SageManifolds: raising and lowering Lorentzian indices
In the reference manual of SageManifolds is shown how to calculate the curvature_form (pag. 407 of the reference).
However, the result is shown as $\Omega^a{}_b$, while I'd like to be able of manipulate the $\Omega^{ab}$.
Question: How can the latter be defined?
Thank you.
Since a and b are not tensor indices, but some labels for the set of curvature 2-forms, what do you mean by Omega^{ab} ?
The
aandbindices are indices in the tangent space. Which are raised and lowered with the flat metric (compatible with the signature of the curved spacetime). In my case $\eta = \rm{diag}( -1, 1, 1, 1)$, so $\Omega^1{}_0 = - \Omega^{10}$.