Using of AutomaticRules for Transversality

[Ali rezaie]

I am trying to use AutomaticRules to cancel the Transverse terms from my ansatz, I mean terms like(\partial_a A^{a,b} =0 ).

it does not work properly and does not omit terms like \partial_a \partial_d A^{a,b}. and the same.

I am writing in this way:

 AutomaticRules[H, MakeRule[{CD[-a]@H[a, b], 0}]] 

How can I resolve this problem?

Tell me to attach if it is needed.

Many many thanks, friends.

Ali

[Leo Stein]

Dear Ali,

xTensor is not automatically commuting the derivatives for you—you have to ask it do that, if you want it to. You can manually ask it with CommuteCovDs (to guide Mma step-by-step) or CommuteCovDs (to put the derivatives into a 'canonical' order).

You can imagine why this would be important with a non-flat connection: modifying your example, if we start with ∇_a ∇_d A^{ab}, we should commute covariant derivatives, which introduces Riemann terms; and then the div(A) term will vanish.

Anyway, to get the correct Riemann terms, you should commute covariant derivatives which are contracted onto A as far in as possible. That was discussed in a thread on the mailing list long ago here, and Teake has included it into xTras as the function SortCovDsToDiv.

So, you can impose the divergence-free condition on tensors and get the Riemann terms correct by using SortCovDsToDiv, and then letting your AutomaticRules act to annihilate the divergence terms.

Best,

Leo

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