7.6. THEORY OF DARK MATTER AND DARK ENERGY 483
We remark that Theorem7.30is valid provided the initial valueαis small because the
diagram given by Figure7.16is in a neighborhood ofx=0. However, all physically mean-
ingful central fields satisfy the condition (note that any a black hole is enclosed by a huge
quantity of matter with radiusr> 0 ≫ 2 MG/c^2 ). In fact, the Schwarzschild initial values are
as
(7.6.49) x 1 (r 0 ) =x 2 (r 0 ) =
δ
1 −δ, δ=2 MG
c^2 r 0.
For example see (7.6.29), where theδ-factors are of the orderδ≤ 10 −^1 , sufficient for the
requirements of Theorem7.30.
The most important cases are for galaxies and clusters of galaxies. For these two types of
astronomical objects, we have
galaxy : M= 1011 M⊙, r 0 = 3 × 105 ly,
cluster of galaxies : M= 1014 M⊙, r 0 = 3 × 106 ly.Thus theδ-factors are
(7.6.50) galaxiesδ= 10 −^7 , cluster of galaxiesδ= 10 −^5.
In fact, the dark energy phenomenon is mainly evident between galaxies and between clusters
of galaxies. Hence, (7.6.50) shows that Theorem7.30is valid for both central gravitational
fields of galaxies and clusters of galaxies. The asymptotic repulsion of gravity plays the role
to stabilize the large scale homogeneous structure of the Universe.
7.6.5 Simplified gravitational formula
We have shown that all four fundamental interactions are layered. Namely, each interaction
has distinct attracting and repelling behaviors in different scales and levels. The dark matter
and dark energy represent the layered property of gravity.
In this section, we simplify the gravitational formula (7.6.32) to clearly exhibit the layered
phenomena of gravity.
In (7.6.32) the field functionsuandvcan be approximativelyreplaced by the Schwarzschild
solution (7.6.30). Since 2MG/c^2 ris very small forr>r 0 as indicted in (7.6.29) and (7.6.50),
the formula (7.6.32) can be expressed as
(7.6.51) F=mMG
[
−
1
r^2−
r
δr 0φ′′]
, r>r 0.By the field equation (7.6.26), we have(7.6.52) R=Φ forr>r 0 ,
whereRis the scalar curvature, and
Φ=gμ νDμ νφ=e−v[
−φ′′+1
2
(u′−v′)φ′+2
rφ′