Classical Electromagnetism and Special Relativity - Covariant Formulation in Vacuum - Maxwell's Equations in Tensor Form

Maxwell's Equations in Tensor Form

Using these tensors, Maxwell's equations reduce to:

Maxwell's equations (Covariant formulation)

where the partial derivatives may be written in various ways, see 4-gradient. The first equation listed above corresponds to both Gauss's Law (for β = 0) and the Ampère-Maxwell Law (for β = 1, 2, 3). The second equation corresponds to the two remaining equations, Gauss's law for magnetism (for β = 0) and Faraday's Law ( for β = 1, 2, 3).

These tensor equations are manifestly-covariant, meaning the equations can be seen to be covariant by the index positions. This short form of writing Maxwell's equations illustrates an idea shared amongst some physicists, namely that the laws of physics take on a simpler form when written using tensors.

By lowering the indices on Fαβ to obtain F_αβ (see raising and lowering indices):

the second equation can be written in terms of F_αβ as:

where is the contravariant Levi-Civita symbol. Notice the cyclic permutation of indices in this equation: .

Another covariant electromagnetic object is the electromagnetic stress-energy tensor, a covariant rank-2 tensor which includes the Poynting vector, Maxwell stress tensor, and electromagnetic energy density.