Issue with xAct/xCoba performance on Maxwell-Chern-Simons

[Jing Ma]

Hi there,

I'm developing a notebook aimed at: (1) deriving the 5D Einstein-Maxwell-Chern-Simons equations with a negative cosmological constant and (2) solving them to obtain a planar Reissner-Nordstrom black hole solution.

I have based my work on a notebook by Reggie Bernardo titled "Einstein_Maxwell_Klein_Gordon_theory.nb." My modified version, "AdS5-CS-RN_xAct.nb," functions as expected when the Chern-Simons term is omitted from the Lagrangian (L = LEH + LM), with an approximate running time of 6 minutes.

However, when I include the Chern-Simons term (L = LEH + LM + LCS), xAct successfully derives the field equations, but xCoba appears to hang indefinitely during the evaluation on the given background.

Any feedback would be greatly appreciated.

[Jing Ma]

Edit: The title of my notebook is "Einstein-Maxwell-CS.nb"

[Leo Stein]

Dear Jing,

I attach below your notebook, that I modified, to show an example of how to get xCoba to do the calculation you want. Here are a few tips about some issues you may want to keep in mind.

1. In general, split your calculation up into multiple cells, so that if somebody wants to change some code, they can run part of the notebook, inspect outputs, try out some algebra, etc., rather than only being able to run the whole calculation.

2. If you have a tensor which will *always* be replaced with some other expression, there is no need to use DefTensor to define it; this was the case for F[a,b] in the notebook you sent.

3. However for your specific calculation, I did want to go back to using F wherever possible, instead of using A. There will be fewer terms in the EOMs in terms of F rather than using A.

4. As a general rule, Mathematica is not especially clever, and we need to hold its hand -- finding ways to make the algebra as simple as possible at every stage necessary. Hence simplifying the EOMs in terms of F rather than A, and deciding that I wanted all indices on epsilon to be upstairs.

5. When you have the abstract epsilon tensor and want to go into a basis, you have to replace it with that basis's eta *symbol*, using either epsilonToetaUp to epsilonToetaDown.

6. Generally, contractions with epsilons or etas have a huge number of terms, so you should always work hard to reduce the number of terms. Afterwards I realized I could have worked harder — one of the terms in the EOM was *dF, so I should have made xTensor show that dF=d^2A=0 and get rid of this term. But, this email is already too long, so I leave that as an exercise ;)

Best

Leo

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