Four-current

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

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:

$D\; cdot\; J\; =\; partial\_a\; J^a\; =\; frac\{partial\; rho\}\{partial\; t\}\; +\; nabla\; cdot\; mathbf\{j\}\; =\; 0$

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.

$partial\_a\; J^a\; =\; sum\_\{i=0\}^\{3\}\; partial\_a\; J^a$

Sometimes, the above relation is written as

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

In general relativity, the continuity equation is written as:

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

where the semi-colon represents a covariant derivative.

See also

• Noether's theorem
• Covariant formulation of classical electromagnetism