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.