Lie derivative

Defining the derivative of a function ${\displaystyle f\colon M\to {\mathbb {R} }}$ on a manifold is problematic because the difference quotient ${\displaystyle \textstyle (f(x+h)-f(x))/h}$ cannot be determined while the displacement ${\displaystyle x+h}$ is undefined.

The Lie derivative of a function ${\displaystyle f\colon M\to {\mathbb {R} }}$ with respect to a vector field ${\displaystyle X}$ at a point ${\displaystyle p\in M}$ is the function

${\displaystyle ({\mathcal {L}}_{X}f)(p)=\lim _{t\to 0}{\frac {f(P(t,p))-f(p)}{t}}\colon M\to {\mathbb {R} },}$

where ${\displaystyle P(t,p)}$ is the point to which the flow defined by the vector field ${\displaystyle X}$ maps the point ${\displaystyle p}$ at time instant ${\displaystyle t.}$ In the vicinity of ${\displaystyle t=0,}$ ${\displaystyle P(t,p)}$ is the unique solution of the system

${\displaystyle {\frac {d}{dt}}P(t,p)=X(P(t,p))}$

of first-order autonomous (i.e. time-independent) differential equations in the tangent space ${\displaystyle T_{P(t,p)}M}$, with ${\displaystyle P(0,p)=p.}$

For a coordinate chart ${\displaystyle (U,\varphi )}$ on the manifold ${\displaystyle M,}$ and ${\displaystyle x\in U,}$ let ${\displaystyle d\varphi _{x}\colon T_{x}U\to T_{\varphi (x)}{\mathbb {R} }^{n}\cong {\mathbb {R} }^{n}}$ be the tangent linear map. The above system of differential equations is more explicitly written as a system

${\displaystyle {\frac {d}{dt}}\varphi (P(t,p))=d\varphi _{P(t,p)}X(P(t,p))}$

in ${\displaystyle {\mathbb {R} }^{n},}$ with the initial condition being ${\displaystyle \varphi (P(0,p))=\varphi (p).}$ It is easily verifiable that the solution ${\displaystyle P(t,p)}$ is independent from the choice of coordinate chart.

Setting ${\displaystyle {\mathcal {L}}_{X}f=\nabla _{X}f}$ identifies the Lie derivative of a function with the directional derivative.

If X and Y are both vector fields, then the Lie derivative of Y with respect to X is also known as the Lie bracket of X and Y, and is sometimes denoted ${\displaystyle [X,Y]}$. There are several approaches to defining the Lie bracket, all of which are equivalent. We list two definitions here, corresponding to the two definitions of a vector field given above:

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 ${\displaystyle X}$ on a smooth manifold ${\displaystyle M,}$ let ${\displaystyle \Gamma _{X}^{t}:M\to M}$ be the corresponding local flow and ${\displaystyle \Gamma _{X}^{0}}$ the identity map. Since ${\displaystyle \Gamma _{X}^{t}}$ is a local diffeomorphism, for each ${\displaystyle t}$ and ${\displaystyle p\in M,}$ the inverse

${\displaystyle \left(d_{p}\Gamma _{X}^{t}\right)^{-1}:T_{\Gamma _{X}^{t}(p)}M\to T_{p}M}$

of the differential ${\displaystyle \left(d_{p}\Gamma _{X}^{t}\right)}$ extends uniquely to the homomorphism

${\displaystyle h_{p}^{t}:T\left(T_{\Gamma _{X}^{t}(p)}M\right)\to T(T_{p}M)}$

between the tensor algebras of the tangent spaces ${\displaystyle T_{\Gamma _{X}^{t}(p)}M}$ and ${\displaystyle T_{p}M.}$ Likewise, the pullback map

${\displaystyle \left(\Gamma _{X}^{t}\right)_{p}^{*}:T_{\Gamma _{X}^{t}(p)}^{*}M\to T_{p}^{*}M}$

lifts to a unique tensor algebra homomorphism

${\displaystyle h_{p}^{t}:T\left(T_{\Gamma _{X}^{t}(p)}^{*}M\right)\to T(T_{p}^{*}M).}$

For every ${\displaystyle t,}$ there is, consequently, a tensor field ${\displaystyle h_{p}^{t}Y}$ of the same valence as ${\displaystyle Y}$'s.

