Partial derivative in xTensor

[Stephanie Baines]

Hello,

I am just starting with xTensor and wanted to calculate the partial derivative of the determinant of a metric (I'm looking into Kaluza-Klein reduction and need to compute this). I have done it by hand and agree with my reference but I get a sign difference when using xTensor.

I have uploaded my mathematica file and the document with the reference formula (Eq A.7 page 176). I think I am misunderstanding the partial derivative in xTensor.

Thank you for your help!

Stephanie

[Jose]

Hi,

I think xTensor's result is consistent with the formula you mention. They are just written slightly differently. You need the identity:

PD[-c][g[a, b]] == - g[a, d] g[b, e] PD[-c][g[-d, -e]]

which adds the missing sign.

In fact, xTensor canonicalizes the LHS into the RHS of that identity automatically.

Cheers,

Jose.

[Lyudmil Antonov]

How can I make xTensor to not automatically canonicalize the metric partial derivatives or at least to delay the canonicalization? For some calculations I need the partial derivatives to be in the form PD[-c][g[a, b]]  order to simplify some expressins by contraction. For example, trying to calculate the Landau-Lifshitz pseudotensor in terms of partial derivatives of metric densities in this notebook

[Lyudmil Antonov]

I solved the problem using Hold but isn't there some more elegant solution?