A continuity equation

A continuity equation expresses conservation of a quantity. This is a general concept, let's do a simple example first: conservation of number of cars on a motor way.

Suppose you're standing on a hill above a tunnel on a motor way, and counting the number of cars entering the tunnel on one side, and at the same time, the number leaving the tunnel on the other side.

If $n$ is the number density of cars in the tunnel (in cars per km say), and $\Delta l$ is the length of the tunnel, then the number of cars in the tunnel is obviously $N=n \Delta l$. The change in $N$ between two times is

$$\Delta N=[n(l,t+\Delta t)-n(l,t)] \Delta l\approx {\partial n(l,t)\over\partial t} \Delta l \Delta t .$$ (8.13)

Assuming there is no other exit, this is also the change between the number of cars entering and leaving the tunnel,

$$\Delta N=[n(l,t)\dot l(l,t)-n(l+\Delta l,t)\dot l(l+\Delta l,t)]\... ...t\approx -{\partial n(l,t) \dot l(l,t)\over \partial l} \Delta l \Delta t .$$ (8.14)

The first term $n(l,t)\dot l(l,t) \Delta t$ is the number of cars entering the tunnel in time $\Delta t$, the second term is how many are leaving in that time. $\dot l(l,t)$ obviously denotes the speed of the cars.

Combining these two equation, we get the following continuity equation:

$${\partial n(l,t)\over \partial t} + {\partial n(l,t) \dot l(l,t)\over \partial l} = 0 .$$ (8.15)

A little bit of thought shows that, in a N-dimensional case (spaceships through a worm-hole?), the more general equation would be

$${\partial n(\hbox{$\bf l$},t)\over \partial t} + {\partial n(\hbo... ...},t) \dot\hbox{$\bf l$}(\hbox{$\bf l$},t)\over \partial \hbox{$\bf l$}} = 0 ,$$ (8.16)

so there is a partial derivative $\partial/\partial l_i$ for every Cartesian coordinate $l_i$, and $n$ is multiplied by $\dot l_i$ in the second term, which is the velocity of the spaceships in the $i$-direction.