Chemistry - Four-current

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

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

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:

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

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

Categories: Relativity | Electromagnetism