Let's have the metric for a 3-sphere: $$ dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right). $$ I tried to calculate Riemann or Ricci tensor's components, but I got problems with it.
In the beginning, I got an expressions for Christoffel's symbols: $$ \Gamma^{i}_{ii} = \frac{1}{2}g^{ii}\partial_{i}g_{ii} = 0, $$
$$ \Gamma^{i}_{ji} = \frac{1}{2}g^{ii}\partial_{j}g_{ii}, $$
$$ \Gamma^{k}_{ll} = -\frac{1}{2}g^{kk}\partial_{k}g_{ll}, $$
$$ \Gamma^{k}_{lj} = \Gamma^{k}_{lk}\delta^{k}_{j} + \Gamma^{k}_{jk}\delta^{k}_{l} + \Gamma^{k}_{jj}\delta^{j}_{l} = 0. $$
The Ricci curvature must be $$ R_{lj}=\frac{2}{R^{2}}g_{lj}. $$ But when I use definition of Ricci tensor,
$$ R_{lj}^{(3)} = \partial_{k}\Gamma^{k}_{lj} - \partial_{l}\Gamma^{\lambda}_{j \lambda} + \Gamma^{k}_{j l}\Gamma^{\sigma}_{k \sigma} - \Gamma^{k }_{l \sigma}\Gamma^{\sigma}_{jk}, $$ I can't associate an expression (if I didn't make the mistakes)
$$ R_{lj}^{(3)} = \partial_{j}\Gamma^{j}_{lj} + \partial_{l}\Gamma^{l}_{jl} + \partial_{k}\Gamma^{k}_{ll}\delta^{l}_{j} - \partial_{l}\Gamma^{k}_{jk} - \Gamma^{k}_{jk}\Gamma^{j}_{lj} + \Gamma^{k}_{lk}\Gamma^{l}_{jl} + \Gamma^{\sigma}_{k \sigma}\Gamma^{k}_{ll}\delta^{l}_{j} - \Gamma^{k}_{jk}\Gamma^{k}_{lk} - \Gamma^{l}_{jl}\Gamma^{j}_{lj} - \Gamma^{l}_{kl}\Gamma^{k}_{ll}\delta^{l}_{j} - \Gamma^{j}_{ll}\Gamma^{l}_{jj} - \Gamma^{k}_{ll}\Gamma^{l}_{kl} = $$
$$ = \partial_{j}\Gamma^{j}_{lj} + \partial_{l}\Gamma^{l}_{jl} + \partial_{k}\Gamma^{k}_{ll}\delta^{l}_{j} - \partial_{l}\Gamma^{k}_{jk} - \Gamma^{k}_{jk}\Gamma^{j}_{lj} + \Gamma^{k}_{lk}\Gamma^{l}_{jl} + \Gamma^{\sigma}_{k \sigma}\Gamma^{k}_{ll}\delta^{l}_{j} - \Gamma^{k}_{jk}\Gamma^{k}_{lk} - \Gamma^{l}_{jl}\Gamma^{j}_{lj} - 2\Gamma^{l}_{kl}\Gamma^{k}_{ll}\delta^{l}_{j} - \Gamma^{j}_{ll}\Gamma^{l}_{jj}, $$ where there is a summation only on $k, \sigma$, with an expression for the metric tensor.
Maybe, there are some hints, which can help?
No comments:
Post a Comment