Double Einstein summation simplification

Question

Suppose that we are given the EFEs for the SEM tensor $$T_{\mu\nu}=\frac{1}{8 \pi}G_{\mu\nu},$$ where the Einstein tensor $G_{\mu\nu}$ is defined to be $$G_{\mu\nu}=R_{\mu\nu}-\frac12Rg_{\mu\nu}.$$ The Ricci tensor is given by $$R_{\mu\nu}=\partial_\lambda\Gamma^{\lambda}_{\mu\nu}-\partial_\mu\Gamma^{\lambda}_{\nu\lambda}+\Gamma^{\lambda}_{\lambda\sigma}\Gamma^{\sigma}_{\mu\nu}-\Gamma^{\lambda}_{\mu\sigma}\Gamma^{\sigma}_{\nu\lambda}.$$ Furthermore, suppose that we already know a metric which is given by $$ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}=-e^{2 \phi}dt^2+\left(1-\frac{b}{r}\right)^{-1}dr^2+r^2d \Omega^2,$$ where $$d \Omega^2=d\theta^2+\sin^2(\theta).$$ Given this metric we can calculate all the Christoffel symbols for it by the expression $$\Gamma_{\mu\nu}^{\lambda}=\frac12g^{\lambda\sigma}\left(\partial_{\nu}g_{\sigma\mu}+\partial_{\mu}g_{\sigma\nu}-\partial_{\sigma}g_{\mu\nu}\right).$$ This leads to the discovery that the only non-zero Christoffel symbols for the metric are $$\Gamma_{rr}^{r},\Gamma_{r \theta}^{\theta},\Gamma_{r \phi}^{\phi},\Gamma_{\theta\theta}^{r},\Gamma_{\theta\phi}^{\phi},\Gamma_{\phi\phi}^{r}\Gamma_{\phi\phi}^{\theta}.$$ Given all the non-zero christoffel symbols we can contract and expand the expression $$R_{\mu\nu}=\partial_\lambda\Gamma^{\lambda}_{\mu\nu}-\partial_\mu\Gamma^{\lambda}_{\nu\lambda}+\Gamma^{\lambda}_{\lambda\sigma}\Gamma^{\sigma}_{\mu\nu}-\Gamma^{\lambda}_{\mu\sigma}\Gamma^{\sigma}_{\nu\lambda}.$$ The first 2 contractions are easy to write out although the second 2 contractions $\Gamma^{\lambda}_{\lambda\sigma}\Gamma^{\sigma}_{\mu\nu}$ and $\Gamma^{\lambda}_{\mu\sigma}\Gamma^{\sigma}_{\nu\lambda}$ yield 16 summed terms, most of which will equal $0$ and vanish; however, I am not aware of a way to distinguish which of the 16 terms will vanish without having to write out the massive expression. Is there a general way to find out which combinations of $\sigma$ and $\lambda$ yield non-zero products of Christoffel symbols or do I explicitly have to write out all of the 16 terms and cancel one by one?

Answer 1

There are some other approaches to computing the curvatures that do not involve dealing in detail with the Christoffel symbols. For example, Wald's General Relativity presents on Sec. 3.4b a method employing an orthonormal basis (tetrads) that sometimes can simplify computations considerably. Later on, he exemplifies this method on Sec. 6.1 by finding the Ricci tensor for a spherically symmetric metric (just like yours) with arguably much less computation than usually necessary when doing the calculations with Christoffel symbols. I believe the Newman–Penrose formalism can also be useful on some situations, but I'm not used to these techniques.

Apart from these examples, which as far as I know do not employ Christoffel symbols at all, I do not know of any ways of avoiding the computations you mentioned. Usually it comes down to a matter of patience (or doing the computations with Mathematica, that works as well).