# On Lie derivatives of tensor fields

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 $X^a$. In the tangent space at point $P$ the transformation law for $X^a$ is

$X^{\prime a} = \big[\frac{\partial x^{\prime a}}{\partial x^b}\big]_P X^b$

whereas in the tangent space at point $Q$ the transformation law is

$X^{\prime a} = \big[\frac{\partial x^{\prime a}}{\partial x^b}\big]_Q X^b$

If we imagine these two tangent spaces at points $P$ and $Q$ on the manifold separated by distance $\delta u$, the derivative would involve computing

$\lim_{\delta u \rightarrow 0} \frac{[X^a]_P - [X^a]_Q}{\delta u}$

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 $X^a (x)$. This can be used to define streamlines in the manifold (also called a congruence of curves) as the solutions of the ordinary differential equations

$\frac{dx^a}{du} = X^a (x(u))$

where $u$ 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 $x^i$ to denote the $i$-th coordinate, suppose the vector field is $X = (1, x^2(u))$. Then $X^1 = 1$, $X^2 = x^2$ and the streamlines for this vector field are obtained by solving simultaneously the differential equations

$\frac{dx^1}{du} = 1$

$\frac{dx^2}{du} = x^2$

Solving the first equation gives

$x^1 = u + c^1$

Solving the second equation gives

$x^2 = e^{c^2}e^u$

Using the solution of the first equation to substitute for $u$ in the solution of the second one we get

$x^2 = e^{c^2} e^{x^1 - c^1} = e^{c^2 - c^1}e^{x^1} \equiv Ce^{x^1}$

Therefore the streamlines of the vector field $(1, x^2(u))$ are the graphs of the equation

$x^2 = Ce^{x^1}$

where $C$ 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, $T^{ab \cdots}_{cd \cdots}(x)$, using the vector field $X^a(x)$. The essential idea is to use the streamlines of the vector field to link the tensor at some point $P$, $T^{ab\cdots}_{cd \cdots}(P)$, with the tensor at some neighbouring point $Q$, $T^{ab\cdots}_{cd \cdots}(Q)$, in such a way that the two will have the same transformation matrix at point $Q$ (with respect to the auxiliary vector field). We can then subtract the two tensors at $Q$ and so define the derivative at $P$ by a limiting process as $Q$ tends to $P$. In all such cases, the technique begins by considering a coordinate transformation from $P$ to $Q$ of the form

$x^{\prime a} = x^a + \delta u X^a(x)$

where $\delta u$ is arbitrarily small. The point $Q$ with coordinates $x^{\prime a}$ lies on the streamline through $P$ which the vector field $X^a(x)$ generates. Differentiating the coordinate transformation we get

$\frac{\partial x^{\prime a}}{\partial x^b} = \frac{\partial x^a}{\partial x^b} + \delta u \ \partial_b X^a$

$= \delta^a_b + \delta u \ \partial_b X^a$

where $\delta^a_b$ is the Kronecker delta and $\partial_b \equiv \frac{\partial}{\partial x^b}$. What we now do is consider the effect of the above coordinate transformation on the tensor field $T^{ab\cdots}_{cd \cdots}$ at the points $P$ and $Q$. In what follows I will employ this general procedure to obtain the Lie derivative formulas with respect to a contravariant vector field $X^a(x)$ in the case of a scalar field $\phi$, a contravariant vector field $Y^a$, a one-form field $Y_a$, and a general mixed tensor field $T^{ab\cdots}_{cd \cdots}$.

The Lie derivative of a scalar field $\phi$

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 $P$ will simply be $\phi(x)$ and the value at the point $Q$ will be

$\phi(x^{\prime}) = \phi(x^c + \delta u X^c)$

We can expand this in a Taylor series about the point $P$ with coordinates $x$ to get the first-order approximation

$\phi(x^{\prime}) \approx \phi(x) + \delta u X^c \ \partial_c \phi(x)$

The Lie derivative of the scalar field with respect to the contravariant vector field $X^a(x)$ is then

$L_X \phi = \lim_{\delta u \rightarrow 0} \frac{\phi(x^{\prime}) - \phi(x)}{\delta u} = X^c \ \partial_c \phi$

