Using SeriesData / O[r] in xAct rules

[ehki...@gmail.com]

Dear all,

I am trying to use SeriesData (or equivalently expressions containing O[r]) inside rules in xAct to speed up asymptotic calculations.
T[-a, -b] == h[-a, -b] + O[r]
Works fine as an equality. However, when I try to turn this into a rule, xAct complains about index inhomogeneity, since the left-hand side carries indices [-A, -B] while O[r] does not. 
My goal is to systematically replace tensors by truncated series expansions using SeriesData, but in a way compatible with xAct’s index structure and pattern matching.
What is the recommended way to implement perturbative series rules in xAct while preserving index homogeneity?

Best regards,
Ehsan  

[Juan Margalef]

Hi Ehsan,

If you want to perform assymptotic analysis using the coordinate r, you need to express your tensors in some coordinate chart and for that, you should check xCoba https://xact.es/xCoba/ and apply the standard Mathematica commands for series.

However, maybe you can get away with what you want to do using xPert https://xact.es/xPert/ Something like this:

DefManifold[M, 4, {a, b, c, d, e, f}]

DefTensor[T[-a, -b], M]
DefParameter[r]
$PerturbationParameter = r

Perturbed[T[-a, -b], 3]

With some rules for higher order terms, you might achieve what you want.

Best,
Juan Margalef