Covariant Derivation of the Ricci Tensor: Einstein's Method Now Online

nobraner
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The full derivation of the covariant derivative of the Ricci Tensor as Einstein did it, is now available on line at

https://sites.google.com/site/generalrelativity101/appendix-c-the-covariant-derivative-of-the-ricci-tensor

For those who wish to study it.
 
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nobraner, Your expressions all have two lower μ indices, which is incorrect. Would you like to raise one of them by inserting a gμν?
 


Bill,

Finally found the time to fix this. One of my biggest weaknesses is ignoring the upper/lower covariant derivative convention. I guess I always think of covariant derivatives as always being covariant.
 
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From $$0 = \delta(g^{\alpha\mu}g_{\mu\nu}) = g^{\alpha\mu} \delta g_{\mu\nu} + g_{\mu\nu} \delta g^{\alpha\mu}$$ we have $$g^{\alpha\mu} \delta g_{\mu\nu} = -g_{\mu\nu} \delta g^{\alpha\mu} \,\, . $$ Multiply both sides by ##g_{\alpha\beta}## to get $$\delta g_{\beta\nu} = -g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \qquad(*)$$ (This is Dirac's eq. (26.9) in "GTR".) On the other hand, the variation ##\delta g^{\alpha\mu} = \bar{g}^{\alpha\mu} - g^{\alpha\mu}## should be a tensor...
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