Mathematical Principles of Theoretical Physics

7.5. THE UNIVERSE 467

Rc

semi-sphere

(a)

Rc

upper semi-sphere

(b)

lower semi-sphere

Figure 7.14: (a) A 3D hemisphere, and (b) a 3D sphere piecing the upper and lower hemi-
spheres together.

space-time curvature is caused by gravitation, a globular universe must be a 3D hemisphere
as shown in Figure7.14(a), and a spherical universe is as shown in Figure7.14(b), which is a
3D sphere piecing the upper and lower hemispheres together.
In cosmology, the globular universe is a black hole, which likes as a 3D hemisphere,
and the spherical universe can be regarded as if there were two hemispheres of black holes
attached together.
We show this version from the cosmological dynamics.
First, by the Newtonian cosmological equation (7.5.14), i.e.

(7.5.45)

( ̇

R

R

) 2

=

8 πG

3

ρ−

kc^2

R^2

.

For a static universe in a black hole with maximal radiusRc, the equation (7.5.45) becomes

(7.5.46) R ̇c= 0 ⇔

8 πG

3

ρ=

kc^2

R^2 c

.

The volume of the hemisphere is

V 0 =

∫ 2 π

0

dθ

∫π

0

dφ

∫π

2

0

R^3 csinθsin^2 ψdψ=π^2 R^3 c,

wherex∈R^4 takes the spherical coordinate:

(7.5.47) (x 1 ,x 2 ,x 3 ,x 4 ) = (Rcsinψsinθcosφ,Rcsinψsinθsinφ,Rcsinψcosθ,Rccosψ).

Thus, the mass density is

(7.5.48) ρ=Mtotal/π^2 R^3 c,

Then it follows from (7.5.46) that

(7.5.49) Rc=
8 GMtotal
3 πc^2

fork= 1.

Remark 7.26.The massMtotalin (7.5.48) contains the energy contributed by the space cur-
vature, i.e.
Mtotal=M+space curved energy,