As discribed in Anthony Zee's "Einstein Gravity in a Nutshell" Ch V Appendix 2, thinking about a vector field $V^{\mu}$ as the velocity field of a fluid in spacetime $V^{\mu}(X(\tau))$ where the worldline $X(\tau)$ takes the role of the trajectory of a fluid parcel, the Lie derivative of a vector (or more general tensor) field $W^{\mu}(x)$ along the direction $V^{\mu}$ can be derived from the mismatch between the "conventional" form applying the ordinary derivative expected (analog to the Eulerian difference in fluid dynamics) difference between W at two points $P(x)$ and $Q(\tilde{x})$ and the (called Lagrangian in fluid dynamics) change of $W^{\mu}(x)$ when following the flow from $x$ to $\tilde{x}$ along the worldline (or trajectory)

\(\mathcal{L}_V W^{\mu}(x) = V^{\nu}(x)D_{\nu}W^{\mu}(x) - W^{\nu}(x)D_{\nu}V^{\mu}(x)\)

where $D_{\nu}$ is the covariant derivative.

In fluid dynamics, a Lagrangian conserved quantity can be described by means of the transport theorem

\(\frac{d}{dt}\int\limits_{\mathcal{G}(t)} A(t)dV = \int\limits_{\mathcal{G}(t)} ( \frac{\partial A(\vec{r,t})}{\partial t} + \nabla\cdot\vec{v}(\vec{r},t)A(\vec{r},t))dV = 0\)

where $\frac{dA}{dt} = \frac{\partial A}{\partial t} + (\vec{v}\nabla))A$ is the material derivative, $\mathcal{G}(t)$ is the volume of a fluid parcel, A(t) is a flow variable, and $\vec{v}(\vec{r},t)$ is the flow field.

The Lie derivative and the second term on the r.h.s. of the transport theorem seem both to describe some kind of correction due to the flow which has to be added to obtain the full derivative seen by an observer moving with the flow. In addition, if the second part on the r.h.s. of the transport theorem vanishes, the quantity $A$ does not change with time which seems similar to the fact that a tensor quantity W does not change with (proper) time if it is Lie transported.

So could the second term in the transport theorem formally be seen as some kind of a Lie derivative along the flow field $\vec{v}(\vec{r},t)$?