Four-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 current density.

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

where

c is the speed of light

ρ the charge density

j the conventional current density.

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

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

where D is an operator called the four-gradient and given by (1/c ∂/∂t, ∇). Sometimes, the above relation is written as

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

In general relativity, the continuity equation is written as:

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

where the semi-colon represents a covariant derivative.

See also

• Noether\'s theorem.

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