Tensors for Physics

18.4 Maxwell Equations in 4D-Formulation 385

As before, the abbreviationγ =( 1 −β^2 )−^1 /^2 is used. In terms of the pertaining
components of theE- andB-fields, (18.65) corresponds to

E′ 1 =E 1 , E′ 2 =γ(E 2 −vB 3 ), E 3 ′=γ(E 3 +vB 2 ), (18.66)

and

B 1 ′=B 1 , B 2 ′=γ(B 2 +vE 3 /c^2 ), B′ 3 =γ(B 3 −vE 2 /c^2 ). (18.67)

The components of the electromagnetic fields perpendicular to the direction of the
relative velocityvare modified and do depend onv. In vector notation, the relations
(18.66) and (18.67) correspond to

E′=γ(E+v×B), B′=γ(B−v×E/c^2 ).

Notice that

E′·B′=E·B, E′·E′−c^2 B′·B′=E·E−c^2 B·B. (18.68)

Dueto(18.47) and (18.48), these transformation properties of the fields are associated
with thescalar invariants of the field tensor,viz.

FikFik= 2 (B^2 −E^2 /c^2 ), det(Fik)=(E·B)^2 /c^2. (18.69)

Relations analogous to (18.65) and (18.66), (18.67) apply for the field tensorHik
and for the field vectorsDandH.

18.4.8 Lagrange Density and Variational Principle

The Maxwell equations can be derived from a variational principle involving a
Lagrange density depending on the relevant scalar invariants. For electric charges
and currents in vacuum, whereFik=μ 0 Hikapplies, the Lagrange densityLis
defined by

L=−

(

JiΦi+

1

4 μ 0

FikFik

)

. (18.70)

Its 4D ‘action’ integral is denoted by

S:=

∫

Ld^4 x. (18.71)

The variational principle states: the action integralSis extremal under a variation
δΦof the 4-potentialΦsuch that