$\equiv X^a \ \partial_a \phi$

We observe that the Lie derivative of the scalar field $\phi(x)$ with respect to the vector field $X^a(x)$ is actually the directional derivative of $\phi$ in the direction of the vector $X^a$. In the differential geometry literature in this area it is common to associate the contravariant vector field $X$ with the linear differential operator $X^a \ \partial_a$ (which operates on any real-valued function $f$ to produce another function $g$) and essentially treat them as the same object. Given a point $P$ on the manifold, one thinks of the partial differential operators $[\partial_a]_P$ as constituting a basis for all the vectors in the tangent space at $P$, so that any vector at $P$ can be written as a linear combination of the $[\partial_a]_P$ in the form

$[X]_P = [X^a]_P [\partial_a]_P$

This is the intuitive justification for treating the vector field $X$ and the linear differential operator $X^a \ \partial_a$ as being the same things. Under this convention, one often sees the Lie derivative of a scalar field $\phi$ with respect to the contravariant vector field $X$ written as

$L_X \phi = X \phi$

The Lie derivative of a contravariant vector field $Y^a$

Under the coordinate transformation from $P$ to $Q$ given by

$x^{\prime a} = x^a + \delta u X^a(x)$

the contravariant vector field $Y^a(x)$ at $P$ is mapped to

$Y^{\prime a}(x^{\prime}) = \frac{\partial x^{\prime a}}{\partial x^b} Y^b(x)$

$= (\delta^a_b + \delta u \ \partial_b X^a) Y^b(x)$

$= Y^a(x) + \delta u \ Y^b \ \partial_b X^a$

The vector already at $Q$, namely $Y^a(x^{\prime})$, has a first-order Taylor series approximation about $x$ of the form

$Y^a(x^{\prime}) = Y^a(x^c + \delta u X^c)$

$\approx Y^a(x) + \delta u \ X^c \ \partial_c Y^a(x)$

The Lie derivative with respect to the vector field $X^a(x)$ is then given by

$L_X Y^a = \lim_{\delta u \rightarrow 0} \frac{Y^a (x^{\prime}) - Y^{\prime a}(x^{\prime})}{\delta u}$

$= X^c \ \partial_c Y^a - Y^b \ \partial_b X^a$

$\equiv X^b \ \partial_b Y^a - Y^b \ \partial_b X^a$

Under the convention of associating the vector field $X$ with the linear differential operator $X^a \partial_a$, one often sees the Lie derivative of a contravariant vector field $Y^a$ with respect to the field $X$ written as

$L_X Y^a = [X, Y]^a$

where $[X, Y] = XY - YX$ is called the Lie bracket (or commutator) of the two vector fields $X$ and $Y$. This is a new vector field (and therefore linear differential operator) that can be written alternatively as

$[X, Y] = X(Y^a \partial_a) - Y(X^a \partial_a)$

$= X^b \partial_b (Y^a \partial_a) - Y^b \partial_b (X^a \partial_a)$

$= (X^b \partial_b Y^a - Y^b \partial_b X^a) \partial_a + X^a Y^b(\partial_b \partial_a - \partial_a \partial_b)$

$= (X^b \partial_b Y^a - Y^b \partial_b X^a) \partial_a$

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 $a$-th component of the vector field $[X, Y]$ is the one that appears in the expression of the Lie derivative $L_X Y^a$ above.

The Lie derivative of a one-form (covariant vector) field $Y_a$

Under the coordinate transformation from $P$ to $Q$ given by

$x^{\prime a} = x^a + \delta u X^a(x)$

the one-form (covariant vector) field $Y_a(x)$ at $P$ is mapped to

$Y^{\prime}_a(x^{\prime}) = \frac{\partial x^b}{\partial x^{\prime a}} Y_b(x)$

To work out the transformation matrix here we need to write the coordinate transformation as

$x^{\prime b} = x^b + \delta u X^b(x)$

$\implies$

$x^b = x^{\prime b} - \delta u X^b(x)$

Partially differentiating we get

$\frac{\partial x^b}{\partial x^{\prime a}} = \delta^b_a - \delta u \frac{\partial}{\partial x^{\prime a}} X^b$

$= \delta^b_a - \delta u \ \partial_c X^b \frac{\partial x^c}{\partial x^{\prime a}}$

$= \delta^b_a - \delta u \ \partial_c X^b \big(\delta^c_a - \delta u \frac{\partial}{\partial x^{\prime a}} X^a\big)$

$= \delta^b_a - \delta u \ \partial_a X^b + O((\delta u)^2)$

We can ignore the $O((\delta u)^2)$ terms as they will disappear in the limiting process of the differentiation, so we have

