Elements of Modern Physics
406
2
12
lim
s
r
r
t
→
→ ∞
This explains the name ‘black hole’ given to the object. It should be
mentioned that this result does not take quantum effects into account. It has
been shown by Hawking that quantum effects do allow some radiation to come
out from the black holes so that, strictly speaking, a black hole is not a black
hole.
11.4 KINEMATICS OF THE UNIVERSE
In this section, the metric space of the universe is considered.
It may be expected that the large-scale properties of the universe are not
affected by local variations in the distribution of matter. This assumption, stated
in the form of a principle, greatly simplifies the analysis. The cosmological
principle states that the universe must appear the same to all observers who are
at rest with respect to matter in the neighbourhood. The universe must also be
isotropic. This means that clustering of galaxies is a local irregularity and the
cosmological scales are must larger than the sizes of the galaxies.
Observationally, the distribution of galaxies does appear to be homogeneous
and isotropic though there may be some small deviations. The concept of an
observer who is at rest with respect to local matter, known as a fundamental
observer, is very useful in the determination of the kinematics of the universe.
The history of each fundamental observer being the same, the time coordinate
can be linked with a local property, say the density, and may be taken to be the
proper time of the fundamental observer. It was shown by Robertson and Walker,
that the metric of the universe satisfying the cosmological principle, is of the
form
(∆τ)
2
=
2
2
2
2
2
2
2
2
2
1
( )
( )
( )
sin ( )
1
r
t
r
r
c
Kr
∆
∆ −
+
∆θ +
θ ∆φ
−
(11.58)
where K is the curvature of space. The spatial part of the metric is similar to the
metric of a spherical surface given in Eq. (11.29). The curvature of the space
may be written as
K = 2 ( )
k
R t
(11.59)
where R(t) is called the comic scale factor, with k = 1 for positive curvature,
k = –1 for negative curvature and k = 0 for flat space. The metric may be written
in a more convenient form in terms of the dimensionless variable σ,
406
2
12
lim
s
r
r
t
→
→ ∞
This explains the name ‘black hole’ given to the object. It should be
mentioned that this result does not take quantum effects into account. It has
been shown by Hawking that quantum effects do allow some radiation to come
out from the black holes so that, strictly speaking, a black hole is not a black
hole.
11.4 KINEMATICS OF THE UNIVERSE
In this section, the metric space of the universe is considered.
It may be expected that the large-scale properties of the universe are not
affected by local variations in the distribution of matter. This assumption, stated
in the form of a principle, greatly simplifies the analysis. The cosmological
principle states that the universe must appear the same to all observers who are
at rest with respect to matter in the neighbourhood. The universe must also be
isotropic. This means that clustering of galaxies is a local irregularity and the
cosmological scales are must larger than the sizes of the galaxies.
Observationally, the distribution of galaxies does appear to be homogeneous
and isotropic though there may be some small deviations. The concept of an
observer who is at rest with respect to local matter, known as a fundamental
observer, is very useful in the determination of the kinematics of the universe.
The history of each fundamental observer being the same, the time coordinate
can be linked with a local property, say the density, and may be taken to be the
proper time of the fundamental observer. It was shown by Robertson and Walker,
that the metric of the universe satisfying the cosmological principle, is of the
form
(∆τ)
2
=
2
2
2
2
2
2
2
2
2
1
( )
( )
( )
sin ( )
1
r
t
r
r
c
Kr
∆
∆ −
+
∆θ +
θ ∆φ
−
(11.58)
where K is the curvature of space. The spatial part of the metric is similar to the
metric of a spherical surface given in Eq. (11.29). The curvature of the space
may be written as
K = 2 ( )
k
R t
(11.59)
where R(t) is called the comic scale factor, with k = 1 for positive curvature,
k = –1 for negative curvature and k = 0 for flat space. The metric may be written
in a more convenient form in terms of the dimensionless variable σ,