In
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
, the Lorenz gauge condition or Lorenz gauge, for
Ludvig Lorenz, is a partial
gauge fixing of the
electromagnetic vector potential by requiring
The name is frequently confused with
Hendrik Lorentz, who has given his name to many concepts in this field. The condition is
Lorentz invariant. The condition does not completely determine the gauge: one can still make a gauge transformation
where
is the
four-gradient and
is a
harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the '' fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
scalar function (that is, a
scalar function satisfying
the equation of a
massless scalar field). The Lorenz condition is used to eliminate the redundant spin-0 component in the
representation theory of the Lorentz group. It is equally used for massive spin-1 fields where the concept of gauge transformations does not apply at all.
Description
In
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
, the Lorenz condition is generally
used in
calculation
A calculation is a deliberate mathematical process that transforms one or more inputs into one or more outputs or ''results''. The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm, to t ...
s of
time-dependent electromagnetic fields through
retarded potential
In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light ''c'', so t ...
s.
The condition is
where
is the
four-potential
An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. W ...
, the comma denotes a
partial differentiation
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Par ...
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
Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs unit ...
the condition is
A quick justification of the Lorenz gauge can be found using
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
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 the Ampère–Maxwell equation,
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
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
. Note that the
Coulomb gauge
In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct co ...
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
In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of ...
operator. These equations are not only valid under vacuum conditions, but also in polarized media,
[For example, see ] if
and
are source density and circulation density, respectively, of the electromagnetic induction fields
and
calculated as usual from
and
by the equations
The explicit solutions for
and
– unique, if all quantities vanish sufficiently fast at infinity – are known as
retarded potential
In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light ''c'', so t ...
s.
History
When originally published, Lorenz's work was not received well by
Maxwell
Maxwell may refer to:
People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of ...
. 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
In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct co ...
. 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
Sir Joseph John Thomson (18 December 1856 – 30 August 1940) was a British physicist and Nobel Laureate in Physics, credited with the discovery of the electron, the first subatomic particle to be discovered.
In 1897, Thomson showed that ...
'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).
See also
*
Gauge fixing
References
External links and further reading
;General
*
;Further reading
*
*
**See also
*
*
;History
*
*
{{DEFAULTSORT:Lorenz Gauge Condition
Electromagnetism
Concepts in physics