Chapter 2: Covariance of Maxwell's Equations
Maxwell's four equations collapse into two elegant tensor equations that are manifestly Lorentz covariant. This reveals electromagnetism's intrinsic compatibility with special relativity.
Maxwell's Equations in Tensor Form
Inhomogeneous Equations
\( \partial_\mu F^{\mu\nu} = \mu_0 J^\nu \)
Contains Gauss's law and Ampère-Maxwell law
Homogeneous Equations (Bianchi Identity)
\( \partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0 \)
Contains Faraday's law and ∇·B = 0
Four-Potential Aμ
\( A^\mu = (\phi/c, \vec{A}) \)
and \( F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu \)
The four-potential unifies scalar and vector potentials. Gauge transformations: Aμ → Aμ + ∂μχ leave Fμν unchanged.