Lorenz gauge condition

In electromagnetism, the Lorenz gauge condition or Lorenz gauge (after Ludvig Lorenz) is a partial gauge fixing of the electromagnetic vector potential by requiring <math>\partial_\mu A^\mu = 0.</math> The name is frequently confused with Hendrik Lorentz, who has given his name to many concepts in this field.<ref>Jackson, J.D.; Okun, L.B. (2001), "Historical roots of gauge invariance", Reviews of Modern Physics, 73 (3): 663–680, arXiv:hep-ph/0012061, Bibcode:2001RvMP...73..663J, doi:10.1103/RevModPhys.73.663, S2CID 8285663</ref> The condition is Lorentz invariant. The Lorenz gauge condition does not completely determine the gauge: one can still make a gauge transformation <math>A^\mu \mapsto A^\mu + \partial^\mu f,</math> where <math>\partial^\mu</math> is the four-gradient and <math>f</math> is any harmonic scalar function: that is, a scalar function obeying <math>\partial_\mu\partial^\mu f = 0,</math> the equation of a massless scalar field.

The Lorenz gauge condition is used to eliminate the redundant spin-0 component in Maxwell's equations when these are used to describe a massless spin-1 quantum field. It is also 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 potentials.<ref name="mcdonald">McDonald, Kirk T. (1997), "The relation between expressions for time-dependent electromagnetic fields given by Jefimenko and by Panofsky and Phillips" (PDF), American Journal of Physics, 65 (11): 1074–1076, Bibcode:1997AmJPh..65.1074M, CiteSeerX 10.1.1.299.9838, doi:10.1119/1.18723, S2CID 13703110, archived from the original (PDF) on 2022-05-19</ref> The condition is <math display="block">\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 display="block">\nabla\cdot{\mathbf{A}} + \frac{1}{c^2}\frac{\partial\varphi}{\partial t} = 0,</math> where <math>\mathbf{A}</math> is the magnetic vector potential and <math> \varphi</math> is the electric potential;<ref>Jackson, John David (1999). Classical Electrodynamics (3rd ed.). John Wiley & Sons. p. 240. ISBN 978-0-471-30932-1.</ref><ref>Keller, Ole (2012-02-02). Quantum Theory of Near-Field Electrodynamics. Springer Science & Business Media. p. 19. Bibcode:2011qtnf.book.....K. ISBN 9783642174100.</ref> see also gauge fixing.

In Gaussian units the condition is<ref>Gbur, Gregory J. (2011). Mathematical Methods for Optical Physics and Engineering. Cambridge University Press. p. 59. Bibcode:2011mmop.book.....G. ISBN 978-0-521-51610-5.</ref><ref>Heitler, Walter (1954). The Quantum Theory of Radiation. Courier Corporation. p. 3. ISBN 9780486645582.</ref> <math display="block">\nabla\cdot{\mathbf{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 display="block">\nabla \times \mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t} = - \frac{\partial (\nabla \times \mathbf{A})}{\partial t}</math>

Therefore, <math display="block">\nabla \times \left(\mathbf{E} + \frac{\partial\mathbf{A}}{\partial t}\right) = 0.</math>

Since the curl is zero, that means there is a scalar function <math>\varphi</math> such that <math display="block">-\nabla\varphi = \mathbf{E} + \frac{\partial\mathbf{A}}{\partial t}.</math>

This gives a well known equation for the electric field: <math display="block">\mathbf{E} = -\nabla \varphi - \frac{\partial\mathbf{A}}{\partial t}.</math>

This result can be plugged into the Ampère–Maxwell equation, <math display="block">\begin{align}

 \nabla \times \mathbf{B} &= \mu_0\mathbf{J} + \frac{1}{c^2}\frac{\partial\mathbf{E}}{\partial t} \\
 \nabla \times \left(\nabla \times \mathbf{A}\right) &= \\
 \Rightarrow
 \nabla\left(\nabla \cdot \mathbf{A}\right) - \nabla^2\mathbf{A} &= \mu_0\mathbf{J} - \frac{1}{c^2}\frac{\partial (\nabla\varphi)}{\partial t} - \frac{1}{c^2}\frac{\partial^2 \mathbf{A}}{\partial t^2}. \\

\end{align}</math>

This leaves <math display="block">\nabla\left(\nabla \cdot \mathbf{A} + \frac{1}{c^2}\frac{\partial\varphi}{\partial t}\right) = \mu_0\mathbf{J} - \frac{1}{c^2} \frac{\partial^2 \mathbf{A}}{\partial t^2} + \nabla^2\mathbf{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 makes the left hand side zero and gives the result <math display="block">\Box\mathbf{A} = \left[\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right]\mathbf{A} = -\mu_0\mathbf{J}.</math>

A similar procedure with a focus on the electric scalar potential and making the same gauge choice will yield <math display="block">\Box\varphi = \left[\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right] \varphi = - \frac{1}{\varepsilon_0}\rho .</math>

These are simpler and more symmetric forms of the inhomogeneous 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 display="block">c = \frac{1}{\sqrt{\varepsilon_0\mu_0}}</math> is the vacuum velocity of light, and <math>\Box</math> is the d'Alembertian operator. These equations are not only valid under vacuum conditions, but also in polarized media,<ref>For example, see Cheremisin, M. V.; Okun, L. B. (2003). "Riemann-Silberstein representation of the complete Maxwell equations set". arXiv:hep-th/0310036.</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 display="block">\begin{align}

 \mathbf{E} &= -\nabla\varphi - \frac{\partial \mathbf{A}}{\partial t} \\
 \mathbf{B} &= \nabla\times \mathbf{A}

\end{align}</math>

The explicit solutions for <math>\varphi</math> and <math>\mathbf{A}</math> – unique, if all quantities vanish sufficiently fast at infinity – are known as retarded potentials.

History

When originally published in 1867, 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, which was introduced in Lorenz's paper "On the identity of the vibrations of light with electrical currents". Lorenz's work was the first use of symmetry to simplify 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 name="mcdonald"/>

See also

• Gauge fixing

External links and further reading

General

• Weisstein, E. W. "Lorenz Gauge". Wolfram Research.

Further reading

• Lorenz, L. (1867). "On the Identity of the Vibrations of Light with Electrical Currents". Philosophical Magazine. Series 4. 34 (230): 287–301.
• van Bladel, J. (1991). "Lorenz or Lorentz?". IEEE Antennas and Propagation Magazine. 33 (2): 69. doi:10.1109/MAP.1991.5672647. S2CID 21922455.
  • See also Bladel, J. (1991). "Lorenz or Lorentz? [Addendum]". IEEE Antennas and Propagation Magazine. 33 (4): 56. Bibcode:1991IAPM...33...56B. doi:10.1109/MAP.1991.5672657.
• Becker, R. (1982). Electromagnetic Fields and Interactions. Dover Publications. Chapter 3.
• O'Rahilly, A. (1938). Electromagnetics. Longmans, Green and Co. Chapter 6.

History

• Nevels, R.; Shin, Chang-Seok (2001). "Lorenz, Lorentz, and the gauge". IEEE Antennas and Propagation Magazine. 43 (3): 70–71. Bibcode:2001IAPM...43...70N. doi:10.1109/74.934904.
• Whittaker, E. T. (1989). A History of the Theories of Aether and Electricity. Vol. 1–2. Dover Publications. p. 268.