Electrodynamics in Covariant Form

Manual: General · Subject: Physics

Formulate Maxwell theory with tensors, potentials, gauge symmetry, and Lorentz-covariant field equations.

Covariant electromagnetism

Field tensor formulation

The electromagnetic field strength tensor is $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ . Maxwell’s equations become $\partial_\mu F^{\mu\nu} = \mu_0 J^\nu, \qquad \partial_{[\lambda}F_{\mu\nu]} = 0,$ where $J^\nu$ is the four-current.

Gauge symmetry

The potentials are defined up to $A_\mu \to A_\mu + \partial_\mu \Lambda$ . This redundancy is not merely mathematical convenience; it reflects a local symmetry underlying charge conservation and the structure of the electromagnetic interaction.

Lorentz force law

The covariant equation of motion is $m\frac{du^\mu}{d\tau} = qF^{\mu}{}_{\nu}u^\nu.$ In 3-vector form, it reproduces $\mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B})$ .

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Unification

Electric and magnetic fields are frame-dependent components of a single antisymmetric tensor; what one observer calls $\mathbf{E}$ another may partially call $\mathbf{B}$ .

Which object transforms as a tensor under Lorentz transformations in covariant electrodynamics?

The scalar potential alone The vector potential four-vector

A^\mu

The electric field three-vector

\mathbf{E}

alone The magnetic field magnitude

|\mathbf{B}|

Write Maxwell’s equations in covariant notation.

Component vs covariant viewpoint

3-vector formulation

Separates electric and magnetic fields
Useful for engineering intuition
Less manifestly relativistic

Tensor formulation

Manifest Lorentz covariance
Compact and geometric
Natural for generalization to curved spacetime

What physical principle is associated with charge conservation in covariant form?

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