Lie Derivatives

Lie Derivatives evaluate the change of a tensor field along the flow defined by another vector field.

Algebraic Definition of Lie Derivative

We now give an algebraic definition. The algebraic definition for the Lie derivative of a tensor field follows from the following four axioms: Axiom 1. The Lie derivative of a function is equal to the directional derivative of the function. This fact is often expressed by the formula ${\mathcal{L}}_{Y}f=Y(f)$ Axiom 2. The Lie derivative obeys the following version of Leibniz’s rule: For any tensor fields $S$ and $T$, we have ${\mathcal{L}}_Y(S\otimes T)=({\mathcal{L}}_{Y}S)\otimes T+S\otimes ({\mathcal{L}}_{Y}T).$ Axiom 3. The Lie derivative obeys the Leibniz rule with respect to contraction: ${\mathcal{L}}_X(T(Y_1,\dots ,Y_n))=({\mathcal{L}}_{X}T)(Y_1,\dots ,Y_n)+T(({\mathcal{L}}_X{Y}_1),\dots ,Y_n)+\cdots +T(Y_1,\dots ,({\mathcal{L}}_X{Y}_n))$ Axiom 4. The Lie derivative commutes with exterior derivative on functions: $[{\mathcal{L}}_X,d]=0$

Using the first and third axioms, applying the Lie derivative ${\mathcal{L}}_X$ to $Y(f)$ shows that ${\mathcal{L}}_{X}Y(f)=X(Y(f))-Y(X(f)),$

Lie Derivative of a Function

The Lie derivative of a function $f:M\to \mathbb{R}$ with respect to a vector field $X$ at a point $p\in M$ is the function
$({\mathcal{L}}_{X}f)(p)=\frac{d}{dt}{|}_{t=0}(f\circ {\Phi }_X^t)(p)={\lim }_{t\to 0}\frac{f({\Phi }_X^t(p))-f(p)}{t}$ where ${\Phi }_X^t(p)$ is the point to which the flow defined by the vector field $X$ maps the point $p$ at time instant $t$. In the vicinity of $t=0$, ${\Phi }_X^t(p)$ is the unique solution of the system
$\frac{d}{dt}{|}_t{\Phi }_X^t(p)=X({\Phi }_X^t(p))$ of first-order autonomous (i.e. time-independent) differential equations, with ${\Phi }_X^0(p)=p$. Setting ${\mathcal{L}}_{X}f={\nabla }_{X}f$ identifies the Lie derivative of a function with the directional derivative, which is also denoted by $X(f):={\mathcal{L}}_{X}f={\nabla }_{X}f$.

Lie Derivative in terms of Flow

The Lie derivative is the speed with which the tensor field changes under the space deformation caused by the flow. Formally, given a differentiable (time-independent) vector field $X$ on a smooth manifold $M$, let ${\Phi }_X^t:M\to M$ be the corresponding local flow. Since ${\Phi }_X^t$ is a local diffeomorphism for each $t$, it gives rise to a pullback of tensor fields. For covariant tensors, this is just the multi-linear extension of the pullback map $({\Phi }_X^t{)}_p^{\ast }:T_{{\Phi }_X^t(p)}^{\ast }M\to T_p^{\ast }M,(({\Phi }_X^t{)}_p^{\ast }\alpha )(Y)=\alpha (T_p{\Phi }_X^t(Y)),\alpha \in T_{{\Phi }_X^t(p)}^{\ast }M,Y\in T_{p}M$ For contravariant tensors, one extends the inverse $(T_p{\Phi }_X^t{)}^{-1}:T_{{\Phi }_X^t(p)}M\to T_{p}M$ of the differential $T_p{\Phi }_X^t$. For every $t$, there is, consequently, a tensor field $({\Phi }_X^t{)}^{\ast }T$ of the same type as $T$‘s. If $T$ is an $(r,0)$- or $(0,s)$-type tensor field, then the Lie derivative ${\mathcal{L}}_{X}T$ of $T$ along a vector field $X$ is defined at point $p\in M$ to be ${\mathcal{L}}_{X}T(p)=\frac{d}{dt}{|}_{t=0}(({\Phi }_X^t{)}^{\ast }T{)}_p=\frac{d}{dt}{|}_{t=0}({\Phi }_X^t{)}^{\ast }{T}_{{\Phi }_X^t(p)}={\lim }_{t\to 0}\frac{({\Phi }_X^t{)}^{\ast }{T}_{{\Phi }_X^t(p)}-T_p}{t}.$ The resulting tensor field ${\mathcal{L}}_{X}T$ is of the same type as $T$‘s. More generally, for every smooth 1-parameter family ${\Phi }_t$ of diffeomorphisms that integrate a vector field $X$ in the sense that $\frac{d}{dt}{|}_{t=0}{\Phi }_t=X\circ {\Phi }_0$, one has ${\mathcal{L}}_{X}T=({\Phi }_0^{-1}{)}^{\ast }\frac{d}{dt}{|}_{t=0}{\Phi }_t^{\ast }T=-\frac{d}{dt}{|}_{t=0}({\Phi }_t^{-1}{)}^{\ast }{\Phi }_0^{\ast }T.$