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# density current

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.