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.