Description
In electromagnetism, the Lorenz condition is generally used in calculations of time-dependent electromagnetic fields through retarded potentials. The condition is
where 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:
where is the magnetic vector potential and is the electric potential; see also Gauge fixing.
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 such that . This gives the well known equation for the electric field, . This result can be plugged into another one of Maxwell's equations,
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 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 is the vacuum velocity of light, and 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, if and are source density and circulation density, respectively, of the electromagnetic induction fields and calculated as usual from and by the equations and The explicit solutions for and – unique, if all quantities vanish sufficiently fast at infinity – are known as retarded potentials.
Read more about this topic: Lorenz Gauge Condition
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