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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Calculation of double dual of Riemann tensor

**Physics Forums | Science Articles, Homework Help, Discussion**