Lorenz gauge condition

In electromagnetism, the Lorenz gauge or Lorenz gauge condition are common misnomers for a particular choice of the electromagnetic four-potential $A^a$. The potential is chosen to satisfy the condition $partial\_aA^a=0$, which was first proposed by the Danish physicist Ludvig Lorenz. It is a Lorentz invariant condition. It is frequently called the "Lorentz condition" because of confusion with Hendrik Lorentz, after whom Lorentz invariance is named. The Lorenz condition is often erroneously assumed to fix the gauge, which it apparently does not, as indeed one can make a gauge transformation $A^a\; o\; A^a+partial^af$ with a harmonic scalar function $f$, $partial^apartial\_af=0$, which does not affect the Lorenz condition $partial\_aA^a=0$,:$partial\_aA^a\; opartial\_aleft(A^a+partial^af\; ight)=partial\_aA^a+partial^apartial\_af=0$

The Lorenz condition though restricts the class of functions for gauge transformations to harmonic functions.

The Lorenz condition is used to eliminate the redundant spin-0 component in the (1/2,1/2) representation of the Lorentz group. It is equally used for massive spin-1 fields where the concept of gauge transformations does not apply at all.

Description

In electromagnetism, the Lorenz condition is generally used in calculations of time-dependent electromagnetic fields through retarded potentialsref|McDonald. The condition is

:$partial\_\{a\}A^a\; equiv\; A^a\{\}\_\{,a\}\; =\; 0\; !$

where $A^a$ is the four-potential, the comma denotes a partial differentiation and the repeated index indicates that the Einstein summation convention is being used. The condition has the advantage of being Lorentz invariant. It still leaves substantial gauge degrees of freedom.

In ordinary vector notation and SI units, the condition is:

:$ablacdot\{mathbf\; A\}\; +\; frac\{1\}\{c^2\}frac\{partialphi\}\{partial\; t\}=0.$

where A is the magnetic vector potential and φ is the electric potential; see also Gauge fixing.

In Gaussian units the condition is:

:$ablacdot\{mathbf\; A\}\; +\; frac\{1\}\{c\}frac\{partialphi\}\{partial\; t\}=0.$

It can be shown that the physical information in the Maxwell's equations can be expressed in the operationally simpler and symmetric form:

:$left\; [\; abla^\{2\}\; -\; frac\{1\}\{c^2\}frac\{partial^2\}\{partial\; t^2\}\; ight]\; vec\{A\}\; =\; -mu\_0vec\{J\}$

:$left\; [\; abla^\{2\}\; -\; frac\{1\}\{c^2\}frac\{partial^2\}\{partial\; t^2\}\; ight]\; phi\; =\; -\; frac\{1\}\{epsilon\_0\}\; ho$

History

When originally published, Lorenz's work was not received well by James Clerk Maxwell. Maxwell had eliminated the Coulomb electrostatic force from his derivation of the electromagnetic wave equation since he was working in what would nowadays be termed the Coulomb gauge. The Lorenz gauge hence contradicted Maxwell's original derivation of the EM wave equation by introducing a retardation effect to the Coulomb force and bringing it inside the EM wave equation alongside the time varying electric field. Lorenz's work was the first symmetrizing shortening of Maxwell's equations after Maxwell himself published his 1865 paper. In 1888, retarded potentials came into general use after Heinrich Rudolf Hertz's experiments on electromagnetic waves. In 1895, a further boost to the theory of retarded potentials came after J. J. Thomson's interpretation of data for electrons (after which investigation into electrical phenomena changed from time-dependent electric charge and electric current distributions over to moving point charges). ref|McDonald00

ee also

* Coulomb gauge
* Gauge fixing
* Weyl gauge

External articles, references, and further reading

;General
* Eric W. Weisstein, " [http://scienceworld.wolfram.com/physics/LorenzGauge.html Lorenz Gauge] ".
* Kirk T. McDonald, "The Relation Between Expressions for Time-Dependent Electromagnetic Fields Given by Jefimenko and by Panofsky and Phillips". Dec. 5, 1996
** Ibid.

;Further reading
* L. Lorenz, "On the Identity of the Vibrations of Light with Electrical Currents" Philos. Mag. 34, 287-301, 1867.
* J. van Bladel, "Lorenz or Lorentz?". IEEE Antennas Prop. Mag. 33, p. 69, 1991.
* R. Becker, "Electromagnetic Fields and Interactions", chap. DIII. Dover Publications, New York, 1982.
* A. O'Rahilly, "Electromagnetics", chap. VI. Longmans, Green and Co, New York, 1938. ;History
*R. Nevels, C.-S. Shin, "Lorenz, Lorentz, and the gauge", IEEE Antennas Prop. Mag. 43, 3, pp. 70-1, 2001.
* E. Whittaker, "A History of the Theories of Aether and Electricity", Vols. 1-2. New York: Dover, p. 268, 1989.

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