If ${\displaystyle Y}$ is an ${\displaystyle (r,0)}$- or ${\displaystyle (0,s)}$-type tensor field, then the Lie derivative ${\displaystyle {\cal {L}}_{X}Y}$ of ${\displaystyle Y}$ along a vector field ${\displaystyle X}$ is defined at point ${\displaystyle p\in M}$ to be

${\displaystyle {\cal {L}}_{X}Y(p)={\frac {d}{dt}}{\Biggl |}_{t=0}\left(h_{p}^{t}\left[Y\left(\Gamma _{X}^{t}(p)\right)\right]\right)=\lim _{t\to 0}{\frac {h_{p}^{t}\left[Y\left(\Gamma _{X}^{t}(p)\right)\right]-Y(p)}{t}}.}$

The resulting tensor field ${\displaystyle {\cal {L}}_{X}Y}$ has the same valence as ${\displaystyle Y}$'s.

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

${\displaystyle {\mathcal {L}}_{Y}f=Y(f)}$

Axiom 2. The Lie derivative obeys the following version of Leibniz's rule: For any tensor fields S and T, we have

${\displaystyle {\mathcal {L}}_{Y}(S\otimes T)=({\mathcal {L}}_{Y}S)\otimes T+S\otimes ({\mathcal {L}}_{Y}T).}$

Axiom 3. The Lie derivative obeys the Leibniz rule with respect to contraction:

${\displaystyle {\mathcal {L}}_{X}(T(Y_{1},\ldots ,Y_{n}))=({\mathcal {L}}_{X}T)(Y_{1},\ldots ,Y_{n})+T(({\mathcal {L}}_{X}Y_{1}),\ldots ,Y_{n})+\cdots +T(Y_{1},\ldots ,({\mathcal {L}}_{X}Y_{n}))}$

Axiom 4. The Lie derivative commutes with exterior derivative on functions:

${\displaystyle [{\mathcal {L}}_{X},d]=0}$

If these axioms hold, then applying the Lie derivative ${\displaystyle {\mathcal {L}}_{X}}$ to the relation ${\displaystyle df(Y)=Y(f)}$ shows that

${\displaystyle {\mathcal {L}}_{X}Y(f)=X(Y(f))-Y(X(f)),}$

which is one of the standard definitions for the Lie bracket.

The Lie derivative acting on a differential form is the anticommutator of the interior product with the exterior derivative. So if α is a differential form,

${\displaystyle {\mathcal {L}}_{Y}\alpha =i_{Y}d\alpha +di_{Y}\alpha .}$

This follows easily by checking that the expression commutes with exterior derivative, is a derivation (being an anticommutator of graded derivations) and does the right thing on functions.

Explicitly, let T be a tensor field of type (p, q). Consider T to be a differentiable multilinear map of smooth sections α¹, α², ..., α^p of the cotangent bundle T^∗M and of sections X₁, X₂, ..., X_q of the tangent bundle TM, written T(α¹, α², ..., X₁, X₂, ...) into R. Define the Lie derivative of T along Y by the formula

${\displaystyle ({\mathcal {L}}_{Y}T)(\alpha _{1},\alpha _{2},\ldots ,X_{1},X_{2},\ldots )=Y(T(\alpha _{1},\alpha _{2},\ldots ,X_{1},X_{2},\ldots ))}$

${\displaystyle -T({\mathcal {L}}_{Y}\alpha _{1},\alpha _{2},\ldots ,X_{1},X_{2},\ldots )-T(\alpha _{1},{\mathcal {L}}_{Y}\alpha _{2},\ldots ,X_{1},X_{2},\ldots )-\ldots }$

${\displaystyle -T(\alpha _{1},\alpha _{2},\ldots ,{\mathcal {L}}_{Y}X_{1},X_{2},\ldots )-T(\alpha _{1},\alpha _{2},\ldots ,X_{1},{\mathcal {L}}_{Y}X_{2},\ldots )-\ldots }$

The analytic and algebraic definitions can be proven to be equivalent using the properties of the pushforward and the Leibniz rule for differentiation. The Lie derivative commutes with the contraction.

