Changing the definition of the Riemann and the torsion tensors

[Yusuf Emre BARAN]

Hi everyone,

I am new to xAct, and I have been using a slightly different convention for the Riemann tensor in my calculations. Specifically, I usually work with the following convention:  

Riemanncd[a,-b,-c,-d] // RiemannToChristoffel
PD[c][Christoffelcd[a,-b,-d]]-PD[d][Christoffelcd[a,-b,-c]]+Christoffelcd[a,-e,-c]Christoffelcd[e,-b,-d]-Christoffelcd[a,-e,-d]Christoffelcd[e,-b,-c]

I understand that xAct allows modifying the definition of the Riemann tensor. Could someone kindly guide me on how to set it up so that the Riemann tensor matches the definition above?  

Additionally, I would like to redefine the torsion tensor such that:  

Torsioncd[-a,-b,c] // TorsionToChristoffel
Antisymmetrize[TorsionCd[a, -b, -c], {-b, -c}]

In other words, I would like to have a factor of 1/2 in front. Finally, I would also like xAct to display the torsion tensor with the letter S instead of T.

Thank you for your time and consideration.

Best regards,
Yusuf

[Juan Margalef]

Changing the definition might be too complicated and have unexpected consequences. I would recommend that you define new tensors and then use "IndexSet" to set the new tensors equal to the old ones (with the prefactors or index reordering that you want). You should check the documentation to make sure you use this command properly. As for your last question, you can use "PrintAs" to change the way an xAct object is printed (either when you define it or at any moment later, as explained in the documentation).

Good luck!