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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. Its four components are given by:

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

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.

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

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.

[edit] See also