Lie derivative

In mathematics, the Lie derivative /ˈliː/, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar function, vector field and one-form), along the flow of another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold.

Functions, tensor fields and forms can be differentiated with respect to a vector field. Since a vector field is a derivation of zero degree on the algebra of smooth functions, the Lie derivative of a function $f\,$ along a vector field $X\,$ is the evaluation $X(f)\,$, i.e., is simply the application of the vector field. The process of Lie differentiation extends to a derivation of zero degree on the algebra of tensor fields over a manifold M. It also commutes with contraction and the exterior derivative on differential forms. This uniquely determines the Lie derivative and it follows that for vector fields the Lie derivative is the commutator

$\mathcal{L}_X(Y) = [X, Y]$

It also shows that the Lie derivatives on M are an infinite-dimensional Lie algebra representation of the Lie algebra of vector fields with the Lie bracket defined by the commutator,

$\mathcal{L}_{[X,Y]} = [\mathcal{L}_X ,\mathcal{L}_{Y}].$

Considering vector fields as infinitesimal generators of flows (active diffeomorphisms) on M, the Lie derivatives are the infinitesimal representation of the representation of the diffeomorphism group on tensor fields, analogous to Lie algebra representations as infinitesimal representations associated to group representation in Lie group theory.

Generalisations exist for spinor fields, fibre bundles with connection and vector-valued differential forms.

Definition

The Lie derivative may be defined in several equivalent ways. In this section, to keep things simple, we begin by defining the Lie derivative acting on scalar functions and vector fields. The Lie derivative can also be defined to act on general tensors, as developed later in the article.

The Lie derivative of a function

Note: the Einstein summation convention of summing on repeated indices is used below.

There are several equivalent definitions of a Lie derivative of a function.

• The Lie derivative can be defined in terms of the definition of vector fields as first order differential operators. Given a function ƒ : M → R and a vector field X defined on M, the Lie derivative $\mathcal{L}_X f$of a function ƒ along a vector field $X$ is simply the application of the vector field. It can be interpreted as the directional derivative of f along X. Hence at a point p ∈ M we have

$(\mathcal{L}_{\!X} f)(p) \triangleq X_p(f) \triangleq (Xf)(p).$

By the definition of the differential of a function on M the definition can also be written as

$(\mathcal{L}_{\!X} f)(p) \triangleq \operatorname{d}f_p\, (X_p).$

Choosing local coordinates x^a, and writing :$X=X^a\partial_a$, where the $\partial_a = \frac{\partial}{\partial x^a}$ are local basis vectors for the tangent bundle $TM$, we have locally

$(\mathcal{L}_{\!X} f)(p) =X^a(p)(\partial_a f)(p).$

Likewise $\operatorname{d}f : M \to T^*M$ is the 1-form locally given by $\operatorname{d}f \triangleq \partial_a f \operatorname{d}x^a$. which implies

$(\mathcal{L}_{\!X} f)(p) = \operatorname{d}f_p\, (X_p)= X^a(p)(\partial_b f)(p)\, dx^b(\partial_a) = X^a(p)(\partial_a f)(p)$

recovering the original definition.

• Alternatively, the Lie derivative can be defined as

$\left. (\mathcal{L}_{\!X} f)(p) \triangleq \frac{\operatorname{d}}{\operatorname{d}t} f(\gamma(t)) \right\vert_{t=0}$

where $\gamma(t)$ is any curve on M with $\gamma(0)=p$ and $\gamma'(0)=X_p$ . One such curve is the flow of $X$ through p: the solution of the first-order ordinary differential equation $\frac{d}{dt}\gamma(t)=X_{\gamma(t)}$ , which exists by the Picard–Lindelöf theorem (see also the Frobenius theorem).

The Lie derivative of a vector field

The Lie derivative can be defined for vector fields by first defining the Lie bracket $[X,Y]$ of a pair of vector fields X and Y. There are several approaches to defining the Lie bracket, all of which are equivalent. Regardless of the chosen definition, one then defines the Lie derivative of the vector field Y to be equal to the Lie bracket of X and Y, that is,

$\mathcal{L}_X Y = [X,Y]. \,$

Other equivalent definitions are (here, $\Phi^X_{t}$ is the flow transformation and d the tangent map derivative operator):

$(\mathcal{L}_X Y)_x := \lim_{t \to 0}\frac{(\mathrm{d}\Phi^X_{-t}) Y_{\Phi^X_t(x)} - Y_x}t = \left.\frac{\mathrm{d}}{\mathrm{d} t}\right|_{t=0} (\mathrm{d}\Phi^X_{-t}) Y_{\Phi^X_t(x)}$

$\mathcal{L}_X Y := \left.\frac12\frac{\mathrm{d}^2}{\mathrm{dt}^2}\right|_{t=0} \Phi^Y_{-t} \circ \Phi^X_{-t} \circ \Phi^Y_{t} \circ \Phi^X_{t} = \left.\frac{\mathrm{d}}{\mathrm{d} t}\right|_{t=0} \Phi^Y_{-\sqrt{t}} \circ \Phi^X_{-\sqrt{t}} \circ \Phi^Y_{\sqrt{t}} \circ \Phi^X_{\sqrt{t}}.\,$

The Lie derivative of differential forms

The Lie derivative can also be defined on differential forms. In this context, it 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 antiderivation or equivalently an interior product, after which the relationships fall out as a set of identities.

Let M be a manifold and X a vector field on M. Let $\omega \in \Lambda^{k+1}(M)$ be a (k + 1)-form. The interior product of X and ω is the k-form $i_X\omega$ defined as

$(i_X\omega) (X_1, \ldots, X_k) = \omega (X,X_1, \ldots, X_k)\,$

The differential form $i_X\omega$ is also called the contraction of ω with X. Note that

$i_X:\Lambda^{k+1}(M) \rightarrow \Lambda^k(M) \,$

and that $i_X$ is a $\wedge$-antiderivation. That is, $i_X$ is R-linear, and

$i_X (\omega \wedge \eta) = (i_X \omega) \wedge \eta + (-1)^k \omega \wedge (i_X \eta)$

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

$i_{fX} \omega = f\,i_X\omega$

where $f X$ denotes the product of f and X. The relationship between exterior derivatives and Lie derivatives can then be summarized as follows. As discussed in a previous section, the Lie derivative of an ordinary function f is just the contraction of the exterior derivative with the vector field X:

$\mathcal{L}_Xf = i_X \, df$

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

$\mathcal{L}_X\omega = i_Xd\omega + d(i_X \omega).$

This identity is known variously as "Cartan's formula" or "Cartan's magic formula," and shows in particular that:

$d\mathcal{L}_X\omega = \mathcal{L}_X(d\omega).$

The derivative of products is distributed:

$\mathcal{L}_{fX}\omega = f\mathcal{L}_X\omega + df \wedge i_X \omega$

Properties

The Lie derivative has a number of properties. Let $\mathcal{F}(M)$ be the algebra of functions defined on the manifold M. Then

$\mathcal{L}_X : \mathcal{F}(M) \rightarrow \mathcal{F}(M)$

is a derivation on the algebra $\mathcal{F}(M)$. That is, $\mathcal{L}_X$ is R-linear and

$\mathcal{L}_X(fg)=(\mathcal{L}_Xf) g + f\mathcal{L}_Xg.$

Similarly, it is a derivation on $\mathcal{F}(M) \times \mathcal{X}(M)$ where $\mathcal{X}(M)$ is the set of vector fields on M:

$\mathcal{L}_X(fY)=(\mathcal{L}_Xf) Y + f\mathcal{L}_X Y$

which may also be written in the equivalent notation

$\mathcal{L}_X(f\otimes Y)= (\mathcal{L}_Xf) \otimes Y + f\otimes \mathcal{L}_X Y$

where the tensor product symbol $\otimes$ is used to emphasize the fact that the product of a function times a vector field is being taken over the entire manifold.

Additional properties are consistent with that of the Lie bracket. Thus, for example, considered as a derivation on a vector field,

$\mathcal{L}_X [Y,Z] = [\mathcal{L}_X Y,Z] + [Y,\mathcal{L}_X Z]$

one finds the above to be just the Jacobi identity. Thus, one has the important result that the space of vector fields over M, equipped with the Lie bracket, forms a Lie algebra.

The Lie derivative also has important properties when acting on differential forms. Let α and β be two differential forms on M, and let X and Y be two vector fields. Then

• $\mathcal{L}_X(\alpha\wedge\beta) = (\mathcal{L}_X\alpha) \wedge\beta + \alpha\wedge (\mathcal{L}_X\beta)$
• $[\mathcal{L}_X,\mathcal{L}_Y]\alpha:= \mathcal{L}_X\mathcal{L}_Y\alpha-\mathcal{L}_Y\mathcal{L}_X\alpha=\mathcal{L}_{[X,Y]}\alpha$
• $[\mathcal{L}_X,i_Y]\alpha=[i_X,\mathcal{L}_Y]\alpha=i_{[X,Y]}\alpha,$ where i denotes interior product defined above and it's clear whether [·,·] denotes the commutator or the Lie bracket of vector fields.

Lie derivative of tensor fields

More generally, if we have a differentiable tensor field T of rank $(q,r)$ and a differentiable vector field Y (i.e. a differentiable section of the tangent bundle TM), then we can define the Lie derivative of T along Y. Let, for some open interval I around 0, φ:M×I → M be the one-parameter semigroup of local diffeomorphisms of M induced by the vector flow of Y and denote φ_t(p) := φ(p, t). For each sufficiently small t, φ_t is a diffeomorphism from a neighborhood in M to another neighborhood in M, and φ₀ is the identity diffeomorphism. The Lie derivative of T is defined at a point p by

$(\mathcal{L}_Y T)_p=\left.\frac{d}{dt}\right|_{t=0}\left((\varphi_{-t})_*T_{\varphi_{t}(p)}\right)=\left.\frac{d}{dt}\right|_{t=0}\left((\varphi_{t})^*T\right)_p.$

where $(\varphi_t)_*$ is the pushforward along the diffeomorphism and $(\varphi_t)^*$ is the pullback along the diffeomorphism. Intuitively, if you have a tensor field $T$ and a vector field Y, then $\mathcal{L}_{Y} T$ is the infinitesimal change you would see when you flow $T$ using the vector field −Y, which is the same thing as the infinitesimal change you would see in $T$ if you yourself flowed along the vector field Y.

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 the directional derivative of the function. So if f is a real valued function on M, then

$\mathcal{L}_Yf=Y(f)$

Axiom 2. The Lie derivative obeys the Leibniz rule. For any tensor fields S and T, we have

$\mathcal{L}_Y(S\otimes T)=(\mathcal{L}_YS)\otimes T+S\otimes (\mathcal{L}_YT).$

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

$\mathcal{L}_X (T(Y_1, \ldots, Y_n)) = (\mathcal{L}_X T)(Y_1,\ldots, Y_n) + T((\mathcal{L}_X Y_1), \ldots, Y_n) + \cdots + T(Y_1, \ldots, (\mathcal{L}_X Y_n))$

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

$[\mathcal{L}_X, d] = 0$

Taking the Lie derivative of the relation $df(Y) = Y(f)$ then easily shows that that the Lie derivative of a vector field is the Lie bracket. So if X is a vector field, one has

$\mathcal{L}_YX=[Y,X].$

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

$\mathcal{L}_Y\alpha=i_Yd\alpha+di_Y\alpha.$

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^q of the cotangent bundle T*M and of sections X₁, X₂, ... X_p of the tangent bundle TM, written T(α¹, α², ..., X₁, X₂, ...) into R. Define the Lie derivative of T along Y by the formula

$(\mathcal{L}_Y T)(\alpha_1, \alpha_2, \ldots, X_1, X_2, \ldots) =Y(T(\alpha_1,\alpha_2,\ldots,X_1,X_2,\ldots))$

$- T(\mathcal{L}_Y\alpha_1, \alpha_2, \ldots, X_1, X_2, \ldots) - T(\alpha_1, \mathcal{L}_Y\alpha_2, \ldots, X_1, X_2, \ldots) -\ldots$

$- T(\alpha_1, \alpha_2, \ldots, \mathcal{L}_YX_1, X_2, \ldots) - T(\alpha_1, \alpha_2, \ldots, X_1, \mathcal{L}_YX_2, \ldots) - \ldots$

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

Coordinate expressions

In local coordinate notation, for a type (r,s) tensor field $T$, the Lie derivative along $X$ is

$\begin{align} (\mathcal{L}_X T) ^{a_1 \ldots a_r}{}_{b_1 \ldots b_s} = & X^c(\partial_c T^{a_1 \ldots a_r}{}_{b_1 \ldots b_s}) \\ & - (\partial_c X ^{a_1}) T ^{c a_2 \ldots a_r}{}_{b_1 \ldots b_s} - \ldots - (\partial_c X^{a_r}) T ^{a_1 \ldots a_{r-1}c}{}_{b_1 \ldots b_s} \\ & + (\partial_{b_1} X^c) T ^{a_1 \ldots a_r}{}_{c b_2 \ldots b_s} + \ldots + (\partial_{b_s}X^c) T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_{s-1} c} \end{align}$

here, the notation $\partial_a = \frac{\partial}{\partial x^a}$ means taking the partial derivative with respect to the coordinate $x^a$. Alternatively, if we are using a torsion-free connection (e.g. the Levi Civita connection), then the partial derivative $\partial_a$ can be replaced with the covariant derivative $\nabla_a$. The Lie derivative of a tensor is another tensor of the same type, i.e. even though the individual terms in the expression depend on the choice of coordinate system, the expression as a whole results in a tensor

$(\mathcal{L}_X T) ^{a_1 \ldots a_r}{}_{b_1 \ldots b_s}\partial_{a_1}\otimes\cdots\otimes\partial_{a_r}\otimes dx^{b_1}\otimes\cdots\otimes dx^{b_s}$

which is independent of any coordinate system.

The definition can be extended further to tensor densities of weight w for any real w. If T is such a tensor density, then its Lie derivative is a tensor density of the same type and weight.

$(\mathcal {L}_X T) ^{a_1 \ldots a_r}{}_{b_1 \ldots b_s} = X^c(\partial_c T^{a_1 \ldots a_r}{}_{b_1 \ldots b_s}) - (\partial_c X ^{a_1}) T ^{c a_2 \ldots a_r}{}_{b_1 \ldots b_s} - \ldots - (\partial_c X^{a_r}) T ^{a_1 \ldots a_{r-1}c}{}_{b_1 \ldots b_s} +$

$+ (\partial_{b_1} X^c) T ^{a_1 \ldots a_r}{}_{c b_2 \ldots b_s} + \ldots + (\partial_{b_s} X^c) T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_{s-1} c} + w (\partial_{c} X^c) T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_{s}}$

Notice the new term at the end of the expression.

Examples

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

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

$(\mathcal {L}_X \phi)=X^a \partial_a \phi$

For a covector field, i.e., a differential form, $A=A_a(x^b)dx^a$ we have:

$(\mathcal {L}_X A)_a=X^b \partial_b A_a+A_b\partial_a X^b$

For a covariant symmetric tensor field $g=g_{ab}(x^c)dx^a\otimes dx^b$ we have:

$(\mathcal {L}_X g)_{ab}=X^c \partial_c g_{ab}+g_{cb}\partial_a X^c+g_{ca}\partial_b X^c$

Generalizations

Various generalizations of the Lie derivative play an important role in differential geometry.

The Lie derivative of a spinor field

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 1972 by Yvette Kosmann. Later, it was provided a geometric framework which justifies her ad hoc prescription within the general framework of Lie derivatives on fiber bundles in the explicit context of gauge natural bundles which turn out to be the most appropriate arena for (gauge-covariant) field theories.

In a given spin manifold, that is in a Riemannian manifold $(M,g)$ admitting a spin structure, the Lie derivative of a spinor field $\psi$ 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:

$\mathcal{L}_X \psi := X^{a}\nabla_{a}\psi -\frac14\nabla_{a}X_{b} \gamma^{a}\,\gamma^{b}\psi\, ,$

where $\nabla_{a}X_{b}=\nabla_{[a}X_{b]}$, as $X=X^{a}\partial_{a}$ is assumed to be a Killing vector field, and $\gamma^{a}$ 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 $X$, but explicitly taking the antisymmetric part of $\nabla_{a}X_{b}$ only. More explicitly, Kosmann's local expression given in 1972 is:

$\mathcal{L}_X \psi := X^{a}\nabla_{a}\psi -\frac18\nabla_{[a}X_{b]} [\gamma^{a},\gamma^{b}]\psi\, = \nabla_X \psi - \frac14 (d X^\flat)\cdot \psi\, ,$

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

Covariant Lie derivative

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.

Nijenhuis–Lie derivative

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:

$\mathcal{L}_K\alpha=[d,i_K]\alpha = di_K\alpha-(-1)^{k-1}i_K \, d\alpha.$

History

In 1931, Władysław Ślebodziński introduced a new differential operator, later called by David van Dantzig that of Lie derivation, which can be applied to scalars, vectors, tensors and affine connections and which proved to be a powerful instrument in the study of groups of automorphisms.

The Lie derivatives of general geometric objects (i.e., sections of natural fiber bundles) were studied by A. Nijenhuis, Y. Tashiro and K. Yano.

For a quite long time, physicists had been using Lie derivatives, without reference to the work of mathematicians. In 1940, Léon Rosenfeld—and before him Wolfgang Pauli—introduced what he called a ‘local variation’ $\delta^{\ast}A$ of a geometric object $A\,$ induced by an infinitesimal transformation of coordinates generated by a vector field $X\,$. One can easily prove that his $\delta^{\ast}A$ is $- \mathcal{L}_X(A)\,$.

See also

• Covariant derivative
• Connection (mathematics)
• Frölicher–Nijenhuis bracket
• Geodesic
• Killing field
• Derivative of the exponential map

References

• See section 2.2.
• See Chapter 0.
• See section 1.6.
• Extensive discussion of Lie brackets, and the general theory of Lie derivatives.
• For generalizations to infinite dimensions.
• For generalizations to infinite dimensions.
• Classical approach using coordinates.