462 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
7.5.2 Globular universe with boundary
If the spatial geometry of a universe is open, then by our theory of black holes developed in
Section7.3, we have shown that the universe must be in a ball of a black hole with a fixed ra-
dius. In fact, according to the basic cosmological principle that the universe is homogeneous
and isotropic (Roos, 2003 ), given the energy densityρ 0 >0 of the universe, by Theorem7.3,
the universe will always be bounded in a black hole of open ball with the Schwarzschild
radius:
Rs=√
3 c^2
8 πGρ 0,
as the mass in the ballBRsis given byMRs= 4 πR^3 sρ 0 /3. This argument also clearly shows
that
there is no unbounded universe.
In addition, since a black hole is unable to expand and shrink, by property (7.5.55) of
black holes, all globular universes must be static.
Globular universe
We have shown that the universe is bounded, and suppose that the universe is open, i.e.
its topological structure is homeomorphic toR^3 , and it begins with a ball. LetEbe its total
energy:
(7.5.29) E=mass+kinetic+thermal+Ψ,
whereΨis the energy of all interaction fields. Let
(7.5.30) M=E/c^2.
At the initial stage, all energy is concentrated in a ball with radiusR 0. By the theory of
black holes, the energy contained in the ball generates a black hole inR^3 with radius
(7.5.31) Rs=
2 MG
c^2,
providedRs≥R 0 ; see (7.3.1) and Figure7.3.
Thus, if the universe is born to a ball, then it is immediatelytrapped in its own black hole
with the Schwarzschild radiusRsof (7.5.31). The 4D metric inside the black hole of the static
universe is given by
(7.5.32) ds^2 =−ψ(r)c^2 dt^2 +α(r)dr^2 +r^2 (dθ^2 +sin^2 θdφ^2 ),
whereψandαsatisfy the equations (7.1.79) and (7.1.80) with boundary conditions:
ψ→ψ ̃, α→α ̃ as r→Rs,whereRsis given by (7.5.31). Also,ψ ̃,α ̃are given via the TOV metric (7.1.40)-(7.1.41):
(7.5.33) ψ ̃=
1
4
(
1 −
r^2
R^2 s)
, α ̃=(
1 −
r^2
R^2 s)− 1
for 0≤r<Rs.