270 14 Rotation of Tensors
(
Pμν,μ(^2 ) ′ν′+P(μν,μ−^2 )′ν′
)
eμ′uν′ =eμuν +1
2
hμhν (h·e)(h·u)−[hμeν(h·u)+hμuν(h·e)],2 i(
P(μν,μ^1 ) ′ν′−Pμν,μ(−^1 )′ν′)
eμ′uν′ =[hμ(h×e)ν+hν(h×e)μ](h·u)(14.63)
+[hμ(h×u)ν+hν(h×u)μ](h·e),
4 i(
P(μν,μ^2 ) ′ν′−Pμν,μ(−^2 )′ν′)
eμ′uν′ ={
uμ(h×e)ν+uν(h×e)μ+eμ(h×u)ν+eν(h×u)μ}
−
{
[hμ(h×e)ν+hν(h×e)μ](h·u)
+[hμ(h×u)ν+hν(h×u)μ](h·e)}
.
Direct application of the equation (14.31) defining the fourth rank projectors in terms
of the second rank tensors and use of symbolic notation, leads to
P(±^1 ):(eu+ue)=1
2
[e‖u⊥+u‖e⊥+u⊥e‖+e⊥u‖] (14.64)∓i
2[e‖utr+u‖etr+utre‖+etru‖],P(±^2 ):(eu+ue)=1
4
[e⊥u⊥+u⊥e⊥−utretr−etrutr] (14.65)∓
i
4[e⊥utr+u⊥etr+utre⊥+etru⊥].Hereeanduare two arbitrary unit vectors which, in special cases, may be parallel or
perpendicular to each other. The parts of a vectorewhich are parallel, perpendicular
and transverse with respect toh, are defined by
e‖=P‖·e, e⊥=P⊥·e, etr=H·e=h×e.Due toPμν,μ(±^2 )′ν′δμ′ν′=0, the expressions(eu+ue)in the equations above may
be replaced by 2eu.Furthermore, the resulting dyadics in (14.64) and (14.65)
are automatically traceless. This is not the case for the corresponding expression
involving the projectorP(^0 )=P(^0 )P(^0 )+P(^1 )P(−^1 )+P(−^1 )P(^1 ). Here one has
P(μν,μ^0 ) ′ν′δμ′ν′=Pμν(^0 )+Pμν(^1 )+Pμν(−^1 )=δμν. Thus one obtains
Pμν,μ(^0 ) ′ν′ 2 eμ′uν′ =e‖μu‖ν+e‖νu‖μ (14.66)+1
2
[e⊥μu⊥ν+u⊥μe⊥ν+utrμetrν+eμtrutrν]−2
3
e·uδμν.