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I was reading Einsteins 1916 original paper on GR, the "The foundation of the general theory of relativity". There are some derivation that he did but I didn't quite understand. It would be nice if someone can give me some direction or some guidance on it.

Here is the link to the paper. http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_GRelativity_1916.pdf

My question originated from chapter #18, from the derivation of equation (57). He says that we can get (57) by by multiplying partial derivative ∂g^μν/∂x_σ with equation (53), but I tried and didn't quite get it. I know that this (57) amounts to showing the left side of the multiplied (53) to be zero (or this is wrong?) but I can't get it. One idea is that it can be factorisation by product rule of derivation. By multiplying (53) by a partial derivative one get two first degree derivative in one term, and I cannot figure out how to factorise them into a degree one derivative by product rule.

Cheers!

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closed as off-topic by Kyle Kanos, ZeroTheHero, peterh, David Hammen, Jon Custer May 30 '17 at 0:56

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    $\begingroup$ Might be useful to include the actual equations, rather than forcing us to download a paper, read it and then assess your thoughts. $\endgroup$ – Kyle Kanos May 28 '17 at 22:06
  • $\begingroup$ The LHS of (53) times $g^{\mu\nu}_{\sigma}$ is not 0 since the LHS times $g^{\mu\nu}_{\sigma}$ is not (57). I would try to work out $\partial_\alpha (50)$ and then compare it to $g^{\mu\nu}_{\sigma}$(53). Keeping $\sqrt{-g}=1$ in mind at all times. In Príncipe this computation is a special case of $\nabla_\mu T^\mu_\nu=0$ as (57b) suggests. In fact the LHS of (53) is the Ricci tensor. Using $R=-\kappa T$, the covariant derivative of (53) and the Second Bianchi identity is an easier way to show (57b). $\endgroup$ – M. J. Steil May 29 '17 at 22:11
  • $\begingroup$ Thanks for the feedback! I'll try to improve it next time. $\endgroup$ – Rocky Wong Jun 5 '17 at 16:16