Mathematical Principles of Theoretical Physics

470 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

By (7.5.56), we have

Mr=

4 π

3

r^3 ρ=

r^3

R^3

M.

Then it follows from (7.5.57) that

(7.5.58) r=

√

R

Rs

R.

Actually, in general for a ballBrin a universe with radiusR<Rs, if its massMrsatisfies
(7.5.57) then it will form a black hole, and its radiusrsatisfies that

r≤

√

R

Rs

R.

In particular, there must exist a black hole whose radiusris as in (7.5.58). Thus, we derive
the conclusion (7.5.55).
Based on (7.5.55) we can deduce that if the Universe were born to a Big-Bang andcontin-
uously expands, then it would contain many black holes with smaller ones being embedded
in the larger ones. In particular, the Universe would contain a huge black hole whose radius
ris almost equal to the cosmic radiusRs. This is not what we observed in our Universe.
The second difficulty of modern cosmology concerns with the Hubble Law (7.5.1), which
is restated asv=HR, wherec/H=Rs. Consider a remote object with massM 0 , then the
observed massMobseris given byMobser=√M^0
1 −vc^22

. Consequently, the corresponding gravita-

tional forceFto the observer with massmis

F=−

mMobserG

r^2

=−

mM 0 G

r^2

√

1 −v

2

c^2

=−

mM 0 G

r^2

√

1 −H

2

c^2 r

2

=−

mM 0 G

r^2

√

1 −r

2

R^2 s

.

It is clear then that asr→Rs,F→ −∞.This is clearly not what is observed.
In conclusion, we have rigorously derived the following newtheory of cosmology:

Theorem 7.27.Assume a) the Einstein theory of general relativity, and b) the principle of
cosmological principle that the universe is homogeneous and isotropic. Then the following
assertions hold true for our Universe:

1) All universes are bounded, are not originated from a Big-Bang, and are static; and

2) The topological structure of our Universe can only be the 3D sphere such that to each

observer, the corresponding equator with the observer at the center of the hemisphere

can be viewed as the black hole horizon.

Theorem 7.28.If we assume only a) the Einstein theory of general relativity, and b’) the
universe is homogeneous. Then all universes can only be either a 3D sphere as given in
Theorem7.27, or a globular universe, which is a 3D open ball BRsof radius Rs, forming the
interior of a black hole with Rsas its Schwarzschild radius. In the later case, the Universeis
also static, is not originated from a Big-Bang, and the matter fills the entire Universe. Also,
the following assertions hold true: