Splitting Perturbation from a Lagrangian

[charles mazuet]

Hello,

I start from a certain lagrangian containing covariant derivative of a tensor field X[-a,-b], for example L=Sqrt[detG[]] CD[-a][X[-b,-c]] D[a][X[b,c]], where X is symmetric and a background field "no perturbations".

I am looking for the first and second perturbation order of this lagrangian with respect to the metric only and then to isolate the Tensor, Vector and Scalar sectors.

First I used DefMetric, SetSlicing, DefMetricField[h,dg,h] and DefProjectedTensor[X[-a,-b], TensorProperties->{"SymmetricTensor"},SpaceTimeOfDefinition->{"Background"}].
Then I can do Perturbed[L,1]//ExpandPerturbation etc..

But here I obtain an expression with both dg[1,a,b] and h[1,a,b], and only the 4D covariant Derivatives
This is a thing i don't understand.

But after that I would like to project perturbations on a flat FLRW background, in order to split my perturbations on the various sectors, but here i end up again with both dg^[1,a,b] and h^[1,a,b], prime derivatives and the 4d covariant derivatives of dg and h.

I suppose, if things had worked normally, i should have terms like X ' [a,b] X ' [-a,-b] + E' [a,b] E ' [-a,-b] X[c,d] X[-c,-d] etc...

Could you please indicate me how to get rid of that problem and have a good splitting.

Thank you for your help.

Charles

[Cyril Pitrou]

Dear Charles,

Obinna Umeh will maybe answer you something different. It would be simpler if you send the notebook to try to see what you want to do.

In your case it is not clear what is your tensor H[a,b]. I assume it is something which is unperturbed, but still one needs to know for which positioning of indices it is unperturbed.

Because if you raise an index with the metric then it gets a perturbation because of metric perturbations...

I any case the idea is to define a formal tensor which is general on which you can define an action and do its perturbation.

You also define the formal perturbation of this tensor.

Then you can define projected tensors which you are going to use to decompose in 1+3 this formal tensor and its perturbation.

And you are going to define rules to replace the formal tensor or its perturbation by something which involves projected tensors, that is tensors which are spatial, possibly transverse and tracefree etc...

I have assumed in the notebook attached you want only a background value for the tensor X[-a,-b].

Because you use Sqrt[something] it can get slighlty messy in xAct. So I replaced it with a formal function f[],

that eventually I specialize to being Sqrt[], knowing that f'=1/(2f). Otherwise xPand which was not really thought to handle this case loses time forever in useless computations.

I hope this helps. Eventually you will need to do integrations by parts but this is another story.

I am not very satisfied with this answer... The notebook is way too slow.

Regards,

Cyril

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