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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 charge current density. Its four components are given by:
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:
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.
Sometimes, the above relation is written as
In general relativity, the continuity equation is written as:
where the semi-colon represents a covariant derivative.
See also
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