Contents

Four-current

4D analogue of electric current density

Articles about
Electromagnetism
Electricity Magnetism Optics History Computational Textbooks Phenomena
Electrostatics Charge density Conductor Coulomb law Electret Electric charge Electric dipole Electric field Electric flux Electric potential Electrostatic discharge Electrostatic induction Gauss law Insulator Permittivity Polarization Potential energy Static electricity Triboelectricity
Magnetostatics Ampère law Biot–Savart law Gauss magnetic law Magnetic dipole Magnetic field Magnetic flux Magnetic scalar potential Magnetic vector potential Magnetization Permeability Right-hand rule
Electrodynamics Bremsstrahlung Cyclotron radiation Displacement current Eddy current Electromagnetic field Electromagnetic induction Electromagnetic pulse Electromagnetic radiation Faraday law Jefimenko equations Larmor formula Lenz law Liénard–Wiechert potential London equations Lorentz force Maxwell equations Maxwell tensor Poynting vector Synchrotron radiation
Electrical network Alternating current Capacitance Current density Direct current Electric current Electrolysis Electromotive force Impedance Inductance Joule heating Kirchhoff laws Network analysis Ohm law Parallel circuit Resistance Resonant cavities Series circuit Voltage Waveguides
Magnetic circuit AC motor DC motor Electric machine Electric motor Gyrator–capacitor Induction motor Linear motor Magnetomotive force Permeance Reluctance (complex) Reluctance (real) Rotor Stator Transformer
Covariant formulation Electromagnetic tensor Electromagnetism and special relativity Four-current Four-potential Mathematical descriptions Maxwell equations in curved spacetime Relativistic electromagnetism Stress–energy tensor
Scientists Ampère Biot Coulomb Davy Einstein Faraday Fizeau Gauss Heaviside Helmholtz Henry Hertz Hopkinson Jefimenko Joule Kelvin Kirchhoff Larmor Lenz Liénard Lorentz Maxwell Neumann Ohm Ørsted Poisson Poynting Ritchie Savart Singer Steinmetz Tesla Thomson Volta Weber Wiechert
v t e

Inspecial and general relativity, the four-current (technically the four-current density)^[1] is the four-dimensional analogue of the current density, with units of charge per unit time per unit area. Also known as vector current, it is used in the geometric context of four-dimensional spacetime, rather than separating time from three-dimensional space. Mathematically it is a four-vector and is Lorentz covariant.

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[edit]

Using the Minkowski metric ${\displaystyle \eta _{\mu \nu }}$ofmetric 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 volume charge density;
• j is the conventional current density;
• The dummy index α labels the spacetime dimensions.

Motion of charges in spacetime[edit]

This can also be expressed in terms of the four-velocity by the equation:^[2][3]

${\displaystyle J^{\alpha }=\rho _{0}U^{\alpha }=\rho _{u}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}U^{\alpha }}$

where:

• ${\displaystyle \rho _{u}}$ is the charge density measured by an inertial observer O who sees the electric current moving at speed u (the magnitude of the 3-velocity);
• ${\displaystyle \rho _{0}}$ is “the rest charge density”, i.e., the charge density for a comoving observer (an observer moving at the speed u - with respect to the inertial observer O - along with the charges).

Qualitatively, the change in charge density (charge per unit volume) is due to the contracted volume of charge due to Lorentz contraction.

Physical interpretation[edit]

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 magnetism) in one electromagnetic entity.

Continuity equation[edit]

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 four-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.

Maxwell's equations[edit]

The four-current appears in two equivalent formulations of Maxwell's equations, in terms of the four-potential^[5] when the Lorenz gauge condition is fulfilled:

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

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

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

where μ₀ is the permeability of free space and ∇_α is the covariant derivative.

General relativity[edit]

Ingeneral 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:

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

Quantum field theory[edit]

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]

See also[edit]

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

References[edit]