Mathematical Principles of Theoretical Physics

24 CHAPTER 1. GENERAL INTRODUCTION

Equivalently, the above interpretation of dark matter and dark energy is consistent with
the theory based on the new PID gravitational field equationsdiscovered by (Ma and Wang,
2014e) and addressed in detail in Chapter 4. It is clear now that gravity can display both
attractive and repulsive effect, caused by the duality between theattractinggravitational field
{gμ ν}and therepulsivedual vector field{Φμ}, together with their nonlinear interactions
governed by the field equations. Consequently, dark energy and dark matter phenomena are
simply a property of gravity. The detailed account of this relation is addressed in Section7.6,
based in part on (Hernandez, Ma and Wang, 2015 ).

PID-cosmological model

One of the main motivations for the introduction of the Big-Bang theory and for the ex-
panding universe is that the Friedamann solution of the Einstein gravitational field equations
demonstrated that the Einstein theory must produce a variable size universe; see Conclusions
of Friedmann Cosmology7.23. Also, it is classical that bringing the cosmological constant
Λinto the Einstein field equations will lead to an unstable static universe.

We have demonstrated that the right cosmological model should be derived from the new
gravitational field equations (1.2.7), taking into consideration the presence of dark matter and
dark energy. In this case, based on the cosmological principle, the the metric of a homoge-
neous spherical universe is of the form

(1.9.5) ds^2 =−c^2 dt^2 +R^2

[

dr^2

1 −r^2

+r^2 (dθ^2 +sin^2 θdφ^2 )

]

,

whereR=R(t)is the cosmic radius. We deduce then from (1.2.7) the following PID-
cosmological model, withφ=φ′′:

(1.9.6)

R′′=−

4 πG

3

(

ρ+

3 p

c^2

+

φ

8 πG

)

R,

(R′)^2 =

1

3

( 8 πGρ+φ)R^2 −c^2 ,

φ′+

3 R′

R

φ=−

24 πG

c^2

R′

R

p,

supplemented with the equation of state:

(1.9.7) p=f(ρ,φ).

Note that only two equations in (1.9.6) are independent.
Also, the model describing the static Universe is the equation of state (1.9.7) together
with the stationary equations of (1.9.6), which are equivalent to the form

(1.9.8)

φ=− 8 πG

(

ρ+

3 p

c^2

)

,

p=−

c^4

8 πGR^2

.