Lorenz gauge condition - AbsoluteAstronomy.com

Electromagnetism

Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

, the Lorenz gauge or Lorenz gauge condition is a partial gauge fixing

Gauge fixing

In the physics of gauge theories, gauge fixing denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field...

of the electromagnetic vector potential

Electromagnetic four-potential

The electromagnetic four-potential is a potential from which the electromagnetic field can be derived. It combines both the electric scalar potential and the magnetic vector potential into a single space-time four-vector. In a given reference frame, the first component is the scalar potential and...

. The condition is that

. This does not completely fix the gauge: one can still make a gauge transformation

where

is a harmonic

Harmonic function

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....

scalar function (that is, a scalar function satisfying

, the equation of a massless scalar field

Scalar field theory

In theoretical physics, scalar field theory can refer to a classical or quantum theory of scalar fields. A field which is invariant under any Lorentz transformation is called a "scalar", in contrast to a vector or tensor field...

).

The Lorenz condition is used to eliminate the redundant spin-0 component in the (1/2,1/2) representation of the Lorentz group

Representations of the Lorentz group

The Lorentz group of theoretical physics has a variety of representations, corresponding to particles with integer and half-integer spins in quantum field theory. These representations are normally constructed out of spinors....

. It is equally used for massive spin-1 fields where the concept of gauge transformations does not apply at all.

The Lorenz condition is named after Ludvig Lorenz. It is a Lorentz invariant condition, and is frequently called the "Lorentz condition" because of confusion with Hendrik Lorentz

Hendrik Lorentz

Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...

, after whom Lorentz covariance is named.

Description

In electromagnetism

Electromagnetism

Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

, the Lorenz condition is generally used

Scientific method

Scientific method refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering empirical and measurable evidence subject to specific principles of...

in calculation

Calculation

A calculation is a deliberate process for transforming one or more inputs into one or more results, with variable change.The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm to the vague heuristics of calculating a strategy in a competition...

s of time-dependent

Time-variant system

A time-variant system is a system that is not time invariant . Roughly speaking, characteristics of its output depend explicitly upon time.- Overview :...

electromagnetic field

Electromagnetic field

An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

s through retarded potential

Retarded potential

The retarded potential formulae describe the scalar or vector potential for electromagnetic fields of a time-varying current or charge distribution. The retardation of the influence connecting cause and effect is thereby essential; e.g...

s. The condition is

where

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

Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

units, the condition is:

where A is the magnetic vector potential and

is the electric potential

Electric potential

In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

; see also Gauge fixing

Gauge fixing

.

In Gaussian

Gaussian units

Gaussian units comprise a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs units. It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units...

units the condition is:

It can be shown that with this gauge the physical information in the inhomogeneous part of Maxwell's equations

Maxwell's equations

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

can be expressed in the operationally simpler and symmetric form:

Here

is the vacuum velocity of light, and

is the d'Alembertian operator. Interestingly, and unexpectedly at a first glance, these equations are not only valid under vacuum conditions, but also in polarized media
, if

and

are source density and circulation density, respectively, of the electromagnetic induction fields

and

calculated as usual from

and

by the equations

and

The explicit solutions for

and

- unique, if all quantities vanish sufficiently fast at infinity - are known as retarded potential

Retarded potential

History

When originally published, Lorenz's work was not received well by James Clerk Maxwell

James Clerk Maxwell

James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

. Maxwell had eliminated the Coulomb electrostatic force from his derivation of the electromagnetic wave equation

Electromagnetic wave equation

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum...

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

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

. Lorenz's work was the first symmetrizing

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

shortening of Maxwell's equations after Maxwell himself published his 1865 paper. In 1888, retarded potentials came into general use after Heinrich Rudolf Hertz

Heinrich Rudolf Hertz

Heinrich Rudolf Hertz was a German physicist who clarified and expanded the electromagnetic theory of light that had been put forth by Maxwell...

's experiments on electromagnetic waves. In 1895, a further boost to the theory of retarded potentials came after J. J. Thomson

J. J. Thomson

Sir Joseph John "J. J." Thomson, OM, FRS was a British physicist and Nobel laureate. He is credited for the discovery of the electron and of isotopes, and the invention of the mass spectrometer...

's interpretation of data for electron

Electron

The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s (after which investigation into electrical phenomena

Electrical phenomena

Electrical phenomena are commonplace and unusual events that can be observed which illuminate the principles of the physics of electricity and are explained by them.Electrical phenomena are a somewhat arbitrary division ofelectromagnetic phenomena....

changed from time-dependent electric charge

Electric charge

Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

and electric current

Electric current

Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

distributions over to moving point charges).

External articles, and further reading

General

Eric W. Weisstein, "Lorenz Gauge".