Appendix: Lie Derivative

That of the Lie derivative is a useful concept which lays the road for the more generic concept of the covariant derivative. In many respects, the Lie derivative can be considered as the extension of the directional derivative in a three-dimensional space (i.e. the derivative of a scalar function along a 3-vector) to the directional derivative in a three-dimensional space (i.e. the derivative of a generic tensor along a 4-vector).

Consider therefore a vector field ${\bf V}(x^\mu)$ and the family of curves having ${\bf V}$ as a tangent vector, i.e. (cf. Fig. 1) .

Figure: Schematic diagram illustrating how the Lie derivatives compares two tensors at the point Q.

$$V^\mu(x^{\mu}) \equiv \frac{d x^\mu}{d\tau} ,$$ (45)

where $\tau$ is the parameter chosen for the curve (This operation is equivalent to finding the streamlines out of velocity vector field.).

The basic idea behind the Lie derivative is then that of comparing tensors that are ``dragged'' along a certain curve defined by a vector field and taking the limit for infinitesimal displacements, i.e.

$${\mathcal L}_{_{\bf V}}{\boldsymbol {\it T}}\equiv \lim_{\delta\tau \rightarrow 0} \frac{T^\nu(x^{\mu'})-T^{\nu\;'}(x^{\mu'})}{\delta\tau} ,$$ (46)

where $T^\nu(x^{\mu'})$ is the tensor ${\boldsymbol {\it T}}$ at the point $Q(x^{\mu'}$) and $T^{\nu\;'}(x^{\mu'})$ is the tensor ${\boldsymbol {\it T}}$ ``dragged'' at $Q$. This is illustrate schematically in Fig. 5

The operation of dragging is made through a standard coordinate transformation

$$x^\mu\longrightarrow x^{\mu^\prime}=x^\mu+\delta\tau V^\mu ,$$ (47)

where $\delta\tau V^\mu$ is the change of coordinates along the curve. The matrix for the coordinate transformation $\Lambda^{\mu'}_{  \nu}$ is then expressed as [cf. eq. (47)]

$$\Lambda^{\mu'}_{  \nu} \equiv \frac{\partial x^{\mu'}}{\partial x^\nu}= \delta^\mu_{  \nu}+\delta\tau V^\mu_{  ,\nu} .$$ (48)

so that the tensor ${\boldsymbol {\it T}}$ will transform as

$$T^{\mu'}(x^{\nu\;'}) \equiv \Lambda^{\mu'}_{  \nu} T^{\nu} =$$

$$= (\delta^\mu_{  \nu}+\delta\tau V^\mu_{  ,\nu}) T^{\nu} =$$

$$= T^{\mu}+\delta\tau V^\mu_{  ,\nu}T^{\nu} .$$ (49)

On the other hand, $T^{\mu}(x^\prime)$ can be calculated through a Taylor expansion around the point $Q$, thus yielding

$$T^{\mu}(x^{\mu'}) = T^{\mu}(x^{\nu\;'})=$$

$$= T^{\mu}\left(x^{\nu}+\delta x^{\nu} + {\cal O}((\delta x^{\nu})^2)\right) =$$

$$= T^{\mu}\left(x^{\nu}+\delta\tau V^\nu + {\cal O}(\delta \tau^2) \right) =$$

$$= T^{\mu}(x^{\nu})+ \delta\tau V^\nu \frac{\partial T^{\mu}}{\partial x^{\nu}} + {\cal O}(\delta \tau^2) .$$ (50)

Collecting now expressions (49) and (50) in the definition (46) and taking the limit for infinitesimal displacements on the curve one obtains that the Lie derivative of ${\boldsymbol {\it T}}$ along the vector field V is

$$(\mathcal{L}_{_{\bf V}}{\boldsymbol {\it T}})^\mu = T^\mu_{  ,\nu}V^\nu-V^\mu_{  ,\nu}T^\nu= \mathcal{L}_{_{\bf V}} T^{\mu} .$$ (51)

Note that in addition to the simple derivative along V (i.e. $T^\mu_{  ,\nu}V^\nu$), the Lie derivative (51) also contains a second term (i.e. $V^\mu_{  ,\nu}T^\nu$) providing information on how the coordinates themselves change along the curve with tangent vector V.

For a generic mixed tensor with $m$ covariant components and $n$ contravariant ones (i.e. $T^{\alpha _1\ldots\alpha _m}_{\hskip 1cm\beta _1\ldots\beta _n}$), the Lie derivative will then be expressed as

$${\mathcal{L}}_{_{\bf V}} T^{\alpha _1\ldots\alpha _m}_{\hskip 1cm\beta _1\ldots\beta _n} = T^{\alpha _1\ldots\alpha _m}_{\hskip 1cm\beta _1\ldots\beta _n,\n... ... \nu} T^{\alpha _1\ldots\alpha _{m-1}\nu}_{\hskip 1.5cm\beta _1\ldots\beta _n}-$$

$$V^{  ,\nu}_{\beta_1} T^{\alpha _1\ldots\alpha _m}_{\hskip 1.3cm\n... ...n} T^{\alpha _1\ldots\alpha _m}_{\hskip 1.3cm\beta _1\ldots\beta _{n-1}\nu} .$$ (52)