SageManifolds: raising and lowering Lorentzian indices

asked 2016-08-12 14:10:51 -0600

Dox gravatar image

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.

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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} ?

eric_g gravatar imageeric_g ( 2016-08-17 06:22:46 -0600 )edit

The a and b indices 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 $\Omega^1{}_0 = - \Omega^{10}$.

Dox gravatar imageDox ( 2016-08-19 08:08:07 -0600 )edit