the dual electromagnetic field tensor
Maxwell's equations have a striking near-symmetry: swap electric and magnetic fields in the right way and the source-free equations turn into each other. The dual electromagnetic field tensor is the object that makes this E-B swap precise — it is the field tensor with electric and magnetic roles exchanged.
The dual is defined by contracting the field tensor with the four-dimensional Levi-Civita symbol: (dual F)^(mu nu) = (1/2) epsilon^(mu nu alpha beta) F_(alpha beta). The effect is the duality substitution E goes to cB and B goes to - E/c: wherever F had the electric field, its dual has the magnetic field, and vice versa (with a sign). In terms of the dual, the two homogeneous Maxwell equations — Faraday's law and the absence of magnetic monopoles — combine into the single compact statement d_mu (dual F)^(mu nu) = 0, mirroring the source equation d_mu F^(mu nu) = mu_0 J^nu for the ordinary tensor.
The dual makes the second Lorentz invariant transparent: the contraction (dual F)_(mu nu) F^(mu nu) is proportional to E·B, a pseudoscalar (it flips sign under a mirror reflection), and it is the same in every frame. If E and B are perpendicular in one frame, they are perpendicular in all frames. The duality symmetry would become an exact symmetry of electromagnetism if magnetic monopoles existed — their absence is exactly why d_mu (dual F)^(mu nu) = 0 has a zero on the right instead of a magnetic four-current.
Apply the duality swap E goes to cB, B goes to - E/c to a plane electromagnetic wave, where E and B are already perpendicular and equal in the combination E = cB: the wave maps to another valid wave. The invariant (dual F)F ~ E·B is zero for such a wave, and stays zero in every frame.
The dual (dual F)^(mu nu) = (1/2) epsilon^(mu nu alpha beta) F_(alpha beta) swaps E and B; its invariant with F is ~ E·B.
Electromagnetic duality is an exact symmetry only in the absence of sources (or if magnetic monopoles existed); the electric four-current on the right of the ordinary Maxwell equation, with no magnetic counterpart, breaks it. The dual is a pseudotensor — it depends on the Levi-Civita symbol and flips under a parity (mirror) inversion.