136
6 de Sitter Cosmology
With w i (u) given by (6.40) we find that
η ij ˙
w
i
˙
w
j
= −
6
m
2
κ ,
(6.55)
with κ given by (6.51). Hence the character of the world line x i = w i (u) depends
upon the sign of κ. For x i = w i (u) to be a null geodesic we must have κ = 0 and
¨
w
i
= C(u) ˙
w
i ,
(6.56)
for some real-valued function C(u). Substituting (6.40) into
¨
w
0
− ¨
w
3
= C ( ˙
w
0
− ˙
w
3 ) ,
(6.57)
results in
C =
1
α
dα
du
−
2 α
m
.
(6.58)
Now (6.40) in
¨
w
1
+ i ¨
w
2
= C( ˙
w
1
+ i ˙
w
2 ) ,
(6.59)
along with (6.58), produces
1
β
dβ
du
=
1
α
dα
du
⇒
1
β
dβ
du
=
1
¯
β
d ¯
β
du
.
(6.60)
As a consequence of (6.60) the equation
¨
w
0
+ ¨
w
3
= C ( ˙
w
0
+ ˙
w
3 ) ,
(6.61)
is automatically satisfied. Finally we note that
κ = 0 and
1
β
dβ
du
=
1
¯
β
d ¯
β
du
⇒
1
β
dβ
du
=
1
α
dα
du
,
(6.62)
and
1
β
dβ
du
=
1
¯
β
d ¯
β
du
⇒ Re β = 0 or Im β = 0 or Re β = c 0 Im β ,
(6.63)
for some real number c 0 . We can summarise the results here in the theorem of Tran
and Robinson [5, 6]: (1) > 0 ⇒ κ > 0 ⇒ intersecting null hyperplanes;
(2) < 0 ⇒ κ > 0 or κ < 0 or κ = 0 with (i) κ > 0 ⇒ intersecting