If you don't like indexes, look away now. I got these terms from a tensor calculus program as part of a the transformed F-P Lagrangian.(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\begin{align}

{g}^{b a}\,{g}^{d e}\,{g}^{f c}\,{X}_{a,b c}\,{X}_{d,e f}\\

+{g}^{b a}\,{g}^{c f}\,{g}^{e d}\,{X}_{a,b c}\,{X}_{d,e f}\\

+{g}^{b a}\,{g}^{c e}\,{g}^{d f}\,{X}_{a,b c}\,{X}_{d,e f}\\

+{g}^{a b}\,{g}^{c e}\,{g}^{d f}\,{X}_{a,b c}\,{X}_{d,e f}

\end{align}

[/tex]

I think I can substitute ##g^{pq}## with ##g^{qp}## without harm. Also ##,{X}_{p,q r}={X}_{p,r q}## so I can exchange ##q## and ##r##. But can I do this if ##q## and ##r## are in different ##g##'s (like swapping ##e## and ##f## in the fourth term) ?

If these gymnastics are allowed then the terms are equal and there is a good simplification.

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# Tensor calculus problem

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