Lorenz gauge condition

The Lorenz gauge, by Ludvig Lorenz, is a special calibration of the electromagnetic potentials. It is often incorrectly referred to as Lorentz gauge and Hendrik Antoon Lorentz attributed, after the Lorentz transformation is named. The Lorenz gauge is the same as in the static case with the Coulomb gauge.

Preliminary

An electromagnetic field composed of an electric field and an H-field. It may be described together with the scalar ( electric ) potential this by specifying the vector potential. The description of the electromagnetic field by potentials is not unique, ie there is a so-called gauge freedom. These additional freedoms can be utilized to match the equations of the problem and simplify by using a calibration is introduced. Such is the Lorenz - calibration, which is often used for analysis of general electromagnetic waves.

The Lorenz gauge, Relativistic invariance

The gauge freedom of the electrodynamic potentials is to the effect exploits the fact that the sum of the divergence of the vector potential and the first partial derivative of the scalar potential with respect to time t yields zero. Depending on whether one uses the Gaussian or the SI unit system, you have the time derivative of the scalar field will be divided by c or c2. The following is the cgs system and also the four-vector notation and the Einstein summation convention will be used. Aμ consolidates the both potentials and together.

Thus going from the four-dimensional formula of the inhomogeneous Maxwell equations

And the field strength tensor

The following expression is produced:

Using the Lorenzeichung the wave equations arising in Four-dimensional ( with the D' Alembertoperator ):

It is thus to solve the differential equation separately for each component of the potential or current. The Lorenz gauge, like any verification the property to allow the physically measurable fields unchanged.

Solution of the latter equation are the so-called retarded potentials of four

This means that the relativistic invariance of Maxwell's equations is explicitly at the same time.

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

Notation by means of differential forms

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

Where the Hodge star operator, the exterior derivative and the potential shape, or shorter than the Koableitung