Hi all, I encounter a technical problem about tensor calculation when studying general relativity. I think it should be proper to post it here.(adsbygoogle = window.adsbygoogle || []).push({});

Riemann curvature tensor has Bianchi identity: [itex]R^\alpha[/itex][itex]_{[\beta\gamma\delta;\epsilon]}=0[/itex]

Now given double (Hodge)dual of Riemann tensor: G = *R*, in component form:

[itex]G^{\alpha\beta}[/itex][itex]_{\gamma\delta}=1/2\epsilon^{\alpha\beta\mu\nu}R_{\mu\nu}[/itex][itex]^{\rho\sigma}1/2\epsilon_{\rho\sigma\gamma\delta}[/itex]

Show that the Bianchi identity can be simply written in terms of divergence of G as

[itex]\nabla\cdot G=0[/itex].

In component form:

[itex]G_{\alpha\beta\gamma}[/itex][itex]^{\delta}[/itex][itex]_{;\delta}=0[/itex]

PS: [tex]\nabla[/tex] and ";" represent covariant derivative in abstract and component form respectively.

I've never done such calculation and is overwhelmed by so much super- and subscripts. Can anyone show me step by step how to get the final answer from the beginning? Thanks very much.

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# Calculation of double dual of Riemann tensor

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