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Compute curvature tensor constructed by a shifted connection

Posted 1 month ago

I'm using xAct package to compute the curvature tensor from a specific connection, of the form $\Gamma^a_{bc}=\left\{^{\:a}_{bc}\right\} + S^a_{\;bc}$, where the first term is the Levi Civita connection constructed from the metric of the manifold, and the second term is the shift tensor which depends on a vector field $f_k$ : $S^a_{\;bc}=(\delta^a_bf_c+\delta^a_cf_b-g_{bc}f^a)$. I would like to find the curvature tensor, Ricci tensor, and eventually the Ricci scalar related to this connection. However, i'm having a hard time trying to apply Simplification to the curvature tensor, and RicciScalar[] does not give me the desired result.

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POSTED BY: Ayberk Cakmak
3 Replies
Posted 1 month ago

Thanks a lot, seems to be working. But I still don't understand why Riccicd[] is not automatically computed.

POSTED BY: Ayberk Cakmak

Riccicd[-a, -b] is a symbolic object, like Riccilc[-a, -b]. You have a rule that relates two tensors, the Christoffels, but xTensor will not attempt to compute relations for other tensors. This would be difficult, and depend on assumptions about whether there are metrics, torsions, etc. You need to guide the computations.

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