Lowering the indices of a variation

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Rafael Carrasco Carmona

Mar 15, 2023, 9:16:08 AMMar 15
to xAct Tensor Computer Algebra
Hello, everyone

I write this post because I have a problem when computing variations with 'xPert'. My problem is that I need to compute a variation of a tensor which can be always expressed as something proportional to the variation of the metric. The problem that I found is that Mathematica gives me one term proportional to \Delta g^{\mu\nu} and another to \Delta g_{\mu\nu}. In order to write everything in terms of the latter term I added an automatic rule as follows:

AutomaticRules[g, MakeRule[{Perturbation[g[\[Mu], \[Nu]],1], -g[\[Mu], \[Alpha]] g[\[Nu], \[Beta]] Perturbation[ g[-\[Alpha], -\[Beta]], 1]}]]

However, this line does not solve the problem, since it also changes the sign of Perturbation[g[\[Mu],\[Nu]],1]. Do you know any way to fix this? Thank you so much in advance.

Kind regards,


Apr 18, 2023, 2:43:01 AMApr 18
to xAct Tensor Computer Algebra

Can you provide a complete set of inputs (or even better a small notebook) where we can see an example of the problem you are referring to? Otherwise it's quite difficult to guess what could be happening.

Regarding signs: if Delta(g) = h and ig = g^-1then Delta(ig) = - ig . h . ig, with a minus sign in front. Is this the sign you see?

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