Definitions

# density current

Any current in either a liquid or a gas that is kept in motion by the force of gravity acting on small differences in density. A density difference can exist between two fluids or between different parts of the same fluid. Density currents flow along ocean and lake bottoms, because the water entering is colder, saltier, or contains more suspended sediment and thus is denser than the surrounding water. Density currents are a factor in water pollution, as the industrial discharge of large amounts of polluted or heated water can generate density currents that affect neighbouring human or animal communities.

In special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density, or indeed any conventional charge current density.

$J^a = left\left(c rho, mathbf\left\{j\right\} right\right)$

where

c is the speed of light
ρ the charge density
j the conventional current density.
a labels the space-time dimensions

In special relativity, the statement of charge conservation (also called the continuity equation) is that the Lorentz invariant divergence of J is zero:

$D cdot J = partial_a J^a = frac\left\{partial rho\right\}\left\{partial t\right\} + nabla cdot mathbf\left\{j\right\} = 0$

where D is an operator called the four-gradient and given by (1/c ∂/∂t, ∇). The summation convention has been used, so that the space-time dimensions are implicitly summed over. i.e.

$partial_a J^a = sum_\left\{i=0\right\}^\left\{3\right\} partial_a J^a$

Sometimes, the above relation is written as

$J^a\left\{\right\}_\left\{,a\right\}=0,$

In general relativity, the continuity equation is written as:

$J^a\left\{\right\}_\left\{;a\right\}=0,$

where the semi-colon represents a covariant derivative.