Four-current

Electromagnetism

Electricity Magnetism
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Electrodynamics Lorentz force law Electromagnetic induction Faraday's law Lenz's law Displacement current Magnetic potential Maxwell's equations Electromagnetic field Electromagnetic radiation Maxwell tensor Poynting vector Liénard–Wiechert potential Jefimenko's equations Eddy current London equations Mathematical descriptions of the electromagnetic field
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Covariant formulation Electromagnetic tensor (stress–energy tensor) Four-current Electromagnetic four-potential
Scientists Ampère Coulomb Faraday Gauss Heaviside Henry Hertz Lorentz Maxwell Tesla Volta Weber Ørsted

In special and general relativity, the four-current (technically the four-current density)^[1] is the four-dimensional analogue of the electric current density, which is used in the geometric context of four-dimensional spacetime, rather than three-dimensional space and time separately. Mathematically it is a four-vector, and is Lorentz covariant.

Analogously, it is possible to have any form of "current density", meaning the flow of a quantity per unit time per unit area, see current density for more on this quantity.

This article uses the summation convention for indices, see covariance and contravariance of vectors for background on raised and lowered indices, and raising and lowering indices on how to switch between them.

Definition

Using the Minkowski metric $\eta_{\mu\nu}$ of metric signature (+−−−), its four components are given by:

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.

Motion of charges in spacetime

Physical interpretation

Charges (free or as a distribution) at rest will appear to remain at the same spatial position for some interval of time (as long as they're stationary). When they do move, this corresponds to changes in position, therefore the charges have velocity, and the motion of charge constitutes an electric current. This means that charge density is related to time, while current density is related to space.

The four-current unifies charge density (related to electricity) and current density (related to electricity magnetism) in one electromagnetic entity.

Continuity equation

Main article: Continuity equation

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

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

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

In general relativity, the continuity equation is written as:

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

where the semi-colon represents a covariant derivative.

Maxwell's equations

Main article: Maxwell's equations

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

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

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

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

where μ₀ is the permeability of free space.

General Relativity

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

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

then

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

References

↑ Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd ed.). Oxford Science Publications. pp. 103–107. ISBN 0-19-853952-5.
↑ Roald K. Wangsness, Electromagnetic Fields, 2nd edition (1986), p. 518, 519
↑ Melvin Schwartz, Principles of Electrodynamics, Dover edition (1987), p. 122, 123
↑ J. D. Jackson, Classical Electrodynamics, 3rd Edition (1999), p. 554
↑ as [ref. 1, p519]

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