Mathematical Principles of Theoretical Physics

7.5. THE UNIVERSE 469

whereMis the cosmic mass of hemisphere, and ̃ris the distance from the light source to the
opposite radial point, andris from the light source to the point. Hence, formula (7.5.52) can
be approximatively written as

(7.5.53) 1 +z=

1

√

α(r)( 1 −R ̃rs)

=

√

2 Rs−r

√

α(r)(Rs−r)

for 0<r<Rs.

where
α( 0 ) = 2 , α(Rs) = 1 , α′(r)< 0.

We see from (7.5.53) that the redshiftz→∞asr→Rs, and, consequently, we cannot see
objects in the opposite hemisphere. Intuitively,α(r)represents the gravitational effect of the
matter in the hemisphere of the observer.

5.Physical conclusions.In either case, globular or spherical, the universe is equivalent
to globular universe(s). It is not originated from a Big-bang, is static, and confined in a black
hole in the sense as addressed above. The observed massMand the implicit massMtotalhave
the relation

(7.5.54) Mtotal= 2 ×

3 π

4

M= 3 πM/ 2 ,

which is derived by (7.5.50) adding the mass of another hemisphere.
The implicit massMtotalof (7.5.54) contains the dark matter. In (Ma and Wang,2014e),
both the dark matterMtotal−Mand the dark energy (i.e. the negative pressure (7.5.44)) are
just a property of gravity.

7.5.4 New cosmology

We start with two difficulties encountered in modern cosmology.
First, if the Universe were born to a Big-Bang and expanded continuously, then in the
expansion process it would generate successively a large number of black holes, whose radii
vary as follows:

(7.5.55)

√

R 0

Rs

R 0 ≤r≤

√

R

Rs

R, R 0 <R≤Rs=

2 MG

c^2

,

whereMis the total mass in the universe,R 0 is the initial radius,Ris the expanding radius,
andris the radius of sub-black holes.
To see this, we consider a homogeneous universe with radiusR<Rs. Then the mass
densityρis given by

(7.5.56) ρ=

3 M

4 πR^3

.

On the other hand, by Theorem7.3, the condition for a ballBrwith radiusrto form a black
hole is that the massMrinBrsatisfies that

(7.5.57)

Mr

r

=

c^2

2 G

.