Check that conjugation 

      G -> Sigma.G.Sigma^T 
      
for area metrics corresponds to conjugation 

      kappa -> Hodge(++++) o kappa o Hodge(++++) 
      
for (2,2)-tensors

SetDirectory["/www/user/fdahl/papers/Conjugation/"];
<<kappaLib.m
<<Petrov.m

We only need the result for skewon-free (2,2)-tensors. However, we will prove the result for all (2,2)-tensors.

kappa = emGeneralKappa["k"];

kappaMat =emKappaToMatrix[kappa];
kappaMat//MatrixForm

Compute Petrov matrix for 

     Hodge(++++) o kappa o Hodge(++++)

metric = DiagonalMatrix[{1,1,1,1}];
hodge=emHodge[metric];

hkh = emCompose[hodge,emCompose[kappa,hodge]];

Petrov[emKappaToMatrix[hkh]]

PetrovConjI=%;

Compute conjugation of Petrov matrix by "Sigma" symbol.

Petrov[kappaMat]

petrovMat = %;
petrovMat//MatrixForm

Sigma = Table[Signature[{aOrd[i][[1]],aOrd[i][[2]],aOrd[j][[1]],aOrd[j][[2]]}],{i,1,6},{j,1,6}];
Sigma//MatrixForm

PetrovConjII=Sigma.petrovMat.Transpose[Sigma];

Check that operations commute

PetrovConjI-PetrovConjII

NotebookPrint[SelectedNotebook[],"/www/user/fdahl/papers/Conjugation/notebooks/Conjugation.pdf"]