468 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
whereMis the mass of the flat space. By the invariance of density,
M/
4 π
3R^3 c=Mtotal/π^2 R^3 c,we get the relation
(7.5.50) Mtotal=
3 π
4M.
With the flat space mass (7.5.50), from (7.5.49) we get the Schwarzschild radiusRs=Rcfor
the cosmic black hole as follows
Rs= 2 GM/c^2.
It means that the globular universe is essentially hemispherically-shaped. In particular the
relation (7.5.50) can be generated to an arbitrary regionΩ⊂R^3 , i.e.
(7.5.51) MΩ;total=
VΩ
|Ω|
MΩ,
whereMΩis the flat space mass inΩ,MΩ;totalis the curved space mass,|Ω|is the volume of
flatΩ,
VΩ=∫Ω√
gdx, g=det(gij),andgij( 1 ≤i,j≤ 3 )is the spatial gravitational metric.
Now, we return to the Friedmann model (7.5.18) withk=1, which has the same form as
that of the globular dynamic equation (7.5.45), and is of the same maximal radiusRcas that
in (7.5.49). Hence, it is natural that a static spherical universe is considered as if there were
two hemispherical black holes attached together. In fact, the static spherical universe forms
an entire black hole as a closed space.
4.Basic problems.Since a static spherical universe is equivalent to two globular universes
to be pieced together along with their boundary, an observerin its hemisphere is as if one is in
a globular universe. Hence, the basic problems – the cosmic edge problem, flatness problem,
horizon problem, and CMB problem– can be explained in the same fashion.
The redshift problem is slightly different, and the gravitational redshift is given by
(7.5.52) 1 +z=
1
√
−g 00 (r),
whereris the distance between the light source and the observer, andg 00 is the time-
component of the gravitational metric.
Due to the horizon of sphere, for an arbitrary point on a spherical universe, its opposite
hemisphere relative to the point plays a similar role as a black hole. Hence, in the redshift
formula (7.5.52),g 00 can be approximatively taken as the Schwarzschild solutionfor distant
objects as follows
−g 00 = 1 −Rs
̃r, Rs=2 MG
c^2, ̃r= 2 Rs−r for 0≤r<Rs,