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(
)
r r r
kwpv
pv
r
=
0
=
+
pk
r
=
w
,
and
(7)
r
which is shown in Figures 6.5(a) and (b):
FIGURE 6.5(A) Spatial and time-like components of k r .
FIGURE 6.5(B)
Null vector k r splitting.
The unitary vector p r only depends on the spatial triad; therefore, it can be
written with common spherical coordinates (see Figure 6.4):
()
()
σ
r
r
r
r
r
p
=
sin
θφ
cos
e
+
sin
θφ
sin
e
+
cos
θ
e
=
p
e
(8a)
()
( )
( )
1
2
3
σ
()
()
γ
γ
r
∴=
pe
p
r
where we have employed the dual base e (γ) r , defined by the following
equation:
 
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