Covariant Formulation of Classical Electromagnetism - Maxwell's Equations in Vacuo

Maxwell's Equations in Vacuo

In a vacuum (or for the microscopic equations, not including macroscopic material descriptions) Maxwell's equations can be written as two tensor equations.

The two inhomogeneous Maxwell's equations, Gauss's Law and Ampère's law (with Maxwell's correction) combine into:

Gauss-Ampère law (vacuum)

while the homogeneous equations - Faraday's law of induction and Gauss's law for magnetism combine to form:

Gauss-Faraday law (vacuum)

where Fαβ is the electromagnetic tensor, Jα is the 4-current, εαβγδ is the Levi-Civita symbol, and the indices behave according to the Einstein summation convention.

The first tensor equation corresponds to four scalar equations, one for each value of β. The second tensor equation actually corresponds to 43 = 64 different scalar equations, but only four of these are independent. Using the antisymmetry of the electromagnetic field one can either reduce to an identity (0 = 0) or render redundant all the equations except for those with λ, μ, ν = either 1,2,3 or 2,3,0 or 3,0,1 or 0,1,2.

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.