Covariant formulation of classical electromagnetism

Covariant formulation of classical electromagnetism
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.
This article uses the classical treatment of tensors and Einstein summation convention throughout and the Minkowski metric has the form . Where the equations are specified as holding in a vacuum, one could instead regard them as the formulation of Maxwell's equations in terms of total charge and current.
For a more general overview of the relationships between classical electromagnetism and special relativity, including various conceptual implications of this picture, see Classical electromagnetism and special relativity.
Covariant objects
Preliminary four-vectors
Lorentz tensors of the following kinds may be used in this article to describe bodies or particles:
four-displacement:
Four-velocity: where γ(u) is the Lorentz factor at the 3-velocity u.
Four-momentum: where is 3-momentum, is the total energy, and is rest mass.
Four-gradient:
The d'Alembertian operator is denoted ,
The signs in the following tensor analysis depend on the convention used for the metric tensor. The convention used here is , corresponding to the Minkowski metric tensor:
Electromagnetic tensor
The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant antisymmetric tensor whose entries are B-field quantities.
and the result of raising its indices is
where E is the electric field, B the magnetic field, and c the speed of light.
Four-current
The four-current is the contravariant four-vector which combines electric charge density ρ and electric current density j:
Four-potential
The electromagnetic four-potential is a covariant four-vector containing the electric potential (also called the scalar potential) ϕ and magnetic vector potential (or vector potential) A, as follows:
The differential of the electromagnetic potential is
In the language of differential forms, which provides the generalisation to curved spacetimes, these are the components of a 1-form and a 2-form respectively. Here, is the exterior derivative and the wedge product.
Electromagnetic stress–energy tensor
The electromagnetic stress–energy tensor can be interpreted as the flux density of the momentum four-vector, and is a contravariant symmetric tensor that is the contribution of the electromagnetic fields to the overall stress–energy tensor:
where is the electric permittivity of vacuum, μ0 is the magnetic permeability of vacuum, the Poynting vector is
and the Maxwell stress tensor is given by
The electromagnetic field tensor F constructs the electromagnetic stress–energy tensor T by the equation:
where η is the Minkowski metric tensor (with signature ). Notice that we use the fact that
which is predicted by Maxwell's equations.
Maxwell's equations in vacuum
In vacuum (or for the microscopic equations, not including macroscopic material descriptions), Maxwell's equations can be written as two tensor equations.
The two inhomogeneous Maxwell's equations, Gauss's Law and Ampère's law (with Maxwell's correction) combine into (with metric):
while the homogeneous equations – Faraday's law of induction and Gauss's law for magnetism combine to form , which may be written using Levi-Civita duality as:
where Fαβ is the electromagnetic tensor, Jα is the four-current, εαβγδ is the Levi-Civita symbol, and the indices behave according to the Einstein summation convention.
Each of these tensor equations corresponds to four scalar equations, one for each value of β.
Using the antisymmetric tensor notation and comma notation for the partial derivative (see Ricci calculus), the second equation can also be written more compactly as:
In the absence of sources, Maxwell's equations reduce to:
which is an electromagnetic wave equation in the field strength tensor.
Maxwell's equations in the Lorenz gauge
The Lorenz gauge condition is a Lorentz-invariant gauge condition. (This can be contrasted with other gauge conditions such as the Coulomb gauge, which if it holds in one inertial frame will generally not hold in any other.) It is expressed in terms of the four-potential as follows:
In the Lorenz gauge, the microscopic Maxwell's equations can be written as:
Lorentz force
Charged particle
Electromagnetic (EM) fields affect the motion of electrically charged matter: due to the Lorentz force. In this way, EM fields can be detected (with applications in particle physics, and natural occurrences such as in aurorae). In relativistic form, the Lorentz force uses the field strength tensor as follows.
Expressed in terms of coordinate time t, it is:
where pα is the four-momentum, q is the charge, and xβ is the position.
Expressed in frame-independent form, we have the four-force
where uβ is the four-velocity, and τ is the particle's proper time, which is related to coordinate time by .
