I want to expand Einstein-Hilbert action for the metric

$$ g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} $$

up to quadratic order in $h_{\mu \nu}$. For this purpose I need to calculate the Ricci tensor at some stage. There is no problem in linear terms of the metric perturbation but there happens to be a problem when I intend to calculate the quadratic terms using

$$ R_{\mu \nu}^{(2)} = \partial_{\alpha} \Gamma^{\alpha} {} _{\mu \nu}^{(2)} - \partial_{\nu} \Gamma^{\alpha} {} _{\mu \alpha}^{(2)} + \Gamma^{\alpha} {} _{\beta \alpha}^{(1)} \Gamma^{\beta} {} _{\mu \nu}^{(1)} - \Gamma^{\alpha} {} _{\beta \nu}^{(1)} \Gamma^{\beta} {} _{\mu \alpha}^{(1)} $$

When I use

$$ \Gamma^{\alpha} {} _{\mu \nu}^{(1)} = \frac{1}{2} \left( \partial_{\mu} h^{\alpha} {} _{\nu} + \partial_{\nu} h^{\alpha} {} _{\mu} - \partial^{\alpha} h_{\mu \nu} \right) $$ $$ \Gamma^{\alpha} {} _{\mu \nu}^{(2)} = -\frac{1}{2}h^{\alpha \beta} \left( \partial_{\mu} h_{\beta \nu} + \partial_{\nu} h_{\mu \beta} - \partial_{\beta} h_{\mu \nu}\right) $$

in the above expression for $R_{\mu \nu}^{(2)}$ I suppose to find

$$ R_{\mu \nu}^{(2)} = \frac{1}{2}\Bigg[ \frac{1}{2}\partial_{\mu}h_{\alpha \beta}\partial_{\nu}h^{\alpha \beta} + \partial_{\beta}h_{\nu \alpha}\left( \partial^{\beta}h^{\alpha}{}_{\mu} - \partial^{\alpha}h^{\beta}{}_{\mu} \right) + h_{\alpha \beta} \left( \partial_{\mu}\partial_{\nu}h^{\alpha \beta} + \partial^{\alpha}\partial^{\beta}h_{\mu \nu} - \partial^{\beta}\partial_{\nu}h^{\alpha}{}_{\mu} - \partial^{\beta}\partial_{\mu}h^{\alpha}{}_{\nu} \right) - \left( \partial_{\alpha}h^{\alpha \beta} - \frac{1}{2}\partial^{\beta}h\right)\left( \partial_{\mu} h_{\nu \beta} + \partial_{\nu} h_{\mu \beta} - \partial_{\beta} h_{\mu \nu}\right) \Bigg] $$

but I do not. I tried it over and over again but somehow I cannot get the correct result. It seems I am missing something but I don't know what. I will be glad if someone can help.

This post imported from StackExchange Physics at 2015-02-13 11:37 (UTC), posted by SE-user sahin