The Lie derivative of an exterior form

The Lie derivative \({L_{v}}\) can be applied to a \({k}\)-form \({\varphi}\) by using the pullback of \({\varphi}\) by the diffeomorphism \({\Phi}\) associated with the flow of \({v}\), i.e. applied to \({k}\) vectors \({w_{I}}\) we define

\(\displaystyle L_{v}\varphi\left(w_{I}\right)\equiv\underset{\varepsilon\rightarrow0}{\textrm{lim}}\frac{1}{\varepsilon}\left[\varphi\left(\mathrm{d}\Phi_{\varepsilon}\left(w_{I}\right)\right)-\varphi\left(w_{I}\right)\right]. \)

The Lie derivative is thus a derivation of degree 0 on the exterior algebra. \({L_{v}\varphi}\) measures the change in \({\varphi}\) as its arguments are transported by the local flow of \({v}\). In the case of a 0-form \({f}\), this is just the differential or directional derivative \({L_{v}f=v(f)=\mathrm{d}f(v)}\).

45.lie-derivative-forms-v3

The above illustrates the Lie derivative for a 1-form \({\varphi}\) with \({\varepsilon=1}\). \({L_{v}\varphi}\) is “the difference between \({\varphi}\) applied to \({w}\) and \({\varphi}\) applied to \({w}\) transported by the local flow of \({v}\),” so above we have \({L_{v}\varphi(w)=2-1=1}\) (valid in the limit \({\varepsilon\rightarrow0}\) if \({\varphi}\) changes linearly in the range shown).

Here and in future figures, we represent a 1-form \({\varphi}\) as a “receptacle” \({\varphi^{\Uparrow}\equiv\varphi^{\sharp}/\left\Vert \varphi^{\sharp}\right\Vert ^{2}}\) which when applied to a vector “arrow” argument \({v}\) yields the number of receptacles covered by the projection of \({v}\) onto \({\varphi^{\sharp}}\), which is the value of \({\varphi(v)}\). This can be seen by recalling that \({\varphi(v)/\left\Vert \varphi^{\sharp}\right\Vert}\) is the length of the projection of \({v}\) onto \({\varphi^{\sharp}}\), so that this projection divided by the length of the receptacle \({\left\Vert \varphi^{\Uparrow}\right\Vert =1/\left\Vert \varphi^{\sharp}\right\Vert }\) recovers the value \({\varphi(v)}\). The advantage of this approach is that values can be calculated from the figure absent a length scale. Another common graphical device is to represent 1-forms as “surfaces” which are “pierced” by the arrows.
Δ The common practice of depicting a 1-form \({\varphi}\) in terms of the associated vector \({\varphi^{\Uparrow}}\) as above has consequences that can be non-intuitive. For example, doubling the value of the 1-form means halving its length in the illustration, i.e. the value of the 1-form can be viewed as the “density” of receptacles. Also, when depicting \({\varphi}\) as changing linearly, the length \({L}\) of the 1-form representation changes like \({L\mapsto L/(1+\varepsilon)}\), which doesn’t appear linear as a vector representation would, whose length changes like \({L\mapsto L(1+\varepsilon)}\).

By using the above definitions of the Lie derivative applied to vectors and 1-forms, we can extend it to tensors by using these definitions for each component. In a holonomic frame, we can obtain an expression for the Lie derivative of a tensor in terms of coordinates

\begin{aligned}L_{w}T^{\mu_{1}\dots\mu_{m}}{}_{\nu_{1}\dots\nu_{n}} & =w^{\lambda}\frac{\partial}{\partial x^{\lambda}}T^{\mu_{1}\dots\mu_{m}}{}_{\nu_{1}\dots\nu_{n}}\\
& -\sum_{j=1}^{m}\left(\frac{\partial}{\partial x^{\lambda}}w^{\mu_{j}}\right)T^{\mu_{1}\dots\mu_{j-1}\lambda\mu_{j+1}\dots\mu_{m}}{}_{\nu_{1}\dots\nu_{n}}\\
& +\sum_{j=1}^{n}\left(\frac{\partial}{\partial x^{\nu_{j}}}w^{\lambda}\right)T^{\mu_{1}\dots\mu_{m}}{}_{\nu_{1}\dots\nu_{j-1}\lambda\nu_{j+1}\dots\nu_{n}}.
\end{aligned}

An Illustrated Handbook