A particularly important class of tensor fields is the class of differential forms. The restriction of the Lie derivative to the space of differential forms is closely related to the exterior derivative. Both the Lie derivative and the exterior derivative attempt to capture the idea of a derivative in different ways. These differences can be bridged by introducing the idea of an interior product, after which the relationships falls out as an identity known as Cartan's formula. Cartan's formula can also be used as a definition of the Lie derivative on the space of differential forms.

Let M be a manifold and X a vector field on M. Let ${\displaystyle \omega \in \Lambda ^{k+1}(M)}$ be a (k + 1)-form, i.e. for each ${\displaystyle p\in M}$, ${\displaystyle \omega (p)}$ is an alternating multilinear map from ${\displaystyle (T_{p}M)^{k+1}}$ to the real numbers. The interior product of X and ω is the k-form ${\displaystyle i_{X}\omega }$ defined as

${\displaystyle (i_{X}\omega )(X_{1},\ldots ,X_{k})=\omega (X,X_{1},\ldots ,X_{k})\,}$

The differential form ${\displaystyle i_{X}\omega }$ is also called the contraction of ω with X, and

${\displaystyle i_{X}:\Lambda ^{k+1}(M)\rightarrow \Lambda ^{k}(M)}$

is a ${\displaystyle \wedge }$-antiderivation where ${\displaystyle \wedge }$ is the wedge product on differential forms. That is, ${\displaystyle i_{X}}$ is R-linear, and

${\displaystyle i_{X}(\omega \wedge \eta )=(i_{X}\omega )\wedge \eta +(-1)^{k}\omega \wedge (i_{X}\eta )}$

for ${\displaystyle \omega \in \Lambda ^{k}(M)}$ and η another differential form. Also, for a function ${\displaystyle f\in \Lambda ^{0}(M)}$, that is, a real- or complex-valued function on M, one has

${\displaystyle i_{fX}\omega =f\,i_{X}\omega }$

where ${\displaystyle fX}$ denotes the product of f and X. The relationship between exterior derivatives and Lie derivatives can then be summarized as follows. First, since the Lie derivative of a function f with respect to a vector field X is the same as the directional derivative X(f), it is also the same as the contraction of the exterior derivative of f with X:

${\displaystyle {\mathcal {L}}_{X}f=i_{X}\,df}$

For a general differential form, the Lie derivative is likewise a contraction, taking into account the variation in X:

${\displaystyle {\mathcal {L}}_{X}\omega =i_{X}d\omega +d(i_{X}\omega ).}$

This identity is known variously as Cartan formula, Cartan homotopy formula or Cartan's magic formula. See interior product for details. The Cartan formula can be used as a definition of the Lie derivative of a differential form. Cartan's formula shows in particular that

${\displaystyle d{\mathcal {L}}_{X}\omega ={\mathcal {L}}_{X}(d\omega ).}$

The Lie derivative also satisfies the relation

${\displaystyle {\mathcal {L}}_{fX}\omega =f{\mathcal {L}}_{X}\omega +df\wedge i_{X}\omega .}$

For clarity we now show the following examples in local coordinate notation.

For a scalar field ${\displaystyle \phi (x^{c})\in {\mathcal {F}}(M)}$ we have:

${\displaystyle ({\mathcal {L}}_{X}\phi )=X(\phi )=X^{a}\partial _{a}\phi }$.

