Tensors for Physics

14.1 Rotation of Vectors 261

reflects that thePμν(m)are ‘eigen-tensors’ of the tensorHμν. On the other hand,Hμν
can be represented as a linear combination of the projection tensors, viz.

Hμν=

∑^1

m=− 1

imPμν(m). (14.12)

Some additional formulas involving second rank projection operators are:

hνhμaν =

(

2

3

Pμν(^0 )+

1

2

Pμν(^1 )+

1

2

Pμν(−^1 )

)

aν, (14.13)

hνhκhμbνκ =

(

3

5

Pμν(^0 )+

8

15

Pμν(^1 )+

8

15

Pμν(−^1 )

)

bνκhκ, (14.14)

whereaνandbμνare a vector and a second rank tensor.

14.1.3 Rotation Tensor for Vectors

With the help of the projection tensors, the rotation tensor for vectors can now be
expressed as

Rμν(φ)=

∑^1

m=− 1

Pμκ(m)(exp[φH])κν=

∑^1

m=− 1

exp[imφ]Pμν(m). (14.15)

Decomposition into real and imaginary parts leads to

Rμν(φ)=Pμν(^0 )+cosφ

(

Pμν(^1 )+Pμν(−^1 )

)

+sinφi

(

Pμν(^1 )−Pμν(−^1 )

)

. (14.16)

Notice that

Pμν(^0 )=hμhν≡Pμν‖, Pμν(^1 )+Pμν(−^1 )=δμν−hμhν≡Pμν⊥, (14.17)

correspond to the projection tensors onto the direction parallel and perpendicular to
h, denoted byP‖andP⊥, respectively. Furthermore, one has

i

(

Pμν(^1 )−Pμν(−^1 )

)

=Hμν. (14.18)

Thus the rotation tensor also reads

Rμν(φ)=hμhν+sinφεμλνhλ+cosφ(δμν−hμhν). (14.19)