Four-current
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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 current density.
- <math>J^a = \left(c \rho, \mathbf{j} \right)</math>
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:
- <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.
See also Noether's theorem.es:Corriente-cuatro th:ความหนาแน่นกระแสสี่มิติ
