 Lie derivative of a covariant tensor field

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.