In electromagnetism, the Lorenz gauge or Lorenz gauge condition is a partial gauge fixing of the electromagnetic vector potential. The condition is that
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The Lorenz condition is used to eliminate the redundant spin-0 component in the (1/2, 1/2) 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.
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.
Description
In electromagnetism, the Lorenz condition is generally used in calculations of time-dependent electromagnetic fields through retarded potentials. The condition is
where
In ordinary vector notation and SI units, the condition is:
where
In Gaussian units the condition is:
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:
Therefore,
Since the curl is zero, that means there is a scalar function
This leaves,
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
A similar procedure with a focus on the electric scalar potential and making the same gauge choice will yield
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
History
When originally published, Lorenz's work was not received well by 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 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 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).