Raising/Lowering dummy indices to obtain cancellations

[Richard Kleeman]

I am new to xAct and am using it at present to perform a Legendre transform on the GR EM Lagrangian to get the ADM Hamiltonian form. As you can see from the attached notebook on the final line of output (apart from the undefines) the final form after using ToCanonical and Simplify contains multiple terms which will cancel if dummy indices are reversed in their upper and lower positions in some of the terms. Is there a way of making xTensor do this dummy index repositioning? ToCanonical doesn't seem to be doing this...

[Juan Margalef]

You have to use ContracMetric:

expr /. ARules // ContractMetric // Simplification

A few additional comments:

1. Be careful, you are writing ^{-1} which creates a list. Compare:

2^(-2)
2^{-2}
2^{-2,-1}

2. Instead of the substitutions rules you have used, it is safer to use MakeRule:

FRules = MakeRule[{F0[i], gr[i] - ADOT[i]}]
ARules = MakeRule[{ADOT[i],  gr[i] + Shift[-s] F[s, i] +   Lapse[] \!\(TraditionalForm\`\[CapitalPi]\)[i] Gam[]^(-1)}]

3. Instead of 0.25, you can write 1/4 to avoid the 1. that you see in your result.

[Richard Kleeman]

That worked thanks!