Electrodynamics in Covariant Form

Manual: General · Subject: Physics

Presents Maxwell’s equations, gauge structure, and electromagnetic field tensors in relativistic notation.

Maxwell Theory as a Relativistic Field Theory

Field Tensor

The electromagnetic field is encoded in the antisymmetric tensor $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ . Its components contain both electric and magnetic fields in a frame-dependent decomposition.

Covariant Maxwell Equations

In compact form, Maxwell’s equations become $\partial_\mu F^{\mu\nu} = \mu_0 J^\nu,\qquad \partial_{[\alpha}F_{\beta\gamma]} = 0.$ The second equation is the Bianchi identity.

Gauge Potential

The four-potential $A^\mu = (\phi/c, \mathbf{A})$ is not unique: gauge transformations $A_\mu \to A_\mu + \partial_\mu \chi$ leave $F_{\mu\nu}$ unchanged. This redundancy is a structural feature of electromagnetism.

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Unified View

Electric and magnetic fields are not separate entities in special relativity; they are frame-dependent components of a single field tensor.

What is the advantage of writing Maxwell’s equations covariantly?

It removes the need for charges and currents It makes Lorentz invariance manifest It forces all fields to become scalar It eliminates gauge freedom

Write the field tensor definition in terms of the four-potential.

Component vs Covariant Electrodynamics

3-Vector Form

Separate electric and magnetic equations
Less symmetry-transparent
Useful for engineering intuition

Tensor Form

Uses $F_{\mu\nu}$ and $J^\mu$
Lorentz covariance is explicit
Best for relativistic field theory and GR

The homogeneous Maxwell equations are compactly represented by:

\partial_\mu F^{\mu\nu} = \mu_0 J^\nu

\partial_{[\alpha}F_{\beta\gamma]} = 0

E = mc^2

\mathbf{p} = m\mathbf{v}

Covariant Reformulation Workflow

1

Step 1: Introduce the four-potential $A^\mu$ .
2

Step 2: Build $F_{\mu\nu}$ from antisymmetric derivatives.
3

Step 3: Rewrite Maxwell’s equations in tensor form.
4

Step 4: Verify gauge invariance and Lorentz covariance.

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