Tensors for Physics

264 14 Rotation of Tensors

Notice thatm 1 +m 2 assumes the five valuesm=− 2 ,− 1 , 0 , 1 ,2. The fourth rank
tensor obeys the eigenvalue equation

∏^2

m=− 2

(H−im 1 )= 0 , (14.30)

with these five eigenvalues form. The corresponding eigen-tensors are

Pμν,μ(m)′ν′=

∑^1

m 1 =− 1

∑^1

m 2 =− 1

Pμμ(m^1 ′)Pνν(m′^2 )δ(m,m 1 +m 2 ), (14.31)

where theδ(m,m 1 +m 2 )=1form=m 1 +m 2 , andδ(m,m 1 +m 2 )=0, for
m=m 1 +m 2. In terms of these projectors, the spectral decomposition ofHreads

Hμν,μ′ν′=

∑^2

m=− 2

imP(μν,μm)′ν′. (14.32)

14.2.3 Fourth Rank Rotation Tensor

By analogy to the rotation of a vector, cf. (14.3), the rotation of a second rank tensor
by the finite angleφis given by

A′μν=(exp[φH])μν,μ′ν′Aμ′ν′≡Rμν,μ′ν′(φ)Aμ′ν′, (14.33)

with the fourth rank rotation tensor

Rμν,μ′ν′(φ)=

∑^2

m=− 2

exp[im]Pμν,μ(m)′ν′. (14.34)

Decomposition into real and imaginary parts yields, by analogy to (14.16)

Rμν,μ′ν′(φ)=P(μν,μ^0 ) ′ν′+

∑^2

m= 1

[

cos(mφ)

(

Pμν,μ(m)′ν′+Pμν,μ(−m)′ν′

)

+sin(mφ)i

(

Pμν,μ(m)′ν′−Pμν,μ(−m)′ν′

)]

. (14.35)

The comparison of the formulas for the rotation of second rank tensor with those for
the rotation of a vector indicates how the general case of a rotation of a tensor of
rankcan be treated.