Four-current

Inspecialandgeneral relativity,thefour-current(technically thefour-current density)^[1]is thefour-dimensionalanalogue of thecurrent density,with units of charge per unit time per unit area. Also known asvector current,it is used in the geometric context offour-dimensionalspacetime,rather than separating time fromthree-dimensionalspace. Mathematically it is afour-vectorand isLorentz covariant.

This article uses thesummation conventionfor indices. Seecovariance and contravariance of vectorsfor background on raised and lowered indices, andraising and lowering indiceson how to switch between them.

Definition

Using theMinkowski metric $\eta _{\mu \nu }$ ofmetric signature(+ − − −),the four-current 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 thespeed of light;
$ρ$ is thevolume charge density;
$j$ is the conventionalcurrent density;
Thedummy index $α$ labels thespacetime dimensions.

Motion of charges in spacetime

This can also be expressed in terms of thefour-velocityby the equation:^[2]^[3]

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

where:

$\rho _{u}$ is thecharge densitymeasured by an inertial observer O who sees theelectric currentmoving at speed $u$ (the magnitude of the3-velocity);
$\rho _{0}$ 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 toLorentz contraction.

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 magnetism) in one electromagnetic entity.

Continuity equation

In special relativity, the statement ofcharge conservationis that theLorentz invariantdivergence ofJis 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 thefour-gradient.This is thecontinuity equation.

In general relativity, the continuity equation is written as:

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

where the semi-colon represents acovariant derivative.

Maxwell's equations

The four-current appears in two equivalent formulations ofMaxwell's equations,in terms of thefour-potential^[5]when theLorenz gauge conditionis fulfilled:

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

where $\Box$ is theD'Alembert operator,or theelectromagnetic field tensor:

\nabla _{\ Alpha }F^{\ Alpha \beta }=\mu _{0}J^{\beta }

whereμ₀is thepermeability of free spaceand ∇_αis thecovariant derivative.

General relativity

Ingeneral 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 }

Quantum field theory

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

References

^Rindler, Wolfgang (1991).Introduction to Special Relativity(2nd ed.). Oxford Science Publications. pp. 103–107.ISBN 978-0-19-853952-0.
^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]
^Cottingham, W. Noel; Greenwood, Derek A. (2003).An introduction to the standard model of particle physics.Cambridge University Press. p. 67.ISBN 9780521588324.
^Marshak, Robert E.(1993).Conceptual foundations of modern particle physics.World Scientific Publishing Company. p.20.ISBN 9789813103368.
^Gershtein, S. S.; Zeldovich, Y. B. (1956),Soviet Phys. JETP,2576.
^Thomas, Anthony W.(1996). "CVC in particle physics".arXiv:nucl-th/9609052.

[1] Rindler, Wolfgang (1991).Introduction to Special Relativity(2nd ed.). Oxford Science Publications. pp. 103–107.ISBN 978-0-19-853952-0.

[2] Roald K. Wangsness, Electromagnetic Fields, 2nd edition (1986), p. 518, 519

[3] Melvin Schwartz, Principles of Electrodynamics, Dover edition (1987), p. 122, 123

[4] J. D. Jackson, Classical Electrodynamics, 3rd Edition (1999), p. 554

[5] s [ref. 1, p519]

[6] Cottingham, W. Noel; Greenwood, Derek A. (2003).An introduction to the standard model of particle physics.Cambridge University Press. p. 67.ISBN 9780521588324.

[7] Marshak, Robert E.(1993).Conceptual foundations of modern particle physics.World Scientific Publishing Company. p.20.ISBN 9789813103368.

[8] Gershtein, S. S.; Zeldovich, Y. B. (1956),Soviet Phys. JETP,2576.

[9] Thomas, Anthony W.(1996). "CVC in particle physics".arXiv:nucl-th/9609052.

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