Lie derivative for tensors

This generalize the Lie derivative of forms and the Lie derivative of vector fields

This includes the previous section. For any smooth vector field $\lambda^a$, any smooth

tensor field $\alpha_{b_{1} \ldots b_{s}}^{a_{1} \ldots a_{r}}$ and any torsion free affine connection $\nabla$:

$$ \begin{aligned} \mathcal{L}_{\lambda} \alpha_{b_{1} \ldots b_{s}}^{a_{1} \ldots a_{r}}=& \lambda^{n} \nabla_{n} \alpha_{b_{1} \ldots b_{s}}^{a_{1} \ldots a_{r}}+\alpha_{n b_{2} \ldots b_{s}}^{a_{1} \ldots a_{r}} \nabla_{b_{1}} \lambda^{n} \\ &+\ldots+\alpha_{b_{1} \ldots b_{s-1} n}^{a_{1} \ldots a_{r}} \nabla_{b_{s}} \lambda^{n} \\ &-\alpha_{b_{1} \ldots b_{s}}^{n a_{2} \ldots a_{r}} \nabla_{n} \lambda^{a_{1}}-\ldots-\alpha_{b_{1} \ldots b_{s}}^{a_{1} \ldots a_{r-1} n} \nabla_{n} \lambda^{a_{r}} \end{aligned} $$

(see \cite{malament} page 53).

In particular:

\begin{itemize}

\item $\mathcal{L}_{\lambda} \alpha^{a}=\lambda^n \nabla_n \alpha^{a}-\alpha^n \nabla_n \lambda^a$

\item $\mathcal{L}_{\lambda} \alpha_{b}=\lambda^n \nabla_n \alpha_{b}+\alpha_n \nabla_b \lambda^n$

\item $\mathcal{L}_{\lambda} g_{ab}=\lambda^n \nabla_n g_{ab}+g_{nb} \nabla_a \lambda^n+g_{an} \nabla_b \lambda^n$

\end{itemize}

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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