Mathematical Principles of Theoretical Physics

484 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

In view of (7.6.52), we obtain that

φ′′=−evR+

2

r

φ′+

1

2

(u′−v′)φ′

Again by the Schwarzschild approximation, we have

(7.6.53) φ′′=

(

2

r

+

δr 0

r^2

)

φ′−R.

Integrating (7.6.53) and omittinge±δr^0 /r, we derive that

φ′=−r^2

[

ε+

∫

r−^2 Rdr

]

,

whereεis a constant. Thus (7.6.51) can be rewritten as

(7.6.54) F=mMG

[

−

1

r^2

+

r

δr 0

R+

(

1 +

2 r

δr 0

)(

εr+r

∫ R

r^2

dr

)]

.

The solutions of (7.6.34) can be Taylor expanded. Hence by (7.6.33) we see that

u′(r) =

1

r^2

∞

∑

k= 0

ak(r−r 0 )k.

By (7.6.46), the gravitational forceFtakes the following form

F=

1

r^2

∞

∑

k= 0

bkrk, b 0 =−mMG.

In view of (7.6.54), it implies thatRcan be expanded as

R=

ε 0

r

−ε 1 +O(r),

and by Physical Conclusion7.29,ε 0 andε 1 are two to-be-determined free parameters. Insert-
ingRinto (7.6.54) we obtain that

(7.6.55) F=mMG

[

−

1

r^2

−

k 0

r

+φ(r)

]

forr>r 0.

wherek 0 =^12 ε 0 , and

φ(r) =ε 1 +k 1 r+O(r), k 1 =ε+

ε 1

δr 0

.

The nature of dark matter and dark energy suggests that

k 0 > 0 , k 1 > 0.

Based on Theorem7.30,φ(r)→0 asr→∞, and (7.6.55) can be further simplified as in the
form forr 0 <r<r 1 ,

(7.6.56) F=mMG

[

−

1

r^2

−

k 0

r

+k 1 r

]

,