Tensors for Physics

36 3 Symmetry of Second Rank Tensors, Cross Product

3.1.5 Fourth Rank Projections Tensors

The decomposition (3.4) of a second rank tensorAinto its isotropic (i=0), its
antisymmetric (i =1), and its symmetric traceless (i =2) partsA(i)can also

be accomplished by application of fourth rank projection tensorsPμνμ(i)′ν′on the
componentsAμ′ν′according to

A(μνi)=Pμνμ(i)′ν′Aμ′ν′. (3.12)

Here pairs of subscripts are used like one index. Furthermore, notice thatA(μν^1 )≡Aasyμν

andA(μν^2 )≡ Aμν. The projection tensors are defined by

Pμν,μ(^0 )′ν′:=

1

3

δμνδμ′ν′, Pμνμ(^1 )′ν′:=

1

2

(δμμ′δνν′−δμν′δνμ′), (3.13)

and

Pμνμ(^2 )′ν′≡Δμν,μ′ν′:=

1

2

(δμμ′δνν′+δμν′δνμ′)−

1

3

δμνδμ′ν′. (3.14)

In the applications presented later, the symbolΔ...is preferred overP...(^2 ).
The projection tensors have the properties

Pμναβ(i) P

(j)

αβμ′ν′=δ

ijP(i)

μνμ′ν′, (3.15)

whereδijis the Kronecker symbol, being equal to 1, wheni=jand 0 wheni=j,
and they obey the ‘sum rule’ or ‘completeness relation’

Pμν,μ(^0 )′ν′+Pμν,μ(^1 )′ν′+Pμν,μ(^2 )′ν′=δμμ′δνν′. (3.16)

The contractionν′=νof the projectors yields

Pμν,μ(^0 ) ′ν=

1

3

δμμ′, Pμν,μ(^1 ) ′ν=δμμ′, Pμνμ(^2 )′ν≡Δμν,μ′ν=

5

3

δμμ′. (3.17)

The subsequent complete contraction, corresponding toμ′=μ, gives the numbers of
the independent components of the isotropic, antisymmetric and symmetric traceless
parts of second rank tensor in 3D, viz.:

Pμν,μν(^0 ) = 1 , Pμν,μν(^1 ) = 3 , Pμνμν(^2 ) ≡Δμν,μν= 5. (3.18)

Generalized Delta-tensors of rank 2which, when applied to tensors of rank,
project out the symmetric traceless part of that tensor, will be introduced later.