Charge continuum
The density of force due to electromagnetism, whose spatial part is the Lorentz force, is given by
and is related to the electromagnetic stress–energy tensor by
Conservation laws
Electric charge
The continuity equation:
expresses charge conservation.
Electromagnetic energy–momentum
Using the Maxwell equations, one can see that the electromagnetic stress–energy tensor (defined above) satisfies the following differential equation, relating it to the electromagnetic tensor and the current four-vector
or
which expresses the conservation of linear momentum and energy by electromagnetic interactions.
Covariant objects in matter
Free and bound four-currents
In order to solve the equations of electromagnetism given here, it is necessary to add information about how to calculate the electric current, Jν Frequently, it is convenient to separate the current into two parts, the free current and the bound current, which are modeled by different equations;
where
Maxwell's macroscopic equations have been used, in addition the definitions of the electric displacement D and the magnetic intensity H:
where M is the magnetization and P the electric polarization.
Magnetization–polarization tensor
The bound current is derived from the P and M fields which form an antisymmetric contravariant magnetization-polarization tensor
which determines the bound current
Electric displacement tensor
If this is combined with Fμν we get the antisymmetric contravariant electromagnetic displacement tensor which combines the D and H fields as follows:
The three field tensors are related by:
which is equivalent to the definitions of the D and H fields given above.
Maxwell's equations in matter
The result is that Ampère's law,
and Gauss's law,
combine into one equation:
The bound current and free current as defined above are automatically and separately conserved
Constitutive equations
Vacuum
In vacuum, the constitutive relations between the field tensor and displacement tensor are:
Antisymmetry reduces these 16 equations to just six independent equations. Because it is usual to define Fμν by
the constitutive equations may, in vacuum, be combined with the Gauss–Ampère law to get:
The electromagnetic stress–energy tensor in terms of the displacement is:
where δαπ is the Kronecker delta. When the upper index is lowered with η, it becomes symmetric and is part of the source of the gravitational field.
Linear, nondispersive matter
Thus we have reduced the problem of modeling the current, Jν to two (hopefully) easier problems — modeling the free current, Jνfree and modeling the magnetization and polarization, . For example, in the simplest materials at low frequencies, one has
where one is in the instantaneously comoving inertial frame of the material, σ is its electrical conductivity, χe is its electric susceptibility, and χm is its magnetic susceptibility.
The constitutive relations between the and F tensors, proposed by Minkowski for a linear materials (that is, E is proportional to D and B proportional to H), are:
where u is the four-velocity of material, ε and μ are respectively the proper permittivity and permeability of the material (i.e. in rest frame of material), and denotes the Hodge star operator.
Lagrangian for classical electrodynamics
Vacuum
The Lagrangian density for classical electrodynamics is composed by two components: a field component and a source component:
In the interaction term, the four-current should be understood as an abbreviation of many terms expressing the electric currents of other charged fields in terms of their variables; the four-current is not itself a fundamental field.
The Lagrange equations for the electromagnetic lagrangian density can be stated as follows:
Noting
the expression inside the square bracket is
The second term is
Therefore, the electromagnetic field's equations of motion are
which is the Gauss–Ampère equation above.
Matter
Separating the free currents from the bound currents, another way to write the Lagrangian density is as follows:
Using Lagrange equation, the equations of motion for can be derived.
The equivalent expression in vector notation is:
See also
Covariant classical field theory
Electromagnetic tensor
Electromagnetic wave equation
Liénard–Wiechert potential for a charge in arbitrary motion
Moving magnet and conductor problem
Inhomogeneous electromagnetic wave equation
Proca action
Quantum electrodynamics
Relativistic electromagnetism
Stueckelberg action
Wheeler–Feynman absorber theory
Notes and references
Further reading
The Feynman Lectures on Physics Vol. II Ch. 25: Electrodynamics in Relativistic Notation
Concepts in physics
Electromagnetism
Special relativity