Differential geometry provides a number of ways of extending the familiar notion of the derivative of a real-valued function to enable differentiation of various types of tensor fields that `live’ on manifolds, such as scalars, contravariant vectors, one-forms (also known as covariant vectors) and mixed tensors. The problem that needs to be overcome in such cases is the fact that partial differentiation of tensors is generally not tensorial, i.e., the result is not itself a tensor. The reason is that the process of differentiation involves subtracting tensors living in different tangent spaces on a curved manifold, so their difference does not transform in the same way as either of the tensors individually. For example, consider a contravariant vector field . In the tangent space at point the transformation law for is
whereas in the tangent space at point the transformation law is
If we imagine these two tangent spaces at points and on the manifold separated by distance , the derivative would involve computing
and the difference in the numerator will clearly not transform like either of the contravariant vectors individually since their transformation matrices are evaluated at different points. The derivative will not therefore be a tensor itself.
The usual way of getting around this is to introduce some kind of auxiliary field on to the manifold which provides a link between the two tensors, thus enabling them to transform in the same way (with respect to the auxiliary field) when they are subtracted. In the present note I want to explore the particular concept of the Lie derivative of a tensor field (named after the mathematician Marius Sophus Lie, 1842-1899) which employs this auxiliary field approach by introducing a contravariant vector field on to the manifold. To this end, suppose we define a vector field . This can be used to define streamlines in the manifold (also called a congruence of curves) as the solutions of the ordinary differential equations
where is a parameter determining the position on a given streamline. The equations encapsulate the fact that each point on a streamline has a tangent vector belonging to the vector field.
Example Using the notation to denote the -th coordinate, suppose the vector field is . Then , and the streamlines for this vector field are obtained by solving simultaneously the differential equations
Solving the first equation gives
Solving the second equation gives
Using the solution of the first equation to substitute for in the solution of the second one we get
Therefore the streamlines of the vector field are the graphs of the equation
where is a constant. Some of the streamlines are shown in the figure below.
Now suppose we want to find the Lie derivative of a tensor field, , using the vector field . The essential idea is to use the streamlines of the vector field to link the tensor at some point , , with the tensor at some neighbouring point , , in such a way that the two will have the same transformation matrix at point (with respect to the auxiliary vector field). We can then subtract the two tensors at and so define the derivative at by a limiting process as tends to . In all such cases, the technique begins by considering a coordinate transformation from to of the form
where is arbitrarily small. The point with coordinates lies on the streamline through which the vector field generates. Differentiating the coordinate transformation we get
where is the Kronecker delta and . What we now do is consider the effect of the above coordinate transformation on the tensor field at the points and . In what follows I will employ this general procedure to obtain the Lie derivative formulas with respect to a contravariant vector field in the case of a scalar field , a contravariant vector field , a one-form field , and a general mixed tensor field .
The Lie derivative of a scalar field
Not surprisingly this is the easiest case to deal with since scalars are invariants: the values of a scalar field defined over a manifold do not change under a change in the coordinate system being used. The value of the scalar field at the point will simply be and the value at the point will be
We can expand this in a Taylor series about the point with coordinates to get the first-order approximation
The Lie derivative of the scalar field with respect to the contravariant vector field is then
We observe that the Lie derivative of the scalar field with respect to the vector field is actually the directional derivative of in the direction of the vector . In the differential geometry literature in this area it is common to associate the contravariant vector field with the linear differential operator (which operates on any real-valued function to produce another function ) and essentially treat them as the same object. Given a point on the manifold, one thinks of the partial differential operators as constituting a basis for all the vectors in the tangent space at , so that any vector at can be written as a linear combination of the in the form
This is the intuitive justification for treating the vector field and the linear differential operator as being the same things. Under this convention, one often sees the Lie derivative of a scalar field with respect to the contravariant vector field written as
The Lie derivative of a contravariant vector field
Under the coordinate transformation from to given by
the contravariant vector field at is mapped to
The vector already at , namely , has a first-order Taylor series approximation about of the form
The Lie derivative with respect to the vector field is then given by
Under the convention of associating the vector field with the linear differential operator , one often sees the Lie derivative of a contravariant vector field with respect to the field written as
where is called the Lie bracket (or commutator) of the two vector fields and . This is a new vector field (and therefore linear differential operator) that can be written alternatively as
where the last equality follows from the fact that the second term in the penultimate line will always vanish by Young’s Theorem (equality of cross-partials). Therefore the -th component of the vector field is the one that appears in the expression of the Lie derivative above.
The Lie derivative of a one-form (covariant vector) field
Under the coordinate transformation from to given by
the one-form (covariant vector) field at is mapped to
To work out the transformation matrix here we need to write the coordinate transformation as
Partially differentiating we get
We can ignore the terms as they will disappear in the limiting process of the differentiation, so we have
Again taking a first-order Taylor series approximation about at the point we get that
Then the Lie derivative of the one-form field with respect to the contravariant vector field is obtained as
The Lie derivative of a mixed tensor field
A good prototype for this case is the Lie derivative of the simplest type of mixed tensor field, the rank-2 tensor of type represented as . We will therefore consider this case first and then use it to extrapolate to a general mixed tensor field of type represented as .
Under the coordinate transformation
the mixed tensor field transforms as
Under a first-order Taylor series approximation about , the tensor at can be written
The Lie derivative of with respect to the contravariant vector field is then
We observe that the contravariant index contributes a term of the form while the covariant index contributes a term of the form .
Now consider the general mixed tensor . The first-order Taylor series approximation of about gives
Therefore the first-term of the Lie derivative will be . This is of the same form as the first term of . Conveniently, it turns out that from then on each contravariant and covariant index in will contribute terms like the corresponding terms we saw above in . Therefore the Lie derivative of the general mixed tensor field with respect to the contravariant vector field will be of the form