Hence for the scalar field ${\displaystyle \phi (x,y)=x^{2}-\sin(y)}$ and the vector field ${\displaystyle X=\sin(x)\partial _{y}-y^{2}\partial _{x}}$ the corresponding Lie derivative becomes

$${\displaystyle {\begin{alignedat}{3}{\mathcal {L}}_{X}\phi &=(\sin(x)\partial _{y}-y^{2}\partial _{x})(x^{2}-\sin(y))\\&=\sin(x)\partial _{y}(x^{2}-\sin(y))-y^{2}\partial _{x}(x^{2}-\sin(y))\\&=-\sin(x)\cos(y)-2xy^{2}\\\end{alignedat}}}$$

For an example of higher rank differential form, consider the 2-form ${\displaystyle \omega =(x^{2}+y^{2})dx\wedge dz}$ and the vector field ${\displaystyle X}$ from the previous example. Then,

$${\displaystyle {\begin{aligned}{\mathcal {L}}_{X}\omega &=d(i_{\sin(x)\partial _{y}-y^{2}\partial _{x}}((x^{2}+y^{2})dx\wedge dz))+i_{\sin(x)\partial _{y}-y^{2}\partial _{x}}(d((x^{2}+y^{2})dx\wedge dz))\\&=d(-y^{2}(x^{2}+y^{2})dz)+i_{\sin(x)\partial _{y}-y^{2}\partial _{x}}(2ydy\wedge dx\wedge dz)\\&=\left(-2xy^{2}dx+(-2yx^{2}-4y^{3})dy\right)\wedge dz+(2y\sin(x)dx\wedge dz+2y^{3}dy\wedge dz)\\&=\left(-2xy^{2}+2y\sin(x)\right)dx\wedge dz+(-2yx^{2}-2y^{3})dy\wedge dz\end{aligned}}}$$

Some more abstract examples.

${\displaystyle {\mathcal {L}}_{X}(dx^{b})=di_{X}(dx^{b})=dX^{b}=\partial _{a}X^{b}dx^{a}}$.

Hence for a covector field, i.e., a differential form, ${\displaystyle A=A_{a}(x^{b})dx^{a}}$ we have:

${\displaystyle {\mathcal {L}}_{X}A=X(A_{a})dx^{a}+A_{b}{\mathcal {L}}_{X}(dx^{b})=(X^{b}\partial _{b}A_{a}+A_{b}\partial _{a}(X^{b}))dx^{a}}$

The coefficient of the last expression is the local coordinate expression of the Lie derivative.

For a covariant rank 2 tensor field ${\displaystyle T=T_{ab}(x^{c})dx^{a}\otimes dx^{b}}$ we have:

$${\displaystyle {\begin{aligned}({\mathcal {L}}_{X}T)&=({\mathcal {L}}_{X}T)_{ab}dx^{a}\otimes dx^{b}\\&=X(T_{ab})dx^{a}\otimes dx^{b}+T_{cb}{\mathcal {L}}_{X}(dx^{c})\otimes dx^{b}+T_{ac}dx^{a}\otimes {\mathcal {L}}_{X}(dx^{c})\\&=(X^{c}\partial _{c}T_{ab}+T_{cb}\partial _{a}X^{c}+T_{ac}\partial _{b}X^{c})dx^{a}\otimes dx^{b}\\\end{aligned}}}$$

If ${\displaystyle T=g}$ is the symmetric metric tensor, it is parallel with respect to the Levi Civita connection (aka covariant derivative), and it becomes fruitful to use the connection. This has the effect of replacing all derivatives with covariant derivatives, giving

${\displaystyle ({\mathcal {L}}_{X}g)=(X^{c}g_{ab;c}+g_{cb}X_{;a}^{c}+g_{ac}X_{;b}^{c})dx^{a}\otimes dx^{b}=(X_{b;a}+X_{a;b})dx^{a}\otimes dx^{b}}$

A definition for Lie derivatives of spinors along generic spacetime vector fields, not necessarily Killing ones, on a general (pseudo) Riemannian manifold was already proposed in 1971 by Yvette Kosmann.[4] Later, it was provided a geometric framework which justifies her ad hoc prescription within the general framework of Lie derivatives on fiber bundles[5] in the explicit context of gauge natural bundles which turn out to be the most appropriate arena for (gauge-covariant) field theories.[6]

