Mathematical Principles of Theoretical Physics

7.6. THEORY OF DARK MATTER AND DARK ENERGY 471

1) The cosmic observable mass M and the total mass Mtotal, which includes both M and

the non-observable mass due to the space curvature energy, satisfy the following rela-

tion

(7.5.59) Mtotal=

{

3 πM/2 for the spherical structure,

3 πM/4 for the globular structure.

The difference Mtotal−M can be regarded as the dark matter;

2) The static Universe has to possess a negative pressure to balance the gravitational at-

tracting force. The negative pressure is actually the effect of the gravitational repelling

force, also called dark energy; and

3) Both dark matter and dark energy are a property of gravity, which is reflected in both

space-time curvature, and the attracting and repulsive gravitational forces in different

scales of the Universe. This law of gravity is precisely described by the new gravita-

tional field equations (4.4.10); see also (Ma and Wang,2014e).

We end this section with three remarks and observations.

First, astronomical observations have shown that the measurable massMis about one
fifth of total massMtotal. By (7.5.59), for the spherical universe,

Mtotal= 4. 7 M.

This relation also suggest that the spherical universe casefits better the current understanding
for our Universe.

Second, due to the horizon of sphere, for an arbitrary point in a spherical universe, its
opposite hemisphere relative to the point is as if it is a black hole. Hence the main contribution
to the redshifts is from the effect of the black hole, as explicitly given by (7.5.53).

Third, in modern cosmology, the view of expanding universe was based essentially on
the Friedmann model and the Hubble Law. The observations canaccurately measure the
distances and redshifts for some galaxies, which allowed astronomers to get both measured
and theoretical data, and their deviation led to the conclusion that the expanding universe is
accelerating. The misunderstanding comes from the perception that the Doppler redshift is
the main source of redshifts.

7.6 Theory of Dark Matter and Dark energy

7.6.1 Dark energy and dark matter phenomena

1.Dark matter and Rubin rotational curve.In astrophysics, dark matter is an unknown form
of matter, which appears only participating in gravitational interaction, but does not emit nor
absorb electromagnetic radiations.