Tensors for Physics

14.5 Additional Formulas Involving Projectors 271

This expression is also traceless, notice thatetrμutrμ=eμ⊥u⊥μandeμ‖uμ‖+e⊥μu⊥μ=e·u.
The application of the fourth rank projectors onto a symmetric traceless tensor yields
a symmetric traceless tensor. By symmetry,

Pμν,μ(^0 ) ′ν′ 2 eμ′uν′ =chμhν

is expected, with a proportionality factorc. Multiplication of this equation byhμhν
and use of (14.66) yieldsc=^32 [ 2 h·eh·u−^23 e·u]. This is in accord with

Pμν,μ(^0 ) ′ν′=

3

2

hμhν hμ′hν′,

as already implied by (14.55).

Application of the fourth rank projectorP(m)on hμhκaκν,whereaμνis an
irreducible second rank tensor, yields

Pμν,μ(m)′ν′hμ′hκaκν′=

(

1

3

−

m^2

6

)

Pμν,μ(m)′ν′aμ′ν′. (14.67)

Furthermore,

hμ 2 ···hμHμ() 1 μ 2 ···μ(),ν 1 ν 2 ···νaν 1 ν 2 ···ν=i

(

Pμ(^11 )μ′

1

−Pμ(− 1 μ^1 )′

1

)

Aμ′ 1 , (14.68)

with
Aμ′ 1 =hμ′ 2 ···hμ′aμ′ 1 μ′ 2 ···μ′, (14.69)

whereaμ′ 1 μ′ 2 ···μ′is an irreducibleth rank tensor.