Combining xTensor with xCoba to get explicit equations of motion

[Héloïse]

Hi everyone,

I am considering a specific class of scalar-tensor Horndeski theories. I have obtained the equations of motion in tensorial form with xTensor, and would now like to evaluate them explicitly for a given diagonal metric in a given set of coordinates with xCoba.

I have followed two different approaches to do so, but both "fail":

1) In Approach1.nb (see attached), some tensorial expressions remain after evaluating it (like Christoffel symbols and basis vectors). What am I doing wrong here?

2) In Approach2.nb (see attached), the evaluation is just taking so long that I have to abort and do no know whether this approach is working or not. Is this approach correct? If yes, can I do some further simplifications to make the calculation faster?

More generally, I am looking for a standardised framework to combine xTensor with xCoba for such calculations, something I have not found in xAct documentation.

Thank you in advance for your detailed help!

Best,

Héloïse

[Armando Roque]

Dear Héloïse,

I have checked your different approaches. In particular, I am more familiar with the second one; in fact, it is the method I used when working with higher-order covariant derivatives of a field. I am confident that the way you computed the covariant derivative could be improved. Unfortunately, I do not have much time and did not attempt this myself. However, I did improve the methodology for evaluating the metric (I simply separated it into a list of elements). On my computer (MacBook Pro M3 Pro), it takes less than 30 seconds, which I believe is sufficient for your purposes.

Best regards,

Armando

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