Charts and Hodge star

[Marc]

Hi all,

I would like to use charts with xTerior and got some trouble with the Hodge star operator acting on a tensor. I have tried two ways, the first one worked, the second, however, not.

Suppose we have tensor Q and we know its components for a given chart "flrw". If I first evaluate the components for this chart, then apply the Hodge operator, I can find the final correct result:

Q[-a, -b]
ToValues@TableOfComponents[TraceBasisDummy@ToBasis[flrw]@%, flrw]
Hodge[metricg]@%
% /. metricg -> FLRWmetric
ExpandHodgeDual[%, dx[Man], FLRWmetric]
ToValues@%

However, I have some expressions that contain terms like Hodge[metricg][Q[-a,-b]]. If I then try

Hodge[metricg][Q[-a, -b]]
% /. metricg -> FLRWmetric
ToBasis[flrw]@%
ToValues@TableOfComponents[%, flrw]
ToBasis[flrw]@%

I do not obtain the same result. Specifically, in the third line xAct is doing a general basis transformation

e_a^c e_b^d (Hodge[FLRWmetric][Q[-c, -d]])

but does not replace Q[-a,-b] with the known components in the chart flrw.

Do you haven an idea how to replace Q[-a,-b] in the second case with the components?

Thanks a lot!

[Alfonso García-Parrado Gómez-Lobo]

Hi,

could you send a notebook with a summary of these computations?

Thanks.

Alfonso.

[Marc]

Hi Alfonso,

Thanks for your answer. I've attached a notebook.

Thanks!

[Alfonso García-Parrado Gómez-Lobo]

Hi,

I'm sending you back your notebook indicating how the computation has to
be done.

Alfonso.

[Marc]

Hi Alfonso,

Great, thank you very much!