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 o ...
, the Lorenz gauge condition or Lorenz gauge, for
Ludvig Lorenz
Ludvig Valentin Lorenz (; 18 January 1829 – 9 June 1891) was a Danish physicist and mathematician. He developed mathematical formulae to describe phenomena such as the relation between the refraction of light and the density of a pure transp ...
, is a partial
gauge fixing
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 ...
of the
electromagnetic vector potential by requiring
The name is frequently confused with
Hendrik Lorentz
Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the Lorent ...
, who has given his name to many concepts in this field. The condition is
Lorentz invariant
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
. The condition does not completely determine the gauge: one can still make a gauge transformation
where
is the
four-gradient In differential geometry, the four-gradient (or 4-gradient) \boldsymbol is the four-vector analogue of the gradient \vec from vector calculus.
In special relativity and in quantum mechanics, the four-gradient is used to define the properties ...
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'', t ...
scalar function (that is, a
scalar function
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity ( ...
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
The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representati ...
. 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 o ...
, the Lorenz condition is generally
used
Used may refer to:
Common meanings
*Used good, goods of any type that have been used before or pre-owned
*Used to, English auxiliary verb
Places
*Used, Huesca, a village in Huesca, Aragon, Spain
*Used, Zaragoza, a town in Zaragoza, Aragon, Spain ...
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 field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
s 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 th ...
s.
The condition is
where
is the
four-potential, the comma denotes a
partial differentiation and the repeated index indicates that the
Einstein summation convention
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
is being used. The condition has the advantage of being
Lorentz invariant
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
. It still leaves substantial gauge degrees of freedom.
In ordinary vector notation and
SI units, the condition is
where
is the
magnetic vector potential
In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic ...
and
is the
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
; see also
gauge fixing
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 ...
.
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 uni ...
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 Mi ...
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 th ...
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
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous fo ...
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
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical 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
Heinrich Rudolf Hertz ( ; ; 22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's equations of electromagnetism. The unit ...
's experiments on
electromagnetic wave
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) ...
s. In 1895, a further boost to the theory of retarded potentials came after
J. J. Thomson's interpretation of data for
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
s (after which investigation into
electrical phenomena
This is a list of electrical phenomena. Electrical phenomena are a somewhat arbitrary division of electromagnetic phenomena.
Some examples are:
* Biefeld–Brown effect — Thought by the person who coined the name, Thomas Townsend Brown, to ...
changed from time-dependent
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
and
electric current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The movi ...
distributions over to moving
point charge
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
s).
See also
*
Gauge fixing
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 ...
References
External links and further reading
;General
*
;Further reading
*
*
**See also
*
*
;History
*
*
{{DEFAULTSORT:Lorenz Gauge Condition
Electromagnetism
Concepts in physics