Tensors for Physics

18.4 Maxwell Equations in 4D-Formulation 383

18.4.4 The Homogeneous Maxwell Equations

From the definitionFik=∂kΦi−∂iΦk,cf.(18.55) follows

∂nFik+∂iFkn+∂kFni= 0. (18.58)

The left hand side of this equation is identical to zero unless all three indices(n,i,k)
are different. The case( 1 , 2 , 3 )corresponds to∇μBμ =0, the cases( 2 , 3 , 4 ),
( 3 , 1 , 4 ),( 1 , 2 , 4 )are equivalent to the induction law

εμνλ∇νEλ=−

∂Bμ

∂t

,

cf. (7.57). Thus (18.58) is the 4D formulation of the homogeneous Maxwell equa-
tions, which are a consequence of the field tensor being given in terms of the 4-
potential by (18.55).
The dual field tensor, cf. (18.42), is

F ̃ik=^1

2

εikmnFmn=εikmn∂mΦn. (18.59)

One has

∂kF ̃ik=εikmn∂k∂mΦn= 0 ,

since∂k∂mis symmetric under the interchange ofkandm, while the epsilon-tensor is
antisymmetric. Thus the homogeneous Maxwell equations (18.58) are equivalent to

∂kF ̃ik= 0. (18.60)

18.4.5 The Inhomogeneous Maxwell Equations

By analogy to (18.57), the four-dimensionalH-tensor is defined by

Hik:=

⎛

⎜

⎜

⎝

0 H 3 −H 2 cD 1

−H 3 0 H 1 cD 2

H 2 −H 1 0 cD 3

−cD 1 −cD 2 −cD 3 0

⎞

⎟

⎟

⎠. (18.61)

The field tensorHikhas the same form asHik, just with the opposite sign of the
terms involvingD 1 ,D 2 ,D 3.