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In [[electromagnetism]], the '''Lorenz gauge''' or '''Lorenz gauge condition''' is a partial [[gauge fixing]] of the [[electromagnetic four-potential|electromagnetic vector potential]]. The condition is that <math>\partial_\mu A^\mu=0</math>. This does not completely fix the gauge: one can still make a gauge transformation <math>A^\mu\to A^\mu+\partial^\mu f</math> where <math>f</math> is a [[harmonic function|harmonic]] scalar function (that is, a [[scalar function]] satisfying <math>\partial^\mu\partial_\mu f=0</math>, the equation of a [[scalar field theory|massless scalar field]]).
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The Lorenz condition is used to eliminate the redundant spin-0 component in the (1/2,1/2) [[Representations of the Lorentz group|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.
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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]], after whom Lorentz covariance is named.<ref>{{Citation|last1 = Jackson|first1 = J.D.|author1-link = John David Jackson (physicist)|last2  = Okun|first2 = L.B.|author2-link = Lev Okun|title = Historical roots of gauge invariance|journal = [[Reviews of Modern Physics]]|volume = 73|issue = 3|pages = 663–680|year = 2001|doi = 10.1103/RevModPhys.73.663|arxiv = hep-ph/0012061|bibcode = 2001RvMP...73..663J}}</ref>
 
==Description==
In [[electromagnetism]], the Lorenz condition is generally [[Scientific method|used]] in [[calculation]]s of [[Time-variant system|time-dependent]] [[electromagnetic field]]s through [[retarded potential]]s.<ref name=mcdonald>{{Citation |authorlink1=Kirk T. McDonald |first1=Kirk T. |last1=McDonald|title=The relation between expressions for time-dependent electromagnetic fields given by Jefimenko and by Panofsky and Phillips |journal=[[American Journal of Physics]] |volume=65 |issue=11 |pages=1074–1076 |year=1997 |doi=10.1119/1.18723 |bibcode = 1997AmJPh..65.1074M }} and {{cite web|title=pdf link|url=http://www.physics.princeton.edu/~mcdonald/examples/jefimenko.pdf|accessdate=1 June 2010}}.</ref> The condition is
 
:<math>\partial_{\mu}A^\mu \equiv A^\mu{}_{,\mu} = 0 \!</math>
 
where <math>A^\mu</math> 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:
 
:<math>\nabla\cdot{ \vec{A}} + \frac{1}{c^2}\frac{\partial\varphi}{\partial t}=0.</math>
 
where <math>\vec{A}</math> is the [[magnetic vector potential]] and <math>\,\varphi</math> is the [[electric potential]]; see also [[Gauge fixing]].
 
In [[Gaussian units|Gaussian]] units the condition is:
 
:<math>\nabla\cdot{\vec{A}} + \frac{1}{c}\frac{\partial\varphi}{\partial t}=0.</math>
 
A quick justification of the Lorenz gauge can be found using [[Maxwell's equations]] and the relation between the magnetic vector potential and the magnetic field:
 
:<math>\nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}=\nabla\times-\frac{\partial\vec{A}}{\partial t}</math>
 
Therefore,
 
:<math>\nabla\times(\vec{E}+\frac{\partial\vec{A}}{\partial t})=0</math>
 
Since the curl is zero, that means there is a scalar function <math>\,\varphi</math> such that <math>-\nabla\,\varphi=\vec{E}+\frac{\partial\vec{A}}{\partial t}</math>. This gives the well known equation for the electric field, <math>\vec{E}=-\nabla\,\varphi-\frac{\partial\vec{A}}{\partial t}</math>. This result can be plugged into another one of Maxwell's equations,
 
:<math> {\nabla\times\vec{B}}={\nabla\times(\nabla\times\vec{A})}={\nabla(\nabla\cdot\vec{A})-\nabla^2\vec{A}}={\mu_0\vec{J}+\frac{1}{c^2}\frac{\partial\vec{E}}{\partial t}}={\mu_0\vec{J}-\frac{1}{c^2}\nabla\frac{\partial\varphi}{\partial t}-\frac{1}{c^2}\frac{\partial^2\vec{A}}{\partial t^2}}</math>
 
This leaves,
 
:<math>\nabla(\nabla\cdot\vec{A}+\frac{1}{c^2}\frac{\partial\varphi}{\partial t})=\mu_0\vec{J}-\frac{1}{c^2}\frac{\partial^2\vec{A}}{\partial t^2}+\nabla^2\vec{A} </math>
 
