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.
The convention used here is (+ − − −), corresponding to the Minkowski metric tensor:
The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant antisymmetric tensor whose entries are B-field quantities.
The four-current is the contravariant four-vector which combines electric charge density ρ and electric current density j:
In the language of differential forms, which provides the generalisation to curved spacetimes, these are the components of a 1-form
{\displaystyle T^{\alpha \beta }={\begin{pmatrix}\varepsilon _{0}E^{2}/2+B^{2}/2\mu _{0}&S_{x}/c&S_{y}/c&S_{z}/c\\S_{x}/c&-\sigma _{xx}&-\sigma _{xy}&-\sigma _{xz}\\S_{y}/c&-\sigma _{yx}&-\sigma _{yy}&-\sigma _{yz}\\S_{z}/c&-\sigma _{zx}&-\sigma _{zy}&-\sigma _{zz}\end{pmatrix}}\,,}
where η is the Minkowski metric tensor (with signature (+ − − −)).
The two inhomogeneous Maxwell's equations, Gauss's Law and Ampère's law (with Maxwell's correction) combine into (with (+ − − −) metric):[3]
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.
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.
(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.)
In the Lorenz gauge, the microscopic Maxwell's equations can be written as:
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.
where pα is the four-momentum, q is the charge, and xβ is the position.
where uβ is the four-velocity, and τ is the particle's proper time, which is related to coordinate time by dt = γdτ.
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
which expresses the conservation of linear momentum and energy by electromagnetic interactions.
In order to solve the equations of electromagnetism given here, it is necessary to add information about how to calculate the electric current, Jν.
Maxwell's macroscopic equations have been used, in addition the definitions of the electric displacement D and the magnetic intensity H:
The bound current is derived from the P and M fields which form an antisymmetric contravariant magnetization-polarization tensor [1] [5] [6][7]
the constitutive equations may, in vacuum, be combined with the Gauss–Ampère law to get:
When the upper index is lowered with η, it becomes symmetric and is part of the source of the gravitational field.
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 Lagrange equations for the electromagnetic lagrangian density
