My solution is entirely different to Roberto's (or Laura's).

I think they've both missed the key relationship between time-evolution and Lie derivatives, namely that "D" here is just ; i.e. that the time evolution of a volume is found by "dragging" it along the vector field (where = proper time)!

Note that in the next chapter, on p.484 and in exercise [20.15], the evolution of a volume element is explicitly stated to be its Lie derivative with respect to time (albeit in a rather different context).