higher covariant derivatives of riemann in xCoba

[Ilaria Brivio]

Hi,

I am trying to calculate higher covariant derivatives of the riemann tensor in a specific coordinate chart using xTensor and xCoba, but I can't seem to find an efficient way to do it.

I need to compute up to the 4th covariant derivative of the riemann in a 4D manifold and I really need something that can be fast.

I attach a notebook with the approach I've taken so far (up to the 3rd cov D it's manageable) and something else I tried, but I'm not really convinced that it can be a solution.

could you perhaps suggest some other approach? I would be ok with computing only specific components of the 4th covariant derivative, but one of the riemann indices must be raised.

Thank you very much in advance!

Ilaria

[Ilham Prasetyo]

Hi

I added into your file here "approach 3" which contains my answer. The modified file is attached.

Schematically is as follows

1. I use MetricCompute like your "approach 2", but I make the result for Riemann into kompRiemann which is inputted into another tensor called compRiemann. In this stage, I don't use your reduce function which simplify the expression a lot. 

2. Then, I calculate the 1 covariant derivative of Riemann then put it into kompDRiemann and input it into compDRiemann. To check it, I just called kompDRiemann and use your reduce function. 

3. Repeate the steps again to into kompDDRiemann and input it into compDDRiemann, then into kompDDDRiemann and input it into compDDDRiemann

I find this to be faster and it produce the missing term a2^2 v22 that you mention in the file. Hope it helps.

Best,

Ilham