$Y^{\prime}_a(x^{\prime}) = \frac{\partial x^b}{\partial x^{\prime a}} Y_b(x)$

$= \big( \delta^b_a - \delta u \ \partial_a X^b \big) Y_b(x)$

$= Y_a(x) - \delta u Y_b \ \partial_a X^b$

Again taking a first-order Taylor series approximation about $x$ at the point $Q$ we get that

$Y_a(x^{\prime}) = Y_a(x^c + \delta u X^c)$

$\approx Y_a(x) + \delta u X^c \ \partial_c Y_a(x)$

Then the Lie derivative of the one-form field $Y_a(x)$ with respect to the contravariant vector field $X^a(x)$ is obtained as

$L_X Y_a = \lim_{\delta u \rightarrow 0} \frac{Y_a(x^{\prime}) - Y^{\prime}_a (x^{\prime})}{\delta u}$

$= X^c \partial_c Y_a + Y_b \partial_a X^b$

$= X^b \partial_b Y_a + Y_b \partial_a X^b$

The Lie derivative of a mixed tensor field $T^{ab\cdots}_{cd \cdots}$

A good prototype for this case is the Lie derivative of the simplest type of mixed tensor field, the rank-2 tensor of type $(1, 1)$ represented as $T^a_b(x)$. We will therefore consider this case first and then use it to extrapolate to a general mixed tensor field of type $(p, q)$ represented as $T^{ab \cdots}_{cd \cdots}(x)$.

Under the coordinate transformation

$x^{\prime a} = x^a + \delta u X^a(x)$

the mixed tensor field $T^a_b(x)$ transforms as

$T^{\prime a}_b(x^{\prime}) = \frac{\partial x^{\prime a}}{\partial x^c} \frac{\partial x^d}{\partial x^{\prime b}} T^c_d(x)$

$= (\delta^a_c + \delta u \ \partial_c X^a)(\delta^d_b - \delta u \ \partial_b X^d) T^c_d$

$= (\delta^a_c + \delta u \ \partial_c X^a)(T^c_b - \delta u T^c_d \ \partial_b X^d)$

$= T^a_b(x) - \delta u T^a_d \ \partial_b X^d + \delta u T^c_b \ \partial_c X^a + O((\delta u)^2)$

Under a first-order Taylor series approximation about $x$, the tensor at $Q$ can be written

$T^a_b(x^{\prime}) = T^a_b(x^c + \delta u X^c)$

$\approx T^a_b(x) + \delta u X^c \ \partial_c T^a_b$

The Lie derivative of $T^a_b(x)$ with respect to the contravariant vector field $X^a(x)$ is then

$L_X T^a_b = \lim_{\delta u \rightarrow 0} \frac{T^a_b(x^{\prime}) - T^{\prime a}_b(x^{\prime})}{\delta u}$

$= X^c \ \partial_c T^a_b + T^a_d \ \partial_b X^d - T^c_b \ \partial_c X^a$

We observe that the contravariant index $a$ contributes a term of the form $-T^c_b \ \partial_c X^a$ while the covariant index $b$ contributes a term of the form $T^a_d \ \partial_b X^d$.

Now consider the general mixed tensor $T^{ab \cdots}_{cd \cdots}$. The first-order Taylor series approximation of $T^{ab \cdots}_{cd \cdots}(x^{\prime})$ about $x$ gives

$T^{ab \cdots}_{cd \cdots}(x^e + \delta u X^e)$

$\approx T^{ab \cdots}_{cd \cdots}(x) + \delta u X^e \ \partial_e T^{ab \cdots}_{cd \cdots}(x)$

Therefore the first-term of the Lie derivative will be $X^e \ \partial_e T^{ab \cdots}_{cd \cdots}$. This is of the same form as the first term of $L_X T^a_b$. Conveniently, it turns out that from then on each contravariant and covariant index in $T^{ab \cdots}_{cd \cdots}$ will contribute terms like the corresponding terms we saw above in $L_X T^a_b$. Therefore the Lie derivative of the general mixed tensor field $T^{ab \cdots}_{cd \cdots}(x)$ with respect to the contravariant vector field $X^a(x)$ will be of the form

$L_X T^{ab \cdots}_{cd \cdots} = X^e \ \partial_e T^{ab \cdots}_{cd \cdots}$

$- \ T^{eb \cdots}_{cd \cdots} \ \partial_e X^a \ - \ T^{ae \cdots}_{cd \cdots} \ \partial_e X^b \ - \ \cdots$

$+ \ T^{ab \cdots}_{ed \cdots} \ \partial_c X^e \ + \ T^{ab \cdots}_{ce \cdots} \ \partial_d X^e \ + \ \cdots$