Contraction between Levi-Civita symbol and Riemann tensor

mhob · Dec 22, 2016

How to proof that
ε^μνρσ R_μνρσ =0 ?

Thanks.

ShayanJ · Dec 22, 2016

Use the symmetries of the Riemann tensor!

mhob · Dec 22, 2016

ShayanJ said:

Use the symmetries of the Riemann tensor!

I'm not sure R_μνρσ = R_μ(νρ)σ or not? If so, the problem solved for me.

Thanks.

ShayanJ · Dec 22, 2016

I thought those symmetries would help, but it seems they don't. Start with the first Bianchi identity and contract all the indices with a Levi-Civita symbol. Rearranging the indices of the Levi-Civita symbol in two of the terms will give you what you want.

mhob · Dec 22, 2016

OK, I know use the Bianchi identity
R_μ[νρσ]=0
Thanks!
ShayanJ!

Contraction between Levi-Civita symbol and Riemann tensor

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Contraction between Levi-Civita symbol and Riemann tensor

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