+\partial_\nu \tilde{g}_{\mu \sigma}-\partial_\sigma \tilde{g}_{\mu \nu})$$

$$=\Gamma^\lambda_{\mu \nu }+\frac{k}{2} g^{\lambda \sigma}(\partial_\mu (A_\nu A_\sigma)+\partial_\nu (A_\mu A_\sigma)-\partial_\sigma (A_\mu A_\nu))$$

$$=\Gamma^\lambda_{\mu \nu }+\frac{k}{2} g^{\lambda \sigma}(A_\mu (\partial_\nu A_\sigma -\partial_\sigma A_\nu)+A_\nu (\partial_\mu A_\sigma -\partial_\sigma A_\mu)+A_\sigma (\partial_\mu A_\nu +\partial_\nu A_\mu ))$$

$$=\Gamma^\lambda_{\mu \nu }+\frac{k}{2} g^{\lambda \sigma}(A_\mu F_{\nu \sigma }+A_\nu F_{\mu \sigma}+A_\sigma (\partial_\mu A_\nu +\partial_\nu A_\mu ))$$

According to the paper the $A_\sigma$ term shouldn't be there, but I can't figure out how to make it go away. Anybody have any ideas?

Answer 1

You missed a term in expanding the upper-indexed metric. The full version is below: \begin{align} \tilde{\Gamma}^\lambda_{\mu\nu} & = \frac{1}{2} \tilde{g}^{\lambda X} \left(\partial_\mu \tilde{g}_{\nu X} + \partial_\nu \tilde{g}_{\mu X} - \partial_X \tilde{g}_{\mu\nu}\right) \\ & =\frac{1}{2} \tilde{g}^{\lambda\sigma} \left(\partial_\mu \tilde{g}_{\nu\sigma} + \partial_\nu \tilde{g}_{\mu\sigma} - \partial_\sigma \tilde{g}_{\mu\nu}\right) + \frac{1}{2} \tilde{g}^{\lambda5} \left(\partial_\mu \tilde{g}_{\nu5} + \partial_\nu \tilde{g}_{\nu5} - \partial_5 \tilde{g}_{\mu\nu}\right) \\ & = \frac{1}{2} g^{\lambda\sigma} \left(\partial_\mu \left(g_{\nu\sigma} + k A_\nu A_\sigma\right) + \partial_\nu \left(g_{\mu\sigma} + k A_\mu A_\sigma\right) - \partial_\sigma \left(g_{\mu\nu} + k A_\mu A_\nu\right)\right) \\ & \quad \qquad + \frac{1}{2} \left(-A^\lambda\right) \left(\partial_\mu \left(k A_\nu\right) + \partial_\nu \left(k A_\mu\right) - \partial_5 \left(g_{\mu\nu} + k A_\mu A_\nu\right)\right) \\ & = \frac{1}{2} g^{\lambda\sigma} \left(\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu}\right) + \frac{k}{2} g^{\lambda\sigma} \left(\partial_\mu \left(A_\nu A_\sigma\right) \partial_\nu \left(A_\mu A_\sigma\right) - \partial_\sigma \left(A_\mu A_\nu\right)\right) \\ & \quad \qquad - \frac{k}{2} A^\lambda \left(\partial_\mu A_\nu + \partial_\nu A_\mu\right) \\ & = \Gamma^\lambda_{\mu\nu} + \frac{k}{2} g^{\lambda\sigma} \left(A_\mu \left(\partial_\nu A_\sigma - \partial_\sigma A_\nu\right) + A_\nu \left(\partial_\mu A_\sigma - \partial_\sigma A_\mu\right) + A_\sigma \left(\partial_\mu A_\nu + \partial_\nu A_\mu\right)\right) \\ & \quad \qquad - \frac{k}{2} g^{\lambda\sigma} A_\sigma \left(\partial_\mu A_\nu + \partial_\nu A_\mu\right) \\ & = \Gamma^\lambda_{\mu\nu} + \frac{k}{2} \left(A_\mu F_{\nu\sigma} + A_\nu F_{\mu\sigma}\right). \end{align} This is because we have \begin{cases} \tilde{g}_{\mu\nu} = g_{\mu\nu} + k A_\mu A_\nu \\ \tilde{g}_{\mu5} = k A_\mu \\ \tilde{g}_{55} = k \end{cases} and also \begin{cases} \tilde{g}^{\mu\nu} = g^{\mu\nu} \\ \tilde{g}^{\mu5} = -A_\mu \\ \tilde{g}^{55} = \frac{1}{k} + A_\mu A^\mu. \end{cases}