since the derivative∂ρ∂μAνis symmetric inρ,μ, while theεμνρσis antisymmetric in all its indices,
and may be normalized toε^0123 = 1 (note that because of the Minkowski signature of the metric,
this implies thatε 0123 =−1. Expressed in terms ofEandB, this yields the first group of Maxwell’s
equations, by considering separately the cases whereσ= 0 andσ=iwithi= 1, 2 ,3,
εμνρ^0 ∂ρFμν= 0 ⇔ ∇·B= 0
εμνρi∂ρFμν= 0 ⇔ ∂ 0 B+∇×E= 0 (21.57)It is instructive to give the derivation of these formulas. On the first line, the last index onε
is a time-index, namely 0, so that the other three indices onεmust be space-indices, which we
shall rebaptizei,j,k. Thus, the first line becomesεijk^0 ∂kFij = 0. Using the definition of the
magnetic fieldFij=
∑
mεijmBm, the fact that, with our conventions,εijk (^0) =−εijk, and the double
contraction
εijkεijm= 2δkm (21.58)
we findεijk^0 ∂kFij = 2∂kBk = 2∇ ·B. On the second line, the last index isi, which is space-
like. Thus, one and only one of the indicesμ,ν,ρmust be 0. Collecting all possibilities gives
εjk^0 i∂ 0 Fjk+ 2ε^0 jki∂kF 0 j= 0. Using the definitions of the electric and magnetic fields,ad the above
double contraction formula, we obtain∂ 0 Bi+εijk∂jEk= 0, which gives the formula quoted.
The second group of Maxwell equations are also linear and involve first derivatives of the electric
and magnetic fields and are sourced by the current density. There is only one Lorentz-invariant
combination with these properties, up to an overall constant factor, namely
∂μFμν=−jν (21.59)
Note that, becauseFμν is antisymmetric inμandν, the current density must be conserved,
∂ν∂μFμν =∂νjν = 0. There are actually 4 equations encoded in (21.59), one for each of the
possible values ofν.
∂μFμ^0 =−j^0 ⇔ ∇·E=ρ
∂μFμi=−ji ⇔ ∂ 0 E−∇×B=j (21.60)
In summary, the relativistic form of the two groups of Maxwell’s equations reduce to,
εμνρσ∂ρFμν = 0
∂μFμν = −jν (21.61)
These equations may be derived from an action principle. ConsideringAμ as the fundamental
variable, and definingFμν=∂μAν−∂νAμ, then a suitable action is given by,
S[A] =
∫
d^4 x(
−1
4
FμνFμν+Aμjμ)
(21.62)