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I am completely new to xAct and was wondering if it is possible to tell xAct how to simplify expressions.

Is there, for example, a way (e.g. in form of a supplied rule) to automatically simplify Maxwell equations given as

CD[-b][CD[-a][A[b]]] - CD[-b][CD[b][A[-a]]]

to

CD[-a]@F[a, b]

where

F[a_, b_] := CD[a][A[b]] - CD[b][A[a]] ?

Any help is appreciated.

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1 Answer 1

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You can try making ansatz like the following.

defining the manifold and fields

DefManifold[M,dim,IndexRange[a,z]];
DefMetric[-1,metric[-a,-b],CD,SymbolOfCovD->{";","\[Del]"},PrintAs->"g"];
DefTensor[A[-a],M];
DefTensor[F[-a,-b],M,Antisymmetric[{-a,-b}]];

making ansatz

expr[]=CD[-b]@CD[-a]@A[b]-CD[-b]@CD[b]@A[-a]
ruleFToA=MakeRule[{F[a,b],CD[a]@A[b]-CD[b]@A[a]}];
ansatz[1]=MakeContractionAnsatz[CD[a]@F[b,c],{-a},ConstantPrefix->"a"];
ansatz[2]=MakeContractionAnsatz[CD[a]@CD[b]@A[c],{-a},ConstantPrefix->"b"];
ansatz[0]=(ansatz[1]/.ruleFToA)+ansatz[2]

enter image description here

comparing the constants

SolveConstants[expr[]==ansatz[0]]
ansatz[1]+ansatz[2]/.First@Out[]

enter image description here

now taking $a_1=1$ we get the result that \begin{equation} \nabla_{b}\nabla_{a}A^{b} - \nabla_{b}\nabla^{b}A_{a} \end{equation} can be collected into $\nabla_{b}F_{a}{}^{b}$.

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