In a given spin manifold, that is in a Riemannian manifold ${\displaystyle (M,g)}$ admitting a spin structure, the Lie derivative of a spinor field ${\displaystyle \psi }$ can be defined by first defining it with respect to infinitesimal isometries (Killing vector fields) via the André Lichnerowicz's local expression given in 1963:[7]

${\displaystyle {\mathcal {L}}_{X}\psi$ :=X^{a}\nabla _{a}\psi -{\frac {1}{4}}\nabla _{a}X_{b}\gamma ^{a}\gamma ^{b}\psi \,,}

where ${\displaystyle \nabla _{a}X_{b}=\nabla _{[a}X_{b]}}$, as ${\displaystyle X=X^{a}\partial _{a}}$ is assumed to be a Killing vector field, and ${\displaystyle \gamma ^{a}}$ are Dirac matrices.

It is then possible to extend Lichnerowicz's definition to all vector fields (generic infinitesimal transformations) by retaining Lichnerowicz's local expression for a generic vector field ${\displaystyle X}$, but explicitly taking the antisymmetric part of ${\displaystyle \nabla _{a}X_{b}}$ only.[4] More explicitly, Kosmann's local expression given in 1972 is:[4]

${\displaystyle {\mathcal {L}}_{X}\psi$ :=X^{a}\nabla _{a}\psi -{\frac {1}{8}}\nabla _{[a}X_{b]}[\gamma ^{a},\gamma ^{b}]\psi \,=\nabla _{X}\psi -{\frac {1}{4}}(dX^{\flat })\cdot \psi \,,}

where ${\displaystyle [\gamma ^{a},\gamma ^{b}]=\gamma ^{a}\gamma ^{b}-\gamma ^{b}\gamma ^{a}}$ is the commutator, ${\displaystyle d}$ is exterior derivative, ${\displaystyle X^{\flat }=g(X,-)}$ is the dual 1 form corresponding to ${\displaystyle X}$ under the metric (i.e. with lowered indices) and ${\displaystyle \cdot }$ is Clifford multiplication.

It is worth noting that the spinor Lie derivative is independent of the metric, and hence also of the connection. This is not obvious from the right-hand side of Kosmann's local expression, as the right-hand side seems to depend on the metric through the spin connection (covariant derivative), the dualisation of vector fields (lowering of the indices) and the Clifford multiplication on the spinor bundle. Such is not the case: the quantities on the right-hand side of Kosmann's local expression combine so as to make all metric and connection dependent terms cancel.

To gain a better understanding of the long-debated concept of Lie derivative of spinor fields one may refer to the original article,[8][9] where the definition of a Lie derivative of spinor fields is placed in the more general framework of the theory of Lie derivatives of sections of fiber bundles and the direct approach by Y. Kosmann to the spinor case is generalized to gauge natural bundles in the form of a new geometric concept called the Kosmann lift.

If we have a principal bundle over the manifold M with G as the structure group, and we pick X to be a covariant vector field as section of the tangent space of the principal bundle (i.e. it has horizontal and vertical components), then the covariant Lie derivative is just the Lie derivative with respect to X over the principal bundle.

Now, if we're given a vector field Y over M (but not the principal bundle) but we also have a connection over the principal bundle, we can define a vector field X over the principal bundle such that its horizontal component matches Y and its vertical component agrees with the connection. This is the covariant Lie derivative.

See connection form for more details.

Another generalization, due to Albert Nijenhuis, allows one to define the Lie derivative of a differential form along any section of the bundle Ω^k(M, TM) of differential forms with values in the tangent bundle. If K ∈ Ω^k(M, TM) and α is a differential p-form, then it is possible to define the interior product i_Kα of K and α. The Nijenhuis–Lie derivative is then the anticommutator of the interior product and the exterior derivative:

${\displaystyle {\mathcal {L}}_{K}\alpha =[d,i_{K}]\alpha =di_{K}\alpha -(-1)^{k-1}i_{K}\,d\alpha .}$