Tensors for Physics

386 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations

δS:=

∫

δLd^4 x= 0. (18.72)

It is understood thatδΦis zero at the ‘surface’ of the 4D integration range. Use of

the variational principle (18.72) leads to

(

Ji−(μ 0 )−^1 ∂nFin

)

δΦi= 0 ,

and consequently

Ji=(μ 0 )−^1 ∂nFin. (18.73)

This is the relation (18.62) for the special case whereFik=μ 0 Hikapplies, i.e. for
charges, currents and fields in vacuum. The derivation of (18.73)from(18.72)is
deferred to the Exercise (18.2).
The scalarsJiΦiandFikFikoccurring in the Lagrange density (18.70) are invari-
ant under the parity operationPand the time reversalT, despite the fact that the
quantitiesJi,ΦiandFikhave a well defined symmetry only under the combined
operationPT.
The Lagrange density (18.70) leading to the inhomogeneous Maxwell equations
involves just the first one of the scalar invariants associated with the field tensor, cf.
(18.69). Inclusion of the second scalar invariant, as suggested by Born and Infeld
[207], leads to extended Maxwell equations with terms nonlinear in theEandB
fields, even in vacuum. The resulting electrostatic potential of an electron located at
r=0, no longer diverges forr→0.

18.2 Exercise: Derivation of the Inhomogeneous Maxwell Equations

Derive the inhomogeneous Maxwell equations for fields in vacuum from the varia-

tional principle (18.72) with the Lagrange density (18.70).

18.5 Force Density and Stress Tensor.

18.5.1 4D Force Density

The 4D force densityfiorfiis defined by

fi=JkFki, fi=JkFki. (18.74)

The first three components officorrespond to the 3D force densitykwith

kμ=ρ(Eμ+εμνλvνBλ)=ρEμ+εμνλjνBλ,

whereρis the charge density and the 3D flux density isj=ρv. The fourth component

offiis the power density associated withk, more specifically