Mathematical Principles of Theoretical Physics

212 CHAPTER 4. UNIFIED FIELD THEORY

1.Gravitons and dual gravitons.It is known that as the equations describing field parti-
cles, (4.4.5) characterize the graviton as a massless, neutral bosonic particle with spinJ=2,
and (4.4.6) indicate that the dual vector graviton is a massless, neutral bosonic particle with
J=1. Hence, the gravitational field equations induced by PID and PRI provide a pair of field
particles:

(4.4.7)

tensor graviton: J= 2 ,m= 0 ,Qe= 0 ,

vector graviton: J= 1 ,m= 0 ,Qe= 0 ,

whereQeis the electric charge.
It is the nonlinear interaction of these two field particles in (4.4.7) that lead to the dark
matter and dark energy phenomena.

2.Gravitational force.If we consider the gravitational force only from the Einstein field
equations

(4.4.8) Rμ ν−

1

2

gμ νR=−

8 πG

c^4

Tμ ν,

then by the Schwarzschild solution of (4.4.8), we can derive the classical Newton’s gravita-
tional force as

(4.4.9) F=−

mMG

r^2

,

which is an attracting force generated bygμ ν.
However, with the field equations (4.4.5), we can deduce a revised formula to (4.4.9). Ac-
tually, ignoring the microwave background radiation, the equations (4.4.5) become (Ma and Wang,
2014e):

(4.4.10) Rμ ν−

1

2

gμ νR=−

8 πG

c^4

Tμ ν−∇μ∇νφ,

whereΦν=−∇νφ, andφis a scalar field. In Chapter 7 (see also (Ma and Wang,2014e)),
we are able to derive from (4.4.10) that the gravitational force should be in the form

(4.4.11) F=mMG

[

−

1

r^2

+

c^2

2 MG

Φr−

(

c^2

MG

+

1

r

)

dφ

dr

]

,

whereφis the dual scalar field, representing the scalar potential,and

(4.4.12) Φ=gμ ν∇μ∇νφ.

The first term in the right-hand side of (4.4.11) is the Newton’s gravitational force, and the
second term (4.4.12) represents the repelling force generated by the dual fieldφ, and the third
term

−

(

c^2

MG

+

1

r

)

dφ

dr