[xPert] -> [xCoba] xPress a Tensor Given A Metric

[Benedict]

I'd like to express an abstract tensor I derived using xPert in components of a metric using xCoba. My result seems not to agree with the result I obtain using xPand, so I assume I am not using the correct approach in xCoba.

Here is what I do:

I start with a Lagrangian L and take its second order perturbation:

varL2 = ToCanonical[

  ContractMetric[ExpandPerturbation[Perturbation[L, 2]]]]

I extract the abstract tensorial equation of motion:

vartf2 = 2 (-VarD[pert[LI[1], a, b], cd][varL2]/

         Sqrt[-Detbackground[]] /. {delta[-LI[1], LI[1]] -> 1, 

         delta[-LI[2], LI[1]] -> 0, delta[-LI[1], LI[2]] -> 0} // 

       SeparateMetric[background]) // Expand // 

   ContractMetric // ToCanonical

eomt1[b_] = 

 Simplify[Simplify[

   IndexCollect[cd[a]@vartf2 // ToCanonical, cd[-b][sf[]]] // 

    Simplification, eom0 == 0], eoms1 == 0]

where background is the metric

DefMetric[-1, background[-a, -b], cd, PrintAs -> "g"] and eom0 and eom1 are the scalar equations of motion, which I use to simplify the tensorial part, if such a combination of terms should appear again.

Then, I switch to xCoba and the component part, where I define my actual metric and a chart ch.

To express the Ricci scalar, Jose was so kind to help provide me with the following rules to transition the abstract result into the component-depending result:

rules = {

   background[inds__Symbol] :> Inv[backgroundMetric][inds],

   cd -> bgcd,

   Riccicd[inds__] :> Ricci[bgcd][inds],

   RicciScalarcd[] :> RicciScalar[bgcd][],

   pert[LI[1], inds__] :> perturbation[inds]

   };

With these rules I can express the abstract scalar equations of motion in components of the metric:

ceom0 = Collect[

   ToCanonical[

     eom0 // ExpandPerturbation // SeparateMetric[background], 

     UseMetricOnVBundle -> None] /. rules, \[Epsilon], Simplify] 

These agree with my results from xPand.

Now, I'd like to also express the tensorial equations of motion in a similar manner. But the result seems not to agree with what I get with xPand. xPand gives me a zero/time component and a spatial component (1,2,3 which are "identical" besides the index). Therefore, I tried to extract the time and space component:

tmpeq = Collect[

    ToCanonical[

      eomt1[{0, ch}] // ExpandPerturbation // 

       SeparateMetric[background], UseMetricOnVBundle -> None] /. 

     rules, \[Epsilon], Simplify] // ContractBasis // FullSimplify

Collect[ToCanonical[

      eomt1[{1, ch}] // ExpandPerturbation // 

       SeparateMetric[background], UseMetricOnVBundle -> None] /. 

     rules, \[Epsilon], Simplify] // 

  ContractBasis // FullSimplify

The first expression seems to disagree with what I get with xPand and the second expression runs for hours (I haven't seen a result yet).

Is there something wrong with this last step? Or is there just a better way to re-express an abstract tensor in a given basis?

Thank you very much in advance.