Definitions

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(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.

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