This
tensor, denoted G
μν is in a sense
dual to the
Maxwell Tensor in
electrodynamics. It is defined as G
μν = ½ε
μνρσF
ρσ, where the tensor ε
μνρσ is very similar to
εijk 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 -B1 -B2 -B3)
(B1 0 -E3 E2)
(B2 E3 0 -E1)
(B3 -E2 E1 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
μG
μν=0