Tensors for Physics

420 Appendix: Exercises...

18.2 Derivation of the Inhomogeneous Maxwell Equations from the Lagrange
Density(p.386)
Point of departure is the variational principle (18.72), viz.

δS=

∫

δLd^4 x= 0 ,

with the Lagrange density (18.70). It is understood that the variation ofLis brought
about by a variationδΦof the 4-potential. Thus

δL=−

(

JiδΦi+

1

2 μ 0

FikδFik

)

,

with

δFik=∂kδΦi−∂iδΦk,

cf. (18.55). Due toFik=−Fki, one has

FikδFik= 2 Fik∂kδΦi.

Integration by parts andδΦi=0 at the surface of the 4D integration range, leads to

δS=−

∫

δ

(

Ji−

1

μ 0

∂kFik

)

δΦid^4 x= 0. (A.19)

The 4D integration volume is arbitrary. Thus the integrand has to vanish and one
obtains (18.73), viz.

Ji=(μ 0 )−^1 ∂kFik.

This relation corresponds to the inhomogeneous Maxwell equation (18.62) for fields
in vacuum, whereFik=μ 0 Hikholds true.

18.3 Derive the 4D Stress Tensor(p.388)
The force densityfi=JkFkicf. (18.74), can be rewritten with help of the inhomo-
geneous Maxwell equation (18.62). Where necessary, co- and contra-variant com-
ponents are interchanged with the help of the metric tensor. The first few steps of the
calculation are

fi=JkFki=gkJFki=gk∂mHmFki=gk∂mHmgkk

′

gii

′

Fk′i′.

Due togkgkk

′
=δk′and with the renamingi→kof the summation indices, one
finds

fi=gikFk∂mHm=∂m(gikFkHm)−gikHm∂mFk.