= Lie derivative of a covariant tensor field
{c}
{title2=$\mathcal L_\xi T_{\mu\nu}=\xi^\sigma\partial_\sigma T_{\mu\nu}+T_{\sigma\nu}\partial_\mu\xi^\sigma+T_{\mu\sigma}\partial_\nu\xi^\sigma$}
For a covariant rank-two <tensor field>, pull back by the flow of a vector field $\xi$ and differentiate at zero flow time. Differentiating the evaluation point produces the first term above; differentiating the two coordinate Jacobians produces the remaining terms. In particular this formula gives the change in a <metric tensor> under an infinitesimal coordinate displacement. A perturbation defined relative to a fixed background consequently transforms as $q_{\mu\nu}\mapsto q_{\mu\nu}-\mathcal L_\xi\bar g_{\mu\nu}$ in the passive displacement convention. The formula applies to symmetric tensors as well as differential forms; tensor symmetry does not make the antisymmetric differential-form formula its definition.
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