7.5. THE UNIVERSE 463
Basic problems
A static universe is confined in a ball with fixed radiusRsin (7.5.31), and the ball behaves
like a black hole. We need to examine a few basic problems for astatic universe, including
the cosmic edge, the flatness, the horizon, the redshift, andthe cosmic microwave background
(CMB) radiation problems.
1.The cosmic edge problem.In the ancient Greece, the cosmic-edge riddle was proposed
by the philosopher Archytas, a friend of Plato, who used “what happens when a spear is
thrown across the outer boundary of the Universe?” The problem appears to be very difficult
to answer. Hence, for a long time physicists always believe that the Universe is boundless.
Our theory of black holes presented in Section7.3shows that all objects in a globular
universe cannot reach its boundaryr=Rs. In particular, an observer in any position of the
globular universe looking toward to the boundary will see noboundary due to the openness
of the ball and the relativistic effect near the Schwarzschild surface. Hence the cosmic-edge
riddle is no longer a problem.
2.The flatness problem.In modern cosmology, the flatness problem means thatk= 0
in the FLRW metric (7.5.8)-(7.5.9). It is common to think that the flatness of the universe
is equivalent to the fact that the present energy densityρmust be equal to the critical value
given by (7.5.20). In fact, mathematically the flatness means that any geodesic triangle has
the inner angular sumπ= 180 ◦.
Measurements by the WMAP (Wilson Microwave Anisotropy Probe) spacecraft in the
last ten years indicated that the Universe is nearly flat. Thepresent radius of the Universe is
about
(7.5.34) R= 1026 m.
If the Universe is static, then (7.5.34) gives the Schwarzschild radius (7.5.31), from which it
follows that the densityρof our Universe is just the critical density of (7.5.20):
(7.5.35) ρ=ρc= 10 −^26 kg/m^3.
Thus, we deduce that if the universe is globular, then it is static. In addition, we have shown
that any universe is bounded and confined in a 3D hemisphere ofa black hole or in a 3D
sphere as shown in Figure7.14. Hence as the radius is sufficiently large, the universe is
nearly flat.
3.The horizon problem. The cosmic horizon problem can be simply stated as that all
places in a universe look as the same. It seems as if the staticUniverse with boundary violates
the horizon problem. However, due to the gravitational lensing effect, the light bents around
a massive object. Hence, the boundary of a globular universeis like a concave spherical
mirror, and all lights reaching close to it will be reflected back, as shown in Figure7.13. It
is this lensing effect that makes the globular universe looks as if everywhere is the same, and
is horizontal. In Figure7.13, if we are in positionx, then we can also see a star as if it is in
position ̃y, which is actually a virtual image of the star aty.
4.The redshift problem. Observations show that light coming from a remote galaxy is
redshifted, and the farther away the galaxy is, the larger the redshift is. In astronomy, it is