14.2 Rotation of Second Rank Tensors 263
The fourth rank tensorH is defined by
Hμν,μ′ν′≡εμλμ′hλδνν′+ενλν′hλδμμ′=Hμμ′δνν′+Hνν′δμμ′. (14.24)The infinitesimal rotation of a symmetric tensorSis also described byH,viz.
Sμν′ =Sμν+δφHμν,μ′ν′Sμ′ν′. (14.25)In the following, it is assumed thatSis also traceless, i.e. it is an irreducible second
rank tensor:Sμν= Sμν.The rotated tensorSμν′ is also irreducible. Then one has
Hμν,μ′ν′Sμ′ν′=Hμν,μ′ν′Sμ′ν′,andH is equivalent to
Hμν,μ′ν′= 2 hλμν,λ,μ′ν′, (14.26)
for...see (11.19).
14.2.2 Fourth Rank Projection Tensors
Fourth rank projection tensorsP(m^1 ,m^2 )are defined via products of the second rank
projectors (14.7):
Pμν,μ(m^1 ,m′ν^2 ′)=Pμμ(m^1 ′)Pνν(m′^2 ). (14.27)The fourth rank projectors have the property
Pμν,λκ(m^1 ,m^2 )P
(m′ 1 ,m′ 2 )
λκ,μ′ν′ =P(m 1 ,m 2 )
μν,μ′ν′δ(m 1 m′ 1 )δ(m 2 m′ 2 ). (14.28)The relation
Hμν,μ′ν′Sμ′ν′=∑^1
m 1 =− 1∑^1
m 2 =− 1i(m 1 +m 2 )Pμν,μ(m^1 ,m′ν^2 ′)Sμ′ν′follows the definition ofHand the properties ofH. Insertion ofPinto this equation
and use of (14.28) leads to
Hμν,λκP(λκ,μm^1 ,′mν^2 ′)=i(m 1 +m 2 )Pμν,μ(m^1 ,m′ν^2 ′). (14.29)