Faraday Tensor

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The electric and magnetic field’s, E = (E₁, E₂, E₃) = (E_x, E_y, E_z) and B = (B₁, B₂, B₃) = (B_x, B_y, B_z), can be expressed in terms of the Coulomb potential, F, and the magnetic vector potential, a  = (a₁, a₂, a₃) = (a_x, a_y, a_z), as

and

The 4-potential is defined as

The Faraday tensor, whose components are F^ab, is defined in terms of the components of the 4-potential as 

where

The Faraday tensor is antisymetric since F^ab = -F^ba. It therefore has 6 independent components, three of which are found by evaluating F^0k to obtain

The last three are F¹², F¹³, F²³, where

Eqs. (6) through (8) can be expressed in matrix form as

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