Transverse Traceless Gauge Sometimes Works

[Alex Cassem]

Hello.

I am trying to implement the transverse-traceless gauge on metric perturbations and I can get it to work sometimes only if I have a specific order of operations. 

If I first perturb the equations of motion, and then implement an automatic rule

AutomaticRules[metpert,
 MakeRule[{CD[-a]@metpert[LI[1], a, b], 0}, MetricOn -> All,
  PatternIndices -> All]]

but also at the end of the expression flip covariant derivatives via 

/. {CD[a_][CD[b_][expr_]] :> CD[b][CD[a][expr]]}

I get the right answer but I think it is only by luck since a more careful way, I would assume, would be to use the SortCovDs[] command, but I am unsure how to implement this. I attached an example notebook of showing how this works. (Since I want to work with more complicated systems of course). 

Any help is appreciated, thanks.

Alex. 

[Ilham Prasetyo]

You can use SortCovDs in the definition when doing the perturbation before doing ToCanonical

Collect[ContractMetric[ExpandBackground[einsteinhilbereom, 1]] //
    SortCovDs // ToCanonical, {alpha, beta,
   kappa}] /. {metpert[LI[1], _, -_] -> 0, metpert[LI[1], -_, _] -> 0}

Hope this helps.

Ilham