7.5. THE UNIVERSE 465
Whenv/cis small, (7.5.38) can be approximatively expressed as
(7.5.39) z≃v/c.
In addition, Hubble discovered that the redshift has an approximatively linear relation with
the distance:
(7.5.40) z≃kR, kis a constant.
Thus, the Hubble Law (7.5.1) follows from (7.5.39) and (7.5.40). It is the Hubble Law (7.5.1)
that leads to the conclusion that our Universe is expanding.
However, if we adopt the view that the globular universe is ina black hole with the
Schwarzschild radiusRsas in (7.5.31), the black hole redshift (7.5.37) cannot be ignored. By
(7.5.32) and (7.5.33), the time-componentg 00 for the black hole can approximatively take the
TOV solution asrnearRs:
g 00 =−1
4
(
1 −
r^2
R^2 s)
, forrnearRs.Hence, the redshift (7.5.37) is as
(7.5.41) 1 +z=
√
1 −r 02 /R^2 s
√
1 −r 12 /R^2 sforr 0 ,r 1 <Rs.It is known that for a remote galaxy,r 1 is close to the boundaryr=Rs. Therefore by (7.5.41)
we have
z→+∞ as r 1 →Rs.
It reflects the redshifts observed from most remote objects.If the object is a virtual image as
shown in Figure7.13, then its position is the reflection pointr 1. Thus, we see that even if the
remote object is not moving, its redshift can still be very large.
5.CMB problem.In 1965, two physicists A. Penzias and R. Wilson discovered the low-
temperature cosmic microwave background (CMB) radiation,which fills our Universe, and
it is ever regarded as the Big-Bang product. However, for a static closed Universe, it is the
most natural thing that there exists a CMB, because the Universe is a black-body and CMB
is a result of black-body radiation.
- None expanding Universe. As the energy of the Universe is given, the maximal
radius, i.e. the Schwarzschild radiusRs, is determined, and the boundary is invariant. In fact,
a globular universe must fill the ball with the Schwarzschildradius, although the distribution
of the matter in this ball may be slightly non-homogenous. The main reason is that if the
universe has a radiusRsmaller thanRs, then it must contain at least a sub-black hole with
radiusR 0 as follows
R 0 =√
R
RsR.
In Section7.5.4we shall discuss this topic.