Inverse of perturbed product metric

[Dr. Subeom]

Hi, everyone~

I have a question about xPert. I'm reconstructing the Appendix B result in the paper, "Gravitational perturbations of the Schwarzschild spacetime: A practical covariant and

gauge-invariant formalism", arXiv:gr-qc/0502028. I attached the paper and my xAct code below. Basically, given a perturbed metric, Perturbed[g[-a,-b]], if we index up, i.e  Perturbed[g[a,b]], we obtain the inverse of  Perturbed[g[-a,-b]].

However, in the setting of the product metric in the note, indexing up does not work and can't get the inverse metric. Alternatively, I solved linear equations naively to get the inverses as noted in the code below. Anybody knows better way to derive the inverses?

Thanks.

[Ébano Vitor]

Same problem here. Were you able to solve it in an automatic way? I need to calculate the perturbed Christoffel symbols, but it doesn't work because of this problem

[Ayon Tarafdar (Joy)]

I've run into the same issue with the same paper. I think the paper brings out the 1/r factors and defines the inverse perturbed tensor accordingly: $p^{AB} = \Omega^{AC}p_{CD}\Omega^{DB}$ instead of  $p^{AB} = g^{AC}p_{CD}g^{DB}$ (which is the "standard" way to define it?). xPert does the latter, which is why we're unable to obtain this form.

[Jose]

Hi,

Perturbing the metric and expanding it into blocks are two operations that do not commute, at least as currently implemented in xTensor + xPert. If one is careful to perform these operations in the correct order then the computation can be safely performed. See the attached notebook.

Cheers,

Jose.

[Ayon Tarafdar (Joy)]

Hi Jose,

Thank you so much for your help! I had reached the point where I suspected that I should perturb the product manifold metric and then index into it, but I wasn't sure and didn't understand how to implement it. Your notebook makes everything crystal clear. 

However, I was trying to apply the same code to a 1+2 (spatial) manifold instead, and ran into some interesting problems. When I perturb the Ricci tensor and use your 'expand' function, cov. derivs of one manifold with indices of another appear! The code works exactly like yours when it's a 1+3 system as in the paper, though!

I worked on this a bit to figure out what the problem could be by breaking down the `expand` function step-by-step. It seems like `Simplification`, or more specifically, `ToCanonical` is mixing up the indices - until that step everything works fine. I have attached the notebook and highlighted the steps. I'd be glad if you can find the time to take a look.

[Jose]

Hi,

It seems you have found this issue with Symmetry1D again:

https://groups.google.com/g/xact/c/8ogCOy91qRs/

As a workaround until the next release, deactivate the function as suggested there after loading xTensor (or xPert).

Cheers,

Jose.