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Concept# Covariant formulation of classical electromagnetism

Summary

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 diag(+1, −1, −1, −1). 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.
Lorentz covariance
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.

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