466 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
7.5.3 Spherical Universe without boundary
Bounded universe has finite energy and space, and our Universe is bounded as we have
demonstrated in the last section. Besides the globular universe, another type of bounded
universe is the spherically-shaped corresponding to thek=1 case in the Friedmann model
(7.5.15)-(7.5.17).
A globular universe must be static. With the same argument, aspherical closed universe
have to be static as well. In this subsection, we are devoted to investigate the spherical
cosmology.
1.Cosmic radius.For a static spherical universe, its radiusRcsatisfies thatR ̇c= 0 , R ̈c= 0.By the Friedmann equation (7.5.18), it leads to that
(7.5.42) R^2 c=
3 c^2
8 πGρ.
For a 3D sphere, its volumeVis given by
V= 2 π^2 R^3.Thus,ρ=M/ 2 π^2 R^3 c, and by (7.5.43) we get the radiusRcas
(7.5.43) Rc=
4 MG
3 πc^2.
This value (7.5.43) is also the maximal radius for a (possibly) oscillatory spherical universe.
2.Negative pressure.By (7.5.15) andR ̈c=0, the pressure is negative:(7.5.44) p=−
ρc^2
3
.
In order to resist the gravitational pulling, it is natural that there is a negative pressure in a
static universe, which originates from three sources:
thermal expanding, radiation pressure, and dark energy.These three types of forces are repulsive, and therefore yield the negative pressure as given
by (7.5.44).
In fact, in our Universe both thermal and radiation (microwave radiation) pressures are
very small. The main negative pressure is generated by the socalled “dark energy”. In
(Ma and Wang,2014e), we have shown that the dark energy is the repulsive gravitational
effect for a remote object of great distance. From the field theoretical point view, dark energy
is an effect of the dual gravitational fieldΨμin the PID-induced gravitational field equations
(4.4.10) discovered by the authors.
3.Equivalence.It seems that both spherical and globular geometries are very different.
However, in the following we show that they are equivalent incosmology. In fact, as the