Four-current

In special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density

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

where c is the speed of light, ρ the charge density, and j the conventional current density.

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

<math>D \cdot J = \partial_a J^a = \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0<math>

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

<math>J^a{}_{,a}=0<math>

In general relativity, the continuity equation is written as:

<math>J^a{}_{;a}=0<math>

where the semi-colon represents a covariant derivative.