Lorenz calibration

The Lorenz calibration , according to Ludvig Lorenz , is a special calibration of electromagnetic potentials. It has nothing to do with Hendrik Antoon Lorentz , after whom the Lorentz transformation is named.

In the static case, the Lorenz calibration is identical to the Coulomb calibration .

Preliminary remark

An electromagnetic field consists of an E and an H field . These fields can also be described by specifying the vector potential together with the scalar (electrical) potential .

The description of the electromagnetic field in terms of potentials is ambiguous. H. there is freedom from calibration. These additional freedoms can be used to adapt the equations to the problem and to simplify them by introducing a calibration. One such is the Lorenz calibration, which is often used to calculate electromagnetic waves .

The Lorenz calibration, relativistic invariance

${\displaystyle {\rm {div}}{\vec {A}}+{\frac {1}{c^{2}}}{\frac {\partial }{\partial t}}\phi =0}$( International System of Units (SI) )

${\displaystyle {\rm {div}}{\vec {A}}+{\frac {1}{c}}{\frac {\partial }{\partial t}}\phi =0}$( Gaussian system )

The freedom from calibration of the electrodynamic potentials is used to the effect that the sum of the divergence of the vector potential and the first partial derivative of the scalar potential after time t results in zero. Depending on whether you use the Gaussian or the SI system of units , you have to divide the time derivative of the scalar field by c or c ² . In the following, the cgs system and also the four-vector notation and Einstein's sum convention are used. The four potential is defined by . ${\displaystyle {\vec {A}}}$ ${\displaystyle \phi }$ ${\displaystyle A^{\mu }}$ ${\displaystyle A^{\mu }={\begin{pmatrix}\phi &{\vec {A}}\end{pmatrix}}^{T}}$

${\displaystyle \partial _{\mu }A^{\mu }=0}$

Thus, the four-dimensional formula is based on the inhomogeneous Maxwell equations

${\displaystyle \partial _{\mu }F^{\mu \nu }={\frac {4\pi }{c}}j^{\nu }}$

and the field strength tensor

${\displaystyle F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }}$

the following expression emerges:

${\displaystyle \partial _{\mu }F^{\mu \nu }=\partial _{\mu }\partial ^{\mu }A^{\nu }-\partial _{\mu }\partial ^{\nu }A^{\mu }=\square A^{\nu }-\partial ^{\nu }\partial _{\mu }A^{\mu }={\frac {4\pi }{c}}j^{\nu }}$

Using the Lorenz calibration , the four-dimensional wave equations result (with the D'Alembert operator ): ${\displaystyle \partial _{\mu }A^{\mu }=0}$ ${\displaystyle \square }$

${\displaystyle \square A^{\nu }={\frac {4\pi }{c}}j^{\nu }}$

So you can solve the differential equation for each component of the potential or the current separately. The Lorenz calibration, like any calibration, has the property of leaving the physically measurable fields unchanged.

The solution to the last-mentioned equation is the so-called retarded four-potentials

${\displaystyle A^{\nu }({\vec {r}},t)=\int \,{\frac {j^{\nu }({\vec {r}}',t-{\frac {|{\vec {r}}-{\vec {r}}'|}{c}})}{c\cdot |{\vec {r}}-{\vec {r}}'|}}\,\mathrm {d} ^{3}{\vec {r}}'\,}$

This also makes the relativistic invariance of Maxwell's equations explicit.

Instead of the Lorenz calibration, the Coulomb calibration is often used, which characterizes the electrostatic potential, but in most cases does not bring any simplification.

Notation using differential forms

In the language of differential forms , the Lorenz calibration can be written as

${\displaystyle d\star A=0}$,

in which

• ${\displaystyle d}$the outer derivative
• ${\displaystyle \star }$the Hodge star operator
• ${\displaystyle A=A_{\mu }dx^{\mu }}$ the potential form is

or shorter with the co-derivative than ${\displaystyle \delta }$

${\displaystyle \delta A=0}$.

literature

• Ludvig Lorenz: On the Identity of the Vibrations of Light with Electrical Currents. In: Philosophical Magazine . Series 4, Vol. 34, No. 230, 1867, pp. 287-301 , doi : 10.1080 / 14786446708639882 .
• Robert Nevels, Chang-Seok Shin: Lorenz, Lorentz, and the Gauge. In: IEEE Antennas and Propagation Magazine. Vol. 43, No. 3, June 2001, pp. 70-71, doi : 10.1109 / 74.934904 .
• Adolf Schwab, C. Fuchs, Peter Kistenmacher: Semantics of the Irrotational Component of the Magnetic Vector Potential, A. In: IEEE Antennas and Propagation Magazine. Vol. 39, No. 1, February 1997, pp. 46-51, doi : 10.1109 / 74.583518 .
• Ari Sihvola: Lorenz-Lorentz or Lorentz-Lorenz. In: IEEE Antennas and Propagation Magazine. Vol. 33, No. 4, August 1991, p. 56, doi : 10.1109 / MAP.1991.5672658 .