Tensors for Physics

262 14 Rotation of Tensors

In hindsight, this result is not unexpected for the rotation of a vector. However, the

formal considerations presented here are suitable for a generalization to the rotation

of tensors.

The orthogonal transformation matrixUfor the rotation of the coordinate system,

as introduced in Sect.2.4.2, is related toRbyUμν(φ)=Rμν(−φ). To compare with

(2.41), choosehparallel to the 3-axis, also referred to as to thez-axis.

14.1 Exercise: Scalar Product of two Rotated Vectors
Leta ̃μ=Rμν(φ)aνandb ̃μ=Rμκ(φ)aκbe the Cartesian components of the vectors
aandbwhich have been rotated by the same angleφabout the same axis. Prove that
the scalar productsa ̃·b ̃is equal toa·b.

14.1.4 Connection with Spherical Components.

The complex basis vectorse(m), introduced by (9.16) and employed with the defini-

tion of spherical components, cf. (9.18) and (9.19), are eigenvectors of the projection

tensors, provided thathis chosen parallel to the unit vectore(z), more specifically:

Pμν(m)e(m

′)

ν =δmm′e

(m)

μ , m,m

′=− 1 , 0 , 1. (14.20)

Thus, due to (9.18), application the projector on a vectorayields

Pμν(m)aν=a(m)e(μm), (14.21)

wherea(m)is a spherical component of this vector. Furthermore, the projection tensor

can be expressed by

Pμν(m)=

(

e(μm)

)∗

e(νm), (14.22)

when one choosesh=e(z).

14.2 Rotation of Second Rank Tensors

14.2.1 Infinitesimal Rotation

LetAμν=aμaνbe a second rank tensor composed of the components of the vector

a. The infinitesimal rotation by the angleδφabout an axis parallel toh, as described

by (14.1) with (14.2) implies that the rotated tensorA′μνis

A′μν=Aμν+δφ (Hμμ′δνν′+Hνν′δμμ′)Aμ′ν′=Aμν+δφHμν,μ′ν′Aμ′ν′.

(14.23)