I'm looking at the informal arguements in deriving the EFE equation.(adsbygoogle = window.adsbygoogle || []).push({});

The step that by the bianchi identity the divergence of the einstein tensor is automatically zero.

So the bianchi identity is ##\bigtriangledown^{u}R_{pu}-\frac{1}{2}\bigtriangledown_{p}R=0##

##G_{uv}=R_{uv}-\frac{1}{2}Rg_{uv}##

So I see this if the covariant derivative is a actual tensor itself, such that indices can be lowered and raised i.e. ##\bigtriangledown^{u}G_{uv}=\bigtriangledown^{u}R_{uv}-\frac{1}{2}\bigtriangledown^{u}Rg_{uv}=\bigtriangledown^{u}R_{uv}-\frac{1}{2}\bigtriangledown_{v}R##

So from the 2nd to third equality I've assumed the covariant derivaitve is a tensor.

Is it?

Or is my working incorrect?

Thanks.

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# Vanishing of Einstein tensor from Bianchi identity

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