13.3 Consequences of the Metric
217
The approximate form is rather elegant. Note that we may again use a(t 0 ) = 1 if
desired. Notice also from (13.35) that for positive k we may interpret 1/
√
k as the
maximum distance of a galaxy from us.
Equation (13.36) is also useful for illustrating an interesting geometric concept.
We can carry out in three dimensions the same calculation that led to (13.6) in 2
dimensions, that is the ratio of the circumference C s to the radius R s of a small circle
as illustrated in Fig. 13.5. We obtain for the present three dimensional case
C s
R s
∼ = 2π
1 + k
r
2
6
.
(13.37)
This is the same result as (13.6) although in slightly different notation: in (13.6)
the curvature parameter k is dimensionless and in (13.37) it is an inverse distance
squared. This clearly tells us that, loosely speaking, there is “too little space” around
a given point in the hypersphere and “too much space” in the pseudo-hypersphere.
It is also easy to relate physical distances to the radial coordinate u used in (13.10)
and (13.11), which we leave to the readier in Exercise 13.16.
13.4 De Sitter Space
It is amusing to consider a simple cosmological model based only on ad hoc considerations and not on the gravitational field equations. Without physical justification
we suppose that the Hubble constant is truly constant, H (t) = H 0 . Then we may
integrate (13.26) to obtain the scale factor at any time, which we write as
a(t) = a(t 0 )e
H 0 (t−t0)
.
(13.38)
This is known as the de Sitter model universe. It is a perpetual universe: it has always
existed and always will exist, expanding exponentially forever. It was extremely
small in the distant past, but never had zero size. If we also assume k = 0 the metric
is quite simple
ds
2
= c
2 dt
2
− a(t 0 )
2 e
2H 0 (t−t0)
dr
2
+ r
2
(dθ
2
+ sin
2
θ dϕ
2
)
.
(13.39)
The space-time is called de Sitter space. We will return to de Sitter space in later
chapters. It is intimately related to the actual universe in the distant future, and also
has features in common with the very early inflationary universe. Later we will also
use the field equations and consider spatially non-flat versions with k = 0.
217
The approximate form is rather elegant. Note that we may again use a(t 0 ) = 1 if
desired. Notice also from (13.35) that for positive k we may interpret 1/
√
k as the
maximum distance of a galaxy from us.
Equation (13.36) is also useful for illustrating an interesting geometric concept.
We can carry out in three dimensions the same calculation that led to (13.6) in 2
dimensions, that is the ratio of the circumference C s to the radius R s of a small circle
as illustrated in Fig. 13.5. We obtain for the present three dimensional case
C s
R s
∼ = 2π
1 + k
r
2
6
.
(13.37)
This is the same result as (13.6) although in slightly different notation: in (13.6)
the curvature parameter k is dimensionless and in (13.37) it is an inverse distance
squared. This clearly tells us that, loosely speaking, there is “too little space” around
a given point in the hypersphere and “too much space” in the pseudo-hypersphere.
It is also easy to relate physical distances to the radial coordinate u used in (13.10)
and (13.11), which we leave to the readier in Exercise 13.16.
13.4 De Sitter Space
It is amusing to consider a simple cosmological model based only on ad hoc considerations and not on the gravitational field equations. Without physical justification
we suppose that the Hubble constant is truly constant, H (t) = H 0 . Then we may
integrate (13.26) to obtain the scale factor at any time, which we write as
a(t) = a(t 0 )e
H 0 (t−t0)
.
(13.38)
This is known as the de Sitter model universe. It is a perpetual universe: it has always
existed and always will exist, expanding exponentially forever. It was extremely
small in the distant past, but never had zero size. If we also assume k = 0 the metric
is quite simple
ds
2
= c
2 dt
2
− a(t 0 )
2 e
2H 0 (t−t0)
dr
2
+ r
2
(dθ
2
+ sin
2
θ dϕ
2
)
.
(13.39)
The space-time is called de Sitter space. We will return to de Sitter space in later
chapters. It is intimately related to the actual universe in the distant future, and also
has features in common with the very early inflationary universe. Later we will also
use the field equations and consider spatially non-flat versions with k = 0.