How to avoid Ricci tensors in perturbative expansion of the curvature scalar and the Einstein tensor?

[Nancy Lemon]

Hello everyone,   

  

I’d like to expand perturbations of a Ricci tensor and a Einstein tensor to metric as illustrated in the documentation of the xPert package. However, Ricci tensors and Ricci scalars remain in the default results and can’t be automatically converted even with the ToCanonical and NoScalar commands. Actually I worry it’s not correct to further substitute the remaining Ricci tensors and Ricci scalars with another perturbated expansion, which seems to largely complicate the things. I wonder whether there’re any other methods to clean up the unwanted Ricci tensors and Ricci scalars?  

  

Thank you in advance!  

  

   

  

Nancy 

[Jose]

Hi,

I don't think I understand this question. The perturbation of RiemannCD[-a, -b, -c, d] is returned as a sum of second-order covariant derivatives of the metric. The perturbation of RiemannCD[-a, -b, -c, -d],  with all indices down, which is what you compute in the attached notebook, is then the perturbation of RiemannCD[-a, -b, -c, e] g[-e, -d] and therefore has a term consisting of the product of the background Riemann tensor and  a first-order perturbation of the metric. Is this type of term with explicit Riemann tensors that you are referring to? Those are background tensors, and will be there for general backgrounds. If your background is flat you can eliminate them with something like ... /. _RiemannCD -> 0 .

xPert knows how to perturb the curvature tensors, like Ricci or Riemann. It's much better to perturb them first, and then continue with the computation, instead of converting them into derivatives of Christoffel or the metric and then perturb. The latter will be a harder computation, and will involve canonicalizing expressions involving mixtures of Christoffels and derivatives of the metric, which can get complicated.

If I'm missing the point of your question, please ask again.

Cheers,

Jose.