142
6 de Sitter Cosmology
with
ξ
c = −f
−1
˙
a c − λ
−1 λ ,c −
1
2
f
−1 ˙
f u ,c ,
(6.101)
and
f = ˙
b + ˙
a c x
c
,
(6.102)
and the dot denoting differentiation with respect to u. The fact that u ,a ;b has the
algebraic form of the right hand side of (6.100) confirms that, in the space-time
with metric g a b , u ,a is geodesic and shear-free following the Robinson–Trautman
[3] test for these properties. However we also require u ,a to be expansion-free if
u = constant are to be null hyperplanes in the space-time with metric g a b . This
means that we must have
g
a b
u ,a ;b = −2 λ
−3 f
−1 λ ,b a
b = 0 ⇒ a
0 = 0 .
(6.103)
But since a b a b = 0 this means that we must have a b = 0 and so we get no null
hyperplanes in the de Sitter space-time in this case. Next we choose u(T , x, y, z) to
be given implicitly by
η a b (x
a − w
a
(u))(x
b − w
b
(u)) = 0 .
(6.104)
Thus each u = constant is a null cone in Minkowskian space-time. The vertices
of the null cones u = constant lie on an arbitrary world line x a = w a (u). From
(6.104) it follows that
u ,a =
x a − w a (u)
η c d ˙
w c (x d − w d )
.
(6.105)
These null cones u = constant are generated by expanding, shear-free null geodesics
in Minkowskian space-time. In the de Sitter space-time u = constant will be null
hyperplanes provided u ,a is expansion-free. This is the case if
g
a b
u ,a ;b = 0 ⇒ 1 +
√
2 k T λ ,a (x
a − w
a
) ⇒ w
0
(u) = 0 .
(6.106)
Hence the world line of the vertices of the null cones u = constant in Minkowskian
space-time is space-like. Consequently the null cones must intersect each other and
so the corresponding null hyperplanes in the de Sitter space-time also intersect each
other since they are described by (6.104) too, which now reads
(x
α
− w
α )(x
α
− w
α ) = T
2
=
1
2 k 2 e
−2
√
2 k t .
(6.107)
Using (6.95) we see that (6.107) coincides with (6.94).
6 de Sitter Cosmology
with
ξ
c = −f
−1
˙
a c − λ
−1 λ ,c −
1
2
f
−1 ˙
f u ,c ,
(6.101)
and
f = ˙
b + ˙
a c x
c
,
(6.102)
and the dot denoting differentiation with respect to u. The fact that u ,a ;b has the
algebraic form of the right hand side of (6.100) confirms that, in the space-time
with metric g a b , u ,a is geodesic and shear-free following the Robinson–Trautman
[3] test for these properties. However we also require u ,a to be expansion-free if
u = constant are to be null hyperplanes in the space-time with metric g a b . This
means that we must have
g
a b
u ,a ;b = −2 λ
−3 f
−1 λ ,b a
b = 0 ⇒ a
0 = 0 .
(6.103)
But since a b a b = 0 this means that we must have a b = 0 and so we get no null
hyperplanes in the de Sitter space-time in this case. Next we choose u(T , x, y, z) to
be given implicitly by
η a b (x
a − w
a
(u))(x
b − w
b
(u)) = 0 .
(6.104)
Thus each u = constant is a null cone in Minkowskian space-time. The vertices
of the null cones u = constant lie on an arbitrary world line x a = w a (u). From
(6.104) it follows that
u ,a =
x a − w a (u)
η c d ˙
w c (x d − w d )
.
(6.105)
These null cones u = constant are generated by expanding, shear-free null geodesics
in Minkowskian space-time. In the de Sitter space-time u = constant will be null
hyperplanes provided u ,a is expansion-free. This is the case if
g
a b
u ,a ;b = 0 ⇒ 1 +
√
2 k T λ ,a (x
a − w
a
) ⇒ w
0
(u) = 0 .
(6.106)
Hence the world line of the vertices of the null cones u = constant in Minkowskian
space-time is space-like. Consequently the null cones must intersect each other and
so the corresponding null hyperplanes in the de Sitter space-time also intersect each
other since they are described by (6.104) too, which now reads
(x
α
− w
α )(x
α
− w
α ) = T
2
=
1
2 k 2 e
−2
√
2 k t .
(6.107)
Using (6.95) we see that (6.107) coincides with (6.94).