Dual electromagnetic field tensor (${\tilde{F}}^{\mu \nu }$)

The dual electromagnetic field tensor is obtained by applying the Hodge star operator to the electromagnetic field tensor. For metric signature $(+---)$ and ${ϵ}^{0123}=+1$, define ${\tilde{F}}^{\mu \nu }=\frac{1}{2}{ϵ}^{\mu \nu \rho \sigma }{F}_{\rho \sigma }$. With $F^{0i}=-E_i/c$ and $F^{ij}=-{ϵ}_{ijk}{B}_k$, it has components ${\tilde{F}}^{0i}=-B_i$ and ${\tilde{F}}^{ij}={ϵ}_{ijk}{E}_k/c$. The homogeneous Maxwell equations become ${\partial }_{\mu }{\tilde{F}}^{\mu \nu }=0$.

In oriented Minkowski coordinates with $(+---)$ signature and ${ϵ}^{0123}=+1$, the conventional dual is ${\tilde{F}}^{ab}=\frac{1}{2}{ϵ}^{abcd}{F}_{cd}$. For $F_{0i}=E_i$ and $F_{ij}=-{ϵ}_{ijk}{B}_k$ in $c=1$ units, ${\tilde{F}}^{ab}{F}_{ab}=-4\mathbf{E}\cdot \mathbf{B}$. A definition without the half produces twice this dual and contraction $-8\mathbf{E}\cdot \mathbf{B}$. The factor accounts for summing both orders of each antisymmetric index pair.