This

tensor, denoted G

_{μν} is in a sense

dual to the

Maxwell Tensor in

electrodynamics. It is defined as G

_{μν} = ½ε

_{μνρσ}F

^{ρσ}, where the tensor ε

_{μνρσ} is very similar to

ε_{ijk} in that it is antisymmetric under any change of indicies, is zero for repeated indicies, and has ε

^{0123}=1.

G

_{μν} is then also antisymmetric, and has components:

(0 -B_{1} -B_{2} -B_{3})
(B_{1} 0 -E_{3} E_{2})
(B_{2} E_{3} 0 -E_{1})
(B_{3} -E_{2} E_{1} 0 )

That is, it is just like the Maxwell tensor, with

**B** in place of

**E** and -

**E** in place of

**B**.

Maxwell's Equations imply that d

_{μ}G

^{μν}=0