Four-current

Using the Minkowski metric ${\displaystyle \eta _{\mu \nu }}$ of metric signature (+−−−), the four-current components are given by:

${\displaystyle J^{\alpha }=\left(c\rho ,j^{1},j^{2},j^{3}\right)=\left(c\rho ,\mathbf {j} \right)}$

where c is the speed of light, ρ is the charge density, and j the conventional current density. The dummy index α labels the spacetime dimensions.

In special relativity, the statement of charge conservation is that the Lorentz invariant divergence of J is zero:[4]

${\displaystyle {\dfrac {\partial J^{\alpha }}{\partial x^{\alpha }}}={\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {j} =0}$

where ${\displaystyle \partial /\partial x^{\alpha }}$ is the 4-gradient. This is the continuity equation.

In general relativity, the continuity equation is written as:

${\displaystyle J^{\alpha }{}_{;\alpha }=0\,}$

where the semi-colon represents a covariant derivative.

The four-current appears in two equivalent formulations of Maxwell's equations, in terms of the four-potential:[5]

${\displaystyle \Box A^{\alpha }=\mu _{0}J^{\alpha }}$

where ${\displaystyle \Box }$ is the D'Alembert operator, or the electromagnetic field tensor:

${\displaystyle \partial _{\beta }F^{\alpha \beta }=\mu _{0}J^{\alpha }}$

where μ₀ is the permeability of free space.

In general relativity, the four-current is defined as the divergence of the electromagnetic displacement, defined as

${\displaystyle {\mathcal {D}}^{\mu \nu }\,=\,{\frac {1}{\mu _{0}}}\,g^{\mu \alpha }\,F_{\alpha \beta }\,g^{\beta \nu }\,{\sqrt {-g}}\,}$

then

${\displaystyle J^{\mu }=\partial _{\nu }{\mathcal {D}}^{\mu \nu }}$

The four-current density of charge is an essential component of the Lagrangian density used in quantum electrodynamics.[6] In 1956 Gershtein and Zeldovich considered the conserved vector current (CVC) hypothesis for electroweak interactions.[7][8][9]

• Four-vector
• Noether's theorem
• Covariant formulation of classical electromagnetism
• Ricci calculus