I am trying to expand $$\varepsilon^{{abcd}} R_{{abcd}}$$ by using four identities of the Riemann curvature tensor:(adsbygoogle = window.adsbygoogle || []).push({});

Symmetry

$$R_{{abcd}} = R_{{cdab}}$$

Antisymmetry first pair of indicies

$$R_{{abcd}} = - R_{{bacd}}$$

Antisymmetry last pair of indicies

$$R_{{abcd}} = - R_{{abdc}}$$

Cyclicity

$$R_{{abcd}} + R_{{adbc}} + R_{{acdb}} = 0$$

From what I understand, the terms should cancel out and I should end up with is $$\varepsilon^{{abcd}}R_{{abcd}} = 0$$. What I ended up with was this mess:

$$\begin{array}{l}

\varepsilon^{{abcd}} R_{{abcd}} = R_{\left[ {abcd} \right]} =

\frac{1}{4!} \left( \underset{- R_{{dcab}}}{\underset{+

R_{{cdab}}}{\underset{- R_{{abdc}}}{\underset{{\color{dark green}

+ R_{{badc}}}}{\underset{- {\color{red} {\color{black}

R_{{bacd}}}}}{{\color{blue} R_{{abcd}}}}}}}} +

\underset{{\color{magenta} - R_{{adbc}}}}{\underset{{\color{red} +

R_{{cbad}}}}{\underset{- R_{{cbda}}}{\underset{+

R_{{bcda}}}{{\color{magenta} {\color{black} R_{{dabc}}}}}}}} +

\underset{- R_{{abdc}}}{\underset{+ R_{{dcba}}}{\underset{-

R_{{cdba}}}{\underset{- R_{{dcab}}}{\underset{{\color{dark green}

+ R_{{badc}}}}{\underset{- R_{{bacd}}}{\underset{+

R_{{abcd}}}{R_{{cdab}}}}}}}}} + \underset{+

R_{{dabc}}}{\underset{- R_{{cbda}}}{\underset{{\color{red} +

R_{{cbad}}}}{\underset{- {\color{blue} {\color{black}

R_{{bcad}}}}}{{R_{{bcda}}}}}}} - \underset{-

R_{{bdca}}}{\underset{{\color{blue} + R_{{acdb}}}}{\underset{+

R_{{dbca}}}{\underset{- R_{{dbac}}}{\underset{+

R_{{bdac}}}{R_{{acbd}}}}}}} - \underset{{\color{blue} +

R_{{adbc}}}}{\underset{- R_{{bcda}}}{\underset{+ {\color{blue}

{\color{black} R_{{bcad}}}}_{}}{\underset{{\color{red} -

R_{{cbad}}}}{R_{{adcb}}}}}} - \underset{{\color{black} +

R_{{abcd}}}}{\underset{- R_{{bacd}}}{\underset{{\color{dark green}

+ R_{{badc}}}}{R_{{abdc}}}}} - \underset{{\color{magenta} +

R_{{abcd}}}}{\underset{- R_{{abdc}}}{\underset{{\color{red}

{\color{dark green} + R_{{badc}}}}}{\underset{- {\color{red}

{\color{black} R_{{bacd}}}}}{R_{{cdba}}}}}} \right)

\end{array}$$

where I can get rid of the blue or the purple terms using cyclicity (sorry for colors but it'll be a pain to change it), but I'm stuck because I cant see how I can get all the terms to cancel. The main problem seems to be is that the last term in the cyclicity identity $$\left(R_{{acdb}} \right)$$ can only be acquired from the 5th term $$\left(R_{{acbd}} \right)$$ in the expression i have. After I get rid of 6 terms with cyclicity I was thinking I could get of what remains with some symmetry relationship. Am I going down the wrong path here? Do I need another relationship? Carroll in ``Introduction to General Relativity'' says in eq 3.83 that all I have to do is expand the expression for $$R_{\left[ {abcd}\right]}$$ and mess with the indicies using the 4 identities to proove that it reduces to zero. Thank you for any help.

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