Lie derivative

Motivation

A "naive" attempt to define the derivative of a tensor field with respect to a vector field would be to take the directional derivative of the components of the tensor field with respect to the vector field. However, this definition is undesirable because it is not invariant under coordinate transformations, and is thus meaningless when considered on an abstract manifold. In differential geometry, there are two main notions of differentiation (of arbitrary tensor fields) that are invariant under coordinate transformations: Lie derivatives, and derivatives with respect to connections. The main difference between these is that taking a derivative with respect to a connection requires an additional geometric structure (e.g. a Riemannian metric or just an abstract connection) on the manifold, but the derivative of a tensor field with respect to a tangent vector is well-defined even if it is not specified how to extend that tangent vector to a vector field; by contrast, when taking a Lie derivative, no additional information about the manifold is needed, but it is impossible to talk about the Lie derivative of a tensor field with respect to a single tangent vector, since the value of the Lie derivative of a tensor field with respect to a vector field X at a point p depends on the value of X in a neighborhood of p, not just at p itself.

Definition

The Lie derivative may be defined in several equivalent ways. To keep things simple, we begin by defining the Lie derivative acting on scalar functions and vector fields, before moving on to the definition for general tensors.

The (Lie) derivative of a function

The directional derivative of a function with respect to a vector field is one of the most fundamental concepts in differential geometry, and the Lie derivative of a function is simply defined to be the directional derivative of the function. The precise definition of the directional derivative depends on what formalization one is using for vector fields. Two common formalizations exist:

A vector field on a manifold M can be defined as a function that inputs a point p of M and outputs an element of the tangent space of M at p. Equivalently, a vector field is a section of the tangent bundle of M. In the case where M is a Euclidean space, this definition is equivalent to saying that a vector field is a function that inputs points and outputs vectors.

In this formalization, the directional derivative of a function can be defined using local coordinates as follows: the directional derivative of f with respect to a vector field X at a point p is the number ∂ X f ( p ) = lim t → 0 f ( p + t X ( p ) ) − f ( p ) t The chain rule shows that this definition is independent of the choice of coordinate system. There is also a coordinate-free definition that uses the notion of the differential of a function between manifolds: the directional derivative of f with respect to X at p is ∂ X f ( p ) ≜ d d ⁡ t f ( γ ( t ) ) | t = 0 where γ ( t ) is any curve on M with γ ( 0 ) = p and γ ′ ( 0 ) = X p , where γ ′ denotes the differential of γ . However, this definition is not completely coordinate-free because coordinates are necessary in order to define the differential of a function.

Another definition is that a vector field on a manifold M is a derivation of degree zero on the algebra of smooth functions on M. This definition is usually motivated in terms of the first definition: if X is a vector field according to the first definition, then the map sending a smooth function f to its derivative with respect to X is a vector field according to the second definition. Although it is less intuitively clear than the first definition, the second definition has the advantage that it often easier to work with. In particular, it is much simpler to define the directional derivative of a function using this definition: the directional derivative of f with respect to the vector field X is simply the value X(f) that results from inputting f into X.

The Lie derivative of a vector field

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 [ 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 bracket of X and Y at p is given in local coordinates by the formula

L X Y ( p ) = [ X , Y ] ( p ) = ∂ X Y ( p ) − ∂ Y X ( p ) ,

where ∂ X and ∂ Y denote the operations of taking the directional derivatives with respect to X and Y, respectively. Here we are treating a vector in n-dimensional space as an n-tuple, so that its directional derivative is simply the tuple consisting of the directional derivatives of its coordinates. Note that although the final expression ∂ X Y ( p ) − ∂ Y X ( p ) appearing in this definition does not depend on the choice of local coordinates, the individual terms ∂ X Y ( p ) and ∂ Y X ( p ) do depend on the choice of coordinates.

If X and Y are vector fields on a manifold M according to the second definition, then the operator L X Y = [ X , Y ] defined by the formula

is a derivation of order zero of the algebra of smooth functions of M, i.e. this operator is a vector field according to the second definition.

The Lie derivative of a tensor field

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

( L Y T ) p = d d t | t = 0 ( ( φ − t ) ∗ T φ t ( p ) ) = d d t | t = 0 ( ( φ t ) ∗ T ) p .

where ( φ t ) ∗ is the pushforward along the diffeomorphism and ( φ t ) ∗ is the pullback along the diffeomorphism. Intuitively, if you have a tensor field T and a vector field Y, then 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 equal to the directional derivative of the function. This fact is often expressed by the formula 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 L Y ( S ⊗ T ) = ( L Y S ) ⊗ T + S ⊗ ( L Y T ) . Axiom 3. The Lie derivative obeys the Leibniz rule with respect to contraction: L X ( T ( Y 1 , … , Y n ) ) = ( L X T ) ( Y 1 , … , Y n ) + T ( ( L X Y 1 ) , … , Y n ) + ⋯ + T ( Y 1 , … , ( L X Y n ) ) Axiom 4. The Lie derivative commutes with exterior derivative on functions: [ L X , d ] = 0

If these axioms hold, then applying the Lie derivative L X to the relation d f ( Y ) = Y ( f ) shows that

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

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,

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

( L Y T ) ( α 1 , α 2 , … , X 1 , X 2 , … ) = Y ( T ( α 1 , α 2 , … , X 1 , X 2 , … ) ) − T ( L Y α 1 , α 2 , … , X 1 , X 2 , … ) − T ( α 1 , L Y α 2 , … , X 1 , X 2 , … ) − … − T ( α 1 , α 2 , … , L Y X 1 , X 2 , … ) − T ( α 1 , α 2 , … , X 1 , L Y X 2 , … ) − …

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.

The Lie derivative of a differential form

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. Note that these 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 ω ∈ Λ k + 1 ( M ) be a (k + 1)-form, i.e. for each p ∈ M , ω ( p ) is an alternating multilinear map from ( T p M ) k + 1 to the real numbers. The interior product of X and ω is the k-form i X ω defined as

( i X ω ) ( X 1 , … , X k ) = ω ( X , X 1 , … , X k )

The differential form i X ω is also called the contraction of ω with X. Note that

i X : Λ k + 1 ( M ) → Λ k ( M )

and that i X is a ∧ -antiderivation. That is, i X is R-linear, and

i X ( ω ∧ η ) = ( i X ω ) ∧ η + ( − 1 ) k ω ∧ ( i X η )

for ω ∈ Λ k ( M ) and η another differential form. Also, for a function f ∈ Λ 0 ( M ) , that is, a real- or complex-valued function on M, one has

i f X ω = f i X ω

where f X 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:

L X f = i X d f

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

L X ω = i X d ω + d ( i X ω ) .

This identity is known variously as "Cartan's formula" or "Cartan's magic formula", and can be used as a definition of the Lie derivative of a differential form. Cartan's formula shows in particular that

d L X ω = L X ( d ω ) .

The Lie derivative also satisfies the relation

L f X ω = f L X ω + d f ∧ i X ω .

Coordinate expressions

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

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

( L X T ) a 1 … a r b 1 … b s = X c ( ∂ c T a 1 … a r b 1 … b s ) − ( ∂ c X a 1 ) T c a 2 … a r b 1 … b s − … − ( ∂ c X a r ) T a 1 … a r − 1 c b 1 … b s + ( ∂ b 1 X c ) T a 1 … a r c b 2 … b s + … + ( ∂ b s X c ) T a 1 … a r b 1 … b s − 1 c

here, the notation ∂ a = ∂ ∂ 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 ∂ a can be replaced with the covariant derivative ∇ 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

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.

( L X T ) a 1 … a r b 1 … b s = X c ( ∂ c T a 1 … a r b 1 … b s ) − ( ∂ c X a 1 ) T c a 2 … a r b 1 … b s − … − ( ∂ c X a r ) T a 1 … a r − 1 c b 1 … b s + + ( ∂ b 1 X c ) T a 1 … a r c b 2 … b s + … + ( ∂ b s X c ) T a 1 … a r b 1 … b s − 1 c + w ( ∂ c X c ) T a 1 … a r b 1 … 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 ϕ ( x c ) ∈ F ( M ) we have:

( L X ϕ ) = X a ∂ a ϕ

So less abstractly, consider the scalar field ϕ ( x , y ) = x 2 − sin ⁡ ( y ) and the vector field X = sin ⁡ ( x ) ∂ y − y 2 ∂ x . The corresponding Lie derivative evaluates as

A = A a ( x b ) d x a ( L X A ) a = X b ∂ b A a + A b ∂ a ( X b )

Concretely, consider the 2-form ω = ( x 2 + y 2 ) d x ∧ d z and the vector field X from the previous example. Then,

g = g a b ( x c ) d x a ⊗ d x b ( L X g ) a b = X c ∂ c g a b + g c b ∂ a X c + g c a ∂ b X c

Properties

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

L X : F ( M ) → F ( M )

is a derivation on the algebra F ( M ) . That is, L X is R-linear and

L X ( f g ) = ( L X f ) g + f L X g .

Similarly, it is a derivation on F ( M ) × X ( M ) where X ( M ) is the set of vector fields on M:

L X ( f Y ) = ( L X f ) Y + f L X Y

which may also be written in the equivalent notation

L X ( f ⊗ Y ) = ( L X f ) ⊗ Y + f ⊗ L X Y

where the tensor product symbol ⊗ 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,

L X [ Y , Z ] = [ L X Y , Z ] + [ Y , 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

L X ( α ∧ β ) = ( L X α ) ∧ β + α ∧ ( L X β )

[ L X , L Y ] α := L X L Y α − L Y L X α = L [ X , Y ] α

[ L X , i Y ] α = [ i X , L Y ] α = i [ X , Y ] α , where i denotes interior product defined above and it's clear whether [·,·] denotes the commutator or the Lie bracket of vector fields.

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 ψ 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:

L X ψ := X a ∇ a ψ − 1 4 ∇ a X b γ a γ b ψ ,

where ∇ a X b = ∇ [ a X b ] , as X = X a ∂ a is assumed to be a Killing vector field, and γ 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 ∇ a X b only. More explicitly, Kosmann's local expression given in 1972 is:

L X ψ := X a ∇ a ψ − 1 8 ∇ [ a X b ] [ γ a , γ b ] ψ = ∇ X ψ − 1 4 ( d X ♭ ) ⋅ ψ ,

where [ γ a , γ b ] = γ a γ b − γ b γ a is the commutator, d is exterior derivative, X ♭ = g ( X , − ) is the dual 1 form corresponding to X under the metric (i.e. with lowered indices) and ⋅ 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, 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:

L K α = [ d , i K ] α = d i K α − ( − 1 ) k − 1 i K d α .

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’ δ ∗ 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 δ ∗ A is − L X ( A ) .