478 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
Now we are ready to deduce from (7.6.27) the PID gravitational interaction formula as
follows.
First, we infer from (7.6.27) that
u′+v′=rφ′′
1 +r 2 φ′,
u′−v′=1
1 −r 2 φ′[
2
r(ev− 1 )+rφ′′]
.
Consequently,
(7.6.28) u′=
1
1 −r 2 φ′1
r(ev− 1 )+rφ′′
1 −(r 2 φ′)^2.
It is known that the interaction forceFis given by
F=−m∇ψ, ψ=c^2
2(eu− 1 ).Then, it follows from (7.6.28) that
(7.6.29) F=
mc^2
2
eu[
−
1
1 −r 2 φ′1
r
(ev− 1 )−rφ′′
1 −(r
2 φ′)^2
]
.
The formula (7.6.29) provides the precise gravitational interaction force exerted on an
object with massmin a spherically symmetric gravitation field.
In classical physics, the field functionsuandvin (7.6.29) are taken by the Schwarzschild
solution:
(7.6.30) eu= 1 −
2 GM
c^2 r, ev=(
1 −
2 GM
c^2 r)− 1
,
andφ′=φ′′=0, which leads to the Newton gravitation.
However, due to the presence of dark matter and dark energy, the field functionsu,v,φin
(7.6.29) should be an approximation of the Schwarzschild solution (7.6.30). Hence we have
(7.6.31) |rφ′| ≪1 forr>r 0.
Under the condition (7.6.31), formula (7.6.29) can be approximatively expressed as
(7.6.32) F=
mc^2
2eu[
−
1
r(ev− 1 )−rφ′′]
.
7.6.4 Asymptotic repulsion of gravity
In this section, we shall consider the asymptotic properties of gravity, and rigorously prove
that the interaction force given by (7.6.32) is repulsive at very large distance.