Dummy Index manipulation

[Tanush Reddy Vaka]

Hello Everyone, 

I'm fairly new to xAct, and I'm having trouble figuring out how to simplify the following expression. 

1/2  Christoffelcd[i, -j, -k]   Christoffelcd[  j, -l, -m]   \[CapitalTheta][k, n]  \[CapitalTheta][ l, m] \[Sigma]2[-i, -n] -  1/2  Christoffelcd[i, -j, -k]   Christoffelcd[ j, -l, -m]   \[CapitalTheta][k, l]   \[CapitalTheta][m, n]   \[Sigma]2[-i, -n] -1/2  \[CapitalTheta][i, j]   \[CapitalTheta][ k, l]  \[Sigma]2[-i, -m]  (cd[-j][Christoffelcd[m, -k, -l]]) +1/2  \[CapitalTheta][i, j]   \[CapitalTheta][k, l]  \[Sigma]2[-i, -m]  (cd[-l][Christoffelcd[m, -j, -k]])

I believe this should simplify to just Riccicd[-i, -j]   \[Sigma]2[i, j].

I think GradChristofellToRiemman (from xTras) will work after I write the equation as (some tensor)[-i,-j]  \[Sigma]2[i, j].

I do not see how change \[Sigma]2[-i, -n]  into \[Sigma]2[-i, -j] when j is already a contracted index in the terms that multiply \[Sigma]2[-i, -n]. 

I would greatly appreciate any help.
Tanush

[Thomas Bäckdahl]

Hi!

GradChristoffelToRiemann transforms partial derivatives (PD) of Christoffel symbols to Riemann. Therefore, you have to translate your covariant derivatives to partial derivatives first. Hence

ChangeCovD[christo, cd, PD] // GradChristoffelToRiemann // ToCanonical

However, you should be careful here. A covariant derivative of something that transforms like a tensor makes sense. However, a Christoffel symbol does not transform like a tensor, so to me it is unclear what a covariant derivative of a Chistoffel symbol is.
Perhaps you intended to write PD instead of cd in your original expression?

Regards
Thomas

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[Tanush Reddy Vaka]

Hi Thomas! 

Thank you for the quick response. I now see my elementary mistake.

About the covariant derivative of a Christoffel symbol, I agree with you. One can use the same argument for raising and lowering indices of a Christoffel symbol as well, I suppose, but terms like these do appear when you do say ChangeCovD[PD[-i][PD[-j][\[Sigma]2[i, j]]], PD, cd] or christo //ContractMetric. I don't know if there are ways that prevent this, but for now, I am able to do the required simplifications after using SeparateMetric and changing cd to partial derivatives.

Thank you once again.
Tanush