Assigning indices after Hodge dual computation

[C. Arias]

Hi!

I’m working on defining Killing vectors and tensors using xTerior, and I am having trouble with one particular step.

I need to construct a 3-form

f=⋆h,

where h is a 2-form.
I was able to compute the Hodge dual, but I am not sure how to assign indices to the resulting 3-form in the same way I assigned indices to h. I need this so that I can eventually build objects like

K^ab=f^ac ​f^b_c

Any help is very much appreciated!

Cynthia

[Juan Margalef]

Hi Cynthia!

I think you might have a missunderstanding about how xTerior works. xTerior allows to define forms WITHOUT indices. If you insist on adding them, then you have to think of them as additional indices. For instance:

DefDiffForm[h[-a], M5, 2]

is a two form with one additional index (which can be an index of the tangent bundle, like this case, or any other vbundle). In the xAct language, it is equivalent (but loosing the power and useful commands of xTerior) to:

DefTensor[h[-a,-b,-c],M5,Antisymmetric[{-b, -c}]

Thus, when you write in your code:

DefDiffForm[h[-a, -b], M5, 2, Antisymmetric[{-a, -b}]]

This is equivalent to 

DefTensor[h[-a, -b, -c, -d], M5, GenSet[-Cycles[{1, 2}], -Cycles[{3, 4}]]]

so you are defining a two form in the indices {a,b} with two additional antisymmetric indices {c,d} (or a two form in the indices {c,d} with two additional antisymmetric indices {a,b}). The two pair of indices are "independent". Thus, if I understand correctly your code, you are taking the hodge with respect to the first pair of indices (which are "hidden" within the xTerior philosophy), while assigning values with respect to the second pair. You should define either:

DefDiffForm[h, M5, 2]

and work with xTerior and xCoba or

DefTensor[h[-a, -b], M5, 2, Antisymmetric[{-a, -b}]]

and work with xTensor and xCoba forgetting about xTerior.

I hope this helps!

Best,
Juan Margalef