To have Lorentz invariance, the time derivatives and spatial derivatives must be treated equally (i.e. of the same order). Therefore it is convenient to choose the Lorenz gauge condition, which gives the result
 
:<math>\Box \vec{A}=\left[\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^{2}\right]\vec{A} = \mu_0\vec{J}</math>
 
A similar procedure with a focus on the electric scalar potential and making the same gauge choice will yield
 
:<math>\Box\varphi = \left[\frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^{2}\right] \varphi = \frac{1}{\epsilon_0}\rho\,.</math>
 
These are simpler and more symmetric forms of the inhomogenous [[Maxwell's equations]]. Note that the [[Coulomb gauge]] also fixes the problem of Lorentz invariance, but leaves a coupling term with first-order derivatives.
 
Here <math>c=\frac{1}{\sqrt{\epsilon_0\mu_0}} </math> is the vacuum velocity of light, and <math>\Box</math> 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 <ref>See for example U. Krey, A. Owen, ''Basic Theoretical Physics – A Concise Overview'', Berlin-Heidelberg-New York, Springer 2007.</ref>
, if <math>\rho</math> and <math>\vec J</math> are  source density and circulation density,  respectively,  of the electromagnetic induction  fields <math>\vec E</math> and <math>\vec B\,,</math> calculated as usual from <math>\,\varphi</math> and <math>\vec A</math> by the equations <math>\vec E=-\nabla\varphi -\frac{\partial \vec A}{\partial t}</math> and <math>\vec B=\nabla\times \vec A\,.</math> The explicit solutions for <math>\,\varphi</math> and <math>\vec A</math> – unique, if all quantities vanish sufficiently fast at infinity – are known as [[retarded potential]]s.
 
==History==
When originally published, Lorenz's work was not received well by [[James Clerk Maxwell|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]], which was introduced in Lorenz's paper "On the identity of the vibrations of light with electrical currents". Lorenz's work was the first [[symmetry|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 wave]]s. In 1895, a further boost to the theory of retarded potentials came after [[J. J. Thomson]]'s interpretation of data for [[electron]]s (after which investigation into [[electrical phenomena]] changed from time-dependent [[electric charge]] and [[electric current]] distributions over to moving [[point charge]]s).<ref name="mcdonald"/>
 
==See also==
* [[Gauge fixing]]
* [[Weyl gauge]]
 
==References==
{{reflist}}
 
==External articles, and further reading==
;General
* Eric W. Weisstein, "''[http://scienceworld.wolfram.com/physics/LorenzGauge.html Lorenz Gauge]''".
 
;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, '''2''', p.&nbsp;69, April 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.&nbsp;70–1, 2001.
* E. T. Whittaker, "''[[A History of the Theories of Aether and Electricity]]''", Vols. 1–2. New York: Dover, p.&nbsp;268, 1989.
 
{{DEFAULTSORT:Lorenz Gauge Condition}}
[[Category:Electromagnetism]]
[[Category:Concepts in physics]]

Latest revision as of 23:54, 12 February 2014

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- Entirely All natural - made from environmentally friendly plant and vegetables extracts

- Prepared with pre and probiotics for softer digestive function

- 76 meticulously determined materials, There is nothing omitted

- Each and every scoop gets the antioxidising strength of 2 lbs of green vegetables

- Absolutely no fabricated compounds, no sweetener or man-made sweeteners

- Entirely scrumptious and very an easy task to get ready-just put standard water!

For decades now, just about every big wellness power and federal government health agency are actually suggesting a regular intake of your five to 9 portions of natural green fruit and vegetables to fight degenerative illnesses. Unfortunately, the USDA reports that fewer than 9% of United states grownups follow this advice. Thinking about take in natural food items each day?

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- Natural green fresh vegetables and berries also guide better lower and digestion of food inflammation.

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Athletic Greens is created just with 100% all-natural materials, such as a variety of berries, green vegetables, probiotics, and healthier ingredients. These materials are cautiously hold dry to maintain phytonutrient potency. Go through the change Athletic Greens take. Order your container now!

- Healthy and balanced blue colored-green algae like chlorella and spirulina

- Full-foods sourced nutrients and vitamins yourself can digest

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- Dark green veggie fruit juice such as ingredients from broccoli and spinach

Athletic Greens has surveyed a wide array of well being healthy fruit and veggies and packed by far the most nourishing into an individual delightful scoop. Buy your container currently and initiate emotion far healthier! !

Get your Total day-to-day suggested allowance of fruits and vegetables a single SCOOP!

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- Every scoop comes complete with zero-getting older and wellness enhancing vitamin